CORPORATE FINANCE Politecnico di Milano Matteo Verzicco Table of Contents INTRODUCTION TO CORPORATE FINANCE & FINANCIAL STRUCTURE ................................................... 3 FINANCIAL MARKETS ...................................................................................................................................................... 4 Taxonomy of Financial Markets ................................................................................................................................ 4 Capital Structure ........................................................................................................................................................ 4 Modigliani and Miller’s Model (I) ............................................................................................................................. 5 Modigliani and Miller’s Model (II) ............................................................................................................................ 6 The effect of Risk ...................................................................................................................................................... 10 The dark side of leverage ......................................................................................................................................... 10 EFFECT OF TAXES ON M&M PROPOSITIONS .................................................................................................................. 11 Leverage in practice and objections ........................................................................................................................ 12 Theories on Optimal Debt definition ........................................................................................................................ 14 Pecking Order Theory .............................................................................................................................................. 16 DEBT FINANCING AND BONDS ................................................................................................................................ 19 INTEREST RATE TERM STRUCTURE .................................................................................................................... 24 Expectations ............................................................................................................................................................. 25 Nominal and Real Interest Rates .............................................................................................................................. 26 BONDS .............................................................................................................................................................................. 27 Ratings ...................................................................................................................................................................... 27 Credit Default Swaps ............................................................................................................................................... 29 Currency ................................................................................................................................................................... 29 Maturity .................................................................................................................................................................... 30 Coupons .................................................................................................................................................................... 30 Seniority ................................................................................................................................................................... 31 BONDS EVALUATION ..................................................................................................................................................... 31 Illiquid and risky bonds ............................................................................................................................................ 34 Yield to Maturity....................................................................................................................................................... 35 Duration ................................................................................................................................................................... 35 Volatility ................................................................................................................................................................... 35 BOND TAXATION ........................................................................................................................................................... 36 BOND ISSUING ............................................................................................................................................................... 37 BOND LISTING ............................................................................................................................................................... 38 MINI BONDS................................................................................................................................................................... 38 SHARES ........................................................................................................................................................................... 39 Tax impact on shares................................................................................................................................................ 40 SHARES EVALUATION .................................................................................................................................................... 40 THE DIVIDEND DISCOUNT MODEL (DDM) .................................................................................................................... 42 The Gordon and Shapiro Model............................................................................................................................... 42 Useful share ratios ................................................................................................................................................... 43 Present value of Growth Opportunity ...................................................................................................................... 45 Equity research and Fundamental Analysis............................................................................................................. 48 Capital Asset Pricing Model .................................................................................................................................... 48 Arbitrage Pricing Theory ......................................................................................................................................... 49 Fama and French three-factor model ...................................................................................................................... 49 Technical Analysis.................................................................................................................................................... 50 Behavioral Finance .................................................................................................................................................. 51 STOCK MARKETS ........................................................................................................................................................... 52 INTERACTIONS BETWEEN VALUATION AND FINANCING ............................................................................ 55 ADJUSTED PRESENT VALUE ........................................................................................................................................... 55 WACC APPROACH ......................................................................................................................................................... 58 Miles & Ezzell formula ............................................................................................................................................. 59 Modigliani & Miller formula ................................................................................................................................... 59 DERIVATIVES ................................................................................................................................................................ 62 HISTORY OF DERIVATIVES ............................................................................................................................................. 62 1 FORWARD CONTRACTS .................................................................................................................................................. 62 Forward: Value at time zero .................................................................................................................................... 63 Case with Cash Flows .............................................................................................................................................. 65 FUTURES ........................................................................................................................................................................ 66 SWAP CONTRACTS ......................................................................................................................................................... 69 HEDGING USING FORWARD AND FUTURES ..................................................................................................................... 69 OPTIONS......................................................................................................................................................................... 70 American vs European Options................................................................................................................................ 71 Put-call parity theorem on European options .......................................................................................................... 71 How to evaluate options ........................................................................................................................................... 72 Options evaluation models ....................................................................................................................................... 74 Binomial Model ........................................................................................................................................................ 74 Delta (Hedge Ratio) ................................................................................................................................................. 75 Black & Scholes formula .......................................................................................................................................... 76 Correlations ............................................................................................................................................................. 79 Hedging with options ............................................................................................................................................... 79 DERIVATIVES ON THE ITALIAN EXCHANGE .................................................................................................................... 80 2 Introduction to Corporate Finance & Financial Structure The main objective is to analyse how companies finance their operations. Why finance is necessary? Why do company need to raise money on the market? Basically, the company is seen as a black box, receiving some inputs to generate some outputs that a market is willing to buy with a certain price. Companies engage in their production activities through investments. Investments are seen as resources in the asset side to hold the back of production and operations. OF course, investments are costly, not once but continuously (technology, raw materials, machinery, labour…). Money could be found through external capital market or internal capital. Financial markets are an example of external ones. In financial markets we have mainly two sides: - Companies: needing money today, promising a future value creation able to repay, through dividends or others, the initial effort of the investors. - Investors: professional, private, funds, having money and providing them to companies, expecting back extra profitability by the companies themselves. The expected profitability is balanced against the risk of the business, which of course is different depending on the type of securities we’re considering (German bunds, swiss bunds are very low risk securities, on the other hand, Italian, Turkish bunds are definitely riskier). Investors are willing to take risks as much as expected returns grows in balancing these risks. We could consider another player, which role’s played by Intermediaries, collecting deposits from different types of subjects, investing these money on third securities issued by companies. Typical examples of intermediaries are banks or funds. As said, companies need cash. The immediate ways to create cash are: - Divestment of some existing assets: o Reduction of fixed and long term assets: i.e. business units, subsidiaries are sold to generate cash and liquidity. o Increase of cash - Issue liabilities: this means enlarging the size of the company o Increasing liabilities to provide liquidity o Increase of cash 3 Financial Markets Financial markets are similar to any other markets. There is a demand curve and a demand curve: companies’ demand for capital is matched with the investors’ supply of cash. The more efficient F.Ms are, more easily companies can access to capital and grow. As other markets, F.Ms have some frictions inside them, like Information Asymmetries and Market Imperfections. Transactions on F.Ms are regulated by contracts which are called securities (bonds, shares swaps, futures, options). Taxonomy of Financial Markets The main definition of F.Ms define the existence of: - Primary Markets: direct relationship between the investor on one side and the firm on the other side. The company directly issues securities, selling them to investors. - Secondary Market: there’s an indirect relationship between investors and firms through securities, but the intermediation is played by intermediaries. Intermediators also play a good role to the health of primary market, optimizing the match between the demand and the supply of securities. The timing companies issue security opportunities on the markets is totally subjective, depending on the type of the security and on the need they have in collecting funds. This is because issuing securities on the market is costly (regulations, agreements with banks and post offices, consultants…). On the other hand, financial markets work every day almost 24h per day. In each moment is possible to trade shares and securities. This happens on the secondary markets. This is the main difference between the two: in primary markets, companies directly go on the market issuing securities and raising cash; then, those securities are traded each day on the secondary market (mainly if the company is listed on the stock exchange). On the second market, there’s no flow involving the company, but companies only see a change in shareholders and, eventually, on the value of the shares. Reasons pushing secondary market trades are different. Economically, we can refer on the utility function driving the investors: every day, investors could find optimal for their utility function to buy or to sell shares and securities. Capital Structure Typically, a firm finances its assets through a combination of: - Equity (E): capital provided by shareholders - Debt (D): capital raised on the market by third parties Equity is a title of ownership, through which shareholders have the possibility of wielding power on the firm itself. On the other hand, for shareholders, remuneration is residual, net of all the costs the companies have to bare to run its activities. Profits for shareholders have to be seen in Net Profits, at the bottom of the line (sometimes neither). This could be a positive or negative feature: if Net Profit is low (or even negative), it’s not worth to be a shareholder, it would mean no profits and probably the need of increasing company’s capital; if the company is very efficient, Net Profit for shareholders could be even higher. This directly means that for Equity holders, the risk is higher. Debt is not a title of ownership. The remuneration for debt is contractually fixed at the time of signing the contract; it may not be constant but is ruled by a contract, so it’s known. 4 Which is the best capital structure for a company? Companies have to create an equilibrium between debts and equity. Modigliani and Miller’s Model (I) Modigliani and Milles developed the Proposition I and Proposition II about capital structure irrelevance (also known as irrelevance theory or model). In perfect markets and under some key assumptions, it does not matter which capital structure a firm decides to adopt. This thesis is true only under some very specific conditions, which are very unlikely to be met in the real world: - No Corporate Taxes - There is a unique interest rate (r%) for companies and investors: everyone can borrow having the same interest rate. - Perfectly symmetric information and no transaction costs: every time something in the market occurs, everyone immediately knows. - Firms can choose how to finance their operations with no endogenous limits: firms can raise as much funds as they want, every time. Demonstration is given through two firms, virtually identical, according to M&Ms assumptions. The two companies have the same EBIT, with the unique exception dividing the companies between: - Unlevered (U): company financed only with Equity, so π" = πΈ" - Levered (L): company financed with Equity and Debt, so π% = πΈ% + π·% Keep attention on the fact that, in finance, we do not deal with balance sheet, nominal and accounting values but only with market values. Earnings for the two companies are given by: Earnings (U) = EBIT Earnings (L) : EBIT - r%*Dl If we’re able to demonstrate that the value of the two companies do not change, it means that financing through Equity or Debt is indifferent. Since the two companies are identical, we can say that also operating margin is the same (revenues, material costs, wages, depreciation, external services.. are the same). The only difference is given on Interests and Taxes. Assuming we have two different portfolios for an investor: 1. I) = 5%(E. ) → Profit) = 5%[Earnings(U)] = 5%(EBIT) 2. I@ = 5%(EA ) + 5%(DA ) → Profit @ = 5%[Earnings(L)] + 5%(r ∗ DA ) = 5%(EBIT − r ∗ DA ) + 5%(r ∗ DA ) = 5%(EBIT) So, we have two different portfolios, providing the same outcomes for every value of the operating margin. We are in a no-arbitrage equilibrium situation, so each time we have two different portfolios leading to the same result, they must have the same value even at the beginning. Otherwise, it would not be an equilibrium on the market, because if the value of a portfolio is lower than the other, with the same output, every investor would buy the lowest value portfolio without effectively generating an equilibrium. π%ππ = π%ππ + π%ππ → ππ = ππ + ππ → ππ = ππ 5 Please note that we’re talking about the value of the portfolio, not the value of the investment. It’s true and obvious that for a Levered company the investment would be lower. In the end, it is possible to have two equivalent portfolios in terms of payoffs if investing in the unlevered company, or both in the levered company and in its debt. Investing in its debt would mean acquiring part of its debt and selling it at the same interest rate r%. This leads to no value creation for the market. An alternative demonstration to M&M(I) could be done. In the 4) portfolio, our initial investment will be lower since we borrow money on the market, financing a portion of this portfolio with cash. These two situation would lead to a worst-off solution than the 1) and 2) portfolios, but it’s a no arbitrage equilibrium, so the values of the two portfolios have to be the same. This example shows the motivation why companies leveraging on the market are not creating value in this situation. The equivalence between portfolios 3) and 4) tells us that it’s the same having a company which is levered on the market or investing in a company which is unlevered, but collecting by ourselves debt on the market. Modigliani and Miller’s Model (II) Modigliani & Miller’s Tradeoff Theory of Leverage: as the proportion of Debt in the firm’s capital structure increases, the profitability expected by shareholders increases in a linear function under the assumption that the return of investments is larger than the interest rate on debt (r). We introduce: - πM : expected profitability of equity capital - πN : expected profitability of the company assets These parameters are not accounting values but only market values. For sake of simplicity we consider a company that every year exhibits the same constant operating margin (EBIT). Under this assumption we have: MOPQ We consider π·% + πΈ% = π, so we have in the second step that R RS By developing and collecting M , we get πM as a function of πN . S S TMS = πN . By the formula, we obtain that if the company is unlevered and there’s no debt, πM equals πN , having D which is null offsetting the last term of the equation. 6 R This formula is strictly like the financial leverage formula π ππΈ = π ππΌ + YM Z ∗ (π ππΌ − π) where returns are computed on accounting numbers, while in finance we use market values. The spread between the expected profitability of company assets and the interest rate that is charged for borrowings leads to a kind of extra profit for the shareholder, without investing his own money. This has a very important managerial implication. There are a number of business in which the leverage effect is fundamental: there are some industries, like renewable energy, in which profitability is not really relevant but is at a very low risk. This could lead to higher remuneration and profitability leveraging on debt. The trick is all inside the risk: we’re working with market values and expected profitability. Risk could be led by several business exogenous elements (financial crisis, recessions, new competitors…). In the case expected profitability is lowered, if I have only equity capital, until we have a positive profitability it’s ok. But if we are a levered company and our expected profitability drops under the rate charged for debt, we’re not able in repaying debts and shareholders have to increase equity, making it more expensive, proportionally to the level of debt we have. This aspect underlines the very negative aspect of risk: volatility and uncertainty. When we talk about the return on an asset, most of the time there’s a very high volatility. This graph shows how, linearly, expected profits of equity capital increases as leverage (D/E) increases if expected profits on the assets of the company are higher than the interest rate on debt. If, on the other hand, expected profit on assets is lower than the rate r, there would be value destruction. Despite this, there’s no contradiction between the first and the second M&Ms: a company leveraging on the market can provide its shareholders an higher profitability, even with the same value of investment. Later, are given some examples over Modigliani and Millers’ models (I) and (II). Example When the company is unlevered, we have that πΈ" is equal to the whole value V of the company. We also know that all of the values considered are ‘on the market’, so, given the number of issued shares, we have that πΈ" = π ∗ π. If the company buys back half of the shares, it means that there are just the half of the shares issued on the market. The money borrowed on the market are used to buy back the shares on the market. 7 Considering to keep fixed prices per each share in order to grand the no arbitrage equilibrium situation, the new value of Equity Levered will be πΈ% = π^_` ∗ π. M&M(I) tells us that the two companies have the same value in terms of assets, so 10M. The value of the levered company will be given by the sum of Equity and Debt levered. This means issuing half of the whole value of the company on debt. It’s not surprising that the earnings for a levered company with the same operating margin is lower net of interest rates (Earnings (L)) = 1M€. Both expected profitability for shareholders (πM ) and EPS increase. In this situation, it could be reasonable to think that the value of shares would increase, because of the higher profitability; but it’s not the case. p = 10€ is the only equilibrium value in the sense that is the only price that can lead to have two portfolios with the same value with this leverage. Our initial investment in the two portfolios is the same: 10€. The first portfolio is composed just by one share of the Levered firm, while the second is built by two shares from the Unlevered company, borrowing 10€ with an interest rate of 10%. In this situation, the value of the portfolio in the two strategies is the same as defined in the alternative demonstration of M&M(I). We have a no arbitrage equilibrium: the outcome is the same, with the same investment at the beginning. The trade-off between the increase of profitability and the increase of risk leads to an equilibrium situation in which price is unchanged: the company is not creating value. 8 The equilibrium in this second situation is granted. Of course, profitability is higher in this cases rather than in the first ones, but we’re talking about different situations, since we have different leverages. Here is the definition of M&M(II) about expected profitability growing linearly with leverage. In portfolios number 1) and 2), the leverage is 1, while in case number 3), we have a leverage of 2 (D/E = 20/10); indeed, these three strategies cannot be considered together. Higher profitability in the third scenario is attractive, as a symptom of a more aggressive leverage: leverage is attractive but is so much riskier. 9 The effect of Risk The ultimate reason why the value of the share is constant (albeit the expected profitability increases), is related to risk. A change in EBIT causes a change in the profit more or less pronounced because of the leverage effect. In the real world, the future EBIT can be estimated, but is affected by risk (volatility). Assume that: MOPQ πN = a has a variance (π @ ), r is known and does not have variance, then: The risk of volatility is increasing with the increase of debt. If the debt is zero, the risk of the business is the risk of the shareholder. If we increase debt, risk increases as well more than linearly (exponentially). The dark side of leverage The negative effect of leverage given by the fact that the spread between expected asset profitability is lower than the rate charged for leverage. We could mention several example of companies that got in trouble because of the significant leverage they were exposed to. Most of the times a professional private equity company buys a small company to invest, they strongly deal on debt. LBOs are the main examples of these cases. Case Study: Ferretti SpA 10 Effect of Taxes on M&M Propositions We’re definitely moving out from the assumptions of M&M. If there are taxes there’s an advantage for the levered company for the possibility of deducting interest from the taxable quote. Its possibility in repaying shareholders is exactly higher for an amount equal to the taxes he is not paying due to interests on debt. This would lead to a saving on taxes for the company: debtholders’ repayment is before taxes, while shareholder’s repayment is net of taxes. This effect is reflected also in cash flow distribution; as said, cash-flow distributed to debtholders is taxdeductible, whereas cash-flow distributed to equityholders is not. k is the cost of capital of the company. If tax saving does not depend on the business risk (i.e. the level of debt each year is pre-determined), then k=r. For sake of simplicity, we can assume having year after year a constant D indefinitely in the future: so today I raise debt which will remain constant in the future. We can demonstrate that: Thus, assets for the levered company are more valuable considering their full present value: π% = π" + π‘d ∗ π· As well, with taxes the profitability of assets ππ¨ will depend on D. The main consequence is that introducing taxes in M&M models is good, since it could increase more the profitability in the future on the back of leverage. The larger is the corporate tax rate, the larger is the difference between the value created between the levered and unlevered company. 11 We could also run M&M(II) with taxes. As said before, with taxes, πN∗ depends on leverage (while under the M&M assumptions πN is constant with leverage). This because π% is depending on the company’s leverage. Moreover, we would have to consider taxation for investors, which is a personal burden. This graph represents how the value of the firm increases linearly with the weight of debt as a direct effect of tax shield. The immediate reaction would be for a company to borrow as much as possible in order to increase the value. In reality, this is not happening, mainly because there are some issues we’re not taking into account. Leverage in practice and objections Should we think that it is convenient to increase D as much as we can? Of course not, and this is not what we actually see in the real world. The main reasons because a company does not generally increase indefinitely debt are: - tax savings are limited (if gross income equals 0, the company would not pay taxes, but then we go negative). Tax losses could be postponed year by year when we go negative, but of course there’s an opportunity cost on that postponements, and the negative loss will be present also in the following year, so there’s a limit. - most countries impose restrictions on the deduction of interests (anti thin-capitalization rules) like Italy. If you have an interest to save on taxes and you borrow due to this we would see companies that are very levered. Probably, the risk linked with this kind of structures in the market is too high to be bared. This aspect is critical considering a company which is listed against one which is not. Of course, a company which is not listed in the stock exchange, which has no really incentives in showing high profits, which has not to be really transparent in order to attract investors, has a strong incentive in going for debt. Small companies, like family one, in order to cheat on taxes were able to issue debt, subscribing it directly with shareholders. This trick helped them to pay less taxes, but now there’s a limit on deductibility which is fixed at a maximum of 20% almost in Italy. Moreover, if tax agencies get aware that companies issue debt buying them back through shareholders at interest rates which are out of the market, this is seen as an unfair dividend distribution and so condemned. - when debt is too high, a number of problems arise under the shape of costs 12 When the level of Debt is too high, some costs for the firm arise: 1) Costs of Financial Distress As the debt increases, the probability of a default increases: the company may not be able to pay back debt in the future. It’s the present value of the expected costs a company shall bare in the case of bankruptcy. If π« » π½, when a default occurs, all the assets of the firm are sold and creditors receive the cash collected according to a hierarchy: suppliers – employees – debtholders. This could happen even if the level of the debt grows or the level of investments (V, in general) goes down because of market condition, with a constant level of Debt. Even if in the best situation in which assets are not sold, debt must be re-negotiated. This implies a number of costs: legal fees, liquidation costs (considering costs in terms of net losses from lowered values of asset divestments) , image damages, tight payment conditions imposed by suppliers and other financiers... Such bankruptcy costs are discounted in advance, even if the company at the current moment is not in default (but it’s getting close and the probability increases!). It’s kind of parachute the company wants to hold its back in case of high risk of default. Each time the leverage is too high, either because the debt is going up or because the level of the assets is going down, the market thinks that the probability of bankruptcy of the company is increasing and, moreover, the market will discount in advance that the probability o have the latter costs is increasing and this affect even more the value of the company. 2) Agency Costs Mainly cost referred to the principal-agent problem. The agent is delegated by the principal (in our case the debtholder invests in the company which is managed by equity investors), but the agent may act opportunistically maximizing his interest and not the principal’s interests. The equity holders and performance responsible could act in order to favor themselves at the expenses of debtholders. The main reasons leading to this fact are: - information asymmetry: the principal is not able to totally control the agent (hidden information) or correctly judge his actions (hidden actions). - moral hazard: the agent has an incentive to maximize his utility function rather than the one of the company and of shareholders. Agency costs do generally characterize the relationship between debtholders and equity holders (also between controlling and non-controlling shareholders), but if the company is close to default these problems are more relevant and costly. Normally, if the company is far from the financial distress this effect is tinier. Closeness to financial distress results in a different level of incentives in actors leading to an higher or lower agency cost. We could consider an example in which there’s a company which has a strong risk of default. The value of the company is financed through a vast majority of debt and a very little equity capital. The company could have three main alternatives: 1. Do nothing 2. Investment A: NPV [+10 (100% probability)] 3. Investment B: NPV [-30 (90% probability); +200 (10% probability)] A is the social optimum: the value for shareholder increases and the risk of distress is lower. This is efficient. B has an expected net present value which is negative, leading to a value which is lower to the other alternatives, so completely dominated as a solution: it’s the worst. 13 Mathematically: 1. Investment A, Expected NPV (E(NPV)) = 1 * 10 = 10 E(V) = 400 + 10 = 410 E(D) = D = 390 E(E) = E(V) – E(D) = 410 – 390 = 20 2. Investment B, Expected NPV (E(NPV)) = 0,1 * 200 – 0,9 * 30 = - 7 E(V) = 400 – 7 = 393 E(E) = 0 * 0,9 + 0,1 * 210 = 21 E(D) = E(V) – E(E) = 393 – 21 = 372 If we split the two situations in investment B, we would have a case in which we have bankruptcy with V=370 against 390 of D and a case in which the company gain value up to V=600. Computing Equity, we have 90% probability of having 0 E, while 10% probability of gaining 200, with an E of 210. The expected Equity is 21. Expected equity can be computed like this because the worst case for the shareholder is the one in which shares are valued zero. Share value cannot go negative. As a consequence, the expected value of Debt is given as 372. The best solutions are: - B for shareholders (maximizing attended profits): the tricky aspect is that for a shareholder the expected gain could run to infinity, while losses are limited to zero in case of a default. This is because of the privilege of the limited liability. This strongly affects their incentives. The expected value for shareholders is the higher at the price of increasing the level of risk of debtholders. - A for banks (reduce the probability of distress) Theories on Optimal Debt definition When defining the financial structure of a company, theoretically, we should have an optimal value of debt in order to maximize the value of the firm. π·∗ shall be the optimal amount of Debt that maximizes the value of the firm. That’s the best trade-off between tax savings and cost of debt. The level of the optimal amount of debt depends also from the characteristics of the firm. In the graph the trade-off is expressed presenting three curves: - Value of the firm with no debt (M&MI) The value of the firm of course is not affected by the increase of debt and it keeps constant. - Value of the levered firm with tax shield (M&MII) As debt increases the value of the levered company increases as the effect of the tax shield which is translated into present value. - Actual firm value 14 Defines how really the value of the company moves as a function of the debt increase net of default costs (financial distress and agency): o with low debt levels, it is true that the value of the company increases as the main effect of tax shield o once a certain level is reached, value would slightly increase until π«∗ which is the optimal amount of the debt. o after this level, all of costs mentioned above start affecting the real value of the company, lowering its value and increasing the financial distress costs. All of the models which show how the value of the company is un-linearly linked with the level of financial exposure and leverage, belong to the so called trade-off theories. Theoretically, if profitability grows the benefit of tax saving is more valuable since the company is becoming less risky. Moreover the costs of financial distress would decrease. When there’s a positive cycle in the market, we would see company increasing their leverage; on the other hand if we’re in a recession we would see companies decreasing the leverage. In reality, we see how things go slightly different. Pecking Order Theory explains this. 15 Pecking Order Theory The Pecking order theory is a competing theory and states that the cost of financing increases with asymmetric information. Companies prioritize their sources of financing, first preferring internal financing, and then debt, lastly raising equity as a last choice. Generally, they avoid the market. There’s a hierarchical method to select one option of financing rather than another. Cash, delaying payables, shortening receivables are the very first options chosen by companies to finance. The market is just the second alternative. So Cash, Debt, Equity. Equity is less preferred because when managers issue new equity, investors believe that managers know that the firm is overrated and managers are taking advantage of this over-valuation. This is clearly demonstrated by Arkelof... The market of Lemons (Arkelof) Arkelof presented the effect of asymmetric information on the market through an example which is useful to understand these dynamic. A lemon is a car found defective only after it has been bought, while a plum is a car with a good quality. As an assumption, in the market, buyers can’t distinguish a good quality car (G) from a lemon (B) at priori. Of course, the value of the latter is much lower than the first one: πO βͺ πk . Sellers perfectly know whether they propose a good car or a lemon; of course, they would have an incentive in saying that they sell type-G cars in order to sell them at a higher price. The information given by sellers is not credible, so buyers attribute a medium expected value considering the probability of distribution of one car type or another. (e.g. if probability of having one type or another is 50%a Ta 50%, l @ m is the value attributed from customers to a single product). Buyers then submit an offer at this value. If this value is lower than the minimum price that good cars sellers are willing to accept, then there will be no more incentive for good cars producers to keep producing high quality cars. As a direct effect, good cars would exit from the market. So buyers will start offering πO when buying cars (signaling effect). This situation of course is a Nash equilibrium between consumers and car producers, but suboptimal (and also the worst possible because the market is destroyed and low quality goods have taken the market). All of this happens because there’s no credible information in the market before acquiring. Companies raising capital are like cars: the more information is asymmetric in the market between investors, debtholders and companies, the more investors will have a lower willingness to invest on an averagely lower value of investments themselves. In the end, assuming that a company is going on the market to raise money, it is likely that the company is overvalued. A company which is undervalued would never go on the market to raise money because it would issue debt with a lower value than the one of the company itself. This may be a reason why generally companies try to avoid going on the market. So what the manager of a good quality company could make to convince of the goodness of the company when going on the market? - Making aware the market of having in its b.o.d. some high level figure to monitor - Dispose information to increase transparency - Auditing in order to make sure the market of the goodness of both their business and information disposed 16 All of these solutions are costly, and so signal is costly in order to be credible. This means that good quality companies have to spend their money in order to convince the market of their goodness. The effect of signals in the market is extremely powerful. Market could react extremely positive to a very simple signal. Cattolica, an insurance company, experienced the positive effect of signaling scoring an increase in their market price of about 15% just because the market trusted the acquiring of Berkshire Hathaway (Warren Buffet). There are also some governance issues to consider when dealing with the process of financing in general: 1. Extraction of Private Benefit: we’ve assumed that agents act with the only aim of maximizing the benefit of the company. In reality, it could be that managers and agents in general could work and use company’s resources with the aim of maximizing their own profitability, which is different than the one of the company. This is another application of the principal agent problem, reflecting in problems between shareholders and managers: investors and shareholders want the value of shares, of assets and business to be maximized; managers want to maximize revenues in order to gain power on their position. There’s another expression of the agency problem between majority and minority shareholders. Shareholders, if heterogeneous, could act for their interest rather than for the one of the whole owners board. Controlling shareholders may extract private benefits from the company (i.e. inefficient use of company cash); such danger is discounted by small minority shareholders. Generally, each time there’s a separation between ownership and control there’s an agency problem. Agency Problems are discounted at the investors level: if investors see that there’s this spread between ownership and control they could discount in advance the price they’re willing to pay to acquire shares since they see the risk of private benefit extracting from controllers. 2. Free Cash Flow Moreover, the potential trouble related to agency problem increase as the cash generated by the company increases. According to Jensen, companies with large ‘free’ cash flows are ‘tempted’ by engaging in suboptimal investments; in this case debt may have a ‘discipline’ effect over managers. Raising debt is considered as a signal to the market: if a company raises debt, they ensure the market that they will not undergo to some inefficient investments, since they’ve to be compliant to debtholders. 3. Covenants One of the trouble of debt investors is a too high leverage. Covenants are able to reduce the risk of default. They are often used to reduce costs of debt for example through: minimum interest coverage (keep the ratio between operating margin and interest on debt above a certain threshold), limit on dividends (in order to retain cash), limit on sale, limit on further borrowing, (every general option of calling back the loan if something changes at a business level in the company)... 17 Private Benefit Example To conclude, how do real companies actually behave in terms of debt raising? The situation is different according to the markets we’re working in. Markets in continental Europe, are called as bank-based system since the role of bank is very important. Historically, companies especially in Italy, are used to rely a lot on banks. This is both positive and negative. The main cons is that if you rely on a single source of finance (as SMIs generally do in Italy) your risk exposure is higher. You have to differentiate financial sources. On the other side, there are some studies saying that one of the benefits for bank-based systems is a strong long term perspective, based on long term relationships. ON the other hand, companies exclusively raising on stock exchange are more shirt term oriented. This is one of the reason why sometimes stock exchange companies are cheating in order to increase their prices. One of the reasons linked with this is the alimentation of the agency problem: if managers are paid in stock options, they’ve got an higher incentive in increasing the short term value of the company. US market are mainly equity markets, with listed companies raising money through investors. In Italy we’ve got a very strong regulation on banking activities. Firstly, only banks can lend money to companies. In other countries, the situation is different: there could be other companies able to lend money to the system, such as insurance companies, funds and so on. 18 Debt Financing and Bonds First of all, debt investors are not owners of a firm: they’ve no ownership right and decisional power on the company. Debt investors are entitled to a return that is contractually known through the payment of an interest rate, that represents the percentage rate of profitability on capital. There are different types of interest rates, according to the way the debtholder is paid: - Simple: the interest I is determined by multiplying the interest rate r by the principal K, which is the value of the bond. π° = π · π², on a single-period basis. - Compound: the interest I is calculated on the initial principal and also on the accumulated interests of previous periods. The interest is not paid just at the maturity but in more tranches during the period. I = (r * K) / m (m is the number of times the interests are paid, in each period) Usually, it’s better to have the second type of rate, because money now are more valuable than money in the future. In the offer B), the repayment of the bond is split in a first pre-payment of 2 and the final of 102, with gives the payback of a 4% interest rate over a bond valued 100. But the real rate to discount the two tranches is 4,04%. The yield (internal rate of return) obtained is computed with the inverse discounting formula. We discount the value of cash flows at the end of the year, increased by the nominal interest rate. By doing this, of course, the true yield on the second case is higher than the one in the first case. The main reason behind this example is that time has a value. The value of time is even higher in this second example. 19 In this case, the repayment involves the value of the bond (100 + 2), the interest on the year net of reinvestment (2) and the interests for the other half of the year on the re investment (4%*2/2). We cannot compare annually the same interest rates if they’re compounded on different bases. There’s a relationship between simple and compound interest rates: - r(EQ) is the equivalent interest rate. M is the number of periods in which the rate is compounded. Ex: I buy a car borrowing at an annual interest rate of 7%, but interests have to be paid monthly (M=12). I want to compute the ‘real’ annual cost of capital. Given such difference, the European Union imposes the publication of the ‘equivalent’ cost of capital (APR, Annual Percentage Rate = TAEG, Tasso Annuo Effettivo Globale) each time there is a loan offering (European Directive 2011/90/EU). TAEG (APR) consider both the compounding effect and costs and insurances added to the rate charged on the consumer. Continuously Compounded Rates In reality, no one is offering a continuously compounded interest rate, but it’s useful at a theoretical level to explain the definition of some securities. As seen, we can say that, up to infinite, the compounded formula tends to its exponential form. The capital is continuously compounded and there’s a continuous growth of the money. So, theoretically, we can both compound interest rates in different rates or on a continuous base. 20 Financial Mathematics on DCF Let us recall the DCF methodology. Basically, the first problem we deal with is the comparison between cash flows with different time windows. With DCF methodology we can shift the cash flows as much as we want, finding both present or future values of cash flows. - Compounding: we find the future value where πΆr is the cash to be invested at date 0, r is the interest rate per period, and T is the number of periods over which the cash is invested. - Discounting: we find the present value where πΆs is the cash flow at date T and r is the appropriate discount rate. This means that we can find both the present and future value of single cash flows, but also the one of collections of cash flows. Financial mathematics leads to the definition of some useful equations: - Perpetuity: it is a constant stream of cash flows (value=C) up to infinity. The present value is computed through the geometric series limit of convergence. If r is positive, the denominator in the future becomes lower and lower: this means convergence. The sum to infinity of alpha^q equals 1/(1-q). - Annuity: is a stream of regular payments (value=C) that lasts for a fixed number of periods T. It is more likely to exist as a contract in respect of the first type. To proof it, we see the annuity as the sum of a perpetuity and a ‘lagged’ negative perpetuity. In fact, if T goes up to infinity, we’re in the perpetuity case. We could start thinking about a perpetuity, which would give us a total value of C/r. If we divide the two times windows, the second window is made by negative contribution to define Annuity. The value we lose up to infinity is -C/r which still has to be discounted at the time 0, so -C/r * 1/(1-r)^T. - Growing Perpetuity: perpetuity with C increasing each year at the rate g < r. An example of these kind of contracts could be pensions funds adjusted with inflation; they grow on average with a g ratio reflecting inflation itself. 21 - Growing Annuity: annuity with C increasing each year at the rate g < r. Which is the most general formulation. We provide an example in order to clarify the previous formulas. Ex. We invest in a pension-fund: we pay C=350€ every 3 months for T=20 years in order to benefit from a life-long rent when we retire. Our money is invested in a fund providing an expected annual return r=4% Compute the compounded value FV. Alternative hint: see as the sum of 4 ‘lagged’ annuities, then compound to 20 years: The first two factors are the representation of annuity. The third term refers to the four compounds of the year, while the fourth one leads to the future value. This is a trick since we cannot use formulas which are annual. Another alternative is the adjustment of the interest rate: we define the equivalent interest rate on a quarterly basis. If 4% is the annual one, we have to convert it in a 3-month periods. Keep attention on not using the formula of compounded interest rates, since this is not an annual interest rate but a quarterly based. We just want to assess the rate per quarter if we want to have a 4% yearly one. We assume now that the investor has some fees to pay in order to get the investment. The insurance company charges a fee equal to € 2 on each payment as well as an initial contribution for medical check-up equal to € 100 paid by the customer. The net effective return is computed on the real cash flow for the investor, so we write 350€. 22 When computing the compounded value, on the other hand, we consider just the amount invested, so (350 – 2) €. The nominal return is 4%, but since there are some charges and costs, the net effect comprehensive of all the fees charged leads to a net interest rate of 3,9%. 23 Interest Rate Term Structure Financial markets in general offer a different return to different investment maturities in the future. For example, some days ago this was the equilibrium on return offered on the European market to risk-free investing. This means that, for example: - Investing on a 5-years basis in € leads to an annual yield equal to 0.76%! - Investing on a 15-years basis in € leads to an annual yield equal to 0.31%! This is a static picture disposed periodically. This chart is built on the German government bonds, which are considered as the risk free investment in the eurozone area. The tricky aspect is the fact that returns are negative. Of course, there will be a different situation and equilibrium for each different currency. Since interest rates are negative, it means that after the lending repayment, the government will pay back less money than the one borrowed at the beginning. Why does an investor have to pay to lend money? The situation is extremely complex. A central role is plaid by funds, which have to invest investors’ money on some securities on the market. Moreover, funds have to invest in very low risk securities, so their obliged to select low risk securities on the market and so now they have to bear negative returns on the market. On the other hand, the European Central Bank wants to keep very low interest rates in order to disincentive bond investment despite of the increase of real assets investments by banks, funds and investors. It is typical that the slope is positive as maturity grows. There’s no real expressed correlation between the length of maturity and interest rate, even if the slope is typically positive because the governments pays back more on long run investments. The trends of the graph are mainly related to expectations. Moreover there’s no correlation between the slope of the chart and the future value of interest rates (a positive slope does not mean increasing interest rates). Looking at the past, we take the example of September 2008 rates at the beginning of the financial crisis after the LB default. 24 in the US overnight yields are larger than 2-years yields! This shows a huge uncertainty in the short run, looking at LB experience. Of course, the long run was expected to be less risky. Europe is looking. Very large interest rates were kept due to a strict monetary policy, with the aim of reducing the growth of the economy, of prices and inflation and to reduce the purchasing power of citizens. On the contrary, in 2006 there was a steady growth of the economy. Central Banks keep rates high not to rise inflation. In US, the growth of the economy was led by the booming of real estate markets. Notice the ‘flat’ curve in the US, which is the situation that banks do not like, since their profit is mainly made on the spread between long term interest rates and short term ones. Negative returns are good for borrowers and money asker on the market to reduce prices, they’re good for governments. On the other side they’re negative for commercial institutions, banks, funds which must invest in low risk securities. On the market, today, low risk securities are costly. Why the interest rate term structure (IRTS) may have different shapes? - Theory of market segmentation: the market is divided among investors who want to invest in the long-term or in the short-term (different equilibria). There is not an answer defining the whole curve; there is no direct relationship for interest rates in the short run and interest rates in the long run. - Liquidity preference theory: investors prefer short-term investments, and eventually reinvest, compared to investing in the long-run: so they ask a premium to invest in long-term assets (explains why the curve is rising). This theory explains why the slope is positive. - Expectations theory: the IRTS reflects expectations about the future yields on the market. As said before, remember that different currencies have their own IRTS: - ‘low’ interest countries: Japan, Switzerland, ... - ‘high’ interest countries: Australia, New Zealand, ... Expectations We start assuming that we have the following interest rates for different maturities. What is the expectation of the market at the end of the first year? So, what will be the return on an investment after one year? According to expectations, we could have two different strategies (A and B) that, if markets are in equilibrium, should give the same result: 25 (π΄) 100 ∗ (1 + 4%)@ = 100 ∗ (1 + 3%) ∗ (1 + π)→@ ) (π΅) so π)→@ = 5.01% If suddenly, due to a new expectation on the market, also π)→@ = 3%, everybody would invest at the maturity of 2 years at 4% rather than splitting the investment on two years in order to have an higher result. In such a case investors can create an arbitrage portfolio borrowing at 3% at t = 0 and t = 1 and investing this cash at 4% onto 2 years. This would mean prices starting to increase on the 2 year bonds, while everyone trying to sell the 1 year bonds would reduce prices on that one. Oppositely, interest rates will balance price increases and reductions: interest rates on the 2 year bonds would decrease and interest rate on the 1 year bonds would increase. This would lead to a new equilibrium. When there is the possibility of arbitrage, if the market is efficient the equilibrium is immediately restored (market frictions and transaction costs limit the opportunity). Nominal and Real Interest Rates All of the interest rates we’ve seen, and we’re talking about in the future are nominal interest rates. - Nominal: the return is computed in monetary terms (starting from cash flows in €, $, £, ....); - Real: the return is computed in real terms, which means taking into account purchasing power (and thus inflation); cash flows are expressed in ‘real’ terms, meaning asset terms. The difference between a real term value definition and monetary terms is inflation: the value of a good is affected by inflation, while money is not. Fisher relation links together nominal and real interest rates: In real terms, the advantage that an investor could have by investing in monetary terms is balanced by the way purchasing power is affected by inflation. Wealth of families is affected by real interest rates. 26 Bonds Bonds are the most diffused type of debt security (99.9% of debt is made by bonds). A bond is a certificate showing that a borrower owes a specific sum. To repay the money, the borrower has agreed to make interest and principal payments on designated dates. There are many types of bonds in the capital markets. We can distinguish bonds according to very different parameters; the first, is the issuer. The issuer can be: - private firms, corporations: Corporate Bonds - local or national governments: Sovereign Bonds (‘Govies’) - international organizations (WB, EIB, ...) The type of issuer is crucial because the main discrimination among issuers is the risk of insolvency. In fact bonds may be disregarded and go in default. Ratings Rating process is a synthetic measure of solvency capability; ratings are issued by independent agencies and can be attributed to a single bond or to a firm, in the short-run or long-run. The ‘bigthree’ most famous agencies are Moody’s, Standard&Poor’s and Fitch. Ratings could also be assessed by investors, banks, having an internal rating department. Of course, it is better at the eyes of the market, to be rated by a professional rating company. To be called so, a rating agency has to be authorized by global markets authorities. A rating agency is mainly composed by two very separated actors: analysts and commercial side, which engage with companies, assessing prices and so on. Having a wall in between is fundamental to avoid conflicts of interest (good rates disposed for higher payments). Rating agencies talk with managers, look at balance sheets, assess econometrics databases on forecasts, check out future perspectives both in terms of finance and business. From this pool of information, they try to assess the risk of default. The main aspects that are checked are: - Leverage, NFP - EBITDA, Operating Margin and cash generation The rating is basically built on accounting data (debt and liquidity) and qualitative assessment. The main parameters are: - Probability of Default (PD): the higher is the PD, the higher is the return on capital required. A premium is charged to the most risky issuer. - Recovery Rate (RR): the percentage of outstanding debts that are received by creditors in case of default. Of course it would be better to have an high recovery rate, which means being able to repay debt holders even in case of default. Every rating agency has its own degrees. 27 Larger yields are requested by the market as the rating lowers. Ratings are split into three main categories of bonds, defining their nature. Junk Bonds are more likely to be extremely risky Equity capital: very high returns, with a full risk exposure. Pensions fund, insurance companies and big financial commercial institutions must invest on investment grade bonds. This could be extremely critical for companies or, mainly governments, being evaluated as BB: all of investments on bonds would be sold creating a vicious cycle. On the other hand, we have to consider that nowadays for financial institutions it is critical to invest on investment grade bonds, since very low risk bonds have negative rates. Spread is the extra return the market asks to a non-triple A borrower when paying back the interests on debt. Examples on rated governments are given in the map. Besides, examples of ratings on companies are disposed by Moody’s. 28 Credit Default Swaps Another measure of the expected probability of default is the Credit Default Swap. The CDS is the ‘price’ (in basis points) of hedging against the risk of default of an issuer. The larger is the value, the larger we shall pay for this ‘insurance’, the larger is the perceived risk. It’s a contract that may be signed with financial intermediaries and private banks. It’s similar to an insurance contract. Prices are reported daily by major providers for largest issuers and governments. The interesting thing is to analyse price trends and evolution, rather than prices themselves. Clearly, the lower is the price, the lower is the risk that the market is expecting. For instance, the market is thinking that now UniCredit has the lowest risk of default. Currency Currency is the second parameter to assess a bond. There are several ways to a approach a bond, considering different currencies. There’s a dimension of the risk directly related to currency. Issuers may raise debt in currencies other than the Euro; in such a case the currency risk also arises. The main aspect affected by this kind of risk is returns, since returns will change depending on the fluctuation of the exchange rate. So, why should an issuer, or an investor, issue or invest in different currencies bonds? Raising capital in foreign currency may be useful to: - Diversify risk related to interest rates - Offset the currency risk of assets and liabilities (ALM, asset & liabilities management). For example: a company exports products in Japan (thus collects positive cash flows in Yen) and then issues a bond in Yen (thus pays cash flows in Yen): if the Yen is devaluated, we will have a lower revenue in terms of change, but debt capital will be cheaper; on the other hand, if the Yen evaluates in respect of euros, debt capital will be more costly, but we will have larger revenues. 29 Maturity Another important parameter is the maturity, which is the time in which the bond will be expired. We could have very different maturity lengths. Maturity can be as short as some months (short bonds), or as long as a century. Typically corporate bonds mature in some years. There are also perpetual bonds, meaning only interests are paid, while principal is not repaid. Typically bonds pay back the principal at the final maturity (‘bullet’), especially for corporate and government bonds, but there are also bonds paying back the principal during their life (‘amortizing’) similarly to bank mortgages. SMEs are the companies more willing to issue amortizing bonds, to balance the risk. The principal is – most of the times – the par value (book accounting value) of the single bond (€ 1,000 is the typical size for retail investors, € 50k-100k the typical size for professional investors). Maturity is also a proxy of the bond risk (larger maturities are associated with larger risk of default and larger information asymmetry). Volatility, as the percentage change of the value of the bond on the market when there’s a change in the interest rates on the market, is linked with maturity. Maturity and interest rates are strongly linked with the first derivative. Long term bonds are exposed to a higher risk of volatility. Coupons Periodically, bonds may also pay coupons (‘cedole’): - Coupon Bond: periodically, in correspondence with the date of payment, a percentage of the par value of the bond is paid to creditors; the frequency of payment is normally annual, biannual or quarterly (or other...). The coupon is always computed on the par value - Zero Coupon Bond (ZC): in correspondence with the expiration date the face value is paid (purchase price<face value). It is the simplest bond we can have. The coupon rate can be fixed (i.e. predetermined, constant or not – the latter are for example ‘stepup’ or ‘step-down’ bonds) or floating (determined in the future looking at a benchmark, like interbank rates Euribor / Libor + a spread, or other measures (an index, a commodity... anything that is observable on the market). The correlation may be also negative! E.g. Coupon = 7% - 2*Euribor. The coupon could also be defined through a correlation with stocks on the equity market, commodities, or everything that could be objectively visible on the market. A cap and/or a floor can be introduced in the coupon value. We could have also the case of particular options: - Callable Bond: it is a type of bond that allows the issuer to retain the privilege, but not the obligation, of redeeming the bond at some time (call dates) before the bond reaches its date of maturity. There are also puttable bonds (in such a case the option is granted to the investor). Clearly the option will be exercised when convenient for the issuer (callable) or to the investor (puttable). These kind of bonds are useful for small companies, with a lower bargaining power on the market. This kind of bond could be useful for the issuer to redefine its financial position. For instance, if at a certain time on the market interest rates on debt are cheaper, the issuer could be incentive to repay the bond, issuing a new one with a lower interest rate. Something similar to callable or puttable bonds are covenant, as contract through which the issuer, or the buyer is obliged in doing something when a specific situation occurs on the market. - Convertible Bond: it is a type of bond that the holder can convert into a specified number of shares of common stock of the issuing company (or other securities); obviously the conversion is exercised if the market price of the equity securities is ‘sufficiently’ large. 30 The option is in the hands of the investor, so he will decide whether to do nothing or to exercise the option converting the bond into shares. An investor could have the willingness in exercising the option is shares grew in value. This could be a win-win situation: for the investor since he will experience an upside opportunity (meaning he could have extra profits); on the other side, the company will be happy since there would not be any cash outflow. Seniority Seniority refers to the priority of repayment in the event of default of the issuer. Each security, either debt or equity, that a company issues has a specific seniority – debt has a major seniority than equity, but also different classes of debt bonds can have different seniority; to a higher seniority corresponds a lower return (and risk). Typically, banks have priority. Bonds are generally subordinated with respect to bank debt. Priorities define different levels, which are repaid from the highest to the lowest level in case of defaults: - Pari Passu: no priorities - Senior Bonds: first priority - Junior/Subordinate Bonds: second priority Of course, these types of bonds have different risk exposures. This has to be extremely clear and transparent from the issuer, since, of course, even if a Junior and a Senior are sold both with an AAA credit rating, their risk exposure is completely different. Moreover, bonds could be: - Secured bond: it is a bond with a guarantee that consists in the possibility for the creditor to have a compensation in case of default (i.e. pledge or guarantee) - Unsecured bond: is a bond without a guarantee (and thus a larger offered return); most of the corporate bonds are unsecured Bonds Evaluation Bonds are mainly valuated through a DCF method. Risk free assumptions means we have to look at the lowest interest rate in the area in which we’re investing. Always, the price of bonds on newspapers are built in a different way, by building a proportion with the par value, the discounted price paid and an adimensional par value of 100. With an adimensional price, we can compare different bonds on the market. Since this proportion is so built, the price we pay to acquire the bond is always lower than the par value, so than the price we will be paid back at maturity. 31 If we want to estimate the future value of the zero coupon bond, we would find something like this. This is simply linked with the reason that day by day, the investor is earning the interest rate of this type of security. If suddenly interest rates grow, an efficient market would immediately discount this fact, lowering the value of the bond and so the price. This would not be a good thing for the investor, but, efficiently, this is the only way to keep the same final value at maturity. A lower value with higher interest rates lead to the same principal value at the end. The same thing happens if interest rates decrease: the price of the bond increases, decreasing as the time of the repayment gets closer. What could happen if interest rates are negative? Negative interest rates will lead to an higher price than the par value, this means the investor is paying more than what he’ll receive at maturity. This will result in a decreasing trend of value up to maturity since the market is delivering to the investor a negative return. The longer is the maturity, the larger will be the premium to pay compared with the par value of the bond. We have to notice that, for each period of time we consider we have to select the proper interest rate value. If we have a multi-year period bond, each year will probably have its interest rate if it is floating. In the latter example, we have that the price is over the par value. This means the true price of the bond is computed considering the proportion, leading to a full price of 10.194 €. So, the valuation of a bond depends directly on the fact that it is a Coupon, on the time spans of coupon payments and on the interest rate. Example For example, we could consider a Coupon bond, with the following data: - C% = 3% yearly - r% = 2% We want to assess the value of the Bond immediately before the payment of the coupon at t = 1 and 2, and then immediately after. 32 - At t = 0: πr = 3~1,02 + 103~1,02@ = 101,94 - Immediately before t = 1: π)€• = 3 + 103~1,02 = 103,98 - Immediately after t = 1: π)T• = 103~1,02 = 100,98 - Immediately before t = 2: π@€• = 103 Graphically, the situation can be plotted with price variations. One second before the payment of the Coupon at year one, the value of the bond increase since we’re closer to a payment. The present value is increased since the denominator in discounting is lower. Immediately after the payment, the value of the bond decreases. This vertical slope is a technical phenomenon, occurring after a repayment, otherwise, there would be a free value creation. This happens for every kind of security, as for dividends. On the other hand, at one second from the maturity, we will be willing to pay something close to the par value to acquire the bond, since it is no more profitable in terms of coupons. The growing or lowering value in respect of the par value depends on the fact that the coupon pays more or less than what the market asks through interest rates. In this graph, 5% is exactly what the market wants. The price is very stable and very close to the par value, since the coupon pays each year exactly the opportunity cost of value in the market. A coupon which will pay more than the opportunity cost on the market has an higher value at the beginning, but it will align with the market in the end. This fact is expressed with a coupon higher than the interest rates on the market. The opportunity cost is given by the fact that, our best alternative to the coupon is the market itself with r% as an interest rate. On the other hand, if coupons pay less than the interest rates on the market, the value at time zero is lower than the par value, increasing its value as we move towards the moment of the maturity repayment. The price is not indicative of the conveniences of bonds on the market, since an higher price bond will have an higher value, paying higher coupons. 33 So, in general, we could have two main situation considering the relationship between coupons and interest rates, which define two main trends of bonds value up to their expiring. We can define two theoretical different prices, whether if we take into account or not the value of the coupon repayment with the value of the bond itself. - Dirty Price (‘prezzo tel quel’ πr ): it is the price of a bond that includes the present value of all coupons. It is the price that we actually pay on the market. Clean Price (‘corso secco’): it is the price of a bond that does not include the present value of all coupons, so: Clean Price = πr – accrual The accrual (rateo) is defined as the percentage of value covered between the payment of two coupons considering a specific moment in time. In other words, it is a measure of the next coupon that has already ‘matured’. πππππ’ππ = πΆ ∗ Δπ‘ (ππππ π‘βπ πππ¦ππππ‘ ππ π‘βπ πππ π‘ πππ’πππ) Δπ‘ (πππ‘π€πππ π‘π€π πππ’ππππ πππ¦ππππ‘π ) The clean price is useful in order to compare the price of different bonds with different maturities and coupon structures. The prices displayed on secondary markets are typically ‘clean’ prices. Illiquid and risky bonds In the previous equations we discounted cash flows using a ‘risk-free’ assumption on bonds. Therefore we assumed to work with risk-free bonds. If in the case of a non-risk-free bond, we shall increase the cost of capital r. In particular, the higher risk is expressed through: πs∗ = πs + π πππππ The spread will depend on the rating or risk of the bond. The larger the risk, the larger the spread. The same thing is done with illiquid bonds, meaning bonds not traded on exchanges or rarely traded. The premium for liquidity will be larger, the less liquid the market is. The graph shows historical main European spread rates in respect of the German bund, particularly referring to the sovereign bond crisis of 2011. 34 Yield to Maturity The yield to maturity of a bond (rendimento effettivo) is the rate that satisfies the equation: Q πr = ’ s–r πΆs “1 + π_”” • s + πΉ Q “1 + π_”” • It is a kind of IRR, as the rate allowing to have 0 discounting all of the cash flows in the time window. Despite of this, it is not a preference driver. - If IRTS (Interest Rate Term Structure) is flat à π_”” = π - If IRTS is not flat à π_”” is the average of πs - If IRTS is growing à long term bonds have π_”” higher than short term ones Duration The duration of a bond is the weighted average of the maturities of the coupons and principal payment, a sort of ‘time barycentre’. It represents the time to wait to have money back, assuming the owner of the bond wants to get them back at π‘ = π‘) : Q 1 πΆs ∗ π‘ πΉ∗π π· = ∗ ˜’ + › s (1 + πs ) (1 + πQ )Q πr s–s™ For a zero coupon bond, D = T so duration equals maturity. In general T ³ D. Duration is important not just in finance but in asset and investment management in general. The same computation as a weighted average of the investment could be computed on every project, asset or liability. Volatility Duration is proportional to volatility, which is the percentage change in the market price of a bond when interest rates change: 35 ΔPr π π= r Δr By definition, π has negative value, since when markets interest rates rise, the price decreases and the other way around. A most effective an simpler to get proxy of the value of π is given by the following estimation: πœ ≅ −π· (1 + π_”” ) Where π_”” is a good estimation to compute volatility. Each time we have a bond in our portfolio we’re subject to the so called interest rate risk: interest rates floating result in a change on the price of the bond itself and so on its value. Bond Taxation Bonds from the investor country and not from the issuer one. Bonds could be taxed in three ways: 1. Coupon Taxation: individuals are subject to a withholding tax (in Italy the tax rate is 26% for corporate bonds and 12.5% for domestic and foreign government bonds); 2. Capital Gain/Loss taxation: if there is a capital gain (tax rate of 12.5%/26%); if there is a capital loss, it can be offset with future capital gains within 3 years. 3. Issue Discount: it is the difference between the par value and the initial issue price of the bond. When the issue discount is positive, it is taxed with a tax rate of 12.5%/26%. Taxation is never included in the valuation of a bond as we did until now. Of course, mainly for retail investors, it is important to assess taxation as the result of the fact that different retailers from different countries could have very different taxations. Therefore we can compute the YTM net of taxes, including them in cash flows. In the following pictures is given an example of a typical bond issuing. 36 Bond Issuing Bonds can be issued on the primary market in three ways: 1. Auction: The price is offered by various investors that compete with each other; in this way issuers can learn about the demand function and decide the quantity (π) and the issue price (π). Italian government bonds are placed in this way. The relation between price and quantities reflects exactly the relation between price and quantities for physical products: there will be a producer surplus, which in this case results in the total capital collected by the issuer, and a consumer surplus which is represented by the investor surplus. 2. Private Placement: bonds are offered to a restricted number of investors. Usually a survey between institutional investors is carried out to probe the willingness to invest and the average price. PP is the cheapest way to place a bond and is often used by small firms and corporations. 3. Public Offering: It is similar to a private placement, but instead of targeting a small pool of investor, I target the whole market. It is a solicitation of public savings; in this case the market authority (in Italy CONSOB) requires the publication of the prospectus. PO is the most expensive and risky way also because the public may not appreciate the offer. PO are generally used by companies wanting to raise a huge amount of money and to companies wanting to target retail investors. When issuing a PO, there are multiple costs associated to bare in order to be on the market: o Compliant Costs: institutional approval of the prospectus and of disclosures o Legal Costs: the company needs to be covered in any case o Marketing Costs: the company has to market the offer through advertising on newspaper or online channels 37 Bond Listing Bonds can also be traded to the secondary market (exchanges). Bonds on the secondary market are much more liquid, since they can be sold easily to other investors. MOT is the market for bond retail investors in Italy. Generally, the secondary market of bonds develops both on local and international markets. There are also other alternative trading facilities like TLX. The Italian bond market is one of the most liquid and efficient, given the huge amount of Italian public debt (€ 2.02 trillion). Italian Sovereign bonds, as at June 30th 2019 are (source MEF): - BOT (zero coupon, 3-12 months): € 116.6 billion (5.8% of total) - CTZ (zero coupon, 12-24 months): € 55.0 billion (2.7%) - CCT (floating coupon, 2-5 years): € 135.9 billion (6.7%) - BTP (fixed coupon, 3-30-50 years): € 1,677.6 billion (82.8%) - others (for foreign investors): € 40.1 billion (1.9%) Average weighted duration is about 6.79 years. This means that on average every 6.79 year the full debt of the Italian government is refinanced. Mini Bonds In order to face the credit crunch, EU countries (and Italy above all) introduced new regulations allowing SMEs to more easily collect funds with bond emissions (‘mini- bonds’). Since this moment, particularly SMEs used to fund through banks since they were not able to reach the bond market. Nowadays, it is easier even for a small company to issue debt on the market, resulting in the generation of a new market. Parallelly, a very new regulation made possible for small companies to use crowdfunding not just to raise equity capital but also debt capital, even if only by professional investors or banks. It is interesting that banks actually prefer rather than lending money to small company, to buy bonds from them. This is not really due to financial or accountability accounting, but to the effect of Basel 2 and 3 regulation on capital requirements for loans. Having bonds instead of loans makes these regulation stricter. The Italian Exchange opened a new trading facility for professional investors willing to trade on minibonds (ExtraMOT PRO3), with low fees for issuers. As at today, considering bond issues lower than € 50 million, about 300 Italian SMEs raised capital on the market. Mini-bonds include also commercial papers (‘cambiali finanziarie’) suitable for working capital financing. On average, the maturity of this kind of debt is about 5-6 years, with an average coupon of 5-6%. 38 Shares When a firm needs to raise capital for new investments, one option is to issue equity. Besides debt capital, they represent credit capital for shareholders. Shareholders are residual of the profits of the company: after all of the stake holders and debtholders are paid, net of taxation, shareholders can be paid. Cash flows from equity investments from a shareholder point of view are related to: - Dividend Yield: Dividends are generally paid on a regular basis, even if shareholders may decide to plough back profits and reinvest them into new opportunities. There could be so the opportunity to retain earnings without paying dividends actually. - Capital Gain: The price of shares on the market lead to a gain when they are sold. Differently from bonds, there’s no maturity for shares. The equity capital of a firm may be composed of different classes of shares: 1. Common Shares (‘azioni ordinarie’): they have standard features provided by the law and the company statutes so they are regulated. 2. Preference Shares (‘azioni privilegiate’): they have limiting voting rights – shareholders can vote only in the extraordinary meetings - . On the other hand there is a benefit consisting in a priority in the distribution of dividends, compared to shareholders owning common shares. 3. Saving Shares (‘azioni di risparmio’): they don’t have voting rights and also in this case the benefit consists in a priority in the distribution of dividends, and furthermore in a larger dividend compared to common or preference shares. More generally, there are several degrees of freedom in the opportunities to issue heterogeneous classes of shares: restriction in selling, voting power, priority in case of liquidation of the assets, convertibility into other classes of shares, … the definition of the characteristics of shares are given according to relevant national rules. To give a very simple example, the equity capital of TIM-Telecom Italia is divided into 15,203,122,583 common shares, and 6,027,781,699 saving shares. Saving shares are non-voting shares and offer: - A priority on the distribution of the dividend equal to 5% of the par value of the shares – i.e. 5% of € 0.55 = 0.0275 – . - An extra dividend compared to common shares, equal to 2% of the par value. The par value (‘valore nominale’) is the accounting book value of equity. In June 2019, the last dividend has been distributed: 0 to common shares, € 0.0275 to each saving share (the same happened in 2015, 2016, 2017, 2018). Note that the company registered a loss in 2018, and had to finance the dividend with reinvested profits from the past. Facebook Inc. as of September 30th 2015 had 2,825,921,446 shares of common stock outstanding (2,268,131,435 Class A common stock and 557,790,011 Class B common stock, as instituted in 2009). Class A shares are listed on the New York Stock Exchange NYSE (you can buy them) and they have one vote per share. Class B shares are not listed on a stock exchange and are owned by founders, early investors and managers only, with 10 voting rights per share. Mark Zuckerberg owns 28% of Class B shares only... this means he has anyway the majority of the total votes! In June 2016 the company announced the issue of Class C shares, with no voting right, but the decision has been revoked due to investors’ complaints. 39 Tax impact on shares Both Dividends and capital gains/losses are subject to taxation. In Italy dividends received by domestic investors are taxed differently depending on the investor typology: - Individuals: there is a withholding tax rate equal to 26%; - Unlimited partnerships: 58.14% of the dividend is taxed at the partners’ income tax rate which is the tax rate of the company (41.86% exemption, so 41.86% * 58.14% * Dividend). This limitation is given by the fact that dividends are already net of taxation at a company level; - Corporation: 5% of the dividend is taxed at the corporate income tax rate (95% of exemption). Capital gains/losses taxation depends on the partner type: - Unqualified Partners, ∫small retail investor: Tax rate 26% - Qualified Partners, big institutional investor: After 12 months, anyway, only 95% of the capital gain is taxed (‘participation exemption’, PEX) The TOBIN TAX is a tax introduced in 2013 only for Blue Chips shares (and other financial securities like derivatives) and is due each time shares are traded (tax rate = 0.1%). This tax was introduced in order to avoid speculation. Differently from before, it is not a tax on profit but is a kind of fee to pay to run operations. From January 2017 tax exemptions are granted in Italy to investments in securities held for at least 5 years, under certain restrictions (“PIR, piani individuali di risparmio”). Yet the industry at the moment is stuck because of changes in the regulation decided in 2019 that created troubles to fund managers. Shares Evaluation Shares evaluation is similar to bonds in terms of cash flows, but the difference is that dividends are not contractually driven and profit is residual. To evaluate shares we still use the DCF (Discounted Cash Flows) method, but we cannot use π” as a discount rate. We have to use an appropriate, which is the cost of equity capital, which included a risk premium to π” : π_ = π” + βπ. The risk premium quantity could be defined through different models and assumptions. The most popular model to evaluate the cost of capital is the Capital Asset Pricing Model (CAPM): π_ = π” + π½ ∗ “π¡ − π” • Initially, we assume that dividends π·πΌπs with (t = 1, 2, …, n) are paid annually and π·πΌπ) is paid exactly one year after the evaluation date. We could have even different assumptions over the definition of the premium rate. Standard and Poor’s offers an estimation of the Risk Premium Index, which directly measures the spread of returns of U.S. stocks over long term government bonds. 40 The risk premium changed with high volatility in the last 30 years as a reflection of different risk and return situations. 41 The Dividend Discount Model (DDM) In the Dividend Discount Model the price of a share is equal to the present value of future cash flows – i.e. dividends and expected price – . If we buy a share now and decide to sell it after an year we would sell it at P¢T) : πr = π·πΌπ) + π) π·πΌπ@ + π@ , π€βπππ π) = ππ‘π.. (1 + π) (1 + π) So, substituting the formulas and considering an infinite time horizon without selling shares we have that: ¤ π·πΌπ) π·πΌπ@ π·πΌπs πr = + + β― = ’ (1 + π)s (1 + π) (1 + π)@ ππ π·πΌπs = ¦¦¦¦¦ π·πΌπ → πr = s–) ¦¦¦¦¦ π·πΌπ π Note that under the DDM the price of a share is always depending on the expectations about firm profitability in the future. If dividends are constant, like in the second case, through time, the situation is exactly the one of perpetuity, so we can compute the value of the shares through perpetuity. The market is supposed to change according to changes in the expectations. In particular it is possible to create expectations on the payment of dividends: generally analysts job on the market has the aim of understanding which are the expectations on dividends payment. If the market sees some unprofitability in the future the price is going to decrease, vice versa it could increase. Parallelly, also k is changing considering different volatilities, premium free risks and so on. To provide an example, we want to compute the equilibrium price of a share that presumably will pay this series of dividends: π·πΌπ) = 0.30€ π·πΌπ@ = 0.33€ π·πΌπ¨ = 0.35€ We expect that at T = 3 – after the payment of the dividend –, the price of the equity will be 25€ and k = 12%. Using the DDM method: ¤ πr = ’ s–) π·πΌπs πQ 0.30 0.33 0.35 25 + = + + + = 18.575€ s Q @ ¨ (1 + π) (1 + π) (1.12) (1.12)¨ 1.12 (1.12) The Gordon and Shapiro Model The Gordon and Shapiro Model uses the growing perpetuity formula and assumes that: 1. The dividend growth is constant for the future (πs = π) 2. The dividend growth rate is strictly lower than the discount rate (k > g) πr = π·πΌπ) π·πΌπ@ π·πΌπ) π·πΌπ) ∗ (1 + π) + + β― = + +β― (1 + π) (1 + π)@ (1 + π) (1 + π)@ Through the growing perpetuity formula, we get: πr = π·πΌπ) π·πΌπr ∗ (1 + π) + π−π π−π π€ππ‘β π > π We have to note that if g is larger than k, the sum will not converge but explode. 42 It summarizes the main determinants of stock prices: current profitability, expectations about future growth and cost of capital. Companies could have a good idea of future profits, so it’s not that difficult through exhibits and so on to predict which could be the following years growth of revenues. This definition through growth perpetuity includes all of the factors that generally affect prices: current profitability, expectations on future profitability and risk. Useful share ratios - Earnings per share: πΈπππππππ s π It is a very quick measure of the profitability the company is able to offer to each single shareholder. Book value per share: πΈππs = - π΅πππ ππππ’πs π It is one of the few time in which an accounting value is used. Book value is the issued value of equity including retained profits and everything linked with equity. Payout ratio: π΅s = - ππ s = - π·πΌπs (0 ≤ ππ ≤ 1) πΈππs It could be comprised between zero and one, in the first case when the company not to pay dividends and just to retain earnings while in the second case all of retained earnings are payed as dividends. Typically, companies stay in the middle. Obviously, if there’s a loss there’s no dividend for everyone so the payout ratio will be zero. Ploughed-back ratio: βs = 1 − ππ s - It is the symmetrical value of the payout ratio. Return on equity: π ππΈs = - πΈπππππππ s πΈππs = π΅πππ ππππ’πs€) π΅s€) Growth rate of dividends: π·πΌπs − π·πΌπs€) πs = π·πΌπs€) All of those variables are linked together. We could then rewrite them in order to highlight interest relations. 1. πΈππs = π ππΈs ∗ π΅s€) By definition. 2. π·πΌπs = πΈππs ∗ (1 − βs ) = π ππΈs ∗ π΅s€) ∗ (1 − βs ) 43 The computation is made starting from the definitions of the ploughed-back ratio and of the payout ratio. 3. π΅s − π΅s€) = πΈππs − π·πΌπs = βs ∗ πΈππs Growth is financed by ploughed-back profits. This means that any increase in the accounting value of the equity capital can only be linked to the part of the profits on which we don’t pay dividends and so we reinvest in the company. 4. π(π΅) = O® €O®¯™ O®¯™ = °∗M±²® = βs ∗ π ππΈs O®¯™ Growth increases with h and ROE. We compute the growth rate between two years, basing our computation on book value. The more we retain profits the larger the amount of the growth will be. Assuming that h and ROE are constant in the future: 1. π(πΈππ) = M±²® €M±²®¯™ 2. π(π·πΌπ) = RPa® €RPa®¯™ M±²®¯™ RPa®¯™ = = ³´M∗O® €³´M∗O®¯™ ³´M∗O®¯™ = π(π΅) = β ∗ π ππΈ ()€°)∗M±²® €()€°)∗M±²®¯™ ()€°)∗M±²®¯™ = π(πΈππ) = β ∗ π ππΈ By taking these assumptions, we mean that policy of dividend payout is constant every year since h is constant. Moreover we assume ROE constant. Taking this formulation, we could say that if the company is not reinvesting anything, so there’s no retained earnings, then g is going to be equal to zero. Under these assumptions we can recompute the Gordon Shapiro formula: πr = (1 − β) ∗ πΈππ) (1 − β) ∗ π΅r ∗ π ππΈ π·πΌπ) = = π−π π − β ∗ π ππΈ π − β ∗ π ππΈ π π π ππΈ 1−β ¶∗ (π > β ∗ π ππΈ) π ππΈ π 1−β∗Y π Z Through the final formulation we say that, if the company is not reinvesting any profit so h equals zero, g is equal to zero and the company will not grow. Taking ROE = k, we obtain that the equilibrium price on the market is equal to the book value of shares. No matter if the company pays or not dividends, since the cost of capital is the same as the return offered on the equity capital. πr = π΅r ∗ µ 44 But is growth a sufficient condition for creating value? As h increases, share value increases with a more than linear relation. In particular this picture shows how the shares’ value is related with the profit reinvestment policy (h) and the return on equity (ROE). If h = 0, the company pays back every dividend each year. So, we’ve got a growth which is linearly proportional to ROE / k. In general, the share price rises with the increase in firm profitability and in particular: - When ROE > k it is efficient to retain profits in order to seize investment opportunities. If h grows, so P grows (the limit is given by the value ROE/k = 1/h). By saying that ROE is greater than k, this is a signal of extra profitability for shareholders, which gives higher value to them and their return. Having extra profitability, shareholders are willing to invest retaining profits in order to have even higher returns in the future. - When ROE = k the dividend policy is irrelevant. Every single euro invested in the company gives a profitability which is exactly equal to shareholder’s expected one. - When ROE < k it is not efficient to retain profits because it destroys a part of the firm value. The more h grows the more value is destroyed. Shareholders are willing to have k as a normal return, but the company is able to provide them a lower value. This means that the value of money today is extremely higher for shareholders who prefer to receive dividends in order to reinvest their money in higher return investments on the market. The very worst case is the one in which extra profitability is generated but retained earnings are not reinvested. This signals to shareholder a scenario in which the company has no idea on how to use extra profits. Shareholders could ask for extraordinary dividend on retained earnings in order to make that money meaningful. We demonstrated that g not null is not a sufficient element to say that the company is creating value. It depends on the value of ROE / k. Present value of Growth Opportunity The share price can be split into two different parts, with ROE and h constant. πr = (1 − β) ∗ πΈππ) (1 − β) ∗ π΅r ∗ π ππΈ πΈππ) β ∗ πΈππ) ∗ (π ππΈ − π) = = + π − β ∗ π ππΈ π − β ∗ π ππΈ π π ∗ (π − β ∗ π ππΈ) We’re splitting the value of the equilibrium price of a share on the market in two different components. The first part is a perpetuity which provides the present value of the future profits in absence of growth. This would be the price of shares for the following 12 months, in the case that every year, EPS are distributed totally under the form of dividends. Whereas the second component synthesizes the current value of opportunities for future profitability (PVGO, Present Value of Growth Opportunity). In the specific model, two conditions must be fulfilled so as the value of a share of a firm is greater than its value in the absence of growth: 45 - h>0 ROE > k Generally, for every value in the future of the ROE and every value in the future of the PR, which both can change, the formulation of the two components could be written as: πr = πΈππ) + πππΊπ π In general, for each ROE and h. The firs element is not linked with growth and retained earnings, so it is like there’s no possibility of retain end reinvest earnings, while the second term defines future growth opportunities. Both the terms could be negative, depending on performances. For instance, the first term is negative when profits are negative. Each time we have a company which is not profitable in the short run but has important market capitalization and share price it means that the market sees a huge present value of growth opportunity. The best situation is of course the one in which both the terms are positive. If the first component prevails on the second one the share is commonly indicated as an income stock (or value stock). If the second component prevails on the first one the share is commonly indicated as a growth stock. There’s not a better stock between an income or a value stock, it depends on the case. The main thing is that they’re completely different and handled in completely different ways. As an example, a cash cow company could generally be an income stock company, operating in a scenario in which margins and revenues are high but no growth is seen in the market. On the other hand, fast growing, start up or tech companies operate in high growth businesses. Generally, they could be addressed as growth stock companies. Generally, this kind of stocks are much riskier. The graph shows average annualized total returns on growth or value stocks in different time periods. Different time periods revealed how investing in value stocks have been better than investing in growth stocks or the other way around. A strong issue by analysts is the one of understanding the distinction between the two kind of stocks and foresee on those. Example 46 We assume that h and ROE are constant in the future. With an increase in k, the price of shares will go down probably because of the higher risk exposure, but we’ve no growth for the company. We’re exactly in the point (1; Book Value) in the graph. Other ratios which are very often used in finance are: - P/E Defined as the price of the shares on the market at the beginning of the year divided by the value of the EPS for the following 12 months. This could also be computed as Market Cap / Total Earnings. It is extremely useful to assess the valuation of a company especially in the same industry. - M/B Defined as the ratio between the price of the shares on the market and the book value of each single share (Price to book ratio). It is used to assess the values of intangibles for a company. When this ratio is particularly high it means that the price of shares is significantly larger compared to the accounted equity value. By multiplying by n numerator and denominator the multiple could be defined as P / B, so market cap over book value. This highlights those values which are not accounted related to intangible assets like competences, brands, image, knowledge on the market and so on. This is a reason why we can say that accounting is not the reflection of the value of a company. These parameters could be computed also on a world level on the market. For instance , at a world level, these ratios are the reflection of the possibility of having a bubble on the market. They’re the reflection of the whole market expectations for the future. If the ratios are low it means that the market is expecting to face a recession. These ratios are directly linked with stock pricing we’ve seen before. In particular, they’re proportional to the growth value of companies. For instance, this table shows us how these food companies are priced on the market in respect of their earnings. By comparing P/E with EV/EBITDA, which is a pure operating performance ratio, we can see what influences the price of companies on the market. If two companies have the same or similar operating results, it could be that their price on the market is different due to several aspects. One of those could be the financial structure, or governance issues, intangible and branding weight on the value and so on. 47 Dividend yield is the ratio between the market price of a share and the estimated returns related to that share. Dividend yield is affected by the so called dividend smoothing which is a phenomena in payout definition lead by the management in order to keep dividend growth for shareholders constant in time. This means that when profits are higher, payout ratio is a bit smoothed while when profits are low the payout ratio is enhanced. This has the only aim of convey stability. Equity research and Fundamental Analysis Fundamental analysis looks at the fundamental of a company with the aim of defining the expectations on the company’s returns looking at the capability of the company itself to generate profits and cash in the future. It’s a forward looking methodology. One consequence of this methodology is that, most of the times, what happened in the past is not really relevant in assessing future expectations. Basically, if we believe that markets are efficient, we now that actual market prices are already the reflection of future expectations. Analysts role is the one of making these valuations. Analysts are extremely specialized on specific business areas or countries, since they’re supposed to know everything about their sector and upcoming changes in the sector. The typical cover of an equity research analysis is shown below. It is a kind of abstract or executive summary of the estimations and results of the analysis. The main aim of the analyst is to provide a recommendation whether to buy, sell or hold stocks of the analyzed company depending on his forecasts. Capital Asset Pricing Model Is the main model used in order to compute the expected cost of capital for the equity market. π_ = π” + π½ ∗ (π¡ − π” ) Where π½ is a volatility parameter linked with the specific company. Besides, this parameter is able to explain about 30% of the volatility of shares on the market since estimating it, even if it’s complex, is not really satisfactory. This is the reason why we need to find alternative models to assess volatility of shares on the market. There are different models in parallel to the CAPM. 48 Arbitrage Pricing Theory Arbitrage Pricing Theory (ATP) posits that stock returns are explained also by both firm-specific factors and macro-economic factors, such as: - GDP - Inflation rate - Trade surplus in a single country Despite CAPM which is a single factor model, the ATP takes in to account more variables to explain the expected cost of capital for a company. Fama and French three-factor model As said, the traditional asset pricing model, CAPM, relies only on one variable to describe stock returns (the ‘beta’ vs. market return). In the 1990s, Eugene Fama and Kenneth French introduced a three-variable model. They observe that two classes of stocks generally perform better than the market as a whole: 1. Small caps: SMB (Small minus big portfolio) 2. Stocks with a low Price-to-Book ratio or income stocks: HML (high minus low portfolio) They then added two factors to the CSPM to reflect a portfolio’s exposure to these two more factors: π = πΆ + π·π ∗ ππ + π·π ∗ πΊπ΄π© + π·π ∗ π―π΄π³ So, variables taken into account are: - π½ Size P/B ratio: for a value stock the P/B ratio is lower, while for growth stocks it is larger. We assume to have two twin companies in terms of business, targets, financial structure and operations, with the only difference in size. We should observe that the valuation of the equity capital of the bigger company is larger by a factor x than the smaller company (let’s say 10x). So, EBIT of the large company is 10x the EBIT of the smaller one. IN reality, this is different: the market seems to value more larger companies. In other words, small company are undervalued. The reasons behind this could be linked to a matter of risk, since smaller companies are considered to be riskier. A stronger hypothesis is linked with the point that big institutional investors, hedge funds, prefer to invest in bigger companies. This because big investments by big institutional investors can be held only by large companies having a huge floating cap, in order to have a negligible effect on the demand and the supply of shares of that companies – i.e. acquiring 50 million dollars of shares in a company with 1000 billions market 49 capitalization leads to a negligible effect on the equilibrium of shares demand and their price –. Moreover, small companies on the market are not covered by any analyst since it is to expensive. If we say that small companies are undervalued by the market, it means that the market itself asks for higher returns for small companies on the market, while it is good with lower returns of overvalued big companies. Moreover, the market is happy with lower returns on growth stocks compared with returns gained on value stocks. Carhart adds a fourth variable, which is liquidity. Moreover, a further model on the same wave includes also market momentum. Theoretically, we could consider a regression model including all of the factors that are correlated with the value of stocks. All of these valuation methods are empirical, meaning that these are the results of analysis of past data from different company categories in the US and worldwide market. Performances and relative performances (performances compared to the general market index) have been computed. This examples shows how a model is run in order to make estimation of factors affecting the value of returns on the credit market. In this case, several elements are considered and regression analysis is computed considering correlation factors and t-tests in order to assess the validation of the statistics. For instance, the analysis shows that one month excess returns have a negative correlation with the growth of returns in the following months. On the other hand, a twelve month excess returns has a positive correlation, meaning a future possible growth. All of the other variables are so assessed with different correlation factors. Explaining these correlation with rationality is not always possible. Sometimes correlations are related to rational expectations, sometimes there’s not rational meanings. This results in the growth of importance of technical analysis. Technical Analysis According to the Efficient Market Hypothesis (EMH) by Fama (Nobel Prize 2013), stock prices only reflects expectations about the future. The more markets are efficient, the faster they discount such information. 50 Therefore under this assumption past prices cannot explain future prices. When performing a DCF, or a relative valuation, we do not need to know information about the past or historical data. Technical analysis moves from the opposite idea: past prices and trades can explain future prices. Recurrent trends and figures can predict stock returns. If we’re able to recognize these trends in the market, we are able to predict future movements on the market. Technical analysis is not interested into the company, its fundamentals, strategy, industry matters, but just quantitative data tools. Technical analysis is only related to mobile averages, quantitative tools, ratios and signals exclusively on fluctuations. Typically, a good analysis matches both fundamental analysis, profitability predictions and technical analysis: the truth is always in the middle. Behavioral Finance Criticism to the EMH is also moved by Shiller (Nobel Prize 2013). He posits that financial markets are not efficient, and cognitive / behavioral biases characterize investors. Behavioral finance is a new branch of finance based on these assumptions. The main idea is so that financial markets are driven by people, so people can do mistakes. Behavioral finance is studying the effect on financial markets of cognitive biases and behavioral objectives of people. Cognitive bias result in attitudes of people. For example, when people get capital losses they tend to keep stocks just in order not to admit they were wrong in their valuation. Behavioral finance could also result in technical: if everyone has a bias that trends will repeat, then expectations are self-realized. Examples are: - Limited rationality - Prospect theory - Herding 51 Stock Markets In Italy, these markets are credit ones. The firs market is the MTA (Mercato Telematico Azionario), regulated by the European authorities. 52 On a second level, AiM Italia, is an unregulated market for small companies. Small companies can hardly assess to the MTA due to high costs, requirements and affordability. The main aim of this unregulated market is to help small companies going public due to the good effect going public has. The main determinant why companies list on the stock exchange is because they want to be liquid. For companies listed on the stock exchange it is very fast to buy and sell shares with very quick and low cost services. On the other hand, if we want to buy shares of an unlisted company, it is extremely complex. The stock exchange is a centralized market. It means that all of the bids of people willing to buy or sell shares are centralized and grouped together. For any security which is listed on the stock exchange there’s a so called ‘book’ which is a table on which on one side we have the bid (denaro) of people willing to buy shares and asks (lettera) of people willing to sell shares. From the point of view of the market, the best offer from people willing to buy is the one with the largest price. For instance, on the bid side we could have 74,000 units with investors who are willing to buy them at 6.05€. This is the most generous offer, so the best one at the eyes of the market. On the ask side, the most generous price is the lowest one. For instance, 6.1€. In this situation, nothing would happen since the best bid is lower than the best ask. There’s no match between the two and so no transaction. When an order is filled to the stock exchange the investor can select whether to create an order at a given price or an open order, so with fixed quantities and the best prices possible. We can consider a case in which a new order is issued to buy 30,000 shares (open order). The best price is taken and quantities available at that price are bought and so offset from the book. 53 If more than one investor make the same investment, the book works with a FIFO service. On the other hand, we can consider the effect of a large order to sell 140,000 shares. It’s the very simple low of demand and supply: if there’s a strong demand, the price increases, while if there are a lot of people willing to sell shares suddenly, the price plummets. The difference between the best bid and the best ask price is called the bid-ask spread. Typically, the more liquid shares are, the tinier is the bid-ask spread. Blue chips are characterized by a low bid-ask spread. This means that transaction are low cost and frequent. In facts, the bid-ask spread is a huge cost for investors. Costs could come by the need of selling immediately stocks: the bid-ask spread generate a loss that the investor has to bear in order to sell, since he has to meet a much lower bid. 54 Interactions between valuation and financing Modigliani & Miller posited that under some assumptions financing decisions do not have any effect on firm value. All these assumptions are not true really, so there’s the possibility of creating or destroying value for a company when there’s a decision of buying or paying debt for financing a company. This means that alternative financing policies can have differential effect on the value of corporate assets. Therefore, also the value of a single project will depend on how the initial investment will be financed (equity, debt, leasing, cash available or raise new capital, ....) In order to take into account such effect, two methodologies are generally allowed: 1. The ‘adjusted present value’ (APV) 2. The weighted average cost of capital (WACC) Adjusted Present Value It is a very flexible way to account the effect of financing on the valuation of a company. We are asking how different financing alternative could affect the value of the company. The ‘adjusted’ present value of a project is defined as: APV = NPV (base-case) + NPV (financing) The NPV (net present value) in the ‘base-case’ is computed in the situation in which both the firm and the project are unlevered (= financed only by equity capital, no debt). Cash flows are discounted with the ‘unlevered’ cost of capital k*. We need to define it because the cost of equity capital is not a constant. Even when we’re in the assumptions of M&M we know that k is not a constant but depends on the leverage. The NPV (financing) is related to differential effects of financing choice, only. Example of the ‘differential’ effects are: tax savings, costs to raise capital, contributions, grants, ... The latter cash flows are discounted with the cost of capital k* if they depend on the operating cash flows of the project, with the debt cost of capital πà otherwise. So, if the differential of the financing are not really predictable since they’re risky and they’re linked with the project, we have to discount them with cost of capital. If they’re not risky and they’re predictable, they’re not really linked with the returns of the project and they’ll be discounted with πà . We have to consider that both of the two values could be either positive or negative. We could have a situation in which they’re both positive, meaning that the financing strategy is increasing the value of the project. Or we could have the case in which the project seems to be profitable but we’re out of cash and its expensive. This could discourage the undergoing of the project lowering the profitability of the whole project. This is the typical example of start-ups. We could have also a cans in which the profitability of the project is not positive but the whole value is created through financing due to incentives and value adding financing. 1. For example, we can estimate the expected cash flow for a business project. 55 APV e am le We e ima e he e ec ed ca h fl C f a b ine f ca i al k jec nle e ed Time Ne ca h fl PV ca h fl S m NPV ba e ca e IRR in e nal a e f e c n When the company is not levered the k* equals 12%, we have the whole cash flows as predictable and linked with the project and not with financing. We use 12% for the discounting of every year, assuming that we have 50 in our pocket in order to start the project without financing. CHAPTER INTERACTION BETWEEN This VALUATION FINANCING Of course, the IRR is higher than the cost of capital. means that the project can be run and it is efficient, creating value. Besides, NPV is positive. If we think someone could ask for buying the whole project, the value of the project will be for us at minimum 50 + 7.44 so 57.44, which is the initial investment plus the present value of future cash flows the project is able to generate. Moreover, this is the NPV in the base case, so in which the company owns the 50 amount needed to start the project. 2. We can now consider the case in which the company has not those 50 amount in order to start the project. So this amount has to be funded on the market: we need to raise capital. In the second semester we will see that this is a costly process. Assume that: - Equity to be raised = 50 - Costs and fees = 2% (deductible from taxable base, t = 30%. So we have a positive differential cost even on taxes since we can pay less taxes) Therefore: - PV (financing) = -2% (50) * (1-30%) = - 0.7 - APV = NPV (base case) + PV (financing) = 6.74 The project is still nice and it creates value but the costs of raising capital reduces the value of the project (and profitability). APV e am le 3. We could then consider an alternative in which the same project is partially financed with N a me ha he ame jec i a iall financed i h deb debt. Deb i ed a ime In e e a e d Ta a e n c a e inc me Time Ta a ing in e e PV a a ing di c PV financing NPV ba e ca e APV c n an ime c n d N e ha in e e a e n ded c ed f m ca h fl By doing these computation we’re also assuming theVALUATION companyFINANCING has positive taxable incomes each CHAPTER INTERACTION that BETWEEN year. As we can see, tax savings are discounted with πà . This is because the project is risky. We estimated some cash flows for the projects which are the ones in the base case, but can we be sure that those are really the cash flows that we’ll meet? We’re not really since cash flows are risky. On 56 the contrary, if we borrow for an amount of 20 and the interest rate is contractually given, we can be sure about the amount of the interests we’ll have to pay. Cash flows could be larger or smaller in the future but it’s not the case for financial cash flows since there’s a commitment and agreement with the financing part. We can see how raising debt we have a tax saving that enhances the value of the whole project. In fact, the APV is larger than the NPV in the base case. On the other hand, interests lead to an additional cost, and here are not deducted even if the tax saving is deducted. This is because we’re not looking to the value of the project only for shareholders, but we’re looking at the value of the project for the whole investors and the whole company. Interests are not a cost in the point of view of the project but is a cost to shareholders. APV e am le If we want to provide the point of view of shareholders we could include negative cash flows linked with debtCrepayment. ide h ca h fl a e di ided am g ha eh lde a d deb h lde Time Ne ca h fl ba e ca e Ta a i g i h deb Ca h fl a ailable i h deb Ca h fl deb Ca h fl e i e id al IRR deb IRR e i Val e f he Deb E jec APV f hich I c ea e i jec IRR f ha eh lde i edic ed b M M II i INTERACTION VALUATION FINANCINGand debtholders not in the We’re even considering how cashCHAPTER flows are dividedBETWEEN among shareholders perspective of the project. With these cash flows and interests for debt holders we can compute an IRR of 6%, meaning that we’re paying them exactly what they’re expecting. On the shareholders side, they’re contributing with 30. What shareholders will receive are the cash flows from the project returns minus debt repayments and financial expenditures. IRR on equity is higher than the base case, this is the reflection of M&MII. This leads to a new higher value for the project. We want now to compute the cost of capital for shareholders. Shareholders value their part with a value of 38.69 and they proceed each year the level of cash flows accounted above. 38.69 = 11.16 / (1+π_ ) + 24.16 / (1 + π_ )@ + 17.16 / (1 + π_ )¨ + 1.16 / (1 + π_ )Ä with π_ = 16.8% (now larger than k* = 12%, always due to M&M II). We would not have been able to know the ‘new’ value of the cost of capital to discount the cash flows for shareholders!! (equity-side). In fact we discounted the ‘total’ cash flows (asset-side). There’s always a positive difference between the IRR and the cost of capital, meaning that the project is valuable. We have to note that it was a mistake to discount the cash flow for shareholders with k* = 12% since this value is linked with an unlevered situation. 57 APV e am le 4. Now assume that the debt might be changed during the project life-cycle (and thus tax savings are considered ‘risky’ and linked to the project). This means set with so that be really N thata contracts me haare he deb debtholders migh be cha gedwed cannot i g he jecsure lifeabout c clehow a money d h and when debt will be repaid. This is typical in project financing where both debtholders and a a i g a e c ide ed i k a d li ked he jec shareholders rely on the capability of the project in generating cash flows in the future. Deb i ed a ime I ee ae d Ta a e c a e i c me Time Ta a i g i e e PV a a i g di c PV fi a ci g NPV ba e ca e APV c a ime c k Fi a di c a i gi k ea i h d O he f e a a i g ae k di c ea i h d a d ea i hk BETWEEN VALUATIONtheFINANCING Everything relies on the capabilityCHAPTER of theINTERACTION project in generating forecasted level of cash flows. Banks and debt investors also rely on this, so it is typical for them to make deals on covenants. A typical covenant is the right set by the debtholders to get their principal back in advance through a new equity injection. There can be several reasons according to which sometimes we’re in a situation in which we have to consider the tax revenues for the future risky. The only difference is that in this situation we have to discount tax savings with a different cost of capital which is no more πà . For the first period we know exactly the amount of tax saving, sure about πà is valid for the first year. We don’t know if the following years will have an higher level of cost of capital, so we need to discount Ethe future ec ed year ca htax fl savings a e with he k*. ame a bef e APV e am le Time Ne ca h fl ba e ca e Ta a i g i h deb Ca h fl a ailable i h deb Ca h fl deb Ca h fl e i e id al IRR deb IRR e i Val e f he Deb E jec i APV f hich kE Val e f he jec i l e beca e a a i g ae ik c f ca i al i la ge f ha eh lde m e le e age We have to note that the levered cost of capital for shareholders is a bit larger than before. This is due CHAPTER INTERACTION BETWEEN VALUATION FINANCING to the fact that there is more risk. We find that the levered cost of equity is larger even if the debt level is the same as before. This is due to the fact that the leverage is larger (debt compared to the value of the project) if the value of the project is a bit lower. WACC approach The second approach is to compute an adjusted measure of the cost of capital, the weighted average cost of capital (WACC). 58 ππ΄πΆπΆ = πΈ π· ∗ π_ + ∗ π ∗ (1 − π‘Æ ) π·+πΈ π·+πΈ à E is the market value of equity while D is the value of debt. π_ is the levered cost of equity capital and this is a problem because we can now it only when the project is levered. We have seen that π_ depends on D, so we are stuck... This is a less powerful approach, because we take into account only tax savings on debt (and not other possible differential effects of the financing). It is more specific approach compared to the APV. The APV can be used in a larger range of situation in terms of length, largeness and so on. Using the WACC requires that the WACC is constant through time since fluctuating WACC is too complex. WACC is constant only when the leverage is constant, so we can use this approach in a very limited number of situations. The main point is that, even if the level of debt is the same through years, since the quantity of debt is fixed, but on the other hand the value of the project lowers as time pass by since there are less years to have returns in the future. This means that the leverage level is increasing in different years in the life of the project. This means risk increase. The bank also will be exposed to an higher risk towards the end of the period where the debt has to be repaid. In other words, we know that k* is the unlevered cost of capital, but the definition of the WACC contains the ‘levered’ cost of capital π_ , that depends on D. Moreover, we can use the WACC to discount cash flows only when the leverage is constant (because, indeed, WACC changes according to the leverage) i.e. in the previous example we cannot compute a constant WACC (debt is fixed and the value of the project decreases from time 0 to time 4, since cash flows shrink). In order to compute the WACC, starting from the ‘desired’ value of the leverage, we can use two models. Miles & Ezzell formula This formula is valid and used when: - Debt D is adjusted any time in the future to keep constant the leverage L. This means that the level of debt decreases in time by partial repayments of the principal. 1+π∗ ππ΄πΆπΆ = π ∗ −πΏ ∗ πà ∗ π‘Æ ∗ 1 + πà Recall the value of the leverage as L = D / V = D / (D + E) Modigliani & Miller formula The formula is valid only when we have the following assumptions: - Cash Flows (CF) are constant and perpetual - Debt D is fixed and perpetual, and not linked to the project. This, the leverage is constant, since the value of the project is always CF/WACC, with both the values constant. ππ΄πΆπΆ = π ∗ (1 − πΏ ∗ π‘Æ ) 59 WACC e a Le c ide he e e i M e jec a d a E e e ha he i i ia i e e i be Tofibring an example & Ezzell model, previous a ced i h debon the L Miles Re e be ha weheconsider a e the f he jecproject, V i a and iweeassume i thata thea initial investment will be financed with debt (L=50%). he f he di c ed f e ca h f k e e ed d WACC Mi e E e Ti Ne PV S Va D L c D i e ba a ced each ea kee L c a N e ha a a i g a e c ide ed a e a i e e deb a e c ide ed e ca h f ca h f WACC NPV WACC e f jec Tax savings are not considered to theBETWEEN fact that they’re already accounted in the WACC CHAPTER due INTERACTION VALUATION FINANCING computation. Otherwise it would mean to consider their effect two times. The value of the project decreases every year considering that in every year, the value of the project is exactly the sum of the present value of future cash flows discounted to that specific year. As cash flows moving to the future decrease, the value is decreasing since future cash flows are less. In order to balance the decreasing value of the project, debt has to be adjusted. It means that each year D has to be partially repaid in order to keep L constant. Of course, if future expectations change, even debt will be decreased differentially. This is a solution which is appreciated by debt holders and banks for instance. In this way, the level check he eis constant l i he soathee risk e ca ai isi the h he APV e htime. d ofWe riskcalinked withha leverage for thebbank same during WACC e a Ti e Ne ca h fl PV ca h fl WACC S NPV WACC Val e f jec V D I ee Ta a i g PV a a i g di c PV fi a ci g NPV ba e ca e APV e M e E e Si ce deb i e bala ced di c a a i g e ea i h a d he he e i d i h k k d Once debt is adjusted yearCHAPTER by year,INTERACTION we can BETWEEN compute interests linked with decreasing levels of debt. VALUATION FINANCING Moreover, we have now to add tax savings discounting them with k* and πà . Each time we know the amount of the debt, the best approach is to use the APV. This because if we know the amount of debts it is very easy to compute tax savings. On the contrary, if we know the leverage ratio only, the best solution is to compute the WACC, through which we don’t need to know tax savings but just unlevered and levered cost of capital. We can compute the real cash flows available for the project which are comprehensive also of tax savings. For each period, cash flow available for debtholders is given by the sum between the principal repayment of the period and the interest paid on the previous period debt. 60 WACC e a We ca al e M e fi d he e ec ed ca h fl Time Ne ca h fl D Ta a i g Ca h fl Ca h fl Ca h fl Ca h fl a ailable deb i ci al deb i e e ha eh lde e The c f ca i al f f E e ha eh lde Si ce a f he deb i aid back each ea emembe c ide he eimb eme l i ee ha eh lde kE ca be de i ed f m he WACC kE We can compute the IRR from the cash flows for shareholders, equalizing the sum of discounted cash flows for shareholders with IRR as theINTERACTION discounting to find. CHAPTER BETWEENratio VALUATION FINANCING On the other hand, the value of the project for shareholders at time 0 is 50% of the whole project value, considering the leverage ratio, so 29.34. We’re able to define the levered cost of equity capital, which is given by equalizing 29.34 to the sum of discounted cash flows for shareholders with π_ as the discounting ratio to find. WACC e a A e ha ca h f e a ec hM M a a d e e a Agai We can consider an example with the Modigliani and Miller formulation. betofihave a ced deb We assume constanti h cash flowsL and perpetual and a leverage ratio of 25%. k e e ed d WACC M dig ia i a d Mi e he i i ia i e L c Ti e Ne ca h f S NPV WACC i h e e i Va e f jec V D I ee Ta a i g PV fi a ci g i h e e i d NPV ba e ca e i h e e i k APV CHAPTER INTERACTION VALUATION FINANCINGwe need to know All of variables result to be constant. In order to find outBETWEEN the value of the project, the value of debt. On the other hand, the value of the debt depends on the leverage ratio and so o the value of the project itself. Also in this case we can split the cash flows between cash flows for debt holders and for equity shareholders. Each year only the interest is paid back, so the final cash flow for shareholders is given by the levered cash flows with tax savings plus minus interests to be paid. Remember that the cost of capital for shareholders π_ depends on: - The risk of the project (take into account with beta) - The leverage (‘levered k’ is generally larger than ‘unlevered k’) When you know the value of the debt (constant or not) it’s easier to proceed with APV. If L is not constant, you cannot use WACC. When you know the target leverage of the project (constant) it’s easier to proceed with WACC. In the ‘real’ world neither π_ nor WACC are constant when the financing policy is modified, and this impacts on the project (and firm) value, and on equity risk. 61 e i Derivatives A derivative is a financial contract whose payoffs and values are derived from, or depend on, another asset, namely the underlying asset (Sottostante), that can be real, like a commodity or a raw material, or financial, like interest rates, currencies, bonds, stock market, indexes, shares or any other asset with a visible market value. Derivatives are mainly used to hedge against the risk of the market price of the underlying. In particular they’re undertaken when the underlying is volatile on the market. Derivatives are also traded on the stock exchange and yet derivatives are also used to build portfolios with aggressive returns. In fact, we will see that with derivative contracts it is possible to enhance the risk and return of the underlying asset (and moreover it is possible to lose more money than what we invest in). One of the main points is that derivatives can make the owner of the derivative lose more than the value of the security itself depending on how the contract is set. Derivatives are also known as ‘term contracts’, because they typically regulate a trade on the underlying asset that will occur in the future (at pre-determined conditions) between two parties which are defined in a seller and a buyer. History of derivatives Term agreements for oil presses are documented in the ancient Greece. Besides, during the Renaissance, Dutch investors used derivative contracts to sell tulips. The first speculation bubble of the modern era exploded in 1637 and was original of that kind of contracts. Forward contracts on rice began to be traded in Japan in 1697, while the first derivative contracts were settled on the Chicago Exchange in 1848 (Chicago Board of Trade). Nowadays the national amount of outstanding derivative contracts totals 700$ trillion, almost ten times the world GDP. Broadly speaking, there are two main types of derivatives: - F a dc ac Forward: reciprocally binding contract to purchase / sell at a certain maturity T a given quantity of the underlying at a pre-determined price (delivery price, X). Included in this category, among others, there arecontract futures and A for ard is aswaps. a b d g term agreement for the sale of an a Options: contracts that give one of the parties the right, but not the obligation, to buy or sell a certain future date T at a pre determined price called the de i e ice an asset at a certain time (or within a certain maturity T) at a predetermined price (strike price buincluded er is the g position the seller is a h(options position X). In this category are call options (options tohile buy) and put options to sell). Forward contracts A forward contract is a mutuallybinding term-agreement for the sale of an asset at a certain future date T at a pre-determined price (called the “delivery price” X). The buyer is the “long position”, while the seller is a “short position”. Cash X LONG SHORT Underl ing CHAPTER Forward valuation is based on the definition of the following parameters: o T : time to maturity of the forward contract (in years); o πr : spot price of the underlying at time zero; 62 DERIVATIVES o X : delivery price determined in the forward contract; o π(πÈÉ^Ê ππ π Ë°ÉÌs ): value of the forward contract (in long or short position); o πQ : price at maturity T of the underlying (which is unknown). A forward contract at time T can be settled with the physical delivery of the underlying or through an equivalent payment of cash (‘per contanti’). For ard pa off at time T The pa offs at mat rit are eq al and opposite in sign and It is possible to draw the payoff of the derivative at time T. In particular, The payoffs at maturity are equal and opposite in sign andFor they are: the long position 1. For the “long position”: π = πÈÉ^Ê = πQ − π 2. For the “short position”: π = π Ë°ÉÌs = π − πQ = −π For the short position The long position earns as much as the price becomes greater than the delivery price X, while the holder of the short position earns as much as the price of the underlying asset becomes lower than the delivery price. Note that a forward agreement is a ‘zerosum’ game. CHAPTER Forward: Value at time zero The value of a forward agreement (assume long position) is given by: π = πÈÉ^Ê = πr − π ∗ π €Ì∗Q ππ ππ πππππππ DERIVATIVES πÈÉ^Ê = πr − ππ(π) Where r is the continuously compounded annual interest rate. On the other hand, we have that: π Ë°ÉÌs = ππ(π)−πr The idea is that the value of the derivative today is equal to the actual value of the underlying on the market, net of the discounted expected value of the underlying at time T. Note that both the values can be positive or negative. In order to demonstrate this equation, we consider two different portfolios: 1. Portfolio A: a forward contract in “long position” with value (unknown) + cash equal to the present value of delivery price → πÈÉ^Ê + ππ(π) 2. Portfolio B: the underlying unit, whose value today is → πr Proof At maturity the two portfolios leads to the same (but unknown) result, equal to πQ and then the same payoff. According to the “nonarbitrage” logic, the two portfolios should Por folio A B Toda Fo a df S T Xe T ST X X ST ST A ma i he o po folio lead o he ame b nkno n e 63 pa off Acco ding o he non arbi rage logic hen he ame ho ld ha e he ame al e in an ime befo e al o a ime have the same value in any time before, also at time 0. We may look at the value f as the lump sum that one of the party is available to pay (if positive) or wants to receive (if negative) in order to sign the contract. Interestingly enough, we did not need to introduce any assumption about πQ . E am le oil We could provide an example using as an underlying the price of oil on the market. ⁄ . . There is a certain ⁄ that makes equal to the contract value ⁄ This price is called forward price of oil see ne t slide This price πÎ is called ‘forward price’ of oil. In this example we have been able to find out a value of the delivery price that sets the derivative to CHAPTER DERIVATIVES zero. This is the forward price of the underlying. The forward price is commonly used as delivery price in forward contracts, since none of the counterparts has to pay/receive money to enter in the agreement. Do not get confused between the value of the forward (i.e. the derivative) and the forward price of the underlying. Note also that there is not a unique forward price, but there are several forward prices, for any maturity in the future, for the same underlying. A company that buys oil to run its operations could be a company which is interest in hedging the risk of volatile oil prices in the market by entering a forward deal in a long position. The earnings will be get from the derivative whether the price of the underlying grows. This is the so called off letting effect: so if price of the underline grows, the cost to buy it grows but the company can get profits from derivatives allowing it to buy oil at a lower price than the one of the market. On the other hand, if the price of oil falls, costs to buy it would also be lower but the derivative obliges the party with the long position to pay a price which is higher than the one set on the market. On the other hand, in order to take zero risk, there’s the possibility of using options, even if hedging risk through options is expensive and options are more costly. On the other side, companies selling oils could be interested in assuming the short position, in fact, the lower the price of the underlying on the market will be at time T, the higher will be the proceed from selling the underlying at a price which is higher than the one set by the market itself. On the financial side it is a bet, so the financial effect is that this kind of derivative is no more hedging risks. It is possible to demonstrate that there’s no arbitrage opportunity. 64 Case with Cash Flows The underlying assets in a forward contract might generate positive or negative cash flows before the expiry of the contract (for example: dividends, interests, coupons, costs of carry like insurance or storing). In these cases the non-arbitrage equation must be adjusted (cash flows are differential for the two portfolios). The value of the forward in this case is: π = πÈÉ^Ê = [πr + ππ(πππ π‘π ) − ππ(πππ£πππ’ππ )] − π₯ ∗ π Ì∗Q Not that costs must be added (the holder of a forward contract is favoured). Also the forward price of the underlying will be influenced. If we think about costs, of course the long side of the deal will benefit from not owning the underlying. On the other hand, the long side cannot benefit from revenues coming from the underlying itself within the maturity. Example of forward prices Besides, we find an example on future contracts, traded at the CME in Chicago. CHAP Ethe second DE I A contract I E The first forward contract has corn as underlying, while has gold. The main buyers of these kind of commodity are companies which are exposed to the variation of price of that commodity in their operations. Currency is another interesting example for forward contracts. Currencies are a particular underlying assets, because they pay an interest (the risk free rate). Forward contracts on currencies are also known as ‘currency swap’. Since the risk-free interest rate is generally different among different currencies, the forward exchange rate will be increasing or decreasing according to the difference in the foreign vs. domestic interest rate. This does not imply anything on future values of exchange rates, that fluctuate according to market conditions and expectations. 65 C A company which could be interested in acquiring forwards on currency are companies exposed to currency risk. For instance, an Italian manufacturer selling in Japan, could be interest in covering the risk of Japanese Yen in the future to drop with a future contract in which Japanese Yen are sold at an higher strike price. Somehow, it is like shorting on the Yen: fixing a price to convert Yen into Euro which is higher than the market value of Yen in the future. enc e am le On the other hand, an borrower in CHAP JapanE could be interested in having a strike price in the future DE I A I E which is lower than the market one. If we look at the table, present and forward values of currency swaps are displayed as the result of how banks are willing to set deals. If we look at the dollar swap, for instance, we can see how in the future the contract appreciates the dollar in respect of the euro, making the Euro more favourable. These are not predictions on how future currency exchange will fluctuate, so there’s no creation of any expectation. It is all a matter of equilibrium on the market. Looking again at the dollar, today EUR/USD is 1.2029. We know that now, risk free rates are different between Europe and U.S. In particular, interest rates are higher in the U.S. (we could imagine 0.1% vs 0.2%). Imagine to have a bank which could give us the opportunity to have a forward contract ensuring a future exchange rate equal to 1 between USD and EUR, there would be a strong arbitrage opportunity. Everyone on the market would start to by U.S. dollars, invest them in risk free securities in the U.S. market and then according to the forward contract, exchange USD into EUR with an higher return. This type of arbitrage is called carry trade, as a kind of arbitrage in which an agent tries to proceed working on differences between interest rates and currency exchanges. This is not working in the real market. In fact, the actual value of forwards on currencies are only affected by risk free rates on the market. For instance, we can see how the dollar gets weaker in respect of the euro for long time forwards. Futures Future contracts are particular forward contracts, albeit there are a few differences: - Future contracts are generally traded on exchanges and standardized (for example, there is typically only one delivery each month or each quarter), while forward contracts are typically 66 - private contracts (OTC, ‘over the counter’) and customized according to the needs of the parties. Futures are standardized referring on the type of the underlying asset on the market, on the delivery time, in order to reduce the most feasible contractual and transaction costs on the market and so to make them tradable by everyone. Futures may be bought and sold frequently, while OTC forward contracts are generally kept to maturity (and often a fee must be paid in case of early termination); The payoff of a forward contract is paid at maturity (or at different stages, according to the agreement), while futures are subject to the mark-to-market rule (see next slides). The most diffused futures are: - Futures on interest rates (underlying is a risk-free bond) – for example LIFFE, EUREX - Futures on single stock or equity indexes (for example IDEM on the Italian Exchange). Future contracts on the stock market are only issued in the blue chips and not on small cap companies. - Futures on commodities (metals, gold, oil, ... in Italy AGREX on corn) - Futures on electricity and energy (for example IDEX on the Italian Exchange) Futures, as forwards, are subjected to counterparty risk, occurring when at maturity, the contracts are not met by the parties. In order to avoid the counterparty risk, the market deal with this risk through two pillars: - Mark-to-market: investors are charged a % margin deposit on the notional value of the contract when selling or buying the future, and each day money is credited or debited according to the change in the market price of the underlying (see example in next slide); no other money is needed; - Clearing House: entity responsible for settling trading accounts, clearing trades, regulating delivery and reporting trading data. Clearing houses act as third parties to all futures and options contracts in order to guarantee that the payoffs associated with a future contract will be paid. Whenever an agent on the market wants to enter a future contract he has to open an account at the Clearing House to deposit an amount of money decided by the stock exchange. If we assume to have a long position on an underlying asset which price is increasing, the C.H. transfers money from the short money account to the long position’s one. On the other hand, if the price of the underlying is decreasing favouring the short position, the C.H. transfers money from the short to the long’s account. Every day, the C.H. adjusts the balance of the accounts according to how the market is performing. This means that the final payoff will be the sum of everyday exchanges set by the clearing house. The risk of counterparty is avoided by this kind of guarantee requested by the C.H. In the case in which the account of an agent is empty due to a non-profitable contract, the C.H. requests the agent to inject new money in the account. Otherwise, the agent will be kicked out of the market. On the Italian market we have two futures on the FTSE MIB index, FIB and MINI, built on standardized delivery time as we can see from the chart. The underlying for these futures are the FTSE MIB index. The first point to create the future is to translate the MIB index, which is something not measured in currency, in Euro. The stock exchange defines the multiplier to make this translation. 67 The notional value is computed as the theoretical value of the underlying assts on which the future is built. Besides, the CH defines the quantity to deposit in order to enter the market on the basis of the theoretical value of the underlying. Example: FIB future Margin able Underl ing FTSE MIB inde Mar M l iplier Price of f bor ai aliana i Margin re da No ional al e Margin o be paid o CH Da price goe o a Ne al e of con rac Lo in one da on margin As we can see, we can consider a variation on the value on the market of the underlying. A 1% CHAPTER DERIVATIVES decrease leads to a value of 1,125€ which the CH will withdraw from the accounts of the long positions to deposit them on the accounts of short positions. Moreover, there’s a kind of leverage effect as we can see: the market moving down by 1% leads to a marginal loss of almost 10%. With derivatives, an agent could bet on underlying value growing or decreasing without entering the deal on the underlying asset but just acquiring the derivative. This is the reason why we see this leverage effect. For stock market futures, generally, the multiplier is given by a minimum number of shares to be multiplied by their value on the market. The stock exchange will probably define margins on stock derivatives according to their volatility of the price of shares on the market. The margins on the stock market index are typically lower since volatility on market indexes is lower due to compensations. 68 Contracts S a Swap con ac contract defined as an OTC, as an agreement between two parties to swap a A Swap is a forward stream of cash flows for a defined period of time. In other words, two parties owing ap i an agreemen betheeen o par ie two o assets exchange interest rates they need to a a reampay of on ca those h floassets. for a defined period Firm Fi ed ra e Bank ime EURIBOR + spread It can be seen as the sum of two different can be een a he m of o differen forward contracts (one in LONG position ard con rac one in LONG po i ion and he and the other in SHORT position), and er in SHORT i ion and e al a ed thenpo is evaluated usinghen the isame formulas. ng he ame form la Da a p bli hed each da from bank r e The most common swap is the Interest mo common i where he Ina ere Ra e Rate Swap ap (IRS) fixed interest ap IRS rate hereis a‘swapped’ fi ed inagainst ere arafloating e i interest rate apped again a (typically floa ing theinEURIBOR). ere ra e Companies cope with interest rate risk picall he EURIBOR through these kind of swaps. As an example, a company having a debt with a debtholder based on EURIBOR, could set with a CHAPTER DERIVATIVES bank a swap through which they exchange interests on a fixed amount of money. When the EURIBOR on the market grows, the company will pay to its debtholder a larger interest, but it will even receive an higher interest from the bank in the swap. It is like for the company to translate a floating rate into a fixed rate. Hedging using forward and futures It is now clear that future contracts generate positive or negative payoffs ‘following’ the market price of the underlying. Therefore we may use them to ‘offset’ losses or gains generated by the underlying itself. Perfect hedging is not always possible: - The time-to-delivery may not coincide with our needs (futures have fixed maturities during the year, so it could be necessary to go for an OTC) - The underlying asset may not be the same as the one we have to hedge (e.g. oil qualities: Brent vs. WTI, penny stock, unlisted securities, ...). In this case the solution is to go for the underlying asset which is strongly correlated to the asset the company wants to hedge. Therefore a certain ‘basis-risk’ will survive. We should: - Use derivatives on underlying assets that are significantly correlated with the asset to hedge - Choose expiry dates closer to the hedging objective - Buy more forward contracts, if the volatility (beta) of the asset to hedge is larger than the volatility of the underlying. Assume we have to hedge: • An investment in 10,000 ENI shares = forward / future on 10,000 ENI shares (short) • An investment in a generic portfolio of blue chips = forward / future on the market index (short) • Costs for iron = forward / future on iron/steel/metals (long) • Revenues in Japan = currency forward, short on Japanese Yen * Debt in US dollars = currency forward, long on US Dollar 69 • Gold mining = forward / future on gold (short) Companies typically hedge against these risks. In every accounting report there’s a section devoted Case st stdies Case dies to the hedging on risks and derivatives bought to hedge risk. CHAP E Options CHAP E DE I A I E DE I A I E An option gives the right (but not the obligation) to buy or sell the underlying asset, at time T, at a given (and pre-determined) price X (‘strike price’). In particular: - A ‘Call’ option attributes such an option to the BUYER (long) - A ‘Put’ option attributes such a right to the SELLER (short) Compared to forward contracts, options are not symmetrical: one of the parties has the right to go on with the agreement, or do nothing. Obviously, the option will be exercised only when convenient. Therefore an option will never generate a negative cash flow (worst case 0) for the holder of the option, and has always a positive value. Op ion pa off We indicate -π π π π In order to define an option, we indicate: - πprice theasset underlying asset; current of the price underlofing r : current - π: strike price; strike - price π: time to maturity of the option; - πQ : price of the underlying asset at maturity (unknown); time to maturit of the option - π (ππ πΆ): value of the European (or American) option call; - ofπthe (ππunderl π): value of the (or American) option put. price ing asset at European maturit unkno n CALL OP ION value of European or American optionis:call At πΆmaturity thethe value of a call and put option P of the European American put (π; πΏ − πΊ ) ππ= value πͺ = π¦ππ±(π; πΊπ» − πΏ)or πππ π =option π· = π¦ππ± π» At maturit the value of a call and put option is The true difference in respect of forwards is the fact that an option has not an infinite potential loss for the holder. When the payoff is , , positive, the result is the same as for forwards, while when it is negative, the minimum payoff is zero. CHAPTER DERIVATIVES 70 OP ION So, for a call option, the minimum payoff is zero and never negative. Depending on the type of option (put or call) payoffs are inverted. The main difference of a put option, is that even positive payoffs have a cap which is X. The situation is different if we consider the other side of the deal and the seller of the option. In the case of a call option short it is possible to have a payoff which is negative and potentially illimited. Op ion pa off Holding a call option, if at maturity the market price of the underlying asset is lower than the strike Holding a call option if at maturit the market price of the underl ing asset is lo er than price (πQ < π) the option is not exercised and the payoff is zero, whereas if the market price of the he strike price thethan option is notprice e ercised and pa isoff is ero andhereas if the underlying is higher the strike (πQ > π), thethe option exercised the payoff is equal to πQ −ofπ the as happens a forward contract. In fact, are happy at a lower market price underlin ing is higher than the we strike priceto buy the underlying the option is price than the market price. e ercised and the pa off is equal to as happens in a for ard contract In fact e Tobu define value ing of anatoption we price need to assess it is always are happ to thethe underl a lo er than thethat market price positive. This is an important difference in respect of forwards and In orderistothe sellopposite an option, the Put option futures. the decision seller is always paid. As previously said, f ou sold the options the pathe offssituation at T areis the opposite if we look at the short side. If you sold the options the payoff at T are: −π = −πΆ = −max (0, πQ − π) −π = −π = −max (0, π − πQ ) CHAPTER DERIVATIVES American vs European Options Note that the payoff of a Call option is potentially unlimited, while the payoff of a Put option is limited to X. - If the option (Call or Put) may be exercised only at time T, we have a ‘European’ option. - If the option may be exercised also in any moment before time T, we have an ‘American’ option. The distinction has nothing to do with the markets in which the options are written. An American option will always have no lower price compared to an identical European one. We can say that, even with these differences, the value of American style options are not lower compared to the value of the European corresponding option. This is because owning an American style option gives more opportunities to the investor, which has the opportunity to buy or sell the underlying whenever he wants. Options may be OTC contracts, or also listed on exchanges (‘iso-alpha’ options). Put-call parity theorem on European options The values of a European Put (p) and a European Call (c) option (on the same underlying, with same T and X) are tightly correlated, by the ‘put-call parity theorem’: π + πr = π + ππ(π) In fact, we may show that the two portfolios above at time T have the same value: 71 The al es of a E ropean P t p and a E ropean Call c option on the same nderl ing same T and X are tightl correlated b the p t call parit theorem In fact e ma sho that the t o portfolios abo e at time T ha e the same al e t t T S X Call in esting at risk free rate P t in esting in the nderl ing C PV X S p S X S S X X S S X X S XS X If the nderl ing generates cash flo s e ha e to adj st The demonstration is done just looking at the payoff matrix depending on different cases on the two portfolios. In particular, the value of the portfolios are considered whether the value of the underlying at time T is higher or lower than the strike price, defining the exercising or not of the option. The results in terms of payoffs are the same in both the cases for the two portfolios. CHAPTER We have to note that nothing can be said if the DERIVATIVES options are not on the same underling, or have differences in T or X. The same demonstration could be given graphically, by matching a call option payoff graph with a risk free rate one. For a risk free rate, the return is constant equal to X, while the call (in the first portfolio) has its typical payoff, linearly growing after the strike price X. The result is something similar to a convertible bond, with a positive par value (X) and a growing payoff given by the option. Besides, the situation is different if the underlying generate positive or negative cash flows. If the underlying generates cash flows we have to adjust values on the short side: π + πr + ππ(πππ π‘π ) − ππ(πππ£πππ’ππ ) = π + ππ(π) If there are some costs to be adjusted, the firs portfolio will be the best since the holder will not have to pay those costs, while if there are some revenues generated, the put option is the best portfolio since the revenues are proceeded by the owner of the underlying. How to evaluate options In respect of forward contracts, in which the formulation is easy to define, in the case of options we have to introduce a model with assumptions on the deal. Different models and equation exist according to different assumptions we can take on underlying value in the future. So, we have to make some assumptions on how the price of the underlying asset S will ‘move’ in the future. Despite we have no powerful models to define the value of an option, we could set some limits which are always true: - All options: o their value is always positive, never zero or negative (‘no free lottery’). It is not rational to have a zero value since players would have the incentive to take all of the ‘tickets’ on the market since it is free. It would lead to an arbitrage opportunity. - Call options: 72 it o strictly higher value than πΊπ − π·π½(πΏ) (i.e. the value of the corresponding forward contract), o worth less than πΊπ We can demonstrate it with a no arbitrage opportunity situation. In fact, if on the market the value of a call is equal or lower than the value of the corresponding forward contract, everyone would buy it since there’s an arbitrage opportunity. The demonstration could be given through the building of an arbitrage portfolio. […] In another way, we can compare two situations in which we have a call option and we have just the underlying asset. A call will provide a payoff which is in between [0; S-X], while the value of the asset will be S. Of course, the value of the underlying asset will be always larger than the payoff offered by the call option. This is why the value of the option can never be higher or the same of the spot price of the underlying asset. So, for the value of an option there are two lower bounds, which are zero and the value of the corresponding forward contract, and an upper bound which is the underlying asset value. - Put options: o worth less than X (PV(X) for European put contracts) o worth more than π·π½(πΏ) − πΊπ (i.e. the value of the short forward contract). The opposite can be considered for a put option. At its best, the put option will repay X, at the situation in which the spot price in T of the underlying asset is very close to zero. So, a putt will have a payoff which is X in one case and always lower than X in other cases. We could consider that the upper bound for the value of a put option is the value of a risk free investment in which X is repaid at the end of the period mandatorily. It can never be that the value of a put is the same of the one of a risk free bond delivering X in every case. The situation is a bit different for the American style option. For an American style option, there’s always the opportunity of exercising the option. This makes its value higher than European ones. It could be convenient to have an earlier exercise of a put option for instance, when the payoff is larger, so when the underlying of the asset is very close to zero. Since an American style put option can be exercised in any time before T, the limit is no the present value of X, but it’s X effectively. X has an higher value than PV(X), so the upper bound of an American style option is higher. We can state that the differences between American and European options are relevant in case of put options. Indeed, if we look at a call option, the value of an American style and a European style option are really similar. If the market is in equilibrium, we said that π > πr − ππ(π). Present value of X is necessarily lower than the actual X. This means that, if we are in an American style option, the payoff which will be πr − π will necessarily be lower than any equilibrium value on the market. So, exercising the option before the expiring is not a rational choice for the investor since it gives the lowest payoff. So it is worth to never exercise an American call before the expiring; in case of need of liquidity, it is better to sell the option. The only case in which it could be worth to expire the option is the case in which the underlying generates positive cash flows. Due to this fact, it is worth to say that the value of an American call option is close to the one of an European one since it is likely to be never exercised until T. 73 It is different for put options, since they have a payoff cap. In fact, if American style options give the possibility to exercise the option before the expiring of the deal, whether the price of the underlying asset is low enough to get the maximum profit X, the option should be exercised. This is the reason why American put options are worth more than European ones. Options evaluation models There is no model on the market which is the best one, but there are different models giving different results basing on different assumptions. We can never say that one price is better in respect of another. The most well-known evaluation models are two: 1) Binomial Model: it simply assumes that the value of the underlying asset, at any moment, can take only two values. The distribution of S(t) is discrete and binomial. Bi mial m del 2) Black&Schole Model: it hypothesizes that the function π(π‘) is continuous, and partly stochastic with random walks (the underlying return is distributed as a gaussian function). Co Ross and R binstein proposed a t o state q ick method to com Binomial Model Cox, Ross and Rubinstein (1979) proposed a two-state quick method to compute the value of an of an option Their model ass mes that the ret rn of the nderl ing at time option. Their model assumes that the return of the underlying at time T may have two values only: t o al es onl Typically, one of the two represent an increase in price and the other a decrease, but it is not Each e ent is assigned a risk ne tral probabilit p and pd important compulsory. The good news the is thatobjecti the binomial can be necessaril e trmodel e probabilit exploded to add values. This makes this model a very flexible one. CHAPTER DERIVATIVES To solve the model, each event is assigned a ‘risk-neutral’ probability πâ and πà (important: this is not necessarily the ‘objective’ true probability). Risk neutral probability is enough to define the value of the option. The solution is given by three steps: 1) First, determine the payoff of the option in each of the two outcomes Π1 and Π2 2) Compute the risk neutral probabilities (discounted with the risk free rate) 3) Compute the value V of the option at time zero, which is composed by the discounting of payoffs weighted on probabilities 74 this Example 1 The price of a diamond πr today is equal to 300€. Assume that at time 2 the price πQ may be equal to 400€ (+33.33%, u = 1.33) or 250€ (-16.67%, d = 0.83). We want to find out the value of a European call option (time to exercise 2 years, strike price 350€) on the diamond. Assume that the annual risk free rate is equal to 2%. 1) Payoffs at time 2: Πâ = 50€ (400-350) , Πà = 0€ (not use the option) 2) Probabilities are defined by the simul equations: πâ ∗ 400€ + πà ∗ 250€ = 300€ ä (1 + π)@ πâ + πà = 1 Where with the firs equation we define that the average weight of the two price variations has to be 1 and with the second equation that the sum of the two risk neutral probabilities has to be 1. Risk neutral probabilities are so called by the definition given in the first equation, where we discount prices on the risk free rate. In this case the discounting rate is squared since we’re assuming to discount a two years in the future value. Please note that risk neutral probabilities are not the probabilities that the price will be at the lower or at the higher value; they’re different. We’re computing probabilities for a risk neutral investor on the market. By solving the system we get: πâ = 41.41%; πà = 58.59%. 3) The value of the call can then be defined by weighting payoffs on risk neutral probabilities: π= 41.41% ∗ 50€ + 58.59% ∗ 0€ = 19.9€ (1 + π)@ Example 2 If we consider the example of a put option rather than a call, we have: 1. Compute different payoff i.e. Πâ = 0€, , Πà = 100€ and run the binomial valuation (or) 2. Apply the put-call parity formula ππ’π‘ = πΆπππ – πr + ππ(π) = € 19.90 − € 300 + € 350 /1.02@ = € 56.31 Delta (Hedge Ratio) This parameter is important in hedging with options. δ (‘delta’) is an important parameter, because represents the ratio between the option value and the underlying price. It is computed as: Π) − Π@ =πΏ (π’ − π) ∗ πr The δ is telling how many units of the underlying asset we are able to hedge with one single option. It is called ‘hedge ratio’ because if we want to hedge 1 unit of the underlying, we need 1/δ options. Note that in forward and future contracts the hedge ratio was equal to 1 (π% = πr − ππ(π)). We can see that in options the δ is always lower than one, and this means that if we want to hedge the risk that the price goes up or down by 1, we need more than one options. 75 We can compute the delta for the previous examples. Example 1 Delta = 50 / ((1.33-0.83)*300) = 1/3 Note that a portfolio made up with: (i) the diamond, and (ii) selling 3 call options, will be risk- free: Case u: S = €400 payoff of 3 calls = - €50 * 3 = - €150 Total €250 Case d: S = €250 payoff of 3 calls = 0 Total €250 Example 2 Delta (put) = -2/3 Black & Scholes formula Black and Scholes (1973) developed a well-known formula to compute the value of a European call option, with only 5 variables: - The spot price of the underlying (S) - The strike price (X) - The risk free rate (r) - Expiration (T) - Volatility of the underlying return (σ) Their model assumes that the underlying return is distributed normally and so it’s continuously distributed, and the price follows a stochastic process (‘brownian motion’). The Brownian motion defines the path of underlying prices as stochastic variables, performing a random walk in every point in time with a certain trend. Returns on the random walk are normally distributed. So it is like to say that: π(π‘ + π) = πΌ ∗ π(π‘) + ππππππ πππππππππ‘. Schole form la deri a ion The B&S formula valuates the option as the composition of a certain value of the underlying asset minus a debt component. π = πr ∗ π(π) ) − π ∗ π €Ì∗Q ∗ π(π@ ) π) = π@ = π π@ ln Y πr Z + µπ + 2 ¶ ∗ π π ∗ √π π π@ ln Y πr Z + µπ − 2 ¶ ∗ π π(π) = π ∗ √π 1 √2 ∗ π à ) ñ ∗ ï π €@∗ð ππ₯ €¤ Example chapter Compute the value of European call and putDERIVATIVES options on a share (price today S0 = € 7), time to maturity T = 3 years, strike price X equal to € 10. Risk free rate = 3%. Standard deviation σ equal to 19%. 76 We obtain: π) = -0.6458 π(π) ) = 0.2592 = delta (in fact, in the B&S formula it is multiplying πr ) π@ = -0.9749 π(π@ ) = 0.1648 Call = € 0.3082 computed through the B&S formulation Put (apply put-call parity) = € 2.4475 There is a quicker opportunity to compute the value of a call option through tables. We need to know two variables: - S/PV(X) - π ∗ √π To find the value of the call as a percentage of the spot price S. Even if this method is quicker, we’re not having the value of the delta. Black Sch le f m la al e If we want to compute the value of the options for different spot prices, we obtain something like this: in the mone CALL PUT The hedge a i i n hing el e han he l e f he c e at the mone o t of the mone If we look at the call option, when the spot price is close to zero the option is said to be ‘out of the DERIVATIVES money’. The probability of using the option is very chapter low. The price of the call option can be close to zero when the underlying asset is close to zero, but it is never zero. If the spot price goes much higher than the strike price, the value of the option is larger. We say the option to be in the money, and the probability that the option will be exercised is close to 100%.In this case, the difference in the value of the forward contract and the option will be very low. Limits in the value of the call are set as the underlying price (upper bound) and the value of the forward (lower bound). When the spot price is close to strike price, the option is ‘at the money’. We are uncertain in thinking that the option will be exercised or not. The same reasoning can be made on put options. We have to note that we’re dealing with European style call options, and this model on put option is true only for European ones, while in the call option case it was the same (almost) between European and American ones. If the spot price is larger than the strike price, we’re ‘out of the money’: the probability of exercising the option is very low. When the spot price is close to zero, the value of the put option has to be lower than the present value of X; we’re ‘in the money’ since the probability of exercising the option is close to 100%. The hedge ratio is nothing else than the slope of the curve: - When the option is out of the money, the slope is very close to zero, meaning that we need a large set of potions to cover the underlying asset price variation. 77 - When the option is in the money, the slope is close to 1, so we need something less than one option to cover the change in the price of the underlying asset. For the put option of course is the same, but with a value comprised between 0 and – 1. Since the value and the slope of options change with the underlying price, it is a dynamic strategy: when the underlying asset’s price changes, we need to change our portfolio to hedge risk. As time goes on and the deadline approaches, the value of the option will get closer to the actual payoff that will be received. The difference between the option curve and the payoff is the time value. The longer is the maturity, the larger is the value of the call option. This is due to the fact that, the longer the maturity, the higher is the probability that the underlying price will grow. Time alue P CALL Time alue is negati e for in the mone European put options INTRINSIC VALUE PAYOFF TIME VALUE TIME VALUE INTRINSIC VALUE PAYOFF On the contrary, for a put option, when the option is out of the money the effect is the same. The time Option value value of payoff at strike time to zero as - T is the value has the exactlyIntrinsic same meaning the call option one. Thevalue longergoes the maturity, thet higher probability of underlying prices to go down. chapter . DERIVATIVES On the other hand, when we’re in the money, the longer is the maturity, the higher is the probability that the price of the underlying can grow. IF price grow, payoff is reduced. 78 Correlations The following table summarizes the relationship between the value of an option and the most relevant variables. If the underlying pays cash flows or charges costs, compute the value of the European option using: π ∗ = πr + ππ(πππ π‘π ) − ππ(πππ£πππ’ππ ) ela ion table summarizes hip between the ption and the most bles. ing pays cash flows sts, compute the uropean option osts PV revenues It is interesting to see how volatility positively affects the value of the options. This is one out of all the other financial instruments we saw in which there’s a positive correlation between volatility and the value of the security. If our underlying asset is very risky, this means that the probability of a change of price in the future is very high. We can take advantage of the positive part of the risk avoiding losses on its negative part. Hedging with options Hedging with options allows to take advantage of the ‘positive’ part of the volatility, but this has a cost (meanwhile forward contracts where essentially free). Moreover the ‘delta’ is always different from 1, and is changing whenever the spot price of the underlying changes. Example πππππ πr = $89 → π΅π’π¦ππ’π‘πππ‘ππππ Assume = 1%, T = 6 months, X = $90, sigma=9% Put value = $ 2.552 ; Delta = - 0.526 Therefore we need 1,000/0.526 options = 1,901 options (hedging cost = $ 4,853). In fact if Walt Disney stock falls suddenly to $ 85, the put value goes to $ 5.196: we lose $ 4,000 from shares but we earn (5.196 – 2.552) * 1,901 = $ 5,025 from options (and delta goes to -0.784). 79 De i a i e Derivatives on the Italian exchange he I alia E cha ge IDEM IDEX Deri a i es on financial sec ri ies F res on energ I al and German •F res and op ions E ropean on FTSE MIB inde es • F res and op ions American E ropean on single bl e chips •Di idend f res on bl e chips CHAPTER 80 DERIVATIVES AGREX F res on corn
