finance-1-week-7

This document was downloaded from https://efr.nl on 10/18/2019

Use of this document is intended for: Chu, Eileen - Student number: 504051

EFR summary

finance 1, 2018-2019 lectures 1 to 14

weeks 1 to 7



details: subject: finance 1 IBEB 2018-2019 teacher: Laurens Swinkels and N.L. Van der Sar date of publication: 14-12-2018

© This summary is intellectual property of the Economic Faculty association Rotterdam (EFR). All rights reserved. The content of this summary is not in any way a substitute for the lectures or any other study material. We cannot be held liable for any missing or wrong information.

The Erasmus School of Economics is not involved nor affiliated with the publication of this summary.

Economic Faculty association Rotterdam (EFR) 2



finance 1 – IBEB – lecture 1, week 1

Hirshleifer model and Fisher separation theorem

introduction

In finance, individuals face the allocation decision, a financial economic decision for people, deciding on how much to consume and invest in financial and real markets.

Investments in financial assets ( financial market ) at a point in time are called financial investments, but over time these investments can generate cash flows. Real assets and projects ( real market ) are likewise real investments and over time, these can also generate cash flows.

Hirshleifer model without real market

The Hirshleifer model one period covers two points in time, now ( 𝑡 = 0 ) and later ( 𝑡 =

1 ). There is no risk in investing, because full certainty is assumed. Saving and borrowing are possible at the same interest rate in a perfect financial market.

Income received now is either consumed now or kept for later, and income received later is consumed later as well. With the existence of a financial market, current income can be saved and consumed later in time, or future income can be consumed now by borrowing.

When current income is 𝐶𝐹

0

and consumption is more than this, at point 𝐶

0

, one must borrow in order to fulfil its consumption. As a result, future income 𝐶𝐹

1

cannot be fully used to for consumption anymore (because in the second period, the borrowed amount has to be returned, along with the accumulated interest). This results in consumption at 𝐶

1

< 𝐶𝐹

1

. The consumption points have to match the consumption possibility line .

The indifference curves show which combination brings the same utility with combinations of 𝐶

0

and 𝐶

1

. The higher and more to the right the indifference curve is, the higher the utility is as more consumption is possible; it shows the trade-off that maximizes the consumption at both points in time. The consumption point will be where the consumption possibility line touches the highest possible indifference curve.

The total income can be expressed in current terms as present value , or in future terms as 𝐶𝐹

0

𝐶𝐹

0

+

𝐶𝐹

1

1+𝑟

, also known as the

(1 + 𝑟) + 𝐶𝐹

1

( future value ).

Hirshleifer model with real market

The only difference between the Hirshleifer model with the real market and the model without the real market will be how much is invested in real markets. The return curve shows how successive real investment generates revenue and the steepness of the curve declines showing diminishing marginal revenues, thus at one point the additional revenue is not sufficing investment anymore. When the marginal revenue is equal to the interest rate , one is indifferent between investing in the real market and investing in the financial market.

After a real investment, the consumption possibility line shifts to the right, the possibilities increase, and a higher utility can be reached. Investing in both the financial



Use of this document is intended for: Chu, Eileen - Student number: 504051 and the real market allows for a higher utility. Real markets make sure profitable projects will be used, and the financial market enables individuals to spread their income for consumption in different periods of time.

Fisher separation theorem

The Fisher separation theorem states that the decision to invest in real markets (real investment) is taken separately from the consumption decision. The optimal consumption point is where the marginal utility of consumption now and in the future are equal to the interest rate . In this separation theorem, every individual will invest if the net present value ( NPV ) is greater than 0.

Note : For an application of the financial markets on pensions, check slides 60-62 of

Lecture 1.


company

A company has on the assets side the projects they have invested in that generate cash flows . On the liabilities side a company has equity, loans and bonds. Equity holders are paid dividends, and holders of loans and bonds will receive interest. If a company is not able to pay bondholders, the company is bankrupt.

Since the company is owned by equity holders, they prefer the dividend payments to be as high as possible (this is determined by the company’s performance). The chief executive officer’s (CEO) job is to run the company and to maximize the shareholder value . The objectives of the CEO may differ from those of shareholders, which gives rise to the principal-agent problem .

ratios

Finance uses a lot of different ratios are used to judge the performance of a company.

The most important ratios are:

1. Profitability: 𝑛𝑒𝑡 𝑝𝑟𝑜𝑓𝑖𝑡 𝑚𝑎𝑟𝑔𝑖𝑛 = 𝑠𝑎𝑙𝑒𝑠 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑠𝑠𝑒𝑡𝑠

2. Liquidity: 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑟𝑎𝑡𝑖𝑜 = 𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒

; 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 𝑑𝑒𝑏𝑡

3. Leverage: 𝑑𝑒𝑏𝑡 𝑡𝑜 𝑒𝑞𝑢𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 =

, 𝑞𝑢𝑖𝑐𝑘 𝑟𝑎𝑡𝑖𝑜 = 𝑒𝑞𝑢𝑖𝑡𝑦

;

4. Valuation:

𝑃

𝐸 𝑟𝑎𝑡𝑖𝑜 = 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒

. 𝑐𝑎𝑠ℎ+𝑛𝑒𝑎𝑟 𝑐𝑎𝑠ℎ 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠

;

time

When receiving cash now, the value of the cash today is different from the value of the same amount in the future. The present value of a cash flow is equal to:

𝑃𝑉

0

= 𝐶𝐹

𝑇

× (1 + 𝑅

𝑇

) −𝑇

When receiving a series of cash flows, the present value is calculated as:

𝑃𝑉

0

= 𝐶𝐹

1

× (1 + 𝑅

1

) −1 + 𝐶𝐹

2

× (1 + 𝑅

2

) −2 +…+ +𝐶𝐹

𝑇

× (1 + 𝑅

𝑇

) −𝑇

If cash is received now, its future value is:

𝐹𝑉

𝑇

= 𝐶𝐹

0

× (1 + 𝑅

𝑇

) 𝑇




An annual amount (A) which is received for T periods can be expressed in the present value by using the following formula:

𝑃𝑉

𝑃𝑉

0

0

= 𝐴 × (1 + 𝑅)

=

𝐴

𝑅

−1 + 𝐴 × (1 + 𝑅)

× (1 − (1 + 𝑅) −𝑇 ) .

−2 +…+ +𝐴 × (1 + 𝑅) −𝑇

which is equivalent with

As the number of annuity payment periods increases, the present value of the annuity becomes more expensive, which can also be seen in pensions. Due to increased longevity of people, more annuity payments have to be made, thus pensions become

𝐴 more expensive to finance. If T approaches infinity, this means the payment equals

𝑅

.

risk

Risk is dependent on the probability of an accident and whether the individual is risk loving, neutral or averse. The expected value of a future cash flow that depends on a probability can be calculated by:

𝐸{𝐶𝐹

𝑇

} = 𝑝 𝑎

× 𝐴 + 𝑝 𝑏

× 𝐵

Where 𝑝 𝑎

is the probability of receiving payment A and 𝑝 𝑏

is the probability of receiving payment B. The risk neutral value can be found by dividing the expected value by the interest:

𝐸{𝐶𝐹

1+𝑅

𝑇

}

.

law of one price

The Law of One Price adjusts in accordance to investments. In different competitive markets, for the law of one price to hold, equal investment opportunities must exist for trade at the same price at the same time in both markets.

arbitrage

In financial market equilibrium, arbitrage opportunities should be non-existent because of the competitive market for buying and selling. Two conditions should be met for an arbitrage opportunity, this opportunity should never cause a loss and sometimes it generates profit.

finance 1 – IBEB – lecture 3, week 2 investment decision rules

cash flows

Something important to keep in mind is that costs and revenues are different than cash outflows and cash inflows. Costs are incurred, and revenues are generated in a period, while a cash inflow or outflow occurs at a moment in time. This implies that profit is not equal to cash inflow minus cash outflow. Cash flows determine the value. The timing of depreciation influences the NPV of a cash flow, the earlier depreciation the higher NPV.

Loire and Savage established in 1955 three investment questions:

When is an investment attractive?

How to decide between investing in two (mutually exclusive) investments?

How to invest with a limited budget?

when is an investment attractive?




For determining whether an investment is attractive, we can use the NPV rule . If the

NPV is bigger than zero, the project will create value and it is attractive to invest in. But when the NPV is smaller than zero, thus negative, the project will be unattractive because the value will be destroyed. When the NPV equals zero, there will be neither a creation nor a destruction of value.

𝑇

𝑁𝑃𝑉 = ∑

𝐶𝐹

𝑇

(1 + 𝑟) 𝑡 𝑡=0

Some features of the NPV include:

Additivity : if you invest in two projects, 1 and 2, you can calculate the NPV of both projects separately and add these to get the NPV of investing in both projects. The formula is: 𝑁𝑃𝑉(𝐴 + 𝐵) = 𝑁𝑃𝑉(𝐴) + 𝑁𝑃𝑉(𝐵) . If the discount rates are equal, you can as well first sum up the cash flows in each period and afterwards calculate the NPV

Sunk costs : if an amount has already been invested in a project, this has already been paid and does not influence future decisions of the project. When time passes, and new information becomes available, the NPV of a project might become 0 or turn negative, at that point we have to forget about sunk costs and stop investing in the project. Sunk costs are not taken into account because these costs are already incurred

Differential cash flows : if there is an extra investment in a new project, this gives us a cash flow without extra investment and a cash flow with extra investment. The difference between these two cash flows is the differential cash flow, in this way extra investments can be judged apart from the initial investment

An investor decides whether an investment attractive by looking at the NPV. If the NPV is bigger than zero, then they should invest, but when it is equal zero or smaller they will decide not to invest in the project.

how to decide between investing in two (mutually exclusive) investments and how to choose between investments?

When choosing between different investments, there may be some complications with calculating the NPV. Some projects may have a different duration, replacement or inflation. When projects have a different duration , we have to extend the timeline of the projects. For example, if project 1 is from 𝑡 = 0 until 𝑡 = 4 and project 2 is from 𝑡 =

0 to 𝑡 = 3 , we will calculate the NPV for period from 𝑡 = 0 until 𝑡 = 12 . This is because at 𝑡 = 12 we will have invested in project 1 three times and in project 2 four times.

When now comparing the NPV we can make the right decision whether to invest in project 1 or 2.

For a different replacement after the project and different duration we use the same method as for projects with different duration. To establish the optimal replacement strategy, we have to equalize the duration of all projects to determine which project gives the highest NPV. We can use the m-year replacement-investment formula to calculate the NPV:

𝑁𝑃𝑉

∞,𝑚

= 𝑁𝑃𝑉 𝑚

1

× (

1 − 1/(1 + 𝑟) 𝑚

)

In the Fisher-equation we can derive the real interest rate, inflation and nominal interest rate when two of these are given:




(1 + 𝑘) = (1 + 𝑖) × (1 + 𝑟)

Important for deciding on projects, is that for nominal cash flows , we have to use the nominal interest rate , the same holds for real cash flows and the real interest rate. It is not possible to add these two interest rates. Taxes are calculated on nominal cash flows.

how to invest with a limited budget?

If we have to deal with a limited budget when investing, there are three steps to take.

First, we will start by looking at the possible combinations of projects that are available.

When we have all project combinations, we have to cancel out the combinations that are infeasible. Finally, the project combination with the highest NPV is chosen given the budget.

alternative investment decision rules

The pay-back period (PBP) and the internal rate of return (IRR) can both be used to help deciding on investment decisions. They can be useful, but there is still a possibility they give the wrong answer. The NPV is always considered the most favourable method.

The pay-back period (PBP) can be found with the following formula: 𝑛

𝐼 𝑡=0

≤ ∑

𝐶𝐹 𝑡

(1 + 𝑟) 𝑡 𝑡=1

In this formula we assume 𝑟 = 0 , when r is bigger than zero, it is an economic payback-period. The smallest value of n for which the PBP formula holds is the pay-back period. A problem with this method is that only cash flows in the specific period are taken into account, and what happens after that is not considered. Even though the answer is not always correct, this alternative method is used because of uncertainty in the long run, because the future is hard to predict.

The internal rate of return (IRR) is the interest rate at which the NPV of an investment project equals zero. If 𝐼𝑅𝑅 > 𝑟 this indicates that 𝑁𝑃𝑉 > 0 and hence it is attractive to do the project. The IRR can only be deduced by trial and error and the formula is such that: 𝑛

0 = ∑

𝐶𝐹

𝑇

(1 + 𝐼𝑅𝑅) 𝑡 𝑡=0

A few problems with IRR are that it ignores the size of a project, and with longer duration projects where reinvestment is assumed this might be a disadvantage. In case of positive and negative cash flows there is a possibility for two sets of IRRs to exist with positive and negative interest rate when NPV is zero, which may pose problem in deducing in a distinctly defined IRR.

finance 1 – IBEB – lecture 4, week 2 capital budgeting and valuing bonds

capital budgeting




The process of analysing investment opportunities and deciding which one to choose is called capital budgeting . This is analogous to the investment decision rules we discussed in the previous lecture. Capital budgeting involves predicting earnings and costs, computing the free cash flow and dealing with risk and uncertainty in investing.

In capital budgeting, cash is used to facilitate transactions. Therefore, in the beginning we have a NWC (net working capital) increase, and at the end of the investment, this amount is deducted again because the cash can be retrieved back after the project. 𝑛𝑒𝑡 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑠𝑠𝑒𝑡𝑠 − 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 𝑓𝑟𝑒𝑒 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤

= (𝑟𝑒𝑣𝑒𝑛𝑢𝑒 − 𝑐𝑜𝑠𝑡𝑠 − 𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛) × (1 − 𝜏 𝑐

− 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑒𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 − ∆𝑁𝑊𝐶

) + 𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛

In capital budgeting we calculate the present value of the Free Cash Flow as NPV like we previously did. When we have a tax loss carry forward, this amount will be deducted from the pre-tax income of the next period. The tax loss carry forward will reduce the taxable income in a period, and maybe in further periods, until the total tax loss carry forward has been deducted.

If an asset of a firm is unused, there will be an opportunity cost of this idle asset because there is a possibility to sell or rent the assets.

The sunk cost fallacy causes firms to keep investing in a project they once started when the NPV was bigger than zero but now has a negative NPV. Previous money spent on a project should not influence current decisions made whether to keep investing. If the NPV is smaller than zero, it is not attractive to invest.

If there is uncertainty in forecasting investment, we can use three analyses to decide what to do:

Break-even analysis ; this analysis finds the value of a parameter for which the

NPV will be equal to zero. After this we have determine how likely it is that the parameter value is below or above the break-even amount

Sensitivity analysis ; input assumptions in this analysis will be changed, which will affect the NPV. This can help with making investment decisions

Scenario analysis ; in this analysis, the values of parameters will be determined for multiple scenario’s, like a best case and worst case. These scenarios will be compared by their NPV when multiple parameters are changed at the same time

bonds

A bond is a tradable loan, which generally has a fixed time-to-maturity and a fixed cash flow stream. Because of this fixed stream it can be called a fixed income security.

The time-to-maturity has to be longer than one year to have a bond, this is the time at which the last cash flow is received. This time can be a fixed number of years, indefinite or it is possible the length is unknown (callables and extendables).

Not only the length of a bond can vary, the coupons of bonds can be different as well.

The coupon can be a fixed percentage of the principal loan, it can be related to inflation or profit (variable), or there may be no coupon (zero-coupon bonds). In the case that the coupon is related to inflation, the coupon will be higher if inflation is high and lower if inflation is low. If a principal is paid back in one time at the maturity date this is a



Use of this document is intended for: Chu, Eileen - Student number: 504051 bullet payment . A lottery is a payment depending on chance. If each period the same payment occurs, and the principal is paid back partly, then that is an annuity. But it can sometimes be the case for some bonds that the principal will not be paid back- this is a perpetual or consol.

We can value bonds by calculating the present value of all the cash flows the issuer of the bond promises to the holder. To calculate the price of this bond we use the following formula:

𝑇

𝑃 = ∑ 𝑡=1

𝐶𝐹 𝑡

(1 + 𝑟 𝑡

) 𝑡

A bond consists of the coupons and one principal. Each separate cash flow of a bond can be traded separately as a strip. The value of all cash flows should be the same by the law of one price, otherwise arbitrage trade would be possible.

term structures

The term on the horizontal axis is related to the interest on the vertical axis. The most typical case depicts an upward-sloping term structure, because the longer the time until maturity the higher the interest rate. But depending on supply and demand, the term structure can have all kind of shapes.

We can discount with the use of term structures. If we have a cash flow at the moment t, the 𝑅 𝑡

is used to compute to the present value. spot rate , in between 𝑡 = 0 and 𝑡 = 𝑇 .

𝑅

𝑇 is the interest rate, also called

valuing bonds

It is also possible we know the price of a bond, but do not know the term structure. For determining the interest rate, we can do the calculation backwards. For a zero-coupon bond this will be very easy because there will be only one interest rate. But when we want to calculate the value of coupon-bearing bonds this will be harder. When there are for example three years until maturity, this calculation will have three degrees of freedom in the formula to solve. Therefore, we will start with solving the first bond with the shortest maturity of one year, then the bond with two years until maturity and the last bond with three years until maturity. When the interest rate of the 1-year maturity bond is calculated, this rate can be used in in the calculation of the second bond. As a result, we will have the interest rate of the first year and the second year, and by using this in the formula of the third bond, the third-year interest rate can be found. (slides

37-45)

For the determination of the market value or interest rate of a bond we used the following formula:

𝑇

𝑃 = ∑

𝐶𝐹 𝑡

(1 + 𝑟 𝑡

) 𝑡 𝑡=1

The yield is the weighted-average interest rate of the cash flow of a bond. We are able to calculate the yield to maturity if we set the value of the previous formula equal to the new formula:

𝑇

𝑃 = ∑

𝐶𝐹 𝑡

(1 + 𝑦) 𝑡 𝑡=1




spot rates and forward rates

The difference between the spot rate and forward rate is that the spot rate always starts at time 𝑡 = 0 and lasts until a further point in time regardless how far this point is. But forward rates last from one point in time to the next point in time, and thus the forward rate 𝑅

𝑇

is from 𝑡 = 𝑇 − 1 until 𝑡 = 𝑇 . These two rates are not arbitrary, say there is one sport rate from 𝑡 = 0 until 𝑡 = 2 , and another spot rate from 𝑡 = 0 until 𝑡 =

1 is known, then we can calculate the forward rate from 𝑡 = 1 to 𝑡 = 2 by using the following formulas:

(1 + 𝑟

1

) × (1 + 𝑓

(1 + 𝑓

1,2

1,2

) = (1 + 𝑟

) = (1 + 𝑟

2

2

) 2 /(1 + 𝑟

1

) 2

)

In general, for period n:

(1 + 𝑓 𝑛−1,𝑛

) =

(1 + 𝑟 𝑛

(1 + 𝑟 𝑛−1

) 𝑛

) 𝑛−1

A speculative strategy in buying bonds is to buy now a one-year-zero-coupon bond, and the next again. We do not know the value of this yet at 𝑡 = 1 because we do not know what the interest rate will be from 𝑡 = 1 until 𝑡 = 2 . The expected payoff in two years will be: 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑛𝑑 × (1 + 𝑟

1

) × (1 + 𝐸(𝑟

1,2

))

And in theory it will hold for this that 𝐸(𝑟

1,2

) = 𝑓

1,2

.

We have two other theories besides the expectation’s hypothesis. In the liquidity preference theory most investors do prefer short-term investments over long-term investments, this is the reason why the yield curve is upward-sloping, but if different investors prefer different maturities this would mean they would all be on different points of the yield curve; hence the yield curve can take any shape, which is known as the market segmentation theory .

The dirty price is the actual price that has to be paid for a bond. But traders do not look at accrued interest in pricing, thus the dirty price minus the accrued interest will give the clean price of a bond.

finance 1 – IBEB – lecture 5, week 3 bonds and equities

interest rate sensitivity

If we have several bonds, we want to know how the value of a portfolio of bond changes when the interest rate changes. The change in the value of a bond has an effect on the value of the portfolio. We already know that if the interest rates increase, the prices of bonds will decrease. To determine by how much the value changes, we use the duration analysis . This analysis can be applied to pension fund risk management.

We can estimate the value change of a bond by multiplying the maturity with the negative of the interest rate change. This is only an approximation, when the maturity increases the estimation is less precise.




For coupon bearing bonds it is almost the same as for zero-coupon bonds, but now we talk about the average maturity. To calculate the interest rate sensitivity, also called duration , the average maturity of the present value of all cash flows is used. If a bond has a high duration, the bond is very sensitive to interest rate changes.

The solvency is the ability to pay your debts, which can be calculated in the ratio below: 𝑠𝑜𝑙𝑣𝑒𝑛𝑐𝑦 𝑟𝑎𝑡𝑖𝑜 = 𝑎𝑠𝑠𝑒𝑡𝑠 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠

corporate bonds

First, we assumed bonds were riskless, but a corporate bond can go bankrupt.

Therefore, we use a credit spread to define the risk of default. The credit spread uses for its calculation the average default loss and the risk premium of holding credit risk.

If not everything that is promised is paid this will be the default loss.

𝑇

𝑃 = ∑

𝐶𝐹 𝑡

(1 + 𝑟 𝑡

+ 𝑐𝑠) 𝑡 𝑡=1

First, we will calculate the value of a bond without default risk. After this, we will calculate the price of a bond with default risk, which causes the cash flows to be worth less. Thus, we will discount lower cash flows now, resulting in a lower bond value. Now we will set the value of the defaulted bond equal to the cash flows of the bond without risk and determine the corresponding IRR. The risk neutral credit spread is the IRR of the third calculation minus the interest rate used in the first two calculations (slide 22).

If there would be a risk premium in the bond for bearing this risk, it enlarges the credit spread. For all public corporate bonds, the credit spread can be calculated, but the expected default loss and risk premium apart are not. We also have agencies to check the credit quality of firms. Rating agencies base their scores on the probability of default



Use of this document is intended for: Chu, Eileen - Student number: 504051 and their recovery after default. The recovery is how much of the promised cash flows will still be received after default. In the table below the ratings are shown. The high yield bonds are considered risky bonds, but the bonds in green are considered safe

(slide 26).

Source: Corporate Bonds (Swinkels, 2018)

Some investors cannot hold risky bonds, due to regulations for example. Thus, around

BBB- and BB+ a lot of investors are selling and buying bonds. This is because when future predictions of companies change, this can affect their bond rating.

valuing stocks

For bonds, the cash flows were known, they consisted of coupons and a principal. But stock only pays dividends, we get the value of what is left of the earnings of a company, which are not known in advance. 𝑟

𝐸

is the equity cost of capital for the stock.

𝑃 𝑟

0

𝐸

=

= 𝑑𝑖𝑣

1

1 + 𝑟

𝐸 𝑑𝑖𝑣

1

+

𝑃

0

+

𝑃

1

𝑃

1

1 + 𝑟

𝐸

− 𝑃

0

𝑃

0

Where 𝑑𝑖𝑣

1

/𝑃

0

represents dividend yield , and (𝑃

1

− 𝑃

0

)/𝑃

0

the capital gains rate. If we assume there will be no dividend growth, we use the following formula to calculate the value of a stock:

𝑃

0

∞ 𝑑𝑖𝑣

= ∑ 𝑡=1

(1 + 𝑟

𝐸

) 𝑡

= 𝑑𝑖𝑣 𝑟

𝐸

The Gordon growth model assumes a constant dividend growth, g, shown in the formula below:

∞

𝑃

0

= ∑ 𝑡=1 𝑑𝑖𝑣 × (1 + 𝑔) 𝑡−1

(1 + 𝑟

𝐸

) 𝑡

𝑃

0

= 𝑑𝑖𝑣

1 + 𝑟

𝐸

(

1 −

1

1+𝑔

1+𝑟

𝐸

) =

1 + 𝑟

𝐸 𝑑𝑖𝑣

− (1 + 𝑔)

= 𝑟

𝐸 𝑑𝑖𝑣

− 𝑔

But there are some disadvantages in this model. Everything is very dependent on the first dividend forecast, if this is wrong, this has a big impact on the value of the stock.

Also, the expected growth rate will be of impact on the value of the stock. And the growth rate cannot be bigger than the equity cost of capital here. We have two stages for discounting the dividends. On the short run they have an opinion for each year, later on we use the same values in the mature stage, where the Gordon growth model will hold.

A company does not have to pay all earnings, they can keep them as retained earnings, which they can use for new investments. 𝑑𝑖𝑣 𝑡

= 𝐸𝑃𝑆 𝑡

× 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑒 𝑡




𝐸𝑃𝑆 = 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑡 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

When the share count changes, it is more desirable to use to the total payout model which includes the share repurchases and total dividends and thus total earnings are considered instead of per-share basis.

The free cash model is useful as it is resistant to borrowing and thus the price of the stock is valued on the enterprise value, cash, debt and shares outstanding. Such that the price is equal to:

𝑒𝑛𝑡𝑒𝑟𝑝𝑟𝑖𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 + 𝑐𝑎𝑠ℎ – 𝑑𝑒𝑏𝑡 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

For comparing the values of stock, we use the P/E ratio, in which k is used as the dividend payout ratio. 𝑑𝑖𝑣

1

= 𝑘 × 𝐸𝑃𝑆

From the dividend discount model, we know:

1

𝑃

0

= 𝑟 𝑑𝑖𝑣

1

𝐸

− 𝑔

These two formulas can be combined into:

𝑃

0

= 𝑘 × 𝐸𝑃𝑆

1 𝑟

𝐸

− 𝑔

And the P/E ratio thus will be:

𝑃

0

𝐸𝑃𝑆

1

=

𝐸𝑃𝑆 𝑘 × 𝐸𝑃𝑆

1

× (𝑟

𝐸

1

− 𝑔)

= 𝑟

𝐸 𝑘

− 𝑔

For valuing stocks, we can as well use the price-to-book value and enterprise value divided by sales as used in earlier lectures. But these formulas only give relative measures, and the theoretical value from the formula may not be consistent in practice.

The formulas may differ from market values because the market takes into account more aspects than the formula does.

finance 1 – IBEB – lecture 6, week 3 mean-variance analysis

expected utility and mean variance

When investing you chose the project with the highest utility. In the expected utility model, we want to maximize the expected utility under risk with the formula below: max 𝐸𝑈(𝑤) = max ∑ 𝑝 𝑖

𝑈(𝑤 𝑖

)

Where 𝑤 𝑖

are the possible wealth values, 𝑈(𝑤 𝑖 value, and 𝑝 𝑖

) is the utility derived from the wealth

the corresponding probabilities of the event. Two assumptions are made in this model. First, we assume that all possible information is taken into account when making a decision, thus expectations are rational . Secondly, all else equal the lowest risk option will be preferred and thus we assume risk aversion . If we initially invest 𝑤

0 the rate of return is equal to:




𝑅 = (𝑤 − 𝑤

0

)/𝑤

0 max 𝐸𝑈(𝑅) = max ∑ 𝑝 𝑖

𝑈(𝑅 𝑖

)

In the mean variance (MV) analysis we use the mean and the volatility, as these two factors are important for investment decision making. There are two possible justifications. The first one is that returns will have a normal probability distribution in which 𝐸(𝑅) and 𝜎(𝑅) describe the distribution. But in reality, returns are not normally distributed. There may for example be a negative skewness, which means negative outcomes (losses) score higher probabilities, having larger tails at the negative side.

The second justification is that if the utility function will have a quadratic shape, investors will care only about 𝐸(𝑅) and 𝜎(𝑅) . The investment decision is from the possibilities set that lies within the [𝜎(𝑅), 𝐸(𝑅)] interval, in which the highest expected utility portfolio is chosen.

risk and return measures

The risk-free interest rate, 𝑟 𝑓

, indicates the rate for a given period in which someone can lend or borrow money without risk. The value-weighted portfolio of all shares of stock and securities in the whole market is known as the market portfolio , 𝑅

𝑀

. The expected (mean) return can be calculated with the sum of the possible return times the probability for all returns, as in the following formula:

𝐸(𝑅) = ∑ 𝑝 𝑖

𝑅 𝑖

The variance, 𝝈 𝟐 (𝑹) , indicates how much the squared deviation from the mean is, which can be calculated with the formula: 𝜎 2

𝑁

(𝑅) = 𝐸 [(𝑅 − 𝐸(𝑅))

2

] = ∑ 𝑝 𝑖

(𝑅 𝑖

− 𝐸(𝑅))

2 𝑖=1

The standard deviation, 𝝈(𝑅) , which is the square root of the variance, is in Finance is also known as volatility . It measures how returns vary with the spread of the distribution of the return. The bigger the volatility, the higher the risk, which is compensated with a higher return.

The realized return is the actual return that has resulted from the given time period.

The realized return from 𝑡 to 𝑡 + 1 from the investment in a stock, that pays 𝐷𝑖𝑣 𝑡+1 on 𝑡 + 1 and sell the stock at that time for 𝑃 𝑡+1

, can be calculated with:

𝑅 𝑡+1

=

𝐷𝑖𝑣 𝑡+1

𝑃 𝑡

+ 𝑃 𝑡+1

− 1 =

𝐷𝑖𝑣 𝑡+1

𝑃 𝑡

+

𝑃 𝑡+1

𝑃 𝑡

− 𝑃 𝑡

= 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 + 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛 𝑟𝑎𝑡𝑒

If all dividends are used to reinvest again immediately to purchase additional shares of the same stock, the annual realized return will equal:

1 + 𝑅 𝑎𝑛𝑛𝑢𝑎𝑙

= (1 + 𝑅

𝑄1

) × (1 + 𝑅

𝑄2

) × (1 + 𝑅

𝑄3

) × (1 + 𝑅

𝑄4

)

historical returns




The average annual return of an investment is the yearly average of the realized returns, in which 𝑅

𝑇

stands for the realized return of the security in year t. The average annual return for year 1 through T can be calculated with:

𝑅̅ =

1

𝑇

𝑇

∑ 𝑅 𝑡=1 𝑡

In the variance estimate the average realized return is used instead of the mean, because we do not know the mean, and this will be the best estimate.

𝑉𝑎𝑟(𝑅) =

1

𝑇 − 1

𝑇

∑ (𝑅 𝑡 𝑡=1

− 𝑅̅) 2

After this the standard deviation, also known as the volatility estimate, is calculated as the square root of the variance.

The standard error is the standard deviation of the average return. The standard error is calculated with: 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟 = 𝑆𝐷(𝑎𝑛 𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛)/√𝑇

If the standard error is known, a confidence interval of 95%, dependent on the number of observations, can be constructed with the following table (Slide 14):

Source: 95% confidence interval (van der Sar, 2018)

In reality there is no clear relationship between the average return and volatility for individual stocks. But larger stocks tend to have lower volatility than small stocks, and an individual stock will have a higher risk and lower return, than that of a large portfolio.


The covariance between two stocks ( 𝑅

𝐴 and 𝑅

𝐵

) can be calculated with: 𝜎

𝐴𝐵

= ∑ 𝑝 𝑖

(𝑅

𝐴,𝑖

− 𝐸(𝑅

𝐴

)) (𝑅

𝐵,𝑖

− 𝐸(𝑅

𝐵

))

Which is used to compute the correlation between the returns of the two stocks by using: 𝜌

𝐴𝐵

= 𝜎 𝜎

𝐴𝐵

𝐴 𝜎

𝐵

The correlation measures how stocks move together, and thus if they share common risk. If two returns move in a different rhythm, thus independently, the risks are uncorrelated and 𝜌 will be equal to zero. If the 𝜌 is close to one, this means they have a perfect positive correlation.

If two stocks move together at a maximum, this increases the risk. If 𝜌 is close to minus one, this indicates a perfect negative correlation, the two



Use of this document is intended for: Chu, Eileen - Student number: 504051 stocks move in opposite direction, and if you combine these in a portfolio, this neutralizes the risk of a stock which is highly desirable from an investor’s perspective who seeks lower risk.

Common risk is perfectly correlated across all securities, examples being inflation, exchange, term structures. These have influences on all securities in the market and these common risks cannot be avoided. Therefore, these are as well called systematic risks (market/undiversifiable risk). This market wide factor is something we cannot escape from.

Independent risk is not correlated across industries and is only particular to one security of a firm. This does only affect that specific stock and does not have an effect on other stocks. This type of risk is called unsystematic risk (firm-specific/diversifiable risk). A systematic risk cannot be cancelled out by diversification as the risks are corelated.

diversification

There will be no risk premium for diversifiable risk, thus investors will not be compensated for holding firm-specific risk. This is because if they diversify their portfolio, this risk will be eliminated for free. A risk premium will be determined by the systematic risk, which cannot be avoided even when a portfolio is diversified. Which can also be seen below in the graph: when the number of stocks in a portfolio increases, there will be an elimination of diversifiable risk. If we keep increasing the number of stocks, in the end the volatility of the portfolio will be equal to the market correlated risk.

An efficient portfolio, is diversified to the maximum and will only have systematic risk.

The risk of this portfolio cannot be reduced further without decreasing 𝐸(𝑅) . The market risk premium will be equal to 𝐸(𝑅

𝑀

) − 𝑟 𝑓

, in which changes in the value of M are the systematic shocks of the market to the economy (slide 21).

Source: Volatility of an equally weighted portfolio versus the number of stocks (van der Sar, 2018)

measuring systematic risk

To find the risk premium 𝐸(𝑅) − 𝑟 𝑓

of security i, we need to know the systematic risk.

The systematic risk depends on how 𝑅 𝑖 we look at how sensitive 𝑅 𝑖

is to 𝑅

𝑀

moves in comparison with the economy. Thus,

, which means how much the return of a stock is affected by market-wide risk factors.



Use of this document is intended for: Chu, Eileen - Student number: 504051 𝜎 2

𝑀

= 𝐶𝑜𝑣(𝑅

𝑀

, 𝑅

𝑀

) = 𝐶𝑜𝑣 (∑ 𝑥 𝑖

𝑅 𝑖

, 𝑅

𝑀

) = ∑ 𝑥 𝑖

𝐶𝑜𝑣(𝑅 𝑖

, 𝑅

𝑀

) = ∑ 𝑥 𝑖 𝜎 𝑖 𝜎

𝑀

𝐶𝑜𝑟𝑟(𝑅 𝑖

, 𝑅

𝑀

) 𝜎

𝑀

= ∑ 𝑥 𝑖 𝜎 𝑖


, 𝑅

𝑀

)

If there is a one percent change in the return of a market portfolio, we use the beta ( 𝜷 ) to express the expected percentage change in the return of a security. Thus, another formula to look at the correlation of a stock’s return to the market is: 𝛽 𝑖

= 𝜎 𝑖


, 𝑅

𝑀

) 𝜎

𝑀

=

𝐶𝑜𝑣(𝑅 𝑖

, 𝑅

𝑀

) 𝜎 2

𝑀 𝜎

𝑀

= ∑ 𝑥 𝑖 𝜎 𝑖


, 𝑅

𝑀

) = ∑ 𝑥 𝑖 𝛽 𝑖 𝜎

𝑀

If we would have three stocks, i, j and k, their betas can be expressed as: 𝛽 𝑖 𝜎

𝑀 𝛽 𝑗 𝜎

𝑀

= 𝐶𝑜𝑟𝑟(𝑅 𝑖

, 𝑅

𝑀

)𝜎 𝑖

= 𝐶𝑜𝑟𝑟(𝑅 𝑗

, 𝑅

𝑀

)𝜎 𝑗 𝛽 𝑘 𝜎

𝑀

= 𝐶𝑜𝑟𝑟(𝑅 𝑘

, 𝑅

𝑀

)𝜎 𝑘

To clarify, if we have stock i, we will measure the sensitivity of this stock to the market return, and the systematic/market related risk is equal to 𝛽 𝑖 𝜎

𝑀

. Luxurious goods will have high betas, this is because that if the market does not well, people will buy less of these goods, resulting in a lower stock return. Beta only takes into account the systematic risk while sigma overestimates risk that counts.

𝐸(𝑅 𝑖

) − 𝑟 𝑓

= 𝛽 𝑖

(𝐸(𝑅

𝑀

) − 𝑟 𝑓

)

The sum of all stock betas in a portfolio should equal 1, 1 = 𝛽

𝑀


= ∑ 𝑥 𝑖 𝛽 𝑖

.

The portfolio weight is the fraction of a stock in the total portfolio held by an individual.

Thus, if we have n different investments, we call this an n-stock portfolio. 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖 𝑥 𝑖

= 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

, ∑ 𝑥 𝑖

= 1

Also, the expected return of a portfolio , will be equal to the sum of the expected returns of all investments held in the portfolio.

𝐸(𝑅

𝑃

) = ∑ 𝑥 𝑖

𝐸(𝑅 𝑖

)

And the variance of the return of this portfolio can be calculated with: 𝜎 2 (𝑅

𝑃

) = ∑ ∑ 𝑥 𝑖 𝑥 𝑗 𝜎 𝑖𝑗

= ∑ ∑ 𝑥 𝑖 𝑥 𝑗 𝜌 𝑖𝑗 𝜎 𝑖 𝜎 𝑗

For example, if we have a portfolio consisting of two stocks (A and B), we will first calculate the expected return of the portfolio 𝐸

𝑃 is calculated: 𝜎 2

𝑃

= 𝑥 2

𝐴 𝜎 2

𝐴

+ 𝑥 2

𝐵 𝜎 2

𝐵

+ 2𝑥

= 𝑥

𝐴

𝐴

𝐸

𝐴 𝑥

𝐵 𝜌

𝐴𝐵

+ 𝑥 𝜎

𝐴 𝜎

𝐵

𝐵

𝐸

𝐵

. After this the variance




In which 𝑥

𝐴 𝜌 𝜌 𝜌

𝐴𝐵

𝐴𝐵

𝐴𝐵

= +1 : 𝜎

= 0 : 𝜎 2

𝑃

+ 𝑥

𝐵

2

𝑃

= 𝑥

= −1 : 𝜎 2

𝑃

= (𝑥

2

𝐴

= 1 and both 𝑥

𝐴 𝜎

= (𝑥

𝐴

2

𝐴

𝐴 𝜎 𝜎

𝐴

𝐴

+ 𝑥

+ 𝑥 2

𝐵 𝜎

− 𝑥

𝐵

2

𝐵

𝐵 𝜎

𝐵 𝜎

𝐵

) 2

) 2

and 𝑥

𝐵 are bigger than or equal to zero.

We can invest a positive amount into an investment, which is called a long position .

If you invest a negative amount in a stock, this will be a short position . A short sale is a transaction in which you are selling a stock that you do not own, and which you will have to pay back in the future. You use this as a strategy when you expect a stock price to decline, because then you will receive more upfront for the shares than what you need to pay back in the future (Example is shown on slide 32).

The expected return and variance are calculated the same way for a two-stock portfolio. Again 𝑥

𝐴 𝜌

𝐴𝐵

= +1 : 𝜎 2

𝑃

= (𝑥

+ 𝑥

𝐵

𝐴 𝜎

𝐴

= 1

+ 𝑥

) is risk free.

𝐵

holds, but now short sales are allowed. 𝜎

𝐵

) 2

with the portfolio weights 𝑥

𝐴

= 𝜎

𝐵

/(𝜎

𝐵 𝜎 𝜌

𝐴

/(𝜎

𝐴𝐵

𝐴

− 𝜎

= 0 : 𝜎

𝐵

2

𝑃

= 𝑥 2

𝐴 𝜎 2

𝐴

+ 𝑥 2

𝐵 𝜎 2

𝐵 𝜌 𝜎

𝐴𝐵

𝐴

= −1 : 𝜎

/(𝜎

𝐴

+ 𝜎

𝐵

2

𝑃

= (𝑥

𝐴 𝜎

𝐴

− 𝑥

) is risk free.

𝐵 𝜎

𝐵

) 2

with the portfolio weights 𝑥

𝐴

= 𝜎

𝐵

/(𝜎

𝐴

− 𝜎

𝐴

) and 𝑥

𝐵

+ 𝜎

𝐵

) and 𝑥

𝐵


=

=

portfolio theory

normative portfolio theory

In the normative portfolio theory, we start with analysing the characteristics of all individual securities that are relevant for deciding on a portfolio. The second step is to take into account our expectations and derive an objective risk-return trade-off. We will diversify unsystematic risk. We diversify as much as possible and thus spread risk.

The last step is to choose the portfolio that suits our preferences best, our optimal choice.

For the normative portfolio theory, we do have some assumptions. A one-period model is used as planning horizon of the investor. The investor will want to maximize his or her expected utility, which is determined by the expected return and risk of the investment. We assume investors are risk averse and will prefer the highest return possible. In this theory we will have no taxes, no inflation, no transaction costs and no other costs. The information is available for all investors. The investments are infinitely divisible, and investors can short sell without extra costs and unlimitedly.

When choosing an efficient portfolio, we want to choose the lowest risk given a certain level of expected return, or if the risk is given, we will want to maximize the expected return given this amount of risk. The portfolios expected return can be calculated with: 𝑛

And the risk of a portfolio with:

𝐸(𝑅

𝑃

) = ∑ 𝑥 𝑖

𝐸(𝑅 𝑖

) 𝑖=1 𝑛 𝑛 𝜎 2 (𝑅

𝑃

) = ∑ ∑ 𝑥 𝑖 𝑥 𝑗 𝜎(𝑅 𝑖

, 𝑅 𝑗

) 𝑖=1 𝑗=1




combining two securities

When combining two securities, their expected returns and corresponding risks will be combined as well. So, if we have two investments, we will use the formulas below to calculate their combined return and risk.

𝐸(𝑅

𝑃

) = 𝑥

1

𝐸(𝑅

1

) + 𝑥

2

𝐸(𝑅

2

) 𝜎(𝑅

𝑃

) = {𝑥 2

1 𝜎

1

2 + 𝑥 2

2 𝜎 2

2 𝑥

1

+ 2𝑥

1

+ 𝑥

2 𝑥

2 𝜎(𝑅

= 1

1

)𝜎(𝑅

2

)𝜌(𝑅

1

, 𝑅

2

)}

1

2

Source: Effect on volatility and expected return of changing the correlation between

Intel and Coca-Cola stock (Van der Sar, 2018)

The correlation between two stocks will have an effect on the volatility and the expected return of a portfolio. If the correlation is equal to plus one, there will be a straight line between the two investments. But if the correlation between the stocks is minus one, the line between the investment will touch the Y-axis.

If we short sell an amount in investment one, this means we put a negative weight on this stock, and this amount is positively added to the weight of the other investment, as long as 𝑥

1

+ 𝑥

2

= 1 holds. At the end of the period you need to pay back this negative investment, therefore it can be tricky to invest a negative amount. Because if the value of the stock increases, you need to pay back more. Shown in the graph below it is shown that now the investment lines of combined securities are extended. If we invest more to the right than point 2, there will be a long position in stock two and a short position in stock one.

Source: Expected return and risk combinations with short sales (Van der Sar, 2018)

combining three securities




Until now we combined two securities, now we will show how to choose the optimal portfolio with three securities (1, 2 and 3). The graph below shows the risk and expected return for combinations of stocks. For stock one and two, the curved line connecting these two points, we can with a short sell of stock one and long position in stock two reach point D. Like point D, point C is as well on the expected return value of the dashed line. Point C can be reached by combining stock one and three both in a long position. Point C is already better than point D, because of the same expected return but a lower volatility, but B has an ever-lower volatility. Thus, when combining stocks two and three, point B can be reached.

A is the minimum risk portfolio corresponding with the given rate of return by combining all three stocks. Thus, this shows again that by diversifying, investing in all three stocks, the risk can be minimized. The minimum risk frontier is the most line representing portfolio combinations in the graph. These portfolio sets will offer the lowest possible risk for the given returns. This frontier is obtained due to optimal diversification. The

MRP , the minimum risk portfolio, is on the most left point of the frontier The portfolio sets above this point on the line represent the efficient portfolios, and the ones below this point are inefficient because these sets are dominated in terms of expected rate of return by the combinations above. Thus, and efficient portfolio will be located on the minimum risk frontier up from MRP.

Source: Minimum-risk frontier of three securities (Van der Sar, 2018)

adding a risk-free security

It is possible to even get to a more efficient frontier by adding a risk-free security, where the weights satisfy 𝑥 𝑓

+ 𝑥

1

= 1 .

𝐸(𝑅

𝑃

) = 𝑥 𝑓

𝐸(𝑅 𝜎(𝑅

𝑃

𝑃

) = 𝑟 𝑓

{

+ 𝑥 𝑟 𝑓

−

1

𝐸(𝑅

𝐸(𝑅

1

) = [𝑥 2

1

𝑟 𝑓

+ 𝜎 2 (𝑅

1

𝐸(𝑅

1 𝜎(𝑅

1

1

) = 𝑟 𝑓

)]

) − 𝑟 𝑓

)

1

2

+ 𝑥

= |𝑥

1

1 𝜎(𝑅

𝑃

(𝐸(𝑅

1

|𝜎(𝑅

1

), 𝑥

1

)

) − 𝑟 𝜎(𝑅

1

) 𝑓 𝜎(𝑅

𝑃

), 𝑥

1

) − 𝑟 𝑓

≥ 0

< 0

)

When including a risk-free security in a risky portfolio, the line representing the expected return with the volatility goes from the Y-axis, at the risk-free rate, to the risky portfolio. When buying P on margin, this means we short sell the risk-free security. But



Use of this document is intended for: Chu, Eileen - Student number: 504051 in this graph P is still not the portfolio to reach the most efficient outcome when combining with the risk-free security. If P would have been higher on the efficient frontier, it would have been possible to get a steeper line. If a line is steeper, this means for a given risk level, we will earn a higher return. The optimal portfolio will exist of the risk-free security combined with the tangent portfolio, which has the highest Sharpe ratio. In the Sharpe ratio the return-to-volatility ratio is measured:

𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =

𝐸(𝑅

𝑃

) − 𝑟 𝑓 𝜎(𝑅

𝑃

)

(Berk & DeMarzo, 2018)

The tangent line shows the efficient frontier, to the right of this line it will be inefficient, and to the left will not be attainable, thus we have to choose a point on the line.

expectations and preferences

The preferences of an investor are expressed in the formula 𝐸𝑈(𝑅) = 𝑓(𝐸(𝑅), 𝜎(𝑅)) .

This represents the indifference curve. As we know, investors will want to maximise their utility, thus they want to consume on the highest possible indifference curve. The portfolio set that will be chosen lies on the efficient frontier and touches the highest possible indifference curve. The efficient frontier represents the risk-return expectations, and the indifference curves shows the risk-return preferences.

finance 1 – IBEB – lecture 8, week 4 capital asset pricing model

assumptions

We have some assumptions in addition to the previous lecture. There will be a fixed amount of securities in the market, which is the total supply of all tradable securities.

We use the historical information to predict the future. Therefore, we have homogeneous expectations about the return and risk of securities between investors.

All investors are able to buy and sell securities at the same price, and they have the same evaluation period. The individual securities are traded on a market that is in equilibrium.

Because all investors perceive the same efficient portfolio and face the same frontier.

This leads to all investors investing in the same portfolio P and adjusting their investment with the risk-free investment F to match their risk preference. If an investor chooses a point on the capital market line (CML) between the market portfolio and the risk-free investment, he will lend, but if he invests beyond the market portfolio point on the CML, he will borrow.




Source: The efficient frontier line given a risk-free security (Van der Sar, 2018)

The optimal portfolio will be equal to the market portfolio. A security that is not included in the portfolio, will be less preferred by investors. This causes the price of the security to fall, which results in an increase in the expected return to the point where the investment is attractive again for investors. This way the optimal portfolio and the market portfolio stay equal.

capital market line

The capital market line (CML) is the best expected return that can be obtained given the risk. The relationship between risk and expected return for efficient portfolios has the following formula:

𝐸(𝑅

𝑀

) − 𝑟 𝑓

𝐸(𝑅) + 𝑟 𝑓

+ 𝜎(𝑅

𝑀

) 𝜎(𝑅)

The slope of this formula is also called the Sharpe ratio .

Source: Capital Market Line (Van der Sar, 2018)

If you are on the CML, you have invested partly in the market portfolio and partly in the risk-free investment. If you choose an efficient portfolio Q, this means your investment is located on the CML, thus 𝑅

𝑄

= (1 − 𝑥)𝑟 𝑓

+ 𝑥𝑅

𝑀

will hold.

𝐸(𝑅

𝑄

) = (1 − 𝑥)𝑟 𝑓

+ 𝑥𝐸(𝑅

𝑀

) 𝜎(𝑅

𝑄

) = 𝑥𝜎(𝑅

𝑀

) 𝜌

𝑄𝑀

= 1

If a portfolio P is below the CML the portfolio will be inefficient and 𝜌

𝑃𝑀 will not be located on the CML.

< 1 will hold. It




If 𝜌

𝑄𝑀

= 1 , this means the portfolio only contains market risk, and this will be an efficient portfolio. It is not possible to decrease risk by diversifying. Represented in the graph by point Q.

If 𝜌

𝑃𝑀

< 1 , this means the chosen portfolio still contains unsystematic risk which can be diversified and escaped from. This will not be an efficient portfolio. Represented in the graph by point P, where it holds that 𝐸(𝑅

𝑃

) = 𝐸(𝑅

𝑄

) but 𝜎(𝑅

𝑃

) > 𝜎(𝑅

𝑄

) . Thus, the expected returns are equal, but portfolio P is bearing more risk than Q, therefore this is not an efficient portfolio, because the expected return can also be obtained with a lower risk.

We know from earlier lectures that the market risk can be calculated with 𝜎

𝑀

∑ 𝑥 𝑖 𝛽 𝑖 𝜎

𝑀

. The beta of a security or portfolio is calculated with: 𝛽 𝑖

= 𝜎 𝑖 𝜎 𝜌 𝑖𝑀

𝑀

=

If portfolio Q is efficient, the correlation with the market will be equal to one, 𝜌

𝑄𝑀

Which means the beta of Q will be 𝛽

𝑄

= 𝜎

𝑄

/𝜎

𝑀

= 1 .

. The risk premium for bearing the risk of Q, 𝜎

𝑄

= 𝛽

𝑄 𝜎

𝑀

, is equal to 𝐸(𝑅

𝑄

) = 𝑟 𝑓

.

If we talk about portfolio P in the graph, this one is below the CML, thus 𝜌

𝑃𝑀

The beta of P will be equal to 𝛽

P, 𝜎

𝑃 𝜌

𝑃𝑀

= 𝛽

𝑃 𝜎

𝑀

𝑃

= 𝜎

, is equal to 𝐸(𝑅

𝑃

𝑃 𝜌

𝑃𝑀

) = 𝑟 𝑓

.

/𝜎

𝑀

< 1 holds.

. The risk premium for bearing the risk of

An overview is shown in the table below.

Source: Efficient versus Inefficient Portfolios (Van der Sar, 2018)

For all portfolios and securities in an equilibrium the expected return will equal the required return, expressed in formula as:

𝐸(𝑅 𝑖

) = 𝑟 𝑖

= 𝑟 𝑓

+ 𝛽 𝑖

(𝐸(𝑅

𝑀

) − 𝑟 𝑓

)

As a result of the capital asset pricing model, we get the following formula:

𝐸(𝑅) = 𝑟 𝑓

+ (𝐸(𝑅

𝑀

) − 𝑟 𝑓

)𝛽

security market line




The security market line (SML) goes through the coordinates (0, 𝑟 𝑓

) and (1, 𝐸(𝑅

𝑀

)) .

If there will be a negative beta for a security, this means the security does well when the market is not doing well. The expected return of the security will be lower than that of 𝑟 𝑓

, the risk premium is smaller than zero, but this does not mean it is risk free because 𝜎 𝑗

≥ |𝛽 𝑗

|𝜎

𝑀

. This security can be held with other securities to diversify the portfolio, but this does decrease the return.

Source: Security Market Line (Van der Sar, 2018)

Source: Relation between the return of (in)efficient portfolios and betas (Van der Sar,

2018)

The volatility of P common with the market is shown by: 𝜎

𝑃 𝜌

𝑃𝑀

= 𝜎𝐶𝑟𝑜𝑠𝑠(𝑅

𝑃

, 𝑅

𝑀

) = 𝛽

𝑃 𝜎

𝑀

But portfolio P has more risk than the market portfolio thus, 𝜎

𝑃 which is the unsystematic risk of P. If 𝜌

𝑃

− 𝛽

𝑃 𝜎

𝑀

> 𝜎

𝑃 𝜌

𝑃𝑀

if 𝜌

𝑃𝑀

< 1

> 0 holds, meaning that the risk of portfolio P is minus the market related, systematic, its risk is bigger than zero, and the portfolio is inefficient. Thus, the volatility of portfolio P: 𝜎

𝑃

= 𝛽

𝑃 𝜎

𝑀

+ 𝑙𝑖𝑛𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑄𝑃




The CML shows the portfolios which exist of the risk-free investment and the efficient portfolio combined, it represents the highest expected return which is possible given a volatility level. All the stocks and portfolio that are to the right of the CML have some diversifiable risk, on the CML we only have systematic risk.

The SML matches the expected return for each security with the corresponding beta to the market. Because all stocks and portfolios are part of the efficient market portfolio, they should lie on the SML.

If the expected return of one stock increases, and that of another stock decreases, the investors will react to this by changing their investments to make their portfolio efficient again. A change in the expected return of a stock to its required return 𝑟 𝑠

SML is called the stock’s alpha .

according to 𝛼 𝑠

= 𝐸(𝑅 𝑠

) − (𝑟 − 𝑓 + 𝛽 𝑠

(𝐸(𝑅

𝑀

) − 𝑟 𝑓

))

If the alpha of stock A is bigger than zero and that of stock B smaller than zero, the investor is able to get a better portfolio by investing more in A in exchange for less B.

This will increase the Sharpe ratio. But this action of investors causes the price of stock

A to rise and the expected return to decrease, oppositely the price of stock B will fall and the expected return increases. Both A and B will come back to the SML curve where both alphas equal zero again. From the CAPM we can conclude that the market portfolio will be the efficient portfolio as well. The best combinations of risk and expected return are thus located on the CML. The risk premium is proportional to 𝛽 is given by the SML line. The market portfolio can only be inefficient if there are investors who are willing to hold negative alphas due to behaviour that is not like the CAPM.

If borrowers and lenders face different rates, the SML formula will be equal to:

Where 𝑟 ∗ 𝑟 𝑖

= 𝑟 ∗ + (𝐸(𝑅

𝑀

) − 𝑟 ∗ )𝛽 𝑖

denotes the rate between the borrowers and lenders, and thus depends on the amount of people saving or borrowing.

finance 1 – IBEB – lecture 9, week 5 multi-factor model and arbitrage theory

introduction

The return is the expected return plus the systematic factors that influence the return.

If there are no factors and betas influencing the expected return, this means the return is equal to the expected return. We can show this with the formula below:

𝑅 𝑖

= 𝐸(𝑅 𝑖

) + 𝑏 𝑖1

𝐹

1

+. . . +𝑏 𝑖𝑘

𝐹 𝑘

+ 𝜀 𝑖

Here, the risky return is denoted 𝑅 𝑖

, the systematic factors with their factors sensitivities (betas) 𝑏 𝑖1

𝐹

1

, … , 𝑏 𝑖𝑘

𝐹 𝑘

. All firms are affected by the systematic factor, but some more than others. The unsystematic, independent factor, is 𝜀 𝑖

, which is firm specific risk. Thus, the return is the expected return plus the unexpected influence by systematic and unsystematic factors.




In a self-financing portfolio, where we invest long in the market portfolio and short in the risk-free investment, the net wealth invested is zero. This is zero because it does not cost anything to construct this portfolio. But there is a risk that your return will not be like your expectations, because we do not know if 𝑅

𝑀 risk premium 𝐸

𝑀

− 𝑟 𝑓

− 𝑟 𝑓

is equal to 𝐸

𝑀

− 𝑟

will be bigger than zero, because this is systematic risk. 𝑓

. The

Source: Relation between the return of (in)efficient portfolios and beta(Van der Sar,

2018)

An efficient portfolio Q, which is partly the market portfolio and partly the risk-free investment. Q does not contain unsystematic risk and only systematic risk.

𝐸

𝑄

= 𝑟 𝑓

+ 𝛽

𝑄

(𝐸

𝑀

− 𝑟 𝑓

)

The risk premium of Q will be 𝛽

𝑄

(𝐸

𝑀 because we will not be sure if 𝑅

𝑀

− 𝑟 𝑓

− 𝑟 𝑓

) > 0 , with this being the risk premium

is equal to 𝐸

𝑀

− 𝑟 𝑓

.

𝑅

𝑄

= 𝑟 𝑓

+ 𝛽

𝑄

(𝑅

𝑀

− 𝑟 𝑓

)

Portfolio P is inefficient, and contains systematic and unsystematic risk, 𝐸

𝑃

= 𝑟 𝑓

+ 𝛽

𝑃

(𝐸

𝑀

− 𝑟 𝑓

) . The risk premium is 𝛽

𝑃

(𝐸

𝑀

− 𝑟 𝑓

) and is bigger than zero, which is the risk premium for taking systematic risk. P is now exposed to something extra, the 𝜀

𝑃

, which is firm specific. This is not included in the risk premium, because you are not rewarded for having unsystematic risk. Thus, for P we have 𝑅

𝑃

= 𝑟 𝑓

+ 𝛽

𝑃

(𝑅

𝑀

− 𝑟 𝑓

) + 𝜀

𝑃

.

𝑅 𝑖

= 𝑟 𝑓

+ 𝛽 𝑖

(𝑅

𝑀

− 𝑟 𝑓

) + 𝜀 𝑖

The risk-free rate is the return for delay in consumption, the second term indicates the co-movement with the market (systematic risk) and the last term indicates the movement which is not market correlated (unsystematic risk).

single-factor model

𝑅 𝑖

= 𝐸(𝑅 𝑖

) + 𝑏 𝑖1

𝐹

1

+ 𝜀 𝑖

In a single-factor model we look at one risk factor. The risk factor is written as 𝐹

1

𝑅

𝑀

− 𝑟 𝑓

− (𝐸

𝑀

− 𝑟 𝑓

) and the factor risk premium as λ

1

= 𝐸

𝑀

− 𝑟 𝑓

=

. The beta in the single factor model is the sensitivity of 𝑅 𝑖

to 𝐹

1

. The last term of the formula is 𝜀 𝑖

which is called the disturbance term, this is the unsystematic risk and one does not receive a risk premium for holding such risk. If the capital asset pricing model holds we will get the following formula, where λ

0

is the risk-free rate and 𝑏 𝑖1

λ

1

𝐸(𝑅 𝑖

) = λ

0

+ 𝑏 𝑖1

λ

1

the risk premium.




The 𝐹

1

was the systematic risk factor, and the risk being taken, due to market or inflation changes for example, is rewarded in the return. The impact of 𝐹

1

on the returns of securities is expressed in their betas. The 𝜀 is a unique risk factor, this risk is more security specific, and the risk due to this is not given a risk premium. If the covariance of two securities is equal to zero, dependent on 𝐹

1

.

𝐶𝑜𝑣(𝜀 𝑖

, 𝜀 𝑛

) then both 𝑅 𝑖

and 𝑅 𝑛

will be equally

multi-factor model

In a three-factor model, there will be three risk factors, thus we have 𝐹

1 the covariance 𝐶𝑜𝑣(𝜀 𝑖

, 𝜀 𝑛

, 𝐹

2

and 𝐹

3

. If

) = 0 the returns of two different stocks will be dependent only on the three risk factors. Based on the betas of the stocks to the different risk factors, the relevant risk of the securities can be compared. If two stocks will be exposed to the same beta, they expect the same rate of return.

arbitrage pricing theory

The arbitrage pricing theory has several assumptions. First of all, investors are greedy. The security market in which investors operate is perfectly competitive and has free entry and exit. There are no transaction costs. In the theory there are influences on the returns which securities realise due to a collection of occurrences/circumstances. Some occurrences have a systematic nature, unique occurrences have specific natures. Influences of systematic and unique occurrences are processed by investors into a homogenous expected value of the return for each security. There are no systematic possibilities available to investors for arbitrage. In an arbitrage portfolio the net wealth invested is zero, thus it is costless to construct such a portfolio. These portfolios are free from systematic risk and do not contain unique risk. Unique risk can always be eliminated out of a portfolio.

𝑅 𝑖

= 𝐸(𝑅 𝑖

) + 𝑏 𝑖1

𝐹

1

+. . . +𝑏 𝑖𝑘

𝐹 𝑘

+ 𝜀 𝑖

If the returns of securities in the total security market are following the return generating process (RGP) and there is an infinite number of factors and the assumptions apply, then the expected return will be a linear relationship of a constant and the factor sensitivities 𝑏 𝑖𝑗

, with all i values. This will result in the following formula, called the arbitrage pricing relation (ARP) : 𝑘

𝐸(𝑅 𝑖

) = λ

0

+ ∑ 𝑏 𝑖𝑗

λ j 𝑗=1

When we combine two securities, with the portfolio weights 𝑥 𝑖 together be equal to one, we can state:

ad 𝑥 𝑛

, which will

= (𝑥 𝑖

𝐸 𝑖

+ 𝑥 𝑛

𝐸 𝑛

𝑅

𝑃

= 𝑥 𝑖

) + (𝑥 𝑖

𝑅 𝑖 𝑏 𝑖1

= 𝐸(𝑅

𝑃

+ 𝑥

+ 𝑥 𝑛

) + 𝑏

𝑃1

𝐹

1 𝑛

𝑅 𝑛 𝑏 𝑖𝑛

) + (𝑥 𝑖

+ 𝜀

𝑃 𝜀 𝑖

With the return generating process (RGP) 𝑅

𝑃

= 𝐸(𝑅

𝑃

) + ∑ 𝑏

+ 𝑥

𝑃𝑗 𝑛

𝐹 𝑗 𝜀 𝑛

)

+ 𝜀

𝑃

, which exists of:

𝐸(𝑅

𝑃

) = ∑ 𝑥 𝑖

𝐸(𝑅 𝑖

) 𝑏

𝑃𝑗

= ∑ 𝑥 𝑖 𝑏 𝑖𝑗 𝜀

𝑃

= ∑ 𝑥 𝑖 𝜀 𝑖




Source: Risk free and arbitrage portfolios(Van der Sar, 2018)

A self-financing portfolio is constructed by investing long in some stocks and short in other stocks of which the market value is equal. The sum of the portfolio weights together should be zero in an arbitrage portfolio. If a stock P is risk free and the expected return of the stock is bigger than the risk-free investments return, this is an arbitrage opportunity. We can do this by borrowing funds at the risk-free rate, and this amount in investment in P. Thus, our portfolio contains a long position in P and a short position in 𝑟 𝑓

. When 𝑏 𝑖𝑗

= 𝑏 𝑛𝑗

there will be no systematic arbitrage opportunities and the expected returns will be equal, 𝐸(𝑅 𝑖

) = 𝐸(𝑅 𝑛

) .

the APT in practice

In practice the arbitrage pricing theory starts with estimating the betas to factors. The estimation of the factor sensitivities is done by a regression analysis. In risk management a portfolio is chosen per particular exposure, in which the risk is denoted as 𝜎 2

𝑃

= ∑ 𝑏 2

𝑃𝑗 𝜎 2

𝐹𝑗

+ 𝜎 𝜀

2 . After this the factor movements are predicted 𝐹 𝑗

= 𝐸(𝑅

𝑃

) +

∑ 𝑏

𝑃𝑗

𝐸𝐹 𝑗

. In which the first term is expected by the market in the arbitrage pricing relation. And the total of 𝐸(𝑅

𝑃

) + 𝑏

𝑃1

𝐸𝐹

1

is what is expected by people, because this is due to superior info indicating that the 𝐸𝐹

1

will be bigger than zero. A pure factor portfolio is only sensitive to the factor.

There are two types of strategies for self-financing portfolios.

The first is small-minus-big (SMB) in which the investor buys portfolio S, consisting of small stocks, and short sells portfolio B, which has big stocks

𝐹 𝑗

= 𝑅 𝑠

− 𝑅

𝐵

− (𝐸

𝑆

− 𝐸

𝐵

) resulting in λ j

= 𝐸

𝑆

− 𝐸

𝐵

A second strategy is high-minus-low (HML) in which a portfolio is financed by a long position in portfolio H and a short position in portfolio S. In which portfolio

H contains stocks with a high B/M ratio and portfolio S stocks with a low B/M ratio

A factor is a covariance factor if the returns o f all existing securities’ returns show some relation to the change in the risk factor. This means 𝐹 𝑗

is a systematic factor, with

λ j

> 0 . If the factor is related to earnings prospects or financial strength, this means that there is an underlying economic rationale , and it will be a risk factor. But when it is no risk factor, it is a behavioural factor which only lasts for a limited time.




Fama and French is a three-factor model such that the first j portfolio captures the idea of market-wide economic risk, the second j portfolio captures the idea of smallminus-big portfolio and lastly the third j portfolio- high-minus-low portfolio. This model encapsulates both systematic and economic rationale.

In the PR1YR momentum factor of Carhart, we look at the previous years’ winners and losers. The strategy is to have a long position in the previous year winners, and a short position in the previous year losers. There is a systematic factor here, but no economic rationale.

performance measures

investment performance evaluation

We have the investment performance evaluation procedure to gain insights of the results of an investment policy. The result is compared to the benchmark that has been set before, with the goal of optimizing the investment objectives. We will look at ex post, which is usually on a monthly-basis and shows the realized risk and returns. 𝑅

𝑃 measures the average of the 𝑅

𝑃,𝑡

’s, in which a certain period of evaluation, t, is set.

There is a set market index (MI) giving a benchmark and this will be a proxy for our market portfolio.

Sharpe performance measure

The Sharpe performance measures looks somewhat like the first Sharpe ratio we used,

𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = [𝐸(𝑅

𝑃

) − 𝑟 𝑓

]/𝜎(𝑅

𝑃

) , but does have a slight difference. There is a reward per-unit of total risk, still there is no reward for risk arising due to incomplete diversification.

market Model (MM) in excess returns

There is a linear relationship assumed between the returns of P and the MI for a given evaluation period. This relationship is in excess of the risk-free rate.

𝑅

𝑃,𝑡

− 𝑟 𝑓,𝑡

= 𝑎

𝑃

+ 𝑏

𝑃

(𝑅

𝑀𝐼,𝑡

− 𝑟 𝑓,𝑡

) + 𝑒

𝑃,𝑡

Where we use alpha to measure 𝑎

𝑃 𝛼

𝑃

= 𝑅

as:

𝑃

− 𝑟 𝑓

− 𝛽

𝑃

(𝑅

𝑀𝐼

− 𝑟 𝑓

)

And where beta is used to measure 𝑏

𝑃

as: 𝛽 =

𝐶𝑜𝑣(𝑅

𝑃 𝜎 2

− 𝑟 𝑓

(𝑅

𝑀𝐼

, 𝑅

𝑀𝐼

− 𝑟 𝑓

)

− 𝑟 𝑓

)

Treynor performance measure

𝑇

𝑃

=

𝑅

𝑃

− 𝑟 𝑓 𝛽

𝑃

With this performance measure the varying empirically derived slope of the security market line is used which passes through P. The beta is derived from the market model in excess measure.

Jensen performance measure




𝐽

𝑃

= 𝑅

𝑃

− 𝑟 𝑓

− 𝛽

𝑃

(𝑅

𝑀𝐼

− 𝑟 𝑓

)

Comes from 𝐸(𝑅

𝑃

) − 𝑟 𝑓

, where the capital asset pricing model ex ante had the required return 𝑟

𝑃

= 𝑟 𝑓 in which the 𝐽

𝑃

+ 𝛽 𝑝

(𝐸

𝑀

) − 𝑟 𝑓

) . The ex post P version of alpha is denoted with 𝐽 is Jensen’s alpha and where 𝛽

𝑃

𝑃

= 𝛼

𝑃

,

is measured in market model in excess returns.

information ratio or appraisal ratio

In the information ratio/appraisal ratio we look at Jensen’s P per unit of systematic risk, which can be diversified away. Jensen’s P gives us the average realized abnormal return. The term 𝜎(𝑒

𝑃 spread of 𝑒

𝑃,𝑡

.

) is the tracking error of P, known as 𝑇𝐸

𝑃

, and indicates the 𝑒

𝑃,𝑡

= 𝑅

𝐼𝑅

𝑃

=

𝑃,𝑡

− (𝑟

𝑅

𝑃 𝜎(𝑅

𝑃 𝑓,𝑡

+ 𝛽

𝑃

− 𝑅

𝑀𝐼

− 𝑅

𝑀𝐼

)

(𝑅

𝑀𝐼,𝑡

𝑖𝑓 𝛽

𝑃

− 𝑟 𝑓,𝑡

= 1

))

The information ratio is used for measuring portfolio performances of investments.

This is a measure often utilised when one has to measure the performance of the investment portfolio alongside the portfolio manager: specifically, when the portfolio manager is assigned to outperform the market portfolio and/or he/she has to ensure that portfolio P’s deviation from M, measured as the tracking error falls within a certain spectrum.

finance 1 – IBEB – lecture 10, week 5 efficient market hypothesis

informational efficiency

The efficient market hypothesis (EMH) assumes an informationally efficient financial market. Informational efficiency means that when relevant information is available, this is immediately and completely reflected in the prices. The prices in the market are determined by markets marginal costs of supply and demand. Where investors are the market participants supplying wealth, and the firms are market participants who demand the capital of the investors. The competition between investors and firms in the capital market will make sure all positive NPV opportunities are eliminated until the

NPV’s are equal to zero. Due to the competition, the securities are fairly priced, and this means we can trust the market. There is hardly any superior information because everybody is looking for this information. The capital of investors is allocated by the market in an optimal way. This means the money is given to the most productive firms, which is known as allocative efficiency .

It is possible that the net present value in the real market is bigger than zero. This is because an invention can get a valuation which is higher than the initial investment.

On the stock market we have fair pricing, here it is almost impossible to get a positive net present value.

High returns in the market may have several causes, it can be because of luck or high risks taken for the return. But it might be due some investors who have more expertise and are better in interpreting available information leading to higher returns. It is possible as well that an investor knows more than the market does, thus they have an information advantage compared to other investors.




High returns can occur by chance, which is possible as well in diversified portfolios. It can be normal that this high return is expected. The returns can be high for once, which is abnormal. Or the effort put into acquiring information did cost a lot so high returns will be normal for this amount of effort.

efficient market hypothesis

The EMH states that the amount of competition among investors for information determines how efficient the market is. The degree of competition is dependent on the information costs for investors. A net present value that is bigger than zero must be hard to find for investors, because otherwise competition would decrease the net present value until this is equal to zero again.

Arbitrage opportunities will exist when two stocks with equal cash flows have different prices. Inefficiency is existent when the same risks do not result in the same expected returns. The price will not be equal to the future expected cash flows.

three levels of informational efficiency

We know three levels of informational efficiency, but there are two versions. The first version of the three levels was constructed by Fama in 1970:

Weak form : everything from the past is already incorporated in the prices in the weak form. The stock prices are retrieved from historical stock price information.

In technical analysis , trends of history are used to predict the future, but these cannot be considered useful

Semi-strong form : the prices of the stock are based on the publicly available information which is relevant for pricing. The prices adjust to public information

Strong form : not only is the publicly available information reflected in the prices, but also insider information of firms which is not yet public to all investors

The second version of the three levels was established in 1991:

Predictability of returns : all historical information is used, such as characteristics of stocks, past prices with their corresponding returns and other information which can be of use. The fundamental analysis looks at firms and their available information to look at future influences of unexpected occurrences, which can both be at macro and micro level

Event studies : similar events are observed, and of these events they look on how the information affects the market on and after the event date

Private information : this is information that comes to the market because of a price change in the stocks

A model which is not assumed to be right, is the random walk model . In this model it is assumed that all probability parameters of returns are equal, and thus the returns will be unpredictable and random. We can test the random walk hypothesis with a parametric test , in which we use this regression-based autocorrelation test:

𝑅 𝑗,𝑡

= 𝑎 𝑗

+ 𝑏 𝑗

𝑅 𝑗,𝑡−1

+ 𝜀 𝑗,𝑡

The 𝑎 𝑗

denotes the drift, and 𝑏 𝑗

is the first order autocorrelation, but in the random walk hypothesis this is assumed to be equal to zero.

The non-parametric tests look in consecutive changes in prices with the same sign, negative or positive. In the runs test the number of runs in a time is counted. The



Use of this document is intended for: Chu, Eileen - Student number: 504051 number of runs is the amount of sign changes in the given time period, and this is compared with the expected sign changes of a random sample.

In the fair-game model we have unbiased expectations for the returns and prices of stocks which is based at the given information. We assume normal returns, and the efficient market hypothesis is rejected at the moment we systematically get abnormal returns.

When we trade mainly based on historical price changes, we call the buy-and-hold strategy . By a filtering strategy we determine which stocks to buy when the return is above a given percentage, and finance this by short selling the stock that is below this percentage. In the moving average strategy, we do this as well for a given percentage, but this percentage is the return in excess of the moving average. And with a rebalancing strategy we will adjust the portfolio weights regularly.

In the price momentum strategy , we buy the best stocks of the most recent given number of months (3, 6, 9, or 12) and short sell the worst stocks. This portfolio is held for the given number of months, this is a profitable self-financing portfolio.

The value-glamour strategy is a self-financing portfolio as well. We buy the stock of the top ten percent book-to-market ratio and short sell the stocks with the lowest ratio.

This constructed portfolio will be held for a period of time.

The methodology of an economic event consists of several steps explained below:

1. Event: after an event happened, we start studying this event from the date of happening, 𝑡 = 0 . We will have several observations denoted with 𝑗 = 1, … , 𝑁 , the realized returns of the stocks are denoted 𝑅 𝑗,𝑡

.

2. Test period : in the test period we will look at the period around the date of the event, which was 𝑡 = 0 . We will construct an interval which is partly before the event and partly after.

3. Control period : the returns are used to construct a bench mark for the test period. The realized returns will be compared with the test period.

4. Normal returns 𝑹 𝒋,𝟎

: these are benchmark returns, based on the resulting stock information of the control period.

5. Abnormal return 𝒖 𝒋,𝟎 normal return.

= 𝑹 𝒋,𝟎

− 𝑹 𝒋,𝟎

: the difference between the realized and

6. Average abnormal returns 𝑨𝑹

𝟎

: the average of 𝑢̂ 𝑗,0

during the given test period.

7. Test the null hypothesis of no abnormal returns : the test statistics can be calculated with the following formula for each period in time where SE denotes the standard error. 𝑡𝑒𝑠𝑡 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 =

𝐴𝑅 𝑡

𝑆𝐸 𝑡

8. Test result interpretation : if the average residual 𝐴𝑅 𝑡

is not equal to zero.

𝑡 = 0 this means there was an information release if the price change was permanent. Alternatively, it can mean there was no informational release, but a wrong perception was created which can still be rectified later on.




𝑡 > 0 this means the semi-strong efficient market hypothesis is rejected due to an over or under reaction at 𝑡 = 0 .

𝑡 < 0 this gives us two possibilities, if it is about inside information the strong efficient market hypothesis is rejected but this is difficult to prove.

And if it is private information the strong efficient market hypothesis is accepted.

We can use the market model to operationalise the returns:

𝑅 𝑗,𝑡

= 𝑎 𝑗

+ 𝑏 𝑗

𝑅

𝑀,𝑡

+ 𝑢 𝑗,𝑡

Which gives us a linear relationship between 𝑅 𝑗,𝑡 and 𝑅

𝑀,𝑡

. Here 𝑎 𝑗

is stock specific to

𝑅 𝑗,𝑡

and remains constant, and it is not affected by risk. The 𝑏 𝑗

measures the sensitivity to the market movements and 𝑢 𝑗,𝑡 is the error term, which can be different through time t.

The total risk is 𝜎 2 (𝑅 𝑗

) of which the systematic risk is made up for j as 𝑏 𝑗

2 the unsystematic risk of j is equal to 𝜎 2 (𝑢 𝑗

) . 𝜎 2 (𝑅

𝑀

) and

The control period is a time interval which represents the stock under normal circumstances, which are thus not affected by the event. We do not know the parameters 𝑎 𝑗

and 𝑏 𝑗

thus we determine 𝑎̂ 𝑗

and 𝑏̂ 𝑗

with a regression analysis. These two parameters give us the relationship between 𝑅 𝑗,𝑡

and 𝑅

𝑀,𝑡

in the control period.

This is not possible in the test period because this is influenced by the event. The normal return for the test period on 𝑡 = 1 can be calculated with:

𝑅̂ 𝑗,𝑡+1

= 𝑎̂ 𝑗

+ 𝑏̂ 𝑗

𝑅

𝑀,𝑡+1

The abnormal return was defined by: 𝑢̂ 𝑗,𝑡+1

= 𝑅 𝑗,𝑡+1

− 𝑅̂ 𝑗,𝑡+1

Where the null hypothesis states that there were no abnormal returns. For this is now a t-test used. The average abnormal return will equal the average residual:

𝐴𝑅 𝑡+1

=

1

𝑁

∑ 𝑢̂ 𝑗,𝑡+1

The t-statistic is given by:

𝑇𝐴𝑅 𝑡+1

= 𝜎(𝑢̂

𝐴𝑅 𝑡+1 𝑡+1

)/√𝑁

The total effect can be measured over more days as well. We add up the abnormal returns for the security and will take the average of this. The cumulative abnormal return (CAR) , or cumulative average residual over period K to L, is calculated with the formula:

𝐶𝐴𝑅

(𝐾,𝐿)

= 𝐴𝑅

𝐾

+ 𝐴𝑅

𝐾+1

+. . . +𝐴𝑅

𝐿




After an announcement, it is possible that one of these stock paths occur. The price can start rising, or the price adjust until it is equal to its price before the event. If the average stock price drifts up, this means there was an under reaction. If the average stock price shifts down, it was an overreaction to the news and the price adjusts back to its previous level.

Source: Possible average stock price paths (Van der Sar, 2018)

example

Block trades happen when a couple of shares are offered in a block. We have four possible block trades, figure A shows us that after a block trade, the prices are affected for a limited time-period, this is because of the liquidity problem. But the prices adjust after some time again and are not permanently affected. In figure B there is no new information announced, it was already incorporated in the market and thus has no effect on prices. Figure C does give new information to the market, the price is affected, and even though the price adjusts a bit again, the new share price is lower than it was before. In figure D we have a direct and complete reaction to the news, the price remains constant after the news publication.

Source: Example

(Van der Sar, 2018) block trades




In the strong form of efficiency, it was not possible to profit of trading on private information. We can question whether the private information was predictable or if traders have profited by trading with this knowledge.

There are regulations against insider trading , these are people who are from the firm or do have information of the inside of the firm which will be in the news later. These people will have more knowledge about the stocks and can base trade upon this knowledge.

finance 1 – IBEB – lecture 11, week 6 company cost of capital and NPV

value of the firm

The balance sheet shows the financial position of a firm. The balance sheet has on the left side the assets of the firm and on the right side the liabilities. The assets consist of a firm’s cash, inventory, plant & equipment and other investments. The liabilities are made up of the obligations to creditors and stockholder’s equity. This is in formula explained as 𝐴 = 𝐷 + 𝐸 . The assets show what the firm does with its capital and in what way is it invested. The option through which a firm generates its capital is shown by the sources of the capital.

The book value of an asset can be found in the firm’s financial statements. This is determined by the acquisition cost minus the accumulated depreciation of the asset.

The book value of the stockholder’s equity is the firm’s net worth in the form of an accounting measure. The value of the equity is determined by the difference between assets and liabilities, and the two sides must balance. Because this is an accounting measure, it does not give a good true value of the firm’s business operations. This is due to the assets which are not given a good value. The assets are valued by historical costs, and some valuable assets may not be captured by the valuation of the firm. The market value balance sheet covers the assets and liabilities at current market values.

The firm’s market capitalization is equal to the shares outstanding times the market price per share. The capitalization is the amount investors are willing to pay for the shares of a firm. The book values of a balance sheet do not include real values, because intangible assets are not included. The debt holders have unconditional claims , because these debts have to be paid under all circumstances. After the debt holders are paid, the equity holders will be paid, they have conditional claims . The assets’ return, risk and market value determine the equity’s return, risk and market value.

capital structure

The investment decisions of a firm give us the portfolio of investment projects. This portfolio contains the cash, average p roject and speculative project. A firm’s financing decision gives us the portfolio of capital resources, consisting of debt and equity. The way a firm is financed, the proportions of debt, equity and other securities a firm has, determine the capital structure of the firm. Leverage can be measured in the capital structure of the firm by using the following ratios: 𝑑𝑒𝑏𝑡 − 𝑒𝑞𝑢𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 = 𝐷/𝐸 𝑑𝑒𝑏𝑡 − 𝑡𝑜 − 𝑣𝑎𝑙𝑢𝑒 = 𝐷/(𝐸 + 𝐷)




cost of capital

Resources are not obtained by a firm for free, because this amount could have been used to find other projects. Thus, the opportunities cost of using this resource is equal to the value it would have when using the resource for the best possible alternative.

And the opportunity cost of capital is what could have been earned by using the money for an investment with equivalent risk. The 𝑟

𝐷

is the debt cost of capital, and 𝑟

𝐸

the equity cost of capital. The weighted average cost of capital (WACC) denoted with 𝑟

𝑊𝐴𝐶𝐶 can be calculated with: 𝑟

𝑊𝐴𝐶𝐶

=

𝐷

𝐷 + 𝐸 𝑟

𝐷

+

The cost of capital for the securities of a firm, 𝑟

𝐸

𝐷 + 𝐸

𝑊𝐴𝐶𝐶 𝑟

𝐸

= 𝑟

𝐴

, should equal the cost of capital of the firm’s assets, 𝑟

𝐴

.

The assets of a firm generate cash flows with their corresponding risk, which is transferred to the debt and equity of the firm. The debt and equity holders do have claims on the cash flows of the firm’s assets. Debt holders are save in the payments, because the coupons have to be paid regularly, and the principal of these loans has to be repaid at the maturity which is limited to a number of years. Debt holders always get paid first, and after them, the equity holders. Because debt holders have more rights of these payments, the 𝑟

𝐷

is relatively low.

A covenant is a clause which limits the firm’s ability to surpass its obligations towards the debt holders. The stronger the covenant, the less flexibility a firm will have in taking negative-NPV options. Also, covenants reduce the risk of default and ensure the debt holders that the firm will meet the conditions of the contract. These two factors reduce the debt cost of capital demanded by investors to compensate for the risk faced. For the firm, there is a reduced interest payment which it must pay to investors as there is a lower risk of meeting the payment obligations.

In order to formulate an argument for the project’s cost of capital, we must define a few concepts and notations. The expected return of assets is found by taking account of all future cash flows generated by the assets including all associating risks. The weighted average cost of capital (WACC), which is equal to the discount rate of the firm’s assets (r

A

), is the return which debt and equity holders will demand when investing in the firm. From these definitions, the market value of the firm’s assets is equal to all cash flows the assets will generate (consider the cash flows in their present value with the discount rate of the firm’s assets):

𝐴 =

𝐶𝐹

1

1 + 𝑟

𝐴

+

𝐶𝐹

2

(1 + 𝑟

𝐴

) 2

+. ..

project cost of capital

The cost of capital for real projects equals the best expected return of available portfolios with equal risk that are existent in the capital market. This cost of capital depends only on the systematic risk in the efficient capital market. We denote the project’s cost of capital with: 𝑟 𝑝𝑟𝑜𝑗

= 𝑟 𝑓

+ 𝛽 𝑝𝑟𝑜𝑗

[𝐸

𝑀

The company cost of capital is given by:

− 𝑟 𝑓

]



Use of this document is intended for: Chu, Eileen - Student number: 504051 𝑟

𝐴

= ∑ 𝑥 𝑝𝑟𝑜𝑗 𝑟 𝑝𝑟𝑜𝑗

= 𝑟 + 𝛽

𝐴

(𝐸

𝑀

− 𝑟 𝑓

)

It is pointless to diversify your firm in order to reduce risk, because this can be done by shareholders as well and even more efficiently. Diversification does not change the firm value; the total value of a firm will equal the sum of all firm parts. This can be shown by 𝑃𝑉(𝑋𝑌) = 𝑃𝑉(𝑋) + 𝑃𝑉(𝑌) , which assumes that the combination of project X and Y does not affect the others profitability. The average-risk project uses 𝑟

𝐴 𝛽

𝐴

= ∑ 𝑥 𝑝𝑟𝑜𝑗 𝛽 𝑝𝑟𝑜𝑗

. A more or less risky project uses 𝑟 𝑝𝑟𝑜𝑗

because 𝛽 𝑝𝑟𝑜𝑗

≠ 𝛽

𝐴

.

when

The higher the risk of a project which is included in the firm’s assets, the larger the risk premium will be which is added to the risk-free rate.

The higher riskiness of a project or the firm’s assets, the larger the risk premium is added on to the risk-free rate. Which causes the 𝛽

𝐴 a higher 𝑟

𝐴

, and this constitutes of 𝑟

𝐷

and 𝑟

𝐸

to be higher which in turn leads to

thus there is a direct impact on them being high too. Debt will be relatively safe, because debt holders are prioritised in terms of receiving their respective coupons and principal before the stockholders, thus there is low riskiness for the debt. The beta of debt thus is low and fixed, meaning that the cost of capital of debt is low and fixed as well. This finally implies that an increased cost of risk of project or asset is mainly transmitted to shareholders and not debt holders.

We can estimate 𝑟

𝐴

with first determining the market value:

𝐸𝐵𝐼𝑇

𝐴 = ∑

(1 + 𝑟

𝐴

) 𝑡

= 𝐸𝐵𝐼𝑇/𝑟

𝐴 which gives us 𝑟

𝐴

=

𝐸𝐵𝐼𝑇

𝐴

After this we can also estimate 𝑟 𝑝𝑟𝑜𝑗

: 𝑟 𝑝𝑟𝑜𝑗

=

𝐸𝐵𝐼𝑇 𝑝𝑟𝑜𝑗

𝑉 𝑝𝑟𝑜𝑗 𝑟 𝑝𝑟𝑜𝑗

= 𝑟 𝑓

+ 𝛽 𝑝𝑟𝑜𝑗

(𝑅

𝑀

− 𝑟 𝑓

)

Where the cost of capital for a comparable firm is denoted by 𝑟

𝐴 for a comparable project is 𝑟 𝑝𝑟𝑜𝑗

.

and the cost of capital

The constant growth model 𝑃 = 𝐷𝑖𝑣/(𝑟 − 𝑔) , is used to estimate the cost of capital of equity 𝑟

𝐸 estimate

= 𝐷𝑖𝑣/𝑃 + 𝑔 . The market model 𝑅 𝑖 𝛽

𝐸

and 𝑟

𝐸

= 𝑟 𝑓

+ 𝛽(𝐸

𝑀

− 𝑟 𝑓

= 𝑎 𝑖

+ 𝑏 𝑖

𝑅

𝑀𝐼

+ 𝜀 𝑖

is used as well to

) . After this we combine the cost of capital of debt with the cost of capital of equity to find 𝑟

𝐴

= 𝑊𝐴𝐶𝐶 .

certainty-equivalent cash flow

Normally a single discount rate is used for discounting the cash flows in capital budgeting, but the risk of this is not constant the whole time. Therefore, the risky cash flows are converted in a certainty equivalent , 𝐶𝐸 𝑡

, which converts risk. The certainty equivalent is the risk-free cash flow, which is found by discounting risky cash flows into the present value.

𝐶𝐸 𝑡

(1 + 𝑟 𝑓

) 𝑡

=

𝐶𝐹 𝑡

(1 + 𝑟 𝑡

) 𝑡




And this will give us 𝐶𝐸 𝑡 this is 𝐶𝐶 𝑡

= (1 + 𝑟 𝑓

) 𝑡

= 𝐶𝐶

/(1 + 𝑟 𝑡

) 𝑡 𝑡

× 𝐶𝐹 𝑡

and the certainty coefficient corresponding to

. Due to the certainty equivalent we can look separately at the change in value due to time and risk.

The discount rate, 𝑟 𝑡

, will always be bigger than the risk-free rate, which explains that the discount risk compensates for the extra risk in a period an investor is bearing. The later the cash flows, the larger the discounting will be because of more periods being involved.

capital structure in a perfect market

equity and debt financing

In our example we assume the firms U and L are identical firms, but they do differ in their capital structure. The capital structure of a firm indicates whether the firm’s assets are financed with equity or debt. Firm U is an unlevered firm, while Firm L is a levered firm. Their assets will result in the same cash flows for both firms. An unlevered firm will only use their own resources to finance projects. A levered firm instead used debt and sometimes equity to finance the projects.

We will now have multiple costs of capital for the firm. Where 𝑟

𝑈 of unlevered equity, 𝑟

𝐷 of a firm’s assets.

the cost for debt, 𝑟

𝐸

is the cost of capital

for levered equity and 𝑟

𝐴 the cost of capital

The firms U and L have an annual cash flow which is generated by th e firm’s assets, which is their EBIT. Firm L’s assets consist of equity and debt, 𝐴 = 𝐸

𝐿 holders will receive 𝑟

𝐷

𝐷

𝐿

and the equity holders receive 𝑟

𝐸

𝐸

𝐿

+ 𝐷

𝐿

= 𝐸𝐵𝐼𝑇 − 𝑟

𝐷

𝐷

𝐿

. The debt

. If you are an equity holder and hold a share of firm L, this means you are indirectly borrowing because this is a levered firm. It is as well possible to buy a stock of firm U, which was the unlevered firm without debt. But if this is combined with directly borrowing on your own account, 𝐷

𝐿

. The interest paid for the amount borrowed is equal to 𝑟

𝐷

𝐷

𝐿

while receiving EBIT from the share of the unlevered firm. Investors are indifferent between these two investments, because both amounts received will be equal. When investors borrow themselves and buy the share of an unlevered firm, they include leverage into their own portfolio, which is called homemade leverage . There is no difference in investors borrowing on their own account compared to indirectly borrowing. 𝑟

𝐸

𝐸

𝐿

= 𝐸𝐵𝐼𝑇 − 𝑟

𝐷

𝐷

𝐿

= 𝑟

𝑈

𝐸

𝑈

− 𝑟

𝐷

𝐷

𝐿

This as well means the investors are able to change the capital structure of a firm. A change in the capital structure will not change the benefit of investors or the firm value.

Modigliani-Miller proposition I

In the presence of a perfect capital market, the cash flows generated by a firm equals the total value of the firm, and this cannot be affected by the chosen capital structure of the firm. This is because the way a firm is financed does not have any effect on the cash flows the investments of a firm give.

𝑉 𝑈 = 𝑉 𝐿 = 𝐷 + 𝐸

If the capital structure had an impact on the cash flows, then a firm’s investments would generate the most efficient financing choice should be made, resulting in the best possible total value of the firm.

The activities a firm carries out on the real or production market, and the risks belonging to it, are of impact on the firm value. Combinations of securities and splitting



Use of this document is intended for: Chu, Eileen - Student number: 504051 of the cash flows into different streams, will as well not impact the firm value. The

WACC is not dependent on the capital structure chosen by the firm to finance investments. The first proposition separates the financing and investment decision made by a firm, because they do not affect each other.

Modigliani-Miller proposition II

The first Modigliani-Miller proposition states 𝑈 = 𝐷 + 𝐸 . The assets are equal to the unlevered equity, thus 𝐴 = 𝑈 = 𝐷 + 𝐸 , which results again in 𝑟

𝐴

𝐷 𝐸 𝑟

𝐴

= 𝑟

𝑈

= 𝑟

𝐷

+ 𝑟

𝐸

= 𝑊𝐴𝐶𝐶

Where

𝐷+𝐸 equity.

𝐷

𝐴 = 𝑟

𝑈

𝑈 = 𝑟

𝐷

𝐷 + 𝑟

𝐸

𝐸 .

𝐷 + 𝐸 𝐷 + 𝐸

𝐸

and

𝐷+𝐸

denote the fractions of the total firm value financed by debt and

𝐷 𝐸

The formula 𝑟

𝑈

= 𝑟

𝐷

+ 𝑟

𝐸

can be rewritten for 𝑟

𝐸

and this results in 𝑟

𝐸

= 𝑟

𝑈

+

𝐷

𝐸

(𝑟

𝑈

− 𝑟

𝐷

)

𝐷+𝐸 𝐷+𝐸

which gives us the Modigliani-Miller proposition II . The formula means that the levered equity cost of capital is equal to the cost of capital for unlevered equity plus an additional risk premium due to leverage. This premium is the firm’s market value debt-to-equity ratio.

There are three examples:

Equity versus debt financing : here the two firms U and L are the same, except for their capital structure

Debt-to-equity swap : From the exemplified calculation on slides 18-19, we realise that with debt-to-equity swap the WACC is unchanged however, the equity cost of capital declines

Equity-to-debt swap : here the firm has existing equity which is replaced with debt, which explains the name of the swap. In a leveraged recapitalization, which is a change in the capital structure of the firm, a firm will start to borrow funds, and use this to replicate a big number of outstanding shares 𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒 = 𝐸𝐵𝐼𝑇 − 𝑟

𝐷

𝐷 𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 (𝐸𝑃𝑆) = 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 − 𝑡𝑜 − 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑟𝑎𝑡𝑖𝑜 = 𝑃/𝐸𝑃𝑆

levered and unlevered betas

Thus, in a leveraged capitalization a firm will increase its debt and decrease its equity while maintaining the same amount of assets ( 𝐷 + 𝐸 = 𝐴 ). In the new capital structure 𝑟

𝐴

, 𝛽

𝐴

, 𝑟

𝐷

and 𝛽

𝐷

will remain constant, while that is now used for less equity. 𝑟

𝐸

and 𝛽

𝐸

increase due to the business risk

Which we can solve for 𝛽

𝐸 𝛽

𝑈

=

𝐷

𝐷 + 𝐸 𝛽

𝐷

, and leads us to:

𝐷 𝛽

𝐸

= 𝛽

𝑈

+

𝐸

+

(𝛽

𝑈

𝐸

𝐷 + 𝐸

− 𝛽

𝐷 𝛽

)

𝐸

issuing new shares

If a firm issues new shares, there will be no gain or loss for the shareholders if the shares are sold at a fair price. This is because the price of the share has to remain



Use of this document is intended for: Chu, Eileen - Student number: 504051 constant. But we do have earnings dilution in this case, this is because the same amount of earnings now has to be divided among more shares. Which leads us to the dilution fallacy , that if a firm starts issuing equity, this will reduce the value of the existing shares of the shareholders. But here is not taken into account that if a firm starts issuing new shares, the cash earned by a firm will increase, thus a firm will have more assets and higher cash flows. The increases of this will be equal to the dilution of the shares.

With the cash raised from the issuance of shares there are three possibilities, through cash, average project or speculative project. If the assets side increases, the liabilities increase as well. The liabilities increase via equity. The share price remains constant as the financing results in no NPV as such. The new investment generates new cash flows, which increases the market value by the change.

When an amount is used to finance safe new investments, this gives us less risky returns, thus decreasing 𝛽

𝐴

and the cost of capital of assets as well. The cost of capital of debt is constant however, this means the 𝛽

𝐸

and 𝑟

𝐸

will both decrease.

If this amount would be used to finance an average-risk project the cash flows will keep the same amount of risk. As a result, all factors 𝛽

𝐴

, 𝑟

𝐴

𝑎𝑛𝑑 𝑟

𝐷

will all remain constant.

And if an amount of capital is used to finance a high-risk project this will make the cash flows riskier. The increase in the risk of the returns, means 𝛽

𝐴

will increase and the cost of capital of assets as well. Because the cost of capital of debt is constant, this means the 𝛽

𝐸

and 𝑟

𝐸

will both increase.

Due to changes in investments, market values change as a result.

A firm can as well repurchase its own stock. It will use cash to buy back this stock, which lowers the firm value, 𝐴 − ∆ = 𝐷 + 𝐸 − ∆ . This results in riskier cash flows, and 𝛽

𝐴 and 𝑟

𝐴

will increase. And because 𝑟

𝐷 is constant, as well 𝛽

𝐸 and 𝑟

𝐸

will increase.

The earnings per share on average will increase after a share repurchase, but as well there is a higher risk involved now for shareholders, these two exactly offset each other and the price of a share will not increase.

finance 1 – IBEB – lecture 12, week 6 capital structure with corporate tax

Until now we assumed that the capital structure a firm chooses does not influence the value of the firm in a perfect capital market. But in fact, the amount of leverage a firm has does affect the value of the firm via the corporate tax. Firms have to pay tax on their EBIT. The interest expense is the amount of payments made to debt holders, and the left-over cash flows that are paid to debt holders. The income before taxes are deducted is the pre-tax income. The corporate taxes will be the corporate tax rate, 𝜏 𝑐 times the earnings of a firm minus the interest payments, 𝜏 𝑐

(𝐸𝐵𝐼𝑇 − 𝑟

𝐷

,

𝐷) . The net income a firm will have left is the cash flow which can be used to pay equity holders,

(1 − 𝜏 𝑐

)(𝐸𝐵𝐼𝑇 − 𝑟

𝐷

𝐷) .




If the firm would have no leverage, this would mean that the equity holders would have been better off because (1 − 𝜏 𝑐


𝐷

𝐷) < (1 − 𝜏 𝑐

)𝐸𝐵𝐼𝑇 . The cash flows available to all investors will be the payments to debt holders plus the payments made to equity holders, (1 − 𝜏 𝑐


𝐷

𝐷) + 𝑟

𝐷

𝐷 = (1 − 𝜏 𝑐

)𝐸𝐵𝐼𝑇 + 𝜏 𝑐 𝑟

𝐷

𝐷 . The cash flows with debt included in the capital structure results in the firm being better off. The above can be found in an overview in the table below.

Source: Interest Tax Shield (Van der Sar, 2018)

Due to the interest payments that are made to debt holders, 𝑟

𝐷

𝐷 , a smaller part of EBIT can be taxed, which is the tax shield . A firm with leverage will have less equity holders than an unlevered firm, and of course more debt holders. The equity holders of the levered firm will be in favour of those of an unlevered firm due to less corporate tax being deducted from EBIT. Debt holders will only receive the payments they have the right to equal to 𝑟

𝐷

𝐷 . With leverage a firm will be able to generate higher cash flows with their assets than without, because higher cash flows can be paid.

The MM proposition I, which was mentioned earlier, states that the value of firms with leverage will be higher than that without leverage due to tax savings from the debt a firm has.

𝑉 𝐿 = 𝑉 𝑈 + 𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑)

If the debt a firm has issued comes due, it will be refinanced with new debt. This way the debt is permanent, and they use the interest tax shield as a perpetuity.

𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑) = 𝜏 𝑐

𝐷

The market value of debt will be 𝑃𝑉 = 𝑟

𝐷

𝐷/𝑟

𝐷

is the rate used for discounting future interest payments into the present value. And this will give us

𝑉 𝐿 = 𝑉 𝑈 + 𝜏 𝑐

𝐷 .

= 𝐷 where 𝑟

𝐷

Due to the unlevered (U) firm and the levered (L) firm choosing a different capital structure we can conclude the following results. The value of the levered firm 𝑉 𝐿 =

𝑉 𝑈 + 𝜏 𝑐

𝐷 is larger than that of the unlevered firm 𝑉 𝑈 = 𝐸𝐵𝐼𝑇(1 − 𝜏 𝑐

)/𝑟

𝑈

.

The value of the unlevered firm is equal to

𝑉 𝑈 = 𝑉 𝐿 − 𝜏 𝑐

𝐷 = 𝐷 + 𝐸 − 𝜏 𝑐

𝐷 = 𝐸 + (1 − 𝜏 𝑐

)𝐷




Capital holders of a levered firm will have their yearly expected cash flows represented by 𝑉 𝑈 𝑟

𝑈

+ 𝜏 𝑐 𝑟

𝐷

𝐷 = 𝑟

𝐸

𝐸 + 𝑟

𝐷

𝐷 , where 𝑟

𝐸

= 𝑟

𝑈

+

𝐷

𝐸

(1 − 𝜏 𝑐

)(𝑟

𝑈

− 𝑟

𝐷

) .

The weighted average cost of capital (WACC), which is including taxes, will give us the effective after-tax cost of capital to the firm, where the reduction of the interest tax shield is taken into account.

𝐸 𝐷

𝑊𝐴𝐶𝐶 =

𝐸 + 𝐷 𝑟

𝐸

+

𝐸 + 𝐷 𝑟

𝐷

(1 − 𝜏 𝑐

)

When we place the formula of 𝑟

𝐸

into the WACC, this will result in the following:

𝐷

𝑊𝐴𝐶𝐶 = 𝑟

𝑈

(1 −

𝐸 + 𝐷 𝜏 𝑐

)

Meaning that if the leverage of a firm increases, the more the tax advantage of debt is used to lower the WACC.

Source: The WACC with and without Corporate Tax (Van der Sar, 2018)

After this 𝐸 + 𝐷 = 𝑉 𝐿

is again inserted into the previous equation of the WACC giving:

𝑉 𝐿 = 𝐸𝐵𝐼𝑇(1 − 𝜏 𝑐

)/𝑟

𝑈

+ 𝜏 𝑐

𝐷 = 𝐸𝐵𝐼𝑇(1 − 𝜏 𝑐

)/𝑊𝐴𝐶𝐶

The WACC is used for discounting the enterprise value into the present value. The

FCF will not pay attention to the firm’s leverage and only look at the cash flows generated by assets after the tax is paid to the government.

capital structure with corporate tax and personal taxes

The effect of corporate taxes and personal taxes will both result in a lower value of the firm. The financing decision of the firm will have impact and we will look at the cash flows which will be received by the investors after all taxes have been deducted. We will use from now the corporate tax, 𝜏 𝑐

, the personal tax rate on debt income, 𝜏 𝑖 the personal tax rate on the income received by equity, 𝜏 𝑒

.

, and

One can observe the tax advantage by observing the scenario for equity holders, debt holder and in aggregate for the investors as follows:




Source: Adding Personal Taxes: cash flows(Van der Sar, 2018)

The net tax advantage of debt is also called the interest tax shield. The firm value of a levered firm, thus where 𝐷 > 0 will be equal to:

𝑉 𝐿 =

𝐸𝐵𝐼𝑇(1 − 𝜏 𝑟

𝑈 𝑐

)(1 − 𝜏 𝑒

)

+ 𝑟

𝐷

𝐷[(1 − 𝜏 𝑖

𝑉 𝐿 = 𝑉 𝑈 + 𝜏 × 𝐷

) − (1 − 𝜏 𝑟

𝐷

(1 − 𝜏 𝑖

) 𝑐

)(1 − 𝜏 𝑒

)]

The 𝑟

𝑈

used in the first term in the equation in the denominator gives the unlevered cost of capital when cost of capital, 𝑟

𝐷 𝜏 𝑐

(1 − 𝜏 𝑖

, 𝜏 𝑖

and 𝜏 𝑒

are present. The second term is divided by the debt

) . When we had a capital market with just corporate tax where 𝑟

𝐷 was used to discount tax savings 𝜏 𝑐 the return debt holders will receive is equal to (1 − 𝜏 𝑖

𝑉 𝐿 𝑟

𝐷

𝐷

= 𝑉

. Because of corporate and personal taxes,

𝑈 + 𝜏 ∗ 𝐷

)𝑟

𝐷

.




The effective tax advantage of debt can be shown in the formula below if 𝜏 ∗ is bigger than zero, which raises debt and the value of a levered firm.

Source: After-tax investor CF

S resulting from $ 1 in EBIT (Van der Sar, 2018)

The capital needed to let the assets generate money can be done with debt or equity.

And if (1 − 𝜏 𝑖

) > (1 − 𝜏 𝑐

)(1 − 𝜏 𝑒

) means that debt will have an advantage compared to equity when looking at higher after-tax cash flows.

The after-tax cash flows received by the equity holders of levered capital consists of three parts. The first part is the after-tax leverage of an unlevered firm, the second part shows the interest payments which are deducted, and the last part is the effective tax advantage of debt 𝜏 ∗ (1 − 𝜏

(1 − 𝜏 𝑐 𝑖

)𝑟

𝐷

𝐷

)(1 − 𝜏 𝑒

.

)𝐸𝐵𝐼𝑇 − (1 − 𝜏 𝑖

)𝑟

𝐷

𝐷 + 𝜏 ∗ (1 − 𝜏 𝑖

)𝑟

𝐷

𝐷

If the 𝜏 𝑖

is not equal among investors, we start with all firms being unlevered. By issuing debt, firms will try to take the advantage of leverage. This advantage is offset by personal taxes which increases the leverage. At the end where the market is in equilibrium, all the firm’s capital structures will be optimal for the value of the firm. This optimal structure, is after the debt is added there will be no more net tax advantage by adding more debt. Thus, the marginal tax advantage will equal zero. The advantage of issuing more debt is offset by the personal disadvantage.

𝑉 𝐿 = 𝑉 𝑈 + 𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑)

In 1977, when Miller adopted this approach for the economic industry, he did combine all individual firms into this industry. There was assumed to be no optimal debt-toequity ratio for an individual firm, and firm value was independent of its capital structure. But there is in fact an optimal debt-to-equity ratio for the economic industry as whole. This ratio is determined by the personal and corporate taxes which are fixed by the government. The level of debt will be at the point where the following equation holds:

(1 − 𝜏 𝑖

) = (1 − 𝜏 𝑐

)(1 − 𝜏 𝑒

)

This is the level of debt for which the government will have the smallest advantages of the cash flows. This Miller theory is not existent in the book, because the theory is doubted due to a lot of real life differences that came with it. There may not be the typical capital structures which are assumed for this theory, thus there is not a lot of empirical proof. But the view this theory has on the economic industry and the role of



Use of this document is intended for: Chu, Eileen - Student number: 504051 the government on this is good to examine the impact of the tax rates and the changes due to policies.

The 𝜏 ∗

varies across firm, this is due to the decisions of the firm which amount of earnings is shielded from corporate taxes. Decision like depreciation, investment tax credits, carry forwards or operating losses and sometimes possible tax reductions of employee stock options. Firms do not rely completely on the interest tax shield and if the corporate tax does decrease, 𝜏 ∗

is likely to do so as well. (the corporate tax is in fact constant, but the effect on debt does decrease). The variation of 𝜏 ∗ is bigger than the variation in 𝜏 𝑐

, because from investor to investor this may be different. Thus, the tax advantage of debt will therefore be extremely difficult to calculate because it varies.

Source: Net external financing and capital expenditures by U.S. corporations 1975-

2014 (Van der Sar, 2018)

The firm corresponding to this figure shows a preference for increasing their debt to fund investments and repurchase shares, due to which the number of shares outstanding will decrease.

The firm’s industry will be of influence on the leverage decision. Big firms tend to have high leverage ratios and growth industries have large debt reserves compared to their debt. This raises questions because we just studied why carrying debt would benefit a firm. If we assume that EBIT is positive and compare this to 𝑟

𝐷 conclusions. If 𝑟

𝐷

𝐷 < 𝐸𝐵𝐼𝑇 only the part 𝑟

𝐷

𝐷 we can make some

𝐷 will be a tax shield and safe from the corporate tax. If 𝑟

𝐷

𝐷 = 𝐸𝐵𝐼𝑇 the leverage of the firm is at its optimal point where it is not possible to shield more income. When 𝑟

𝐷

𝐷 > 𝐸𝐵𝐼𝑇 the firm carries an excess leverage, this gives a disadvantage, and this should thus be voided. The blue line will be below the red line, because for high levels of interest expense there is a higher probability of excess interest.




Source: Tax savings for different levels of interest (Van der Sar, 2018)

The EBIT of a firm will impact the financing decision of the firm. As we have seen, when there is no taxable income, there will be no debt as well, which is observed in growth firms. Growth will affect the optimal leverage ratio because a higher growth rate will lower 𝐸𝑃𝑆/𝑃 . The lower EBIT/E gives a lower 𝑟

𝐷 ratio decreases 𝐷/(𝐷 + 𝐸) .

𝐷/𝐸 , and the lower debt-to-equity

finance 1 – IBEB – lecture 13, week 7 capital structure in general

introduction

It is possible that the capital structure can affect a firm due to market imperfections.

This may influence the costs of a firm’s financial distress, alter the incentives of managers or give investors signals. This again may reduce the benefits received by using leverage, which results in lower debt levels used in a capital structure. But these consequences may differ per firm and thus the leverage decisions of firms differ a lot as well.

costs of financial distress

In reality, most firms shield one third of their firm’s EBIT from the corporate tax. A higher amount of debt increases the tax benefits of having debt. If the cash flows of a firm are not stable, this causes trouble estimating the EBIT and the appropriate amount of debt. Due to this it is as well possible a firm has trouble with paying the debt obligations that are due, as a result of unstable cash flows. Equity holders only receive dividends if there is money to pay them, but debt holders need to receive their payments because this is their right. The firm should be held to their debt covenants and pay the debt holders the interest and principal payments that are promised. If a firm fails to make any of these payments to debt holders, the debt covenant will be in default .

Whether default would be resultant depends on the money that one can raise through the capital market, taking into account the relative market values of the firm’s assets and liabilities. Thus, it is not dependant on whether the cash flows are less than zero for a temporary phase.




Debt holders will receive certain rights to the assets of the firm, when a default in the firm has occurred. A bankruptcy of a firm shifts ownership of the firm from equity holders to debt holders. This does not have any effect on the total value which is available to all investors.

The cost of capital of debt will be higher than the risk-free rate. The chance of default will be bigger than zero. If debt increases in a firm, the amount of equity decreases.

When debt is low and leverage increases, the cost of capital for equity holders will be higher because risk is shared among more equity holders. But this is compensated by higher returns. But as debt keeps increasing the cost of capital for debt will start to rise as well, which is the same for the cost of equity but less strong.

All effects of financial distress and bankruptcy that create extra costs will be included in the prices. Due to bankruptcy debt holders may not be able to get the full value of assets, because now financial distress costs are included.

The costs of bankruptcy can be divided among two groups, the direct and indirect costs. The direct costs of financial distress include the hiring of professionals from outside. Both the firm and the creditors will hire these professionals. This will decrease the value of the firm’s assets.

Indirect costs of financial distress consist of multiple possible losses. The loss of customers will incur costs because if the value is dependent on the service or product for customers, they will lose customers. But this is not the case if the product, once delivered, is not dependent on the firm and its success any more. The more firms rely on trade credit, the bigger the cost due to loss of suppliers will be. Loss of employees and loss of receivables are both as well of impact on the costs. And fire sales of assets, inefficient liquidation and distress costs to creditors all have effect on the indirect costs of financial distress.

𝑉 𝐿 = 𝑉 𝑈 + 𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑) − 𝑃𝑉(𝑓𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑑𝑖𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑜𝑠𝑡𝑠)

When deciding on the capital structure, the firm will face a trade-off between the tax shield from debt and the financial distress costs.

Source: Optimal leverage with taxes and financial distress costs (Van der Sar, 2018)

The present value of financial distress costs is determined by the three following causes:






The probability of financial distress : the probability of financial distress will increase with the increase of debt relative to the firm’s assets. The risk of a firm’s cash flows and asset values do as well increase the probability



Magnitude of the costs : this will increase the more important the loss of customers, key personal and lack of tangible assets is



The discount rate of distress costs : the discount rate will decrease due to a higher firm beta, because the negative beta of distress costs leads to a lower cost of capital

The trade-off theory helps us understanding that firms will choose a level of debt too low to exploit the full benefits of the tax shield. The optimal level of debt will vary as well across different industries.

Agency costs are costs that are incurred when there are different interests between the holders of a firm. The costs will be created due to the conflicts between debt and equity holders that need to be solved due to the amount of leverage a firm chooses.

And there will be principal agent problems as well due to conflicts between shareholders and managers in a firm, because of different objectives and that the ownership and control is in two different places. This gives us two types of costs; outof-pocket costs are due to the actions taken to make sure the agent’s actions will be in line with the goal of the principal. And the opportunity costs are representing the value that the agent lost by not handling in the best interest of the principal.

Over-investment can happen because of excessive risk-taking and asset substitution.

For shareholders it is not possible to lose more than their investment, thus they have a limited liability .

In a firm, shareholders will be the owners of a firm, but they bet on risk with the money of the debt holders, which is known as risk shifting . Due to this, shareholders will take more risky assets instead of low-risk ones, which is known as the asset substitution problem .

One can consider the example of debt overhang and underinvestment in the case when a firm’s value of asset is lower than that of debt and an evaluation of whether a new investment project should be undertaken is considered (this project generates returns without any risk). In this case, if the project is profitable to the firm as a whole because the NPV to debt and equity holders combined is greater than zero, and when isolated the gain is mainly to debt holders because of the repayment and the NPV of equity holders is negative. In this scenario, shareholders would decline this project and refuse the fresh equity and thus the agency cost of debt reflected is the debt overhang.

When a firm has financial distress, the firm will have to cash out and they will sell equipment or assets for lower prices than normally would have been generated. But these proceeds are used to pay cash dividends. If 𝑟

𝐷

increases the value of D decreases to 𝐷 − ∆ . Due to the wealth transfer and cash dividend to equity holders we have 𝐸 + ∆ − 𝐷𝑖𝑣 .

The leverage ratchet effect is an agency cost of leverage as well. Shareholders may have an incentive to increase leverage even when this does decrease the value of the firm. Or the shareholders will not decrease leverage by buying back, even though this would increase the firm value. If a firm would start issuing more debt, the 𝑟

𝐷

will



Use of this document is intended for: Chu, Eileen - Student number: 504051 increase, wealth will be transferred from debt holders to equity holders. Even though the firm value is reduced by increasing leverage, it does benefit shareholders.

When the debt holders of a firm will lose more than the gain of the equity holders, the net effect of the change in the capital structure is a reduction in the firm’s initial share price.

The managers who make the decisions of the firm may harm debt holders if the decisions taken will affect the value of D and E. It is possible that equity holders will exploit the debt holders of the firm. This is known as the agency costs of leverage .

When financial distress risk is high, this will be more likely to occur.

Managers have the incentive to benefit the equity holders, because they own shares of the firm themselves. Or they will act in the best interest of the equity holders because they will have the decision to hire and fire managers.

Managers may as well make decisions that benefit them, but at the cost of the investors. But because ownership and control are separated this gives a probability of management entrenchment . This mean that if managers run the firm in their own interest, they still face little threat of being fired and replaced. If the equity is not owned by the management of a firm, these costs are called the agency costs of external equity . Agency conflict are more likely to happen when financial distress risk and leverage is low. The agency benefits of leverage happen when it is preferred to use leverage rather than equity financing because of this a firm can be financed more efficiently and effectively in the best interest of equity holders.

If managers have a bigger part of equity, there will be less external equity, as a result they will work harder and consume less corporation perks. Due to this the agency costs will decrease.

Ownership will dilute over time because of the firm that grows. Previous owners will retire, original owners will diversify their portfolios and as there are higher capital needs more equity is issued. But this may increase the conflicts between the interests of managers and shareholders. Sometimes negative NPV investments are taken because managers want to increase the size of a firm and do not pay attention to the profitability of the firm, with this being known as an empire building . Or because of overconfidence the risks of a firm are underestimated and thus a negative NPV decision is taken. Or the managers expect too much and because of their overoptimism they will make a negative investment.

With the free cash flow theory, it is assumed that after a firm has paid all debt holders the payments and positive NPV investments are made the excess of the cash flows are likely to be used for wasteful spending. If the amount of debt increases, cash is assumed to be tight. Cash will be tight because if the amount of debt rises, so will the interest payments, and this causes financial distress threat. Because of less management entrenchment, there will be less job security as well. And because of the high amount of debt, it is closely monitored by the creditors. A leveraged buyout

(LBO) is a buy where large part of the total value is financed by debt and small remaining part is financed by the equity of a small group of investors.

For the optimal level of debt there will be a trade-off between the tax shield of debt against the costs of financial distress and agency costs, which is known as the tradeoff theory .




𝑉 𝐿 = 𝑉 𝑈 + 𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑) − 𝑃𝑉(𝑓𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑑𝑖𝑠𝑡𝑟𝑒𝑠𝑠 𝑐𝑜𝑠𝑡𝑠)

−𝑃𝑉(𝑎𝑔𝑒𝑛𝑐𝑦 𝑐𝑜𝑠𝑡𝑠 𝑜𝑓 𝑑𝑒𝑏𝑡) + 𝑃𝑉(𝑎𝑔𝑒𝑛𝑐𝑦 𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠 𝑜𝑓 𝑑𝑒𝑏𝑡)

At the optimal debt level, denoted with 𝐷 ∗

, the benefits and costs of leverage will balance each other out.

Source: Optimal Leverage with taxes, financial distress, and agency costs (van der

Sar, 2018)

The optimal level of debt varies across firms. A firm with high research and development costs and growing opportunities will not generate high cash flows in the growing period. Because of a low EBIT the tax shield needed is very small thus there is only a little bit of debt. The high human capital in growing firms will cause the financial distress costs to be high. And the agency costs of debt are high, this is the result of risk and new investment opportunities.

For mature firms the optimal level of debt will be higher because there are not that many new investments possible anymore. The stable cash flows of a mature firm will give a higher tax shield. Because tangible assets will have a high liquidation value, the financial distress costs will be low. The optimal level of debt for mature firms will be higher than that of growing firms

Asymmetric information is existent when the managers of a firm for example have superior information compared to investors about the future cash flows of a firm. With asymmetric information the parties will not all have the same information thus the assumptions that securities are fairly priced are not accurate any more. The investors will get this information via press releases, and penalties will be there for when intentionally no or wrong information is given.

The credibility principle states that an action will only be credible if the costs of taking this action would be too costly if the event was not true. When debt is used to signal some kind if information to investors, this is called the signalling theory of debt.

A seller can give free some information via the way he is selling his product. Because the bigger his desire to sell, the higher the probability the product is of low quality.

Adverse selection is when buyers and sellers have different information about the product. The Lemons principle states that when sellers have private information about the product and its value, buyers are likely to ask for lower prices because they do not know of which quality the product will be.




Firms who start issuing new equity will sell these stocks at a discounted price to investors because investors are not willing to pay this much due to the possibility that bad news will be announced. The stock price will decrease due to the issuance of new stocks. But before the firm might have waited for the stock price to rise due to good news that will have been published. But if the managers expect bad news to come, they will start issuing the shares before the share price worsens. Big releases of information to investors will only happen at the moment information asymmetry is small. Due to information asymmetry there is an order of gaining financial resources which is preferred the most:

Retained earnings ; the firm will internally finance the new project

Debt issues ; if the retained earnings of a firm will not be enough to acquire the amount of money needed, external financing is the next step. The issuance of debt is less sensitive to private information of managers due to the covenant

Equity issues ; if both retained earnings and debt capacity are used, the last option will be issuing new equity. Under-pricing will be very likely here because of the negative signal of issuing new equity

finance 1 – IBEB – lecture 14, week 7 dividend policy

introduction

When a firm has a free cash flow, it can take a decision how to use these cash flows.

If there are possible positive NPV opportunities, the cash can be used to reinvest.

Mostly this is the case after a takeover or if we talk about a rapidly growing firm. If there are no profitable investment opportunities, which is often the case for mature firms, they can keep the cash for building up the cash reserves or they use the cash for shareholders. They can pay a dividend to the shareholders or choose to repurchase shares. At the declaration date , known as the announcement date as well, the board of a firm authorizes the dividend that will be paid to shareholders. The stock is said to trade before the ex-dividend date at the cum-dividend price, denoted as 𝑃 𝑐𝑢𝑚

. Investors who buy a stock of this firm after the ex-dividend date will not receive a dividend, known as 𝑃 𝑒𝑥

.

pay dividend with excess cash

Due to 𝑃 𝑒𝑥

no arbitrage opportunity will exist. Investors are not able to purchase a stock just before it goes ex-dividend and sell it right after thus date. If dividends are paid with excess cash, there will be no loss overall for the investors. Due to the dividend payout the value of equity decreases by the amount cash decreases in the balance sheet.

An investor can decide whether or not he or she accepts the money. The money received can be used to reinvest, the investor will purchase additional shares at the ex-dividend price. At the new price after the shares are purchases with the dividends received, the total value of dividends afterwards will be equal to the value before the dividend payment. Before the dividend payment, the old price times the number of shares owned, equals the value of the stocks an investor has, times the ex-dividend price when the dividend proceeds are used to reinvest in new shares. A homemade dividend is when there is no dividend pay-out, but the investor decides to sell shares himself. Example of this can be found on slide 6 till 12.




high dividend, equity issue

By issuing equity and being able to pay a higher dividend because of this, the initial share price will not undergo a change. There will be no benefit to shareholders because of this dividend increase. Example at slides 13 and 14.

As we have seen, buying and selling shares will both be zero-NPV transactions. There is no difference if such an action is done by investors or the firm because the actions a firm can undertake can be replicated or be undone by the investors. The actions that can be taken by investors are zero-NPV transactions as well.

In a perfect capital market, the choice a firm makes regarding the dividend policy will not affect the initial share price.

share repurchase and no dividend

A firm can use the cash it has generated to repurchase outstanding shares today. If a firm does this, no dividends are paid to current shareholders, but in the future the dividend received will be higher. But due to this the price of the stock will not be affected by the share repurchase:

𝑃 𝑟𝑒𝑝

= 𝑃 𝑐𝑢𝑚

= 𝐷𝑖𝑣 + 𝑃

The example can be found in slide 16 to 19. 𝑒𝑥

retaining cash

A firm can choose to retain cash by holding the cash in the bank or by purchasing

Treasury bills, both these options are zero-NPV investments. The money is available the next period again plus maybe some extra interest. If there are new positive NPV investments, these can be taken, or the money can be paid out to shareholders. An

Example can be found in slides 20 and 21.

The Modigliani-Miller pay-out irrelevance proposition states that in a perfect capital market, when a firm decides to invest excess cash into financial securities, the choice of the firm between pay-out and retention will not affect firm value. To shareholders there may be no difference in paying the cash immediately or receiving the cash at a future date after the firm has retained and invested this for a moment in a financial security.

In a perfect capital market, the dividend policy choice of a firm does not influence the share prices determined by dividends. In a world with corporate tax, the dividend is not irrelevant. The cash will be equal to the negative leverage. If holding debt gives a tax advantage, this means that holding cash will give a tax disadvantage. A firm that retains cash to invest this, will take a negative NPV investment, because interest earned by the firm of this investment is taxed.

The 𝜏 𝑔

denotes the capital gains tax rate and 𝜏 𝑑

the dividend tax rate, which are not equal to each other. If 𝜏 𝑔

< 𝜏 𝑑

the personal taxes will discriminate the capital gains and dividends received by the investors, their income. If no arbitrage opportunities are assumed, the expected rate of return 𝑟 ∗ before and after the stock goes ex-dividend must be equal. 𝑟 ∗ 𝑟 ∗

= [(𝑃 𝑒𝑥

= (𝑃 𝑐𝑢𝑚

− 𝑃 𝑛𝑜𝑤

)(1 − 𝜏 𝑔

)/𝑃 𝑛𝑜𝑤

− 𝑃

𝑃 𝑛𝑜𝑤 𝑐𝑢𝑚

)(1 − 𝜏

− 𝑃 𝑒𝑥 𝑔

) + 𝐷𝑖𝑣(1 − 𝜏 𝑑

= 𝐷𝑖𝑣 ×

1 − 𝜏 𝑑

1 − 𝜏 𝑔

)]/𝑃 𝑛𝑜𝑤




The dividend policy and the tax rate of the dividend and capital gains will determine the price drop. If an investor n will be taxed low on his dividends, then:

(𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

)(1 − 𝜏 𝑔

) = 𝐷𝑖𝑣(1 − 𝜏 𝑑

) < 𝐷𝑖𝑣(1 − 𝜏 𝑑,𝑛

)

Only if the dividend pays a high amount, the investor will choose this stock. This is because the after-tax dividend will be bigger than the after-tax capital loss. Investors that are highly taxed on dividend payments will prefer a stock that pays little or no dividend at all.

If every investor chooses the stock that matches their preferences most, we will end up in an equilibrium. The clientele effect means that a chosen dividend policy reflects the preference of the investor clientele. All investors are able to find a stock that will match his or her preferences. If one firm decides to change the dividend policy, this does not mean immediately the equilibrium will not be existent anymore, or this does influence value.

empirical results

There are six empirical known results:

In 1970 Elton and Gruber showed taxes have an impact by the following formula, 𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

= 0,778𝐷𝑖𝑣

The Hong Kong stock exchange, where there are no relevant taxes, showed that 𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

< 𝐷𝑖𝑣 , which invalidates the tax effect

In the Netherlands, there are no relevant taxes. In the US tax rates were cut in

2003, the negative effect of taxes on divided payments and capital gains was strongly reduced. Which gives us (1 − 𝜏 𝑑

)/(1 − 𝜏 𝑔

) ≈ 1 and if this holds, this means 𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

≈ 𝐷𝑖𝑣

We know from earlier:

.

𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

= 𝐷𝑖𝑣 ×

1 − 𝜏 𝑑

1 − 𝜏 𝑔

Which can be rewritten as:

𝑃 𝑐𝑢𝑚

− 𝑃 𝑒𝑥

= 𝐷𝑖𝑣 × (1 − 𝜏 ∗ 𝑑

)

In which: 𝜏 ∗ 𝑑

= 𝜏 𝑑

− 𝜏 𝑔

1 − 𝜏 𝑔

If 𝜏 ∗ 𝑑,𝑛

< 0 holds for an investor, he should buy the stock and receive the dividend. But if 𝜏 ∗ 𝑑,𝑛

> 0 would hold for the investor, he should sell it before it goes ex-dividend, because if he would wait and receive the dividend, it would be heavily taxed

There has been research about imbalances around the ex-dividend date, which resulted in the dividend capture theory . If transactions costs would be absent, investors would be able to trade their shares at the moment of the dividend, thus non-taxed investor will receive the dividend. If this is done in larger volumes, high-tax investors will sell the stocks to low-tax investors just before the dividend, and after the ex-dividend date, the low-tax investors will trade the stock back with the high-tax investors. Even though this would work, a lot of high-tax investors still hold shares at the moment the dividends are paid, thus it is not said that all investors will trade with low-tax investors around this time

If 𝜏 𝑔

< 𝜏 𝑑

holds this means shareholders pay lower taxes at the moment dividends are not paid but the firm repurchases shares. The dividend puzzle is



Use of this document is intended for: Chu, Eileen - Student number: 504051 the fact that even though there is a tax disadvantage when issuing dividends firms continue to do this

Retaining excess cash which is combined with corporate and capital gain taxes will result in a substantial disadvantage when looking at costs

Having cash in a firm can be seen as negative leverage due to the tax disadvantage.

This is known as agency costs of retaining cash and indicates the inefficient use of money which is not in discipline with the market. Holding excess cash is as well in conflict with the interest of shareholders, thus the firm may use the cash for leverage, share repurchases and dividends. These are all known as the agency benefits of dividends .

There is as well a possibility of agency costs of dividends which exist when a firm is not able to cover the cash shortfalls a firm may have in the future. Cash is needed for investments to keep a firm growing, but liquidity is important as well and if there is no liquidity this can cause financial distress.

If the dividends a firm pays out are relatively constant over time, this is called dividend smoothing . It occurs more often that a dividend payment will increase, than that they will cut on dividends. Because of this, a firm signals information via their dividend payments, known as the dividend signalling hypothesis .

Surveys found out that managers prefer a long-term target level for their dividends as a fraction of earnings, and investors prefer stable dividends.

𝐷𝑖𝑣 𝑡

− 𝐷𝑖𝑣 𝑡−1

= 𝑠(𝛿 × 𝐸𝑃𝑆 𝑡

− 𝐷𝑖𝑣 𝑡−1

)

In this formula 𝛿 is the target dividend payout ratio as a percentage of earnings set by the firm. The adjustment rate is set with s and can take on values from 0 to 1. If s is zero this means the dividends are sticky, if s is equal to one, there will be a direct adjustment and if the value of s is in between zero and one there will be partial adjustment. The dividend smoothing mentioned earlier will be close the zero.

If a divided is decreased, this will give a bad signal that the price will decrease as well.

The increase of a dividend is a residual cash flows decision. But it will only be a good idea to increase a dividend if permanent earnings of a firm have increased and if the dividend increase will be preferable for shareholders.

reference list

Swinkels, L. (2018). Lecture 1 [PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/9041645?module_item_id=270800







Use of this document is intended for: Chu, Eileen - Student number: 504051 van der Sar, N.L. (2018). Lecture 6: Mean-Variance Analysis [PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/8791181?module_item_id=254453

Van der Sar, N.L. (2018). Lecture 7: Portfolio Theory [PowerPoint slides].

Retrieved from: https://canvas.eur.nl/courses/25347/files/8791183?module_item_id=254456

Van der Sar, N.L. (2018). Lecture 8: Capital Asset Pricing Model [PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/8791185?module_item_id=254458

Van der Sar, N.L. (2018). Lecture 9: Multi-Factor Model and APT

[PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/8791188?module_item_id=254463

Van der Sar, N.L. (2018). Lecture 9: Performance Measures


Van der Sar, N.L. (2018). Lecture 10: Efficient Market Hypothesis


Van der Sar, N.L. (2018). Lecture 11: Capital Structure in a Perfect

Market [PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/8791196?module_item_id=254466

Van der Sar, N.L. (2018). Lecture 12: Capital Structure with Corporate Tax


Van der Sar, N.L. (2018). Lecture 12: Capital Structure with Corporate Tax and personal taxes [PowerPoint slides]. Retrieved from: https://canvas.eur.nl/courses/25347/files/8791200?module_item_id=254472

Van der Sar, N.L. (2018). Lecture 13: Capital Structure in General


Van der Sar, N.L. (2018). Lecture 14: Dividend Policy [PowerPoint slides].

Retrieved from: https://canvas.eur.nl/courses/25347/files/8791204?module_item_id=254475


finance-1-week-7

EFR summary

weeks 1 to 7

details: subject: finance 1 IBEB 2018-2019 teacher: Laurens Swinkels and N.L. Van der Sar date of publication: 14-12-2018

© This summary is intellectual property of the Economic Faculty association Rotterdam (EFR). All rights reserved. The content of this summary is not in any way a substitute for the lectures or any other study material. We cannot be held liable for any missing or wrong information.

The Erasmus School of Economics is not involved nor affiliated with the publication of this summary.

finance 1 – IBEB – lecture 1, week 1

Hirshleifer model and Fisher separation theorem

introduction

Hirshleifer model without real market

Hirshleifer model with real market

Fisher separation theorem

finance 1 – IBEB – lecture 2, week 1

company

ratios

time

risk

law of one price

arbitrage

finance 1 – IBEB – lecture 3, week 2 investment decision rules

cash flows

when is an investment attractive?

how to decide between investing in two (mutually exclusive) investments and how to choose between investments?

how to invest with a limited budget?

alternative investment decision rules

finance 1 – IBEB – lecture 4, week 2 capital budgeting and valuing bonds

capital budgeting

bonds

term structures

valuing bonds

spot rates and forward rates

finance 1 – IBEB – lecture 5, week 3 bonds and equities

interest rate sensitivity

corporate bonds

valuing stocks

finance 1 – IBEB – lecture 6, week 3 mean-variance analysis

expected utility and mean variance

risk and return measures

historical returns

risk and return measures

diversification

measuring systematic risk

risk and return measures

finance 1 – IBEB – lecture 7, week 4

portfolio theory

normative portfolio theory

combining two securities

combining three securities

adding a risk-free security

expectations and preferences

finance 1 – IBEB – lecture 8, week 4 capital asset pricing model

assumptions

capital market line

security market line

finance 1 – IBEB – lecture 9, week 5 multi-factor model and arbitrage theory

introduction

single-factor model

multi-factor model

arbitrage pricing theory

the APT in practice

performance measures

investment performance evaluation

Sharpe performance measure

market Model (MM) in excess returns

Treynor performance measure

Jensen performance measure

information ratio or appraisal ratio

finance 1 – IBEB – lecture 10, week 5 efficient market hypothesis

informational efficiency

efficient market hypothesis

three levels of informational efficiency

example

finance 1 – IBEB – lecture 11, week 6 company cost of capital and NPV

value of the firm

capital structure

cost of capital

project cost of capital

certainty-equivalent cash flow

capital structure in a perfect market

equity and debt financing