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1. Capital Asset Pricing Model
Capital Asset Pricing Model (CAPM) is one of the theoretical cornerstones in corporate finance. It is based on the modern portfolio theory developed by Markowitz (1952) and demonstrates the relationship between the expected return on a security and the return on the market. The model was independently created by Treynor (1961), Lintner (1965), Sharpe (1964) and Mossin (1966).
The model has found implication in the valuation of investment projects and financial portfolio management. In mathematical terms, the relationship indicated by the CAPM is expressed as follows:
E(Ri) = Rf + β(E(Rm) – Rf) (Bodie et al, 2009; Brealey and Myers, 2003),
Where E(R¬i) is the expected return on an i-th security; Rf is the risk free rate which is often represented by the Treasury bonds or Treasury bills yield; E(Rm) is the expected return on the market, which is often represented by a large stock market index such as S&P 500 and others; beta is the indicator of how sensitive the returns on a security are in relation to the returns on the market. Coefficient beta can also be calculated as follows:
β = cov (Ri, Rm)/var (Rm)
While the model has become popular in corporate finance, its usefulness has been doubted by a number of researchers such as Roll (1977), Shefrin and Statman (2000) and others. The main critiques of the model arise from the following weak assumptions that promoters of CAPM use:
• Normal distribution of security returns. Financial time series have volatility clustering which implies a succession of the periods of high volatility by periods of lower volatility. Furthermore, skewness of distribution is a common pattern in asset returns especially for those that followed a specific constant trend. Well performing stocks would have their returns skewed to the right while the stocks that have performed poorly would have the returns skewed to the left.
• The model assumes that risk can be well represented by standard deviation of returns. Once again, it is valid to argue that such an assumption could be meaningful for a normally distributed time series. Furthermore, there are arguments that other moments besides standard deviation could be used to measure the risk of a security (Shefrin and Statman, 2000).
• Roll (1977) criticized CAPM for suggesting an unrealistic and practically unobservable market returns. CAPM could have been valid if it was possible to estimate the return on all possible investment alternatives that would include not only the equity market but real estate, wines, art, commodities and others. In practice, the stock market index is used as a proxy for market returns which is a great weakness.
There are much more critiques of the model that have been expressed in literature. For this reason, Black, Jensen and Scholes (1972) suggest a number of tests that can be undertaken to evaluate CAPM.
CAPM demonstrates that the expected returns on a security are linearly associated with the security’s beta and are directly proportional to the latter. Traditional tests conducted before Black, Jensen and Scholes (1972) implied running a linear regression to estimate a security beta and running a cross sectional regression to estimate whether the difference between the expected returns and the right hand part of the equation (1) is significantly different from 0 (Lintner, 1965). Black, Jensen and Scholes (1972) argue that such tests may provide misleading results.
Basically, in order to test the applicability of CAPM to real market, one has to collect the returns on all securities traded in a particular market and study the relations between the past returns and the securities betas over time. Black, Jensen and Scholes (1972) investigated all stocks traded on New York Stock Exchange (NYSE). The model (1) that is supposed to be tested should be re-stated as follows in order to run linear regressions.
E(Ri) – Rf = β(E(Rm) – Rf) (2)
This transformation implied deducting the risk free rate from both sides of the equation (1). This does not change the final outcome of the model. Now, according to the traditional CAPM the difference between the expected excess returns on a security and the market risk premium multiplied by the security’s beta should be zero.
γ0 = (E(Ri) – Rf) – β(E(Rm) – Rf)
H0: γ0 = 0 (3)
(3) is the hypothesis that is supposed to be tested. If H0 is true, then CAPM can be accepted as a valid model.
The testing conducted by Lintner (1965) could be criticized for the variability of beta in the sample. Black, Jensen and Scholes (1972) have solved this problem by comprising ten portfolios of all stocks traded on NYSE. The variability of portfolios beta reduced. This would promise more accurate results of the regression that can be expressed by the following equation:
E(Ri) – Rf = γ0 + β(E(Rm) – Rf) (4)
If the intercept γ0 is not found to be statistically different from zero, then the hypothesis that CAPM is valid can be accepted. However, Black, Jensen and Scholes (1972) revealed that the intercept tended to be negative for the assets with high beat and positive for the assets with low beta. Cross sectional regression is then supposed to be run to check the relations between the average excess returns and the estimated betas. The intercept is supposed to be zero if the CAPM holds true and there should be not much variability in the slope of the regression. The tests conducted by Black, Jensen and Scholes (1972) demonstrate that there was high variability in the slope of the regression. In some periods, the slope was found to be steeper than it was predicted by CAPM. Furthermore, the intercept was also found to greatly deviate from zero. Their study can be used as a guide but it is valid to argue that in order to run contemporary tests of CAPM, one should collect more recent data (the data studied by Black, Jensen and Scholes (1972) had the most recent observation dated back to 1966). Furthermore, their methodology does not allow researchers to avoid another limitation, i.e. dealing with the periods of crisis. It can be expected that in the periods of crises stock returns will not behave as they do in normal conditions. Therefore, this problem should be accounted for when making tests of CAPM.
2. Arbitrage Pricing Theory
Arbitrage Pricing Theory (APT) is another popular model used for estimating expected returns on securities. In contrast to CAPM, it has fewer restrictions and limitations. Furthermore, APT implies several betas (Ross, 1976).
APT demonstrates that the expected returns on a security are a linear function of economic factors. Security prices are assumed to be endogenous to the macroeconomic influences. However, it is valid to argue that there could be a bi-directional causality between the stock prices and some of the macroeconomic factors. In mathematical terms, APT is expressed as follows:
Ri = E(Ri) + β1F1 + β2F2 + … + βnF¬n (5),
Where Ri is the actual return on an i-th security; E(Ri) is the expected return on the i-th security; F1…n are the macroeconomic factors that influence the price of the security; beta is the measure of sensitivity of the security returns to the macroeconomic factors (Brealey and Myers, 2003).
It is interesting to note that CAPM could be considered as a specific case of APT if the return on the market is viewed as the macroeconomic factor that affects the returns of the analysed security. So, CAPM had one beta while APT has multiple betas which are equal to the number of factors in the model (Roll and Ross, 1980).
APT has received its name because the model suggests that due to arbitrage the asset prices will be returning to equilibrium. Arbitrage is the process of buying underpriced securities and selling overpriced securities where riskless profit can be expected. If due to the unexpected change of one factor, the asset price becomes mispriced, this mispricing will be resolved by the arbitrage among investors (Burmeister and Wall, 1986).
APT can be tested by findings statistically significant macroeconomic factors that have affected the returns of the security. Such testing was performed by Chen, Roll and Ross (1986).
The original choice of macroeconomic factors should be made theoretically. Chen, Roll and Ross (1986) chose to investigate industrial production index, inflation, terms structure of interest rates, consumption, oil prices, bonds with low rating, government bonds, treasury bills and others. However, it is valid to argue that these variables should not be used in their level form because the APT implies that unexpected changes in macroeconomic factors affect the price and return of an equity security. So, transformations have to be made in order to find these unexpected changes. For example, Chen, Roll and Ross (1986) use the difference between the actual consumer price index growth rate and expected inflation as estimated by Fama and Gibbons (1984) in order to find the unexpected changes in inflation. They also use the risk premium on low rate bonds as the unexpected element. This is found by deducting the yield of the long term government bonds from the yield of the risky bonds.
Equation for the first regression would be as follows:
Ri = α + β1F1 + β2F2 + … + βnF¬n (6)
This regression is run on a limited sample of observations in order to estimate the beta of each factor. The estimated beta are then used as independent variables in a cross sectional regression. The security returns are used as dependent variables in the cross sectional regression. The coefficients estimated using the cross sectional regressions provide the risk premium. These steps are then repeated for different periods in order to create the time series of risk premiums for each asset. The mean of these time series is then subject to t-test which tests for statistical significance of the estimates (Chen, Roll and Ross, 1986). In order to avoid the problem of estimates’ bias, Chen, Roll and Ross (1986) recommend grouping equity securities into portfolios in a manner that was done by Black, Jensen and Scholes (1972) who tested CAPM.
In this testing t-test plays the crucial role in determining statistical significance of the factors. Only those factors that are found to be statistically significant can be used in APT. While the methodology seems simple, it is valid to argue that testing of APT with the method suggested by Chen, Roll and Ross (1986) can generate unviable results if the chosen macroeconomic factors appear to be highly correlated among themselves. In the economy, financial and economic time series can be both endogenous and exogenous. There are rarely purely endogenous or purely exogenous factors. Hence, solving the problem of multicollinearity can be recommended as one of the key elements of effective testing of the APT.
3. European Call and Put Pricing
The pricing of European call and put options are conducted using Black and Scholes (1973) formula. The pricing of the call options would be done as follows:
(7)
Where S is the today’s price of the underlying security; K is the exercise price of the option; (T-t) is the period of maturity; N is the cumulative distribution function; d1 and d2 are estimated as follows:
(8)
(9)
Where r is the risk free rate; σ is standard deviation.
The price of the European put call is estimated using the following formula that is derived from the put-call parity that should hold in the option pricing:
However, these formulas are useful for the cases of no-dividend securities. If a security has a continuous dividend yield, the spot price of the underlying asset should be corrected for the dividend.
Since the dividend is paid before the option expires, the present value of the dividend payment should be deducted from the spot price. If the continuous dividend yield is equal to q, then the present value of the dividend payment would be S(1-e-q(T-t)). The corrected today’s price would be as follows:
S* = S - S(1-e-q(T-t)) = Se-q(T-t)
(T-t) = 3 months = 0.25 years
S = 0.98
q = 3% = 0.03
e = 2.72
S* = 0.98 – 0.98 (1 – 2.72-0.0075) = 0.98 – 0.98 (1 – 0.9925) = 0.973
Now S* will replace S in the equation (7).
d 1 = ((ln(0.973/1.04)+(0.0225+0.182/2))/18×sqrt(0.25) = -0.159
d 2 = -0.159 – 0.18×sqrt(0.25) = -0.249
C(S*,t) = 0.973×N(-0.159) – 1.04×2.72-0.0056×N(-0.249) = 0.0095
P(S*,t) = 2.72-0.0056 – 0.973 + 0.0095 = 0.07
4. Delta, Gamma and Theta in Option Pricing
Delta, Gamma and Theta are the Greeks that are widely used in option evaluation. Delta of an option is the hedge ratio that shows how much an option’s price will increase if the underlying security’s price increases by £1. For a call option, delta is estimated as follows:
Δ = N(d1) (Bodie et al, 2009)
Given the numbers used in the estimating the value of the call option in part 3, N (d1) is equal to 0.44 or 44%. This is the part of the area under the curve of normal distribution up to the value d1.
Delta of a put option is calculated as follows:
Δ = N(d1) – 1
Δ = - 0.56 or - 56%.
These figures imply that if the underlying security increases in value by £1, the price of the call option will increase by £0.44 and the price of the put option will decrease by £0.56.
If delta showed how sensitive the price of the option is to the movements in the value of the underlying asset, gamma indicates how sensitive the delta of an option is to the movements of the price of the underlying asset. It is estimated for both call and put options as follows:
Γ = N`(d1)/Sσ×sqrt(T-t) (Hull, 2009)
N`(d1) is the first derivative of the cumulative distribution function, which is equal to the probability density function that can be expressed as follows:
N`(d1) = (1-(-0.159)2/2)/sqrt(2×3.14) = 0.39
Γ = 0.39 / 0.97×0.18×sqrt(0.25) = 4.5
Finally, theta is the Greek that indicates the sensitivity of the value of an option to the time to maturity. The shorter the time is the lower the price of the option is. For a call option theta is calculated as follows:
Theta (call) = -SN`(d1)σ/2sqrt(T-t) – rKe-r(T-t)N(d2) (Hull, 2009)
Theta (call) = - 0.97×0.39×0.18/2×sqrt(0.25) – 0.0225×1.03×0.4 = -0.078
For a put option theta is calculated as follows:
Theta (put) = -SN`(d1)σ/2sqrt(T-t) + rKe-r(T-t)N(-d2) (Bodie et al, 2009)
Theta (put) = -0.97×0.39×0.18/2×sqrt(0.25) + 0.0225×1.03×0.6 = -0.083
These figures show that both call and put option will lose their value with time.
References
Black, F., Jensen, M. and Scholes, M. (1972) “The Capital Asset Pricing Model: Some Empirical Tests”, Working Paper
Black, F. and Scholes, M. (1973) “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, Vol. 81, No. 3, pp. 637–654
Bodie, Z., Kane, A. and Marcus, A. (2009) Investments, 8th ed., New York: McGraw Hill
Brealey, R. and Myers, S. (2003) Principles of Corporate Finance, New York: McGraw Hill
Burmeister, E. and Wall, K. (1986) “The arbitrage pricing theory and macroeconomic factor measures”, Financial Review, Vol. 21, No. 1, pp. 1–20
Chen, N., Roll, R. and Ross, S. (1986) “Economic Forces and the Stock Market”, Journal of Business, Vol. 59, No. 3, pp. 383–403
Fama, E. and Gibbons, M. (1984) “A Comparison of Inflation Forecasts”, Journal of Monetary Economics, Vol. 13, pp. 327-348
Hull, J. (2009) Options, Futures and Other Derivatives, London: Prentice Hall
Lintner, J. (1965) “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets”, Review of Economics and Statistics, Vol. 47, No.1, pp 13-37
Markowitz, H.M. (1952) “Portfolio Selection”, The Journal of Finance, Vol. 7, No. 1, pp. 77–91
Mossin, J. (1966) “Equilibrium in a Capital Asset Market, Econometrica”, Vol. 34, No. 4, pp. 768–783
Roll, R. (1977) “A Critique of the Asset Pricing Theory’s Tests,” Journal of Financial Economics, Vol.4, pp. 129–176
Roll, R. and Ross, S. (1980) “An empirical investigation of the arbitrage pricing theory”, Journal of Finance, Vol. 35, No. 5, pp. 1073–1103
Ross, S. (1976) “The arbitrage theory of capital asset pricing”, Journal of Economic Theory, Vol. 13, No. 3, pp. 341–360
Sharpe, W. (1964) “Capital asset prices: A theory of market equilibrium under conditions of risk”, Journal of Finance, Vol. 19, No. 3, pp. 425-442
Shefrin, H. and Statman, M. (2000) “Behavioral Portfolio Theory,” Journal of Financial and Quantitative Analysis, Vol. 35, No. 2, pp. 127–151
Treynor, J. (1961) Market Value, Time, and Risk, Unpublished manuscript
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