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Corporate Finance

Central Concepts in Corporate Finance
List the three basic principles on which the course is rooted.
- Firms employ capital to generate cash flows à capital employed denoted as operating assets
- Employed capital has a cost, the cost of capital à it is an investment
- The productivity of the firm’s capital employed determines its value à generating future cash
flows and investment returns
- Addition: the higher the return relative to the cost of capital, the higher the firm value (value
creation spread) & in order to maximize value, financial resources should be invested into high
return projects (efficient internal capital allocation)
List and describe the major differences between proprietorship and corporation.
- owner = manager
- owners are not managers
- unlimited liability
- limited liability
- limited life
- unlimited life
- no access to public markets
- may be closely held or publicly listed
- easy to set up and no reporting
- cost of set up and reporting
- no conflicts of interest
- potential conflicts of interest
- no corporate taxes
- double taxation
List and describe the primary tasks of the CFO.
- Assessment and evaluation of the cash circulation in the company and making sure that there
are no disruptions
- Reporting financial results to investors, internal decision makers and the board
- Management of incentives to maximize the value of the company
What should be the primary objective of managers?
- Maximizing the firm value (translating into maximizing the share price) for investors is the
manager’s most important task
- Focus is clearly on the long run, which means the intrinsic value which is determined by the
sum of all the future expected free cash flows when converted to today’s value
π‘‰π‘Žπ‘™π‘’π‘’ =
+ β‹―+
(1 + π‘€π‘Žπ‘π‘)
(1 + π‘€π‘Žπ‘π‘)
(1 + π‘€π‘Žπ‘π‘)3
FCF = free cash flows generated by operating assets that are available for distribution for all
investors, they are very risky
wacc = average cost of the funds used by the firm to finance its operating assets
- Maximization is under constraints: society and environment
Why do potential conflicts of interest exist between managers and owners in a modern corporation?
- Managers are naturally inclined to act in their own best interest which are not always the same
as the interest of the stockholders
- Managers focus on their own short-term profit
List the key devices used to align the utility of shareholders and managers.
- The corporate governance deals with the ways in which investors assure themselves of getting
a return on their investment
- Managerial compensation packages
- Direct intervention by shareholders
- Threat of firing and takeover
List and describe the three main capital allocation processes in our economies.
- Demanders of capital need capital to fund their investment opportunities, these groups are
willing to pay a rate of return on the capital they borrow
- Suppliers of capital are saving money and invest it into something while looking for a rate of
return on their investment
- Direct transfers: Business à Savers, Securities (stocks, bonds) à Money
- Indirect transfers through investment bankers: Business à Investment Banks à Savers,
Securities à Money
- Indirect transfers through a financial intermediary: Business à Financial Intermediary à
Savers, Business’ Securities à Money, Intermediary’s Securities à Money
Explain the dual role of financial markets, the law of one price and the concept of arbitrage.
- Financial market: place where individuals and organizations wanting to borrow funds are
brought together with those having a surplus of money
- The dual role means, that there are primary markets and secondary markets
- Primary markets: Firm gives securities and issuance to investors, who give the firm cash, so
they provide financing for new ventures and sustain business growth. The financial markets
act as intermediaries between firms that have cash deficit and investors that have cash surplus.
- Secondary markets: Firms have outstanding securities for investors, who give the firm cash, it
is a mechanism for trading outstanding securities. It allows them to value corporate decisions
and assess managerial performance and the investors can vote with their feet. In secondary
markets, you can aggregate information from various sources to value growth opportunities,
which might be an important sorce of learning for managers.
- Law of one price: in competitive and efficient markets, equivalent goods must trade for the
same price. Two similar outcomes cannot have two different prices in an efficient market.
Implication: To make profit in an efficient market, you must take risk.
- Arbitrage: If the law of one price is not respected, there is an arbitarage opportunity (a
situation in which it is possible to make a profit without taking risk and using personal
resources), so it consists in making a risk-free profit by the practice of buying and selling
equivalent assets for which there is a price differential.
Time Value of Money
Draw a timeline illustrating a given set of cash flows.
You have to differentiate between positive CF (inflows) and negative CF (outflows).
List and define the three rules of time travel.
- Only values at the same point in time can be compared or combined. The reasons why you
cannot compare $100 today and $100 next year are inflation, risk of default and the fact that
you could invest today’s money and generate return.
- To move a flow forward in time, you must compound it. Compounding means, that you can
determine the value of today’s investment at a terminal date n.
𝐹𝑉4 = 𝐢 × (1 + π‘Ÿ)4
- To move a cash flow backward in time, you must discount it. Discounting means, that you can
determine today’s value of an amount of money which is due in the future.
𝑃𝑉 = 𝐢 ÷ (1 + π‘Ÿ)4 =
(1 + π‘Ÿ)4
Calculate the future value of the following: a single sum, an uneven stream of cash flows starting
now or in the future, an annuity starting now or in the future and a growing annuity.
- Here, you also have to compound values.
Decompose the future value of a single sum into original, interest on original and interest on
- Original $ amount of money
- Interest on original: n x $ of money x interest rate
- Interest on interest: FVn – Interest on original - original
Calculate the present value of the following: a single sum, an uneven stream of cash flows starting
now or in the future, a perpetuity, an annuity starting now ore in the future, a growing perpetuity
and a growing annuity.
- General
𝑃𝑉 = 9
4;< (1 + π‘Ÿ)
Given cash flows and interest rate, compute the NPV for a series of cash flows.
- NPV allows us to evaluate on an investment decision as it compares the present value of cash
inflows to the present value of cash outflows
- NPV = PV(benefits) – PV(cost) as our general formula
- Accept the projects with a positive NPV because accepting them is equivalent to receiving their
NPV in cash today
- Reject the projects with a negative NPV because accepting them would reduce the wealth of
- When a constant cash flow C occurs at regular intervals forever
𝑃𝑉(𝐢 𝑖𝑛 π‘π‘’π‘Ÿπ‘π‘’π‘‘π‘’π‘–π‘‘π‘¦) =
Growing perpetuities
- When the initial cash flow C is expected to grow at a constant rate g forever
𝑃𝑉(π‘”π‘Ÿπ‘œπ‘€π‘–π‘›π‘” π‘π‘’π‘Ÿπ‘π‘’π‘‘π‘’π‘–π‘‘π‘¦) =
- When a constant cash flow C will occur at regular intervals for a finite number of N periods
𝑃𝑉(π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘–π‘’π‘ ) = × F1 −
(1 + π‘Ÿ)4
𝐹𝑉(π‘Žπ‘‘ π‘‘π‘Žπ‘‘π‘’ 𝑁) = 𝑃𝑉 × (1 + π‘Ÿ)4 = 𝐢 × ((1 + π‘Ÿ)4 − 1)
Growing annuities
- When the cash flow C grows at a constant rate g for a finite number of N periods
1+𝑔 4
𝑃𝑉(π‘”π‘Ÿπ‘œπ‘€π‘–π‘›π‘” π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘–π‘’π‘ ) =
× J1 − F
Valuing Stocks
Define the Capital-Value Theorem.
- This model indicates that the (intrinsic) value of an asset today (V0) is given by the sum of the
discounted cash flows (CF) generated by the asset in the future. It is possible to determine this
value because we know the cash flow stream generated by the asset.
: LM 3
𝑉< = 𝑃𝑉(𝐢𝐹) = 9
(1 + π‘Ÿ)4
- Asset being a financial or real asset, investment horizon being finite or infinite and r being the
discount rate (the higher, the riskier)
- Therefore, the NPV is the intrinsic value of the asset less the amount spend today to acquire
the asset (NPV = V0 – Investment0)
Under which condition is the stock price equal to the intrinsic value?
- In an efficient equilibrium, we assume that a stock price equals its intrinsic value. If it is not in
the equilibrium, the stock price is mispriced.
- Investors have to estimate the intrinsic value in order to estimate which stocks are attractive,
stocks with a market price below the intrinsic value are undervalued and buying an
undervalued stock at the market price today in hope of selling it at the intrinsic value in the
future is a positive NPV decision.
Calculate the total return of a stock, given the dividend payment Div1, the current price Pn and the
previous price P0.
- The expected total return of the stock should equal the expected return oof other investments
available in the market with equivalent risk.
- Here, we have an example for a one-year investor
- Total return =
Dividend Yield + Capital Gain Rate
𝑃* + 𝐷𝑖𝑣< − 𝑃< 𝐷𝑖𝑣* 𝑃* − 𝑃<
π‘ŸN =
Use the dividend-discount model to compute the value of a stock, whether the dividends grow at a
constant rate starting now or in the future.
- Value of a stock is the PV of the future dividend payments: dividend-discount model
- Cash flows CF à Dividends Div and the cash/sales price Pn after selling the asset at the end of
the holding period
- Discount rate r à Cost of equity rE being the required return rate by equity investors
- The intrinsic value will be equal to the present value of the expected cash flows
- We assume a one-year investor, so we derive at this formula for the stock price
𝐷𝑖𝑣* + 𝑃*
𝑃< =
(1 + π‘ŸN )
- Now, we assume a multi-year investor for N years
𝐷𝑖𝑣4 + 𝑃4
𝑃< =
+ β‹―+
(1 + π‘ŸN ) (1 + π‘ŸN )
(1 + π‘ŸN )4
- Now, we hold the stock forever
𝐷𝑖𝑣* + 𝑃*
𝑃< = 9
(1 + π‘ŸN )
The simplest forecast of forecasting a firm’s future dividends is assuming that they will grow
at a constant rate g forever. If the growth rate g now exceeds the cost of equity rE, the formula
leads to a negative stock price, so we can only use the constant growth model if rE > g and g is
expected to be constant forever
π‘ŸN − 𝑔
π‘ŸN =
𝑃< =
Discuss the two key drivers of the growth rate in earnings and in dividends.
- To increase earnings over the next periods, the firm needs to invest into new assets and
therefore you need to reinvest a fraction of earnings (retained earnings for investment
- Dividend: In order to increase the dividend per share a firm can increase its earnings or
increase the fraction of the earnings that is distributed as dividends, so we have to decompose
the earnings per share
- The formula for the dividends per share = earnings per share x payout rate
πΈπ‘Žπ‘Ÿπ‘›π‘–π‘›π‘”π‘ Q
𝐷𝑖𝑣Q =
× π‘‘Q
π‘†β„Žπ‘Žπ‘Ÿπ‘’π‘  π‘‚π‘’π‘‘π‘ π‘‘π‘Žπ‘›π‘‘π‘–π‘›π‘”Q
- The formula for new investment per share = earnings per share x retention rate which will
generate incremental earnings over the next periods
πΈπ‘Žπ‘Ÿπ‘›π‘–π‘›π‘”π‘ Q
× (1 − 𝑑Q )
π‘†β„Žπ‘Žπ‘Ÿπ‘’π‘  π‘‚π‘’π‘‘π‘ π‘‘π‘Žπ‘›π‘‘π‘–π‘›π‘”Q
- Earnings: You can pay them out as dividends or retain and reinvest them into new assets à
important financial decision whether earnings should be allocated or reinvested
- So, our simple model of growth changes because of the new investments: EPSt+1 will be made
of earnings generated from asset in place and incremental earnings from new investments
made at time t; if there were no new investments, there would only be earnings from assets
in place
- Change in EPSt+1= EPSt+1 – EPSt = βˆ†t+1
= new investment per share today x return on new investment
= EPSt x (1-dt) x RONI
- growth between t+1 and t – which is in terms of a stable investment policy and stable dividend
payout as well as retention ratio (dividend and retention policy) the long-term growth rate –
πΆβ„Žπ‘Žπ‘›π‘”π‘’ 𝑖𝑛 𝐸𝑃𝑆QW* 𝐸𝑃𝑆Q × (1 − 𝑑Q ) × π‘…π‘‚π‘πΌ
𝑔* =
= (1 − 𝑑Q ) × π‘…π‘‚π‘πΌ
…. so, g = retention rate x RONI
- therefore, the key drivers are the retention rate as a fraction of reinvested earnings and the
RONI as return of the new investments
Given the retention rate and the return on new investment, calculate the growth rate in dividends,
earnings and share price.
- You can see the calculations above
Describe circumstances in which cutting the firm’s dividend will raise the stock price.
- Cutting the firm’s dividend to increase investment will increase the stock price, if and only if,
the new investments have a positive NPV
- The NPV of the new investment will be positive if, and only if, the investment return is higher
than the cost of equity (positive value creation spread (higher, the better)
- You need profitable growth in order to let the stock price increase, if you cannot guarantee
this, pay out the dividends
- Also, the RONI should be higher than the cost of equity rE
Investment Decision Rules
Define NPV, IRR, payback period and incremental IRR.
- NPV = PV(benefits) – PV(costs), so it is an investment flow depending on the discount rate
measuring the contribution of the project to the value of a firm and shareholder wealth
- IRR is the discount rate such that PV(benefits) = PV(costs), IRR forces the NPV to be 0
- Payback period is the amount of time it takes to recover or pay pack the initial investment and
if the payback period is less than a pre-specified period of time, you accept it and otherwise,
you reject it; this rule is used because it is very simple
- Incremental IRR is the IRR of incremental cash flows that would result from replacing one
project with the other, it tells us the discount rate at which it becomes profitable to switch
Describe the NPV and IRR decision rules, for both stand-alone and mutually exclusive projects.
- NPV stand-alone project: Accept the project if the NPV is positive, reject it otherwise.
- IRR stand-alone project: Accept the project if the IRR is higher than the cost of capital, reject
it otherwise.
- Combination stand-alone projects: usually, they give the same answer, but when the rules
conflict, the NPV rule should be prioritized; they are consistent if the project’s negative CF
precede its positive CF but if the IRR conflicts with the NPV if there are not normal cash flows
- NPV mutually exclusive (among alternatives): Select the project with the highest NPV
- IRR mutually exclusive: You can only use the IRR if the investment projects have the same scale
and the same risk, then, you choose the highest return. An alternative would be taking the
incremental IRR into account where you determine if switching from one project to another is
profitable. The incremental IRR needs to be higher than the cost of capital.
Given cash flows, compute the NPV, payback period and IRR for a given project and the incremental
IRR for a pair of projects.
- NPV = PV(benefits) – PV(costs) while IRR: NPV = 0, so PV(benefits) = PV(costs)
- Payback period: Number of years prior to full recovery + (uncovered cost at start of year/cash
flow during full recovery year)
- Incremental IRR: βˆ†Projects and then calculating the IRR or just look for the crossover point of
the NPV functions
Discuss the reasons IRR can give a flawed decision when assessing stand-alone projects with nonnormal cash flows, and also when comparing mutually exclusive projects with normal cash flows.
- Delayed investments: the investment flow occurs in the future and not today, which is
especially important when all the benefits of an investment occur before the costs, so the NPV
is an increasing function of a discount rate giving us an IRR > cost of capital but often a negative
NPV at the actual discount rate
- Multiple IRRs: If there are several IRRs (NPV function with minimum), IRR rule can’t be applied
- Nonexistent IRR: There is no IRR if the NPV is positive for all values of the discount rate, thus
the IRR rule cannot be used
- In terms of mutually exclusive projects, selecting the project with the highest IRR can lead to
mistakes, because we can only compare projects with the same scale and risk.
Despite the existence of potential flaws with the IRR rules, why do managers use the IRR in their
capital budgeting decision? List and explain two main reasons discussed in class.
- Managers prefer to compare investment returns to hurdle rates and the IRR measures the
(average) return of a project
- The IRR allows to assess the sensitivity of the NPV to any estimation error in the cost of capital
à if the IRR is very close to the cost of capital the NPV rule (take vs. leave) is highly sensitive
to a small change in the cost of capital in this case
Capital Budgeting
What is capital budgeting? And why is it so important?
- Analyzing alternative investments (like potential additions to the firm’s fixed assets) and
decide which ones to accept based on CapEx (positive NPV, high return)
- Important driver of the firm’s future growth and needs to be carefully implemented to have
profitable growth
What are the steps in capital budgeting for a given project?
- Estimating incremental cash flows (inflows & outflows); Assessing riskiness of the cash flows;
which CF do you want to keep?
- Determining the appropriate cost of capital; so, you have to discount future cash flows and if
they are very risky, the cost of capital grows
- Rely on investment decision rules to take or leave the project
The process starts by forecasting the future consequences of the project on the firm’s revenue
/ costs by determining the incremental earnings of the project (the amount by which the firm’s
earnings are expected to change as a result of the new investment) until you derive at the
unlevered net income ((Revenues – Costs – Depreciation) x (1-tax), also called the net
operating profit after tax
The next step is the determination of the project’s free cash flow from the earnings, because
the incremental effect of the project on the firm’s available cash is the project’s free cash flow
(generated by the project after tax, so after investment but before interest payments so it
corresponds to the cash that belongs to investors), so
FCF = (Sales – Costs) x (1 – τC) – CAPEX - βˆ†NWC + (τC x Depreciation)
Having determined the project’s FCF (including salvage and termination value), the next step
is to determine the present values by discounting them
Finally you can come up with the project’s NPV by summing up the discounted FCF from year
0 over the lifetime n of the project; if it is positive, you should decide for the investment and
if there were several alternatives, you should decide for the one investment project with the
highest NPV in order to decide for the least expensive alternative
It is important to know that the NPV depends on estimated value drivers like the base line key
value drivers like quantity, margin, cost of capital and growth rate as well as sensitivity to small
The final step is assessing the importance of parameter uncertainty and identifying the value
drivers of the project
Given a set of facts, identify relevant cash flows for a capital budgeting problem.
- Revenue estimates: sales, per unit price
- Cost estimates: up-front R&D, up-front CapEx (Equipment being depreciated), annual
overhead cost (SGA) and per-unit cost
- Depreciation of assets as cash outflow
- Income tax as cash outflow
- Opportunity costs, effect on SGA
- Project externalities, positive/negative effect on profits (cannibalization) of other product lines
because they change the profit in other product lines, have an effect on Sales and COGS
Explain why opportunity costs must be included in cash flows, while sunk costs and interest expense
must not.
- Opportunity costs: The opportunity cost is the value a resource could have provited in its best
alternative use, so this value is lost if the resource is allocated to the project and therefore,
the value loss should be included in the project’s incremental loss
Sunk costs: They occur regardless of the decision whether or not the investment is undertaken
and therefore, they are not included in the incremental earnings analysis (fixed overhead
expenses because we only include additional overhead expenses that arise because of the
project, past R&D expenditures because this money was spent before investment decision…)
Interest expense: Tax deductibility will be taken into account in the discount rate by relying on
a cost of debt after tax
Explain why tax credit are considered as a positive cash flow for the project in profitable companies.
- Income tax is determined by multiplying the EBIT with the marginal corporate tax rate, if you
have a negative income tax this is equal to a tax credit, which is a positive cash flow for the
firms). If there the EBIT already shows a loss, there will be a tax credit.
Calculate FCF for a given project.
- FCF = (Sales – Costs) x (1 – τC) – CAPEX - βˆ†NWC + (τC x Depreciation)
FCF = (Sales – Costs – Depreciation) x (1 – τC) + Depreciation – CAPEX - βˆ†NWC
- Starting point is the net income
- Add depreciation: non-cash expense, which reduces taxable income, but needs to be added
back when going from unlevered net income to cash flow
- Reduce CAPEX: they are the actual cash outflows when an asset is purchased
- Reduce βˆ†NWC: many projects require investments in NWC (Cash + Inventory + Receivables –
Payables), so an increase in NWC (βˆ†NWCt = NWCt – NWCt-1) because having to maintain cash
balance and inventories is a cash outflow
Illustrate the impact of depreciation expense on cash flows.
- It is a non-cash expense, which reduces taxable income, but needs to be added back when
going from unlevered net income to cash flow
- As it is an accounting flow, we have to add it up again
Given a set of facts, calculate the FCF with salvage value or with terminal (continuation) value.
- Salvage value: assets that are no longer needed have are resale value or salvage value if the
parts are sold for scrap, but sometimes some assets do have a negative liquidation value
After-tax CF from Asset Sale = Sale price – τC x Capital Gain
Capital Gain = Sale Price – Book Value
Book Value: Purchase Price – Accumulated Depreciation
- Terminal (continuation) value: represents the market value of the FCF from the project at all
future dates but it is only added in one respective year for which we calculate it for, we often
have a constant growth perpetuity
π‘‡π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘™ π‘‰π‘Žπ‘™π‘’π‘’ 𝑖𝑛 π‘¦π‘’π‘Žπ‘Ÿ 𝑛 = 𝐹𝐢𝐹4 × F
Use breakeven analysis, sensitivity analysis or scenario analysis to evaluate project risk.
- As the parameters of the capital budgeting model are subject to significant uncertainty and
estimation risk, we have to assess their importance and identify the value drivers
- Break-even analysis: The goal is to determine the break-even level of a key input, which is the
level of an input that forces the NPV of the project to be zero. We do that for each parameter.
If now one parameter exceeds the break-even parameter values, we destroy value
- Sensitivity analysis: The goal is to assess the sensitivity of the project’s NPV to changes in one
of the key parameters/value drivers, holding the other model assumptions constant. So, how
does the NPV change if one parameter is changed?
- Scenario analysis: The goal is to assess the sensitivity of the project’s NPV to changes in several
key parameters/value drivers at the same time while holding the other model assumptions
Capital Structure in a Perfect Capital Market
Define capital structure.
- It refers to the relative proportions of debt and equity that a firm has outstanding and indicates
how the operating cash flows or the value of firm will be shared among the investors à
answers the question how a company is financed
- Net Debt D = Financial debt – Cash, also debtholders provide debt financing and expect a
compensation per unit of financing provided (cost of debt rD)
- Equity E, so shareholders jprovide equity financing and expect a compensation per unit of
funding provided (cost of equity rE)
- The capital structure determines the cost of capital (wacc or the average financing cost of the
invested capital) being a function of leverage (D/E), rE, rD and tax
π‘€π‘Žπ‘π‘ =
π‘ŸN +
π‘Ÿ (1 − π‘‘π‘Žπ‘₯)
𝐸+𝐷 [
Given a capital structure, explain how the total cash flow is allocated to equity and debt investors
and calculate the expected return to equity investors.
- rE is the cost of equity = rF + βE x (rM – rF) = risk-free rate + equity beta x market risk premium
and corresponds to what shareholders are asking in return for € 1 invested in the company, so
it is their expected return, the most important driver is the equity beta βE quantifying the risk
- rD is the cost of debt = rD = rF + βD x (rM – rF ) = risk free rate + default spread and corresponds
to the rate at which the firm can borrow money currently
- When allocating the cash flow, debt investors’ requirements need to be covered first
What is a risk-free debt?
- The cost of debt is only the risk-free rate rF
- βD needs to be zero
List the three conditions that make capital markets perfect.
- Investors and firms can trade the same set of securities at competitive market prices
- There are no taxes, transaction costs or issuance costs associated with security trading
- A firm’s financing decisions do not change the cash flows generated by its investments
Explain why in a perfect capital market the total value of a firm is not affected by its choice of capital
- Equity is riskier than debt and requires a higher rate of return, rE (rF + βE x (rM – rF)) > rD
- Reducing the wacc by substituting cheap debt for expensive equity? It does not make sense!
If we increase the D/E ratio, our equity becomes even more risky and equity holders will
require even higher return because of increasing financial risk. So, cheap debt replaces
expensive equity, but the cost of the remaining equity increases, and the value stays the same.
- Leverage increases the risk of equity even when there is no risk that the firm will default. Thus,
while debt may be cheaper, its use raises the cost of equity. Considering both sources of capital
together, the firm’s average cost of capital with leverage is the same as for the unlevered firm.
What is the implication of MM Proposition I on the cost of capital?
- In a perfect capital market, the total value of a firm is equal to the market value of the total
cash flows generated by its assets and is not affected by its choice of capital structure
- It states, that the firm value is not affected by leverage
VL = EL + D = A = VU
- MM’s first proposition can be used to derive an explicit relationship between leverage and the
cost of equity
Calculate the effect of a change in leverage ratio on the cost of equity (MM Proposition II) and cost
of capital in a perfect market.
- The cost of levered equity is equal to the cost of unlevered equity plus a premium that is
proportional to the market value debt-equity ratio.
- We assume that the unlevered equity is related to the returns of levered equity and returns of
π‘ŸN +
π‘Ÿ = π‘Ÿ] = π‘Ÿ^
𝐸+𝐷 [
So, we solve for rE and derive at MM II
π‘ŸN = π‘Ÿ^ + (π‘Ÿ^ − π‘Ÿ[ )
Where rU is the risk without leverage (economic/operating risk) and the rest of the formula is
the additional risk due to leverage (financial risk)
- So, the cost of equity compensates for operational and financial risk
- Finally, we can derive at the conclusion that in terms of changing the leverage ratio, the wacc
remains unchanged but the rE changes: if you reduce the leverage ratio, the cost of equity
Calculate the asset beta, knowing the equity beta and leverage ratio.
- The asset beta βA is also called the unlevered beta βU and gives a weighted average of the
equity and the debt beta – it is the measure of risk of a firm as it did not have leverage and
therefore accounts for operating risk only
𝛽^ =
𝛽N +
𝐸+𝐷 [
Or you can use the inverted Hamada equation if the debt is risk free
𝛽^ = 𝛽N ÷ F1 + G
- Just for clarification, the βE gives the quantity of risk in the equity of the firm that is
compensated in the market, so whenever the market return changes, the equity return
changes by the same extend multiplied by the βE
- If you try to estimate the unlevered beta, you should base your estimate on unlevered betas
of firms with comparable investments/assets
Illustrate the effect of an increase in leverage on the equity beta.
- Leverage amplifies the market risk of a firm’s assets (βU) raising the market risk of its equity βE,
you can calculate it with the Hamada Equation
How much should a firm borrow?
Explain the effect of interest payments (and corporate tax) on cash flows to investors.
- The tax authority collects taxes on the taxable income as operating agent having a right on CF
- Total CF = CF-to-Debt (interest pay.) + CF-to-Tax (tax auth.) + CF-to-Equity (Equ. Inv.)
- Because corporations pay taxes on their operating cash flows after interest payments, interest
expense reduces the amount of corporate taxes so debt financing sets an incentive to use debt
in order to have tax savings but debt obligations reduce the CF available for equity investors
- In total, the CF available to all investors is increases for a company with leverage as tax savings
create an additional cash flow, the interest tax shield
- When a firm uses debt, the interest tax shield provides a corporate tax benefit each year,
translating into a higher cash flow for investors (CF to investors with leverage = CF to investors
without leverage + interest tax shield)
Still companies do not fully exploit the tax deductibility of interest because an excessive use of
debt incurs other costs like financial distress, bankruptcy costs and agency costs of debt
Calculate the interest tax shield, given the corporate tax rate and interest payments.
- The additional cash flow comes from the interest tax shield (tax savings due to the tax
deductibility of interest expenses)
- It corresponds to the reduction in taxes paid due to the tax deductibility of interest, so it is ITS
= Corporate Tax Rate x Interest Payment
Calculate the present value of the interest tax shield for risk-free debt, fairly priced permanent debt
and firms with target leverage ratio.
- Risk-free debt/interest tax shield: Here, we can value the ITS as an annuity
𝑃𝑉(π‘Žπ‘›π‘›π‘’π‘–π‘‘π‘–π‘’π‘ ) = × F1 −
(1 + π‘Ÿ)4
× F1 −
(1 + π‘Ÿ)4
- Fairly priced permanent debt: We assume, a firm borrows debt D and keeps it permanently,
so we only need the fairly priced permanent debt and a constant tax rate while the debt does
not necessarily need to be risk-free and the interest rate not constant
𝑃𝑉(𝐼𝑇𝑆) = 𝜏b × π·
- Target leverage ratio: The value of this interest tax shield is given by the difference between
VL and VU, so PV(ITS) = VL – VU
𝑉c = 9
Q;* (1 + π‘Žπ‘“π‘‘π‘’π‘Ÿπ‘‘π‘Žπ‘₯ π‘€π‘Žπ‘π‘)
𝑉^ = 9
Q;* (1 + π‘π‘Ÿπ‘’π‘‘π‘Žπ‘₯ π‘€π‘Žπ‘π‘)
If the FCF is growing at a constant rate, we use these formulas
𝑉c = 9
Q;* π‘Žπ‘“π‘‘π‘’π‘Ÿπ‘‘π‘Žπ‘₯ π‘€π‘Žπ‘π‘ − 𝑔
𝑉^ = 9
Q;* π‘π‘Ÿπ‘’π‘‘π‘Žπ‘₯ π‘€π‘Žπ‘π‘ − 𝑔
Calculate the value of a levered firm in the presence of corporate tax.
- The value today of the tax benefit over the firm’s lifetime is equal to the present value of the
stream of future interest tax shields the firm will receive
- In order to value the levered firm, we need to value the unlevered firm and the interest tax
shield stream (in accordance with MM Proposition I)
VL = VU + PV(ITS), so you add up the PV of the ITS to the value of the unlevered firm
(π‘‡π‘Žπ‘₯ π‘…π‘Žπ‘‘π‘’ × πΌπ‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘ƒπ‘Žπ‘¦π‘šπ‘’π‘›π‘‘)4
𝑉c = 𝑉^ + 9
(1 + π‘Ÿ)4
We use this formula because we have to model the future interest payments and tax rates and
then determine the discount rate which accounts for the risk of the interest tax shield
Calculate the weighted average cost of capital with corporate taxes.
- First of all, the wacc decreases due to tax savings
- The effective cost of debt is rD x (1 – τC), so our new wacc formula is
π‘€π‘Žπ‘π‘ =
π‘ŸN +
π‘Ÿ (1 − τb )
𝐸+𝐷 [
π‘€π‘Žπ‘π‘ =
π‘ŸN +
π‘Ÿ[ −
π‘Ÿ τ
𝐸+𝐷 [ b
Pretax wacc
decrease in wacc due to tax savings
Describe the effect of a leveraged recapitalization on the value of equity with corporate tax.
- Leverage recapitalization means, that you borrow debt on a permanent basis in order to fund
the repurchase of outstanding shares and therefore change the right side of the balance sheet
- The value of equity drops after the recap because you borrowed debt in order to increase the
value of the firm, but equity investors still benefit from the recapitalization
- the increasing value of the firm is displayed in the NPV of the company which also leads to an
increasing NPV per outstanding share, so the value of the equity investors’ shares will increase
Describe the effect of bankruptcy and financial distress in a perfect capital market and explain why
MM Proposition 1 is still valid.
- Bankruptcy: can occur if a company heavily uses leverage, equity financing does not carry this
risk because the firm is not legally obliged to pay them
- Financial distress: when a firm has difficulties meeting its debt obligations and can lead a firm
to default
- Default: when a firm fails to make the required interest or principal payments, so debt holders
can liquidize the company or reorganize the assets while the equity investors will gain nothing
- MM Proposition 1: In a perfect capital market, the bankruptcy process is not costly and there
is no additional financial distress cost, so the total cash flow will still be the same because the
value is still independent in a perfect capital market. Bankruptcy simply shifts the ownership
of the firm from equity holders to debt holders without changing the total value available to
all investors à VL = VU
Explain why, in an efficient market, the original shareholders of a firm pay the present value of the
financial distress costs.
- If a product fails, equity holders lose their investment in the levered firm and will not care
about bankruptcy costs, but on the other hand debt holders will provide less debt financing,
so less money is even available to pay dividends, repurchase shares and make investments, so
the difference comes out of the equity à the original shareholders pay the present value of
the costs associated with financial distress
Discuss several direct and indirect costs of bankruptcy and financial distress.
- Bankruptcy is a very complex, time-consuming and costly process, because it imposes direct
and indirect costs
- Direct costs: administrative and legal, around 3-4% of the pre bankruptcy market value of
- Indirect costs: Loss of customers (worrying about resale value etc.), suppliers (disinclined to
put effort into business relationships etc.), employees (difficult to attract new employees etc.)
and fire sales of assets in order to avoid bankruptcy under pressure with less bargaining power
à add up to 20% of the pre bankruptcy market value of assets
What does the trade-off theory imply in terms of debt policy?
- Low levels of debt à low risk of default & increase in leverage à increase in ITS
- Increasing debt à costs of financial distress à reduce the firm value
- Firms should increase their leverage until it reaches the level for which the firm value is
maximized, because at this point the tax savings are perfectly offset by the increased
probability of costs of financial distress
- This is why firms choose debt levels that are too low to fully exploit the ITS (presence of
financial distress costs) and there are differences in the use of leverage across industries
(differences in the magnitude of financial distress costs and volatility of CF)