Fundamentals of Finance

Transcription

1 University of Oulu - Department of Finance Fall 2015

2 What is it about in finance? Finance is basically nothing else but estimating expected future payoffs, and discounting them with an appropriate discount factor. Future payoff is a random variable, describable with an appropriate probability distribution, the estimated expected value of which is typically discounted. Discount factor may be provided in terms of a percentage discount rate, which captures the time-value of money, and includes a risk premium for the uncertainty in the payoff.

3 Probability distribution of the future payoff Lognormal distribution of price Normal distribution of log-return/cash flow 0 0 Probability distribution typically lognormal distribution or normal distribution, descibres the payoff at a certain time in future, is determined on the basis of currently available information, provides expected value of the payoff for further analyses, whereas the variability is captured into the discount rate.

4 Discount factor Discount factor d T discounts a future value to today s value behind a period of T years, expressed in terms of a discretely (typically annually) compounded annual percentage discount rate k, or in terms of a continuously compounded annual discount rate k c. d T = 1 (1 + k) T d T = e kc T Discount rate k consists of the time-value of money i.e. the risk-free rate r, the percentage risk premium p. k = r + p Risk premium p consists of the price of market risk, i.e. the expected market excess return e R m = E(R m) r, the amount of market risk, i.e. the market beta β, k = r + p = r + e R mβ = r + [E(R m) r]β

5 Risk-free rate The risk-free rate may be proxied with the current yield of government bonds, varies with the time-to-maturity in question, but is typically replaced with a flat rate, the current yield of long-term government bonds, for instance. The current yield of a bond is a constand discount rate, which equalizes the present value of future payments, with the current market price of the bond, representing the internal rate of return of the bond.

6 Proxying the risk-free rate We are here! } {{ } The current yield of long-term government bonds y = 0.03 k = r + p = r + e R mβ

7 Price of market risk The price of market risk the reward paid by the market on an average-risk security, may be proxied with the historical equity market premium, the historical average of equity market excess return over government bonds. The return of a bond is the periodic percentage return, consisting of the periodic coupon payments, and the periodic change in the market price of the bond.

8 Proxying the price of market risk We are here! Market return % 36.6% +25.9% +14.8% +2.1% +15.8% Bond return % +20.1% 11.1% +8.4% +16.1% +2.9% Excess return % 56.7% +37.0% +6.4% 14.0% +12.9% } {{ } } {{ } The current yield of long-term government bonds y = 3.0% Long-term historical geometric average of market excess returns e R m = 4.2% k = r + p = r + e R mβ

9 Amount of market risk The amount of market risk the sensitivity of the asset return to the market return, may be proxied with the asset s estimated beta coefficient, representing undiversifiable part in the asset s uncertainty. The beta coefficient the covariance of the asset return with the market return, which is standardized by the variance of the market return, the beta value becoming equal to one for the market risk itself.

10 Proxying the amount of market risk Historical annual returns { }} { Market return % 36.6% +25.9% +14.8% +2.1% +15.8% Bond return % +20.1% 11.1% +8.4% +16.1% +2.9% Excess return % 56.7% +37.0% +6.4% 14.0% +12.9% } {{ } We are here! Long-term historical geometric average of market excess returns e R m = 4.2% k = r + p = r + e R mβ R t = ˆα + ˆβR mt + ɛ t } {{ } The current yield of long-term government bonds y = 3.0% Asset return Market return { }} { } {{ } Historical daily/monthly returns

11 To summarize Risk free rate, r long-term government bond yield for long-horizon evaluations, short-term interest rate for short-horizon evaluations, current yield/rate is a direct future period estimate. Price of market risk, e R m = E(R m) r direct future period estimate is not available, expected market return is replaced with realized returns, bond yield is replaced with realized bond returns. Amount of market risk, β the only asset-specific component in the discount rate, may be estimated from historical returns, if available.

12 Capital investment project current long-term government bond yield is 3% historical market excess return is 5% estimated beta of the project is 1.2 applicable discount rate is k = = 0.09 annual compounding is applied Time in years Initial investment 1000 Expected periodic net cash flow Expected terminal value 100 Total cash flow Discount factor Present value of cash flow Net present value of the project is 36

13 Corporate bond AA-rated 5-year corporate bond with a 5% annual coupon discount rates from the zero-coupon yield curve of AA-rated reference bonds continuous compounding is applied Time in years Coupon payment Face value 1000 Total cash flow Reference bond yields 6.2% 6.6% 6.9% 7.1% 7.2% Discount factor Present value of cash flow Bond value is 903

14 Common stock current long-term government bond yield is 3% historical market excess return is 5% estimated beta of the stock is 1.4 applicable discount rate is k = = 0.10 current dividend is 2 euros expected annual dividend groth rate is 4% annual compounding is applied Time in years Expected dividend payment Discount factor Present value of cash flow An infinite horizon problem...

15 Constant growth model Payoff stream in terms of a convergent infinite geometric series P 0 is the present value of the expected future payoffs, D 0 is the current, already paid, payoff, g is a constant growth rate for the payoff stream (g < k). P 0 = t=1 (1 + g) t D 0 (1 + k) t = (1 + g)d 0 + (1 + g)2 3 D 0 (1 + g) D0 1 + k (1 + k) 2 + (1 + k) S = aq k = a + aq + aq = a 1 q k=0 a = (1 + g)d k q = 1 + g 1 + k P 0 = a 1 q = (1 + g)d 0 k g = D 1 k g

16 Common stock revisited applicable discount rate is k = = 0.10 current dividend is 2 euros expected annual dividend groth rate is 4% annual compounding is applied Time in years Expected dividend payment Discount factor Present value of cash flow P 0 = D 1 k g =

17 Firm valuation current long-term government bond yield is 3% historical market excess return is 5% estimated unlevered asset beta of the firm is 0.8 applicable discount rate is k = = 0.07 non-constant growth in unlevered free cash flow for two years constant 3% growth thereafter annual compounding is applied Time in years Expected unlevered free cash flow (1145) Growth rate 9.2% 11.6% 3.0% 3.0% 3.0% Discount factor Present value of cash flow V = 33765

18 Forward contract current continuously compounded three-month risk-free rate is 2% current asset price is 12 euros continuously compounded expected asset return is 10% delivery price of the forward contract is 11 euros forward contract expires in three months Time in months Expected asset price Expected risk-neutral asset price Expected risk neutral forward contract payoff 1.06 Discount factor Present value of payoff 1.05 Forward contract value is 1.05

19 Option contract current continuously compounded three-month risk-free rate is 2% current asset price is 12 euros continuously compounded expected asset return is 10% asset price volatility is 40% exercise price of the European call option contract is 10 euros option contract expires in three months Time in months Expected asset price Expected risk-neutral asset price Expected risk neutral option contract payoff 2.26 Discount factor Present value of payoff 2.25 European call option contract value is 2.25

20 Summary In financial applications future payoff is described with a probability distribution, expected payoff is converted to today s euros by discounting, payoff s uncertainty is taken into account in the risk premium, risk premium consists of the price and the amount of uncertainty, discount rate consists of the risk-free rate and the risk premium, rates can be expressed with different compounding frequencies, discount factor is a valuation multiplier based on the discount rate.