Investment analysis models so far have focused on individual investors and their preferences for maximising returns while minimising risk. Asset pricing models – also called equilibrium models – take a broader view of the market by examining the attitudes of investors as a whole. Asset pricing models assume that the capital market, i.e., the market for long-term securities, is perfect. A perfect market implies that the market is ‘in equilibrium’, i.e., demand equals supply and each asset has a single market-clearing price. If an asset’s price is temporarily low/high, investors will buy/sell the asset until the forces of supply and demand restore equilibrium.
The best-known asset pricing model is William Sharpe’s capital asset pricing model (CAPM) published in 1964. To simplify the complexity of capital markets, the CAPM assumes that ev- eryone has equal access to the same information and can borrow or lend any amount of money at a risk-free interest rate. It is therefore concluded that all investors will make the same decisions and create identical portfolios using the Markowitz model. Today, such theoretical assumptions are regarded as being too restrictive and unrepresentative of real-world situations, and more realistic asset pricing models have subsequently been developed. The five assumptions are: 1. All assets are marketable. Marketable assets are much easier to quantify than non-marketable
assets such as goodwill, patents, and copyrights.
2. The capital market is perfect, i.e., (i) there are no transaction or tax costs (ii) assets are infinitely divisible, i.e., there is no restriction on the size of the amount to be invested (iii) the same information is freely available to every investor (iv) no single investor can influence the market by buying or selling actions.
3. A risk-free interest rate exists; all investors can borrow or lend any amount at this fixed risk-free interest rate.
4. All investors are risk-averse and seek to maximise expected portfolio returns.
5. All investors have homogeneous expectations, i.e., they make identical portfolio decisions, using the Markowitz model over the same investment horizon.
The Capital Market Line (CML)
The models that were developed earlier in this chapter contained only risky securities. The third CAPM assumption adds a risk-free (riskless) asset to the portfolio. Investors can now choose a set of risky assets (e.g., stocks) as well as a risk-free asset (e.g., fixed-term bank deposit). Consider a portfolio P which consists of risky assets and one risk-free asset with an interest rate of Rf. The expected portfolio return E(Rp) is then
E(Rp)= x E(Ry)+ (1 − x)Rf
where: x= the percentage of the portfolio invested in risky assets (1− x) = the percentage of the portfolio invested in the risk-free asset
E(Ry)= the expected return of the risky-asset portfolio
Assumption 4 states that all investors are risk-averse and seek to maximise their expected portfolio returns, i.e., they prefer to invest in optimal portfolios lying on the efficient frontier. Because all investors make identical portfolio decisions (assumption 5), they will all derive the same efficient frontier of Markowitz. At this stage, all investors will hold their risky assets in the same proportions regardless of their risk preferences. These optimal proportions (or percentages) constitute the ‘market portfolio’, denoted by M in Figure 4.9. The market portfolio, M, is the optimal portfolio of risky assets. It should be noted that the derivation of suitable values for the expected return on the market portfolio, E(RM), and the risk-free interest
rate, Rf, can be problematic.
Because the market is in equilibrium (assumption 2), the market portfolio must include all assets in the market. If an asset was not included in the market portfolio, i.e., implying that no one wanted to buy it, then total demand would not equal total supply. The proportion of each asset, wi, in the market portfolio is given by
wi =
market value of asset i
total market value of all assets in the market
The capital market line (CML) is defined as the line that (i) has an intercept of Rfon the vertical
axis and (ii) is tangential to the efficient frontier of Markowitz at point M (Diacogiannis, Chapter 9). All investors lie on the CML as shown in Figure 4.9. Whatever part of the line they occupy will depend upon investors’ risk preference attitudes. They must decide how much to invest in (i) the risky-asset portfolio M and (ii) the risk-free asset, i.e., they must find a value for x as defined in the E(Rp) equation above.
Very risk-averse investors will put most of their wealth into the risk-free asset while leaving a small percentage in the market portfolio, M. This means that they would choose a point on the line to the left of M such as C. On the other hand, less risk-averse investors will borrow in excess of their current wealth in order to increase their holdings in portfolio M. In this case, they would choose a point to the right of M, e.g., L. At point M, investors put all their wealth into the market portfolio and neither borrow nor lend at the risk-free interest rate.
M E(RP) L Efficient frontier M Slope of CML = a/b C = (E(RM) − Standard deviation, σ E(RM) a b Rf Rf)/σ σ M
EXAMPLE 4.7 Using CAPM to calculate beta and expected return values
Sharpe’s original CAPM model is usually written as
E(Ri)= Rf+ βi[E(RM)− Rf]
where: E(Ri)= the expected return on asset i
Rf= the risk-free rate
E(RM)= the expected return on the market
βi= beta coefficient (or beta) of asset i
The beta coefficient for asset i is defined as
βi = σiM/σM 2
where: σiM= the covariance between asset and market returns Riand RM
σM2= the market variance, i.e., VAR(RM).
The riskless interest rate, Rf, and the ‘market risk premium’ defined as [E(RM)− Rf], are the
same for all assets. However,βi is different for each asset because of the unique covariance
termσiM. It is therefore a measure of the systematic risk of asset i. Beta coefficient,βi , is
usually defined as ‘a measure of the responsiveness of a security or portfolio to the market as a whole’. Assets withβi < 1 are called defensive while those with βi> 1 are aggressive
assets. Whereβi = 1, the asset or security has the same risk as the market portfolio. Data for
market and company share returns given in Table 4.4 are used as input to the CAPM model of Figure 4.10. The risk-free interest rate, Rf, is taken as 4% and the expected market return,
E(RM), as 8%.
The security market line (SML)
The security market line (SML) shown in Figure 4.11 has been constructed using details from Example 4.7. The SML is derived by plotting expected return, E(Ri), along the vertical axis
andβivalues along the horizontal axis, i.e., replacing theσ -values of Figure 4.9 with βi. Since
the expected return of the market portfolio, E(RM) , representsβ = 1 and the vertical-axis
intercept is the risk-free interest rate, Rf, drawing the SML is a straightforward exercise.