Fundamentals of Finance

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Willis Stokes
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1 Finance in a nutshell University of Oulu - Department of Finance Fall 2018

2 Valuation of a future cash-flow Finance is about valuation of future cash-flows is terms of today s euros. Future cash-flow can be considered as a random variable with a probability distribution. Probability distribution describes the probabilities p() of the different values of a random variable. p() E() The present value x 0 of the future cash-flow equals to the discounted value of the expected future cash-flow E(). t = 0 x 0 = d x E() = e kx T E() = e (r+px )T E() t = T The value of the discount factor d x depends on the time-value of money, and the uncertainty in the future cash flow. In terms of an annual discount rate k x, it is about the annual risk-free rate r, the annual cash-flow-specific risk premium p x, and the time in years to the cash-flow, i.e. T.

3 Valuation of the future cash-flow At this point, let us assume that we expect to receive a cash-flow of e5000 two years from now. To be able to calculate the present value x 0 of the cash-flow, we need to identify the risk-free rate r, as well as the cash-flow-specific risk-premium p x. Let us start with the risk-free rate r, which is applicable over the two-year period in question. p() t = 0 E() x 0 = e (r+px )T E() = e (r+px ) = 5000 T = 2

4 Estimation of the risk-free rate The annual risk-free rate r is typically estimated in terms of the annual percentage yield of risk-free (government-issued) zero-coupon bonds. There is no uncertainty in the payments of a government-issued bond. There is no need for any probability distribution. The bond currently trades at e950. p() The value of the discount factor is d f = t = 0 x 0 = 950 = d f 1000 d f = x = 1000 The holder of the bond receives back the e1000 principal at time T = 2. T = 2 We may express the risk-free discount factor in terms of the (two-year) continuously compounded annual risk-free rate r: e r T = d f e r 2 = r = 1 ln = = 2.565% 2

5 Valuation of the future cash-flow Two years from now, we expect to receive a cash-flow of e5000, The two-year risk-free rate is r = 2.565%. Next we need to determine the cash-flow-specific risk-premium p x, which depends on the uncertainty of the cash-flow. p() t = 0 E() x 0 = e (r+px )T E() = e ( px ) = 5000 T = 2

6 Uncertainty in the future cash-flow The risk-premium p x is about, how much reward the market is ready to pay to the uncertainty in the cash-flow. The uncertainty in the cash-flow is a bit difficult issue to analyze. The primary measurement of the uncertainty or variability of a random variable is variance s 2 = E [( E() ) 2 ]. p() s 2 t = T However, from the point of view of the financial market, it is essential, how much the cash-flow contributes the riskiness of a well-diversified portfolio, i.e. the riskiness of the market portfolio. In other words, a part of the variability in the cash-flow is ignored, because diversification of investments enables us to get rid of it, if we just want to. The remaining part of the variability the undiversifiable systematic risk the risk we are not able to get rid of by diversification, determines the size of the risk-premium p x.

7 Uncertainty in the return provided by the cash-flow The riskiness of a portfolio is measured in terms of the return variance of the portfolio, and thus we turn to analyze the return R x provided by the cash-flow over the time-period from t = 0 to t = T : R x = x 0 x 0. Yes, we do not know the current value x 0 of the cash-flow, but let us postpone the issue for a while. Anyway, it is quite obvious, that the variability of return R x mirrors the variability the cash-flow. Thus we are able to measure the variability of the cash-flow by the variance σx 2 of the return provided by the cash-flow: σ 2 x = E[( R x E( R x ) ) 2 ]. However, if the cash-flow is combined as a part of a diversified portfolio m, the most part of the variance risk σ 2 x disappears, and what remains left is the covariance-risk σ xm of the cash-flow with all the other assets in the portfolio: σ xm = E [( Rx E( R x ) )( Rm E( R m) )]. At this point, it is clear that we want to be able to express the discount rate k x in terms of the risk-free rate r, and a risk-premium p x, the size of which depends on the amount of the covariance-risk σ xm involved with the cash-flow : k x = r + p x = r + f (σ xm).

8 Uncertainty in the return provided by the cash-flow We decide to build the functional form of the risk-premium in a way that it contains a common measurement of the unit-price of risk the expected excess return µ m r of the market portfolio over the risk-free rate. The chosen measurement enables us to express the risk-premium in terms of a rate of return, and, furthermore, as the rate of return of such a benchmark portfolio, which is common to all market participants. At the same time we need to adjust our measurement σ xm of the amount of risk to correspond to the above chosen measurement of the price of risk. It is, we multiply the covariance-risk σ xm by a constant c: k x = r + p x = r + ( µ m r ) cσ xm. To find the value of the adjustment factor c, we apply the model to the market-portfolio, the expected return, the required return, and the discount rate of which we know to be equal to k m = µ m. Correspondingly, the amount of the covariance-risk of the market-portfolio equals to σ mm = σm 2, and we get µ m = r + ( µ m r ) cσ 2 m cσ 2 m = 1 c = 1 σm 2. The general form of the models appears now as k x = r + ( µ m r ) σxm σm 2.

9 Capital asset pricing model The model is typically written as k x = r + ( µ m r ) β x, β x = σxm σm 2. In the above Capital Asset Pricing Model the risk-free rate, r, represents the time-value of money, the term µ m r represents the unit-price of market-related undiversifiable risk, the beta, β x, represents the investment-specific amount of market-related systematic risk. When it comes to the value of the beta-coefficient, values near to one indicate a tendency to mirror the market, on average, values greater than one indicate a tendency to follow the market, but with a stronger variability, on average, values lower than one indicate a tendency to follow the market, but with a weaker variability, on average. values near to zero indicate independecy from the market, negative values indicate a tendency to move to the directon opposite to that of the market, on average. In order to be able to apply the model, we need to estimate the unit-price µ m r of the market-related risk, estimate the amount of market-related systematic risk, i.e. the beta-coefficient β x of the investment.

10 Unit-price of market-related risk The unit-price of market-related risk is typically estimated in terms of a long-term historical average excess return of the stock market over the return of a risk-free bond portfolio. The Procedure: Every year, over a long period (say 30 years) of yearly historical data, calculate: the yearly logarithmic return, R m, of the stock market, the yearly logarithmic return, R f, of a portfolio of government-issued risk-free bonds, the yearly excess return, R e = R m R f, of the stock market over the return of the risk-free bond portfolio. The time-series average, R e, of the excess returns serves as an estimate of the expected price of the market risk in terms of an annual percentage rate. In the following numerical examples, let us assume the value of R e = = 4.00%. Arithmetic return: R t = Pt P t = = 0.10 P t 1 10 ln( ) = Logarithmic return: R t = ln P t ln P t 1 = ln 11 ln 10 = e = 0.10

11 Amount of market-related risk When it comes to the estimation of the amount of systematic risk, in our case of an investment project, we do not necessarily have any historical data available for the direct estimation of the beta-coefficient. To get an estimate of the amount of the market-related systematic risk, we use an educated guess of the value of the beta-coefficient, or choose an appropriate benchmark-asset, for which we are able to estimate the beta by available empirical data. Having empirical data available, the beta-coefficient is typically estimated by regressing the asset (excess) returns against the market (excess) returns over a period of historical data: The Procedure: R xt = ˆα x + ˆβ x R mt + ɛ xt, ˆβx = ˆσxm ˆσ m 2. Over a period of an appropriate length (say 5 years) of monthly historical data, calculate the monthly logarithmic (excess) returns, R i, of the asset, calculate the monthly logarithmic (excess) returns, R m, of the stock market, regress the asset (excess) returns against the market returns, use the estimated regression coefficient as the estimate of the systematic risk. In our numerical example, let us assume the value of ˆβ x = 1.20.

12 Valuation of the future cash-flow Two years from now, we expect to receive a cash-flow of e5000, The two-year risk-free rate is r = 2.565%. The historical average market excess return is R e = 4.000%. The estimated beta of the cash-flow is ˆβ x = The applicable discount rate is k x = 7.365%. p() The present value of the cash-flow is e4315. t = 0 E() k x = r + R e β x = x 0 = e kx T E() = e = 5000 T = = = 7.365%

13 Multiple cash-flows Finance in a nutshell The historical average market excess return is R e = 4.000%. The estimated beta of the multiple cash-flow project is ˆβ x = p( 3 ) p( 2 ) T 2 = 2 p( 1 ) 2 E( 2 ) = 300 r 2 = 2.565% T 1 = 1 k x2 =? 1 E( 1 ) = 300 r 1 =? k x1 =? E( 3 ) = 3300 r 3 =? k x3 =? T 3 = 3 3

14 Estimation of the one-year risk-free rate The holder of a one-year government-issued bond receives back the e1000 principal at time T = 1. The bond currently trades at e975. The value of the discount factor is d f = The one-year risk-free rate is r 1 = 2.532%. p( 1 ) t = 0 x 0 = 975 = d f 1000 d f = x 1 = 1000 T = 1 1 e r 1 1 = r = ln = = 2.532%

15 Estimation of the three-year risk-free rate The holder of a three-year government-issued bond receives back the e1000 principal at time T = 3. p( 3 ) The bond currently trades at e925. The value of the discount factor is d f = e r 3 3 = The three-year risk-free rate is r 3 = 2.599%. t = 0 x 0 = 925 = d f 1000 d f = x 3 = 1000 r = 1 ln = = 2.599% 3 T = 3 3

16 Multiple cash-flows Finance in a nutshell The historical average market excess return is R e = 4.000%. p( 3 ) The estimated beta of the multiple cash-flow project is ˆβ x = The one-year risk-free rate is r = 2.532%. The two-year risk-free rate is r = 2.565%. T 2 = 2 p( 1 ) 2 E( 2 ) = 300 r 2 = 2.565% T 1 = 1 1 E( 1 ) = 300 r 1 = 2.532% x The three-year risk-free rate is r = 2.599%. p( 2 ) Market yield curve x x E( 3 ) = 3300 r 3 = 2.599% T 3 = 3 3

17 Multiple cash-flows Finance in a nutshell The historical average market excess return is R e = 4.000%. p( 3 ) The estimated beta of the multiple cash-flow project is ˆβ x = The one-year discount rate is r = 6.932%. The two-year discount rate is r = 6.965%. p( 2 ) T 2 = 2 p( 1 ) 2 E( 2 ) = 300 r 2 = 2.565% T 1 = 1 k x2 = 6.965% 1 E( 1 ) = 300 The three-year discount rate is r = 6.999%. E( 3 ) = 3300 r 3 = 2.599% k x3 = 6.999% T 3 = 3 r 1 = 2.532% k x1 = 6.932% k x1 = r 1 + R e β x = = = 6.932% k x2 = r 2 + R e β x = = = 6.965% k x3 = r 3 + R e β x = = = 6.999% 3

18 Multiple cash-flows Finance in a nutshell The historical average market excess return is R e = 4.000%. The estimated beta of the multiple cash-flow project is ˆβ x = The present value of the cash-flows is e3216. p( 3 ) p( 2 ) T 2 = 2 p( 1 ) 2 E( 2 ) = 300 r 2 = 2.565% T 1 = 1 k x2 = 6.965% 1 E( 1 ) = 300 r 1 = 2.532% k x1 = 6.932% x 0 = e e e E( 3 ) = 3300 r 3 = 2.599% k x3 = 6.999% T 3 = 3 3

19 Infinite cash-flow Finance in a nutshell We may have a case of an infinite cash-flow. Here we assume that the cash-flow continues infinitely with a constant growth rate of g = 5%. p( 2 ) E( 3 ) = T 2 = 2 p( 1 ) 2 E( 2 ) = = (1 + g)e( 1 ) T 1 = 1 = E( 1 ) = x 0 = E( 1) 1 + a k x + (1 + g)e( 1) (1 + a k x ) 2 + p( 3 ) Continues forever... T 3 = 3 = (1 + g)e( 2 ) = (1 + g) 2 E( 1 ) 3 = We are able to express the present value of the cash-flows in terms of an infinite geometric series, which converges if g a k x. 2 3 (1 + g) E(1 ) (1 + g) E(1 ) (1 + a k x ) 3 + (1 + a k x ) = E( 1) a k x g Notice: The discount rate a k x is here in terms of annual compounding!

20 Infinite cash-flow Finance in a nutshell An infinite cash-flow with a constant growth-rate of g = 5%. The long-term government-issued bond yield is r = 2.700%. p( 3 ) Continues forever... p( 2 ) E( 3 ) = T 2 = 2 p( 1 ) 2 E( 2 ) = = (1 + g)e( 1 ) T 1 = 1 1 E( 1 ) = Market yield curve 2.700% T 3 = 3 = (1 + g) 2 E( 1 ) 3

21 Infinite cash-flow Finance in a nutshell An infinite cash-flow with a constant growth-rate of g = 5%. The long-term government-issued bond yield is r = 2.700%. The historical average market excess return is R e = 4.000%. The estimated beta of the cash-flow is ˆβ x = p( 2 ) E( 3 ) = T 2 = 2 p( 1 ) 2 E( 2 ) = = (1 + g)e( 1 ) T 1 = 1 k x = r + R e β x 1 E( 1 ) = a kx = e = = 7.573% The applicable discount rate is a k x = 7.573%. The present value of the cash-flows is e3886. x 0 = E( 1) 1 + a k x + (1 + g)e( 1) (1 + a k x ) 2 + p( 3 ) 2 3 (1 + g) E(1 ) (1 + g) E(1 ) (1 + a k x ) 3 + (1 + a k x ) = E( 1) a k x g Continues forever... T 3 = 3 = (1 + g) 2 E( 1 ) = = =

22 Valuation of a future asset price So far, we have implicitly assumed a cash-flow(s) to follow a normal distribution. Alternatively, we may assume a variable to follow a lognormal distribution. We are still allowed to value the asset in terms of the discounted expected value. p() t = 0 k x = r + R e β x x 0 = e kx T E() 0 E() T

23 Valuation of a future asset price Two years from now, we expect the asset price to be e25.00 The two-year risk-free rate is r = 2.565%. The historical average market excess return is R e = 4.000%. The estimated beta of the asset is ˆβ x = The applicable discount rate is k x = 6.165%. p() The current asset price should be e t = 0 x 0 = e kx T E() = e T = 2 0 E() = k x = r + R e β x = = = 6.165%

24 Infinite dividend-stream An infinite dividend-stream with a constant growth-rate of g = 4%. The long-term government-issued bond yield is r = 2.700%. The historical average market excess return is R e = 4.000%. The estimated beta of the stock is ˆβ x = p( 2 ) T 2 = 2 p( 1 ) 0 2 E( 2 ) = 2.08 = (1 + g)e( 1 ) T 1 = 1 k x = r + R e β x 0 1 E( 1 ) = 2.00 a kx = e = = 6.503% The applicable discount rate is a k x = 6.503%. The present value of the dividends is e x 0 = E( 1) 1 + a k x + (1 + g)e( 1) (1 + a k x ) 2 + p( 3 ) 2 3 (1 + g) E(1 ) (1 + g) E(1 ) (1 + a k x ) 3 + (1 + a k x ) = E( 1) a k x g Continues forever... T 3 = E( 3 ) = = (1 + g) 2 E( 1 ) = = =

25 Corporate bond with default risk A 1000-euro, 3-year corporate bond, with an annual 4% coupon. p( 3 ) The risk-profile of the bond corresponds to that of the BB-rated bonds. 1-year BB-bonds trade at the yield of 4.157%. 2-year BB-bonds trade at the yield of 4.185%. Default risk 3-year BB-bonds trade at the yield of 4.234%. p( 2 ) T 2 = 2 Default risk p( 1 ) 2 x 2 = 40 y 2 = 4.185% T 1 = 1 Default risk 1 x 1 = 40 y 1 = 4.157% x x x The value of the bond is e991. x 0 = e e e x 3 = 1040 y 3 = 4.234% BB-rated bond yield curve T 3 = 3 3

26 Valuation of an option contract* In derivatives pricing, we assume the risk-neutral price of an underlying asset to follow a lognormal distribution. A European call option on the asset pays off an amount of max(0, X ) at the maturity T of the contract. X is a contracted strike price. We draw the payoff max(0, X ) in the same coordinate system, but with a different vertical axis X. p() p() max(0, X ) 0 X X

27 Valuation of an option contract* In derivatives pricing, we assume the risk-neutral price of an underlying asset to follow a lognormal distribution. A European call option on the asset pays off an amount of max(0, X ) at the maturity T of the contract. X is a contracted strike price. We draw the probability-weighted payoff p()max(0, X ) in the same coordinate system, but with a third vertical axis. p() p() max(0, X ) 0 X X p()max(0, X ) p()max(0, X ) We are interested in the probability-weighted payoff p()max(0, X ) of the contract.

28 Valuation of an option contract* In derivatives pricing, we assume the risk-neutral price of an underlying asset to follow a lognormal distribution. A European call option on the asset pays off an amount of max(0, X ) at the maturity T of the contract. X is a contracted strike price. p()max(0, X ) The expected risk-neutral call option payoff equals to the sum of all probability-weighted option payoffs: T E[max(0, X )] = p()max(0, X )d 0 X X The risk-neutral expected payoff E[max(0, X )] is discounted by the risk-free rate r to get the value c 0 of the option. t = 0 c 0 = e rt E[max(0, X )] = e rt p()max(0, X )d X

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