Chapter 8: CAPM. 1. Single Index Model. 2. Adding a Riskless Asset. 3. The Capital Market Line 4. CAPM. 5. The One-Fund Theorem

Transcription

1 Chapter 8: CAPM 1. Single Index Model 2. Adding a Riskless Asset 3. The Capital Market Line 4. CAPM 5. The One-Fund Theorem 6. The Characteristic Line 7. The Pricing Model

2 Single Index Model 1 1. Covariance matrix of a portfolio requires 1 2n(n + 1) parameters. 2. This needs to be simplified. 3. We postulate a linear model for the random return of each asset: R i = a i + β i R m i = 1, 2,..., n, where a i, R m are random variables while the β i s are not. R m is the random return on the market portfolio which is denoted P m

3 Single Index Model 2 The market portfolio is made up of all assets weighted by their market capitalisation. Thus if the market contains n i (oustanding) shares of asset-i with current price π i, the portfolio weights for P m are given by w i = n i π i T market capitalisation of all assets. where T = n i=1 n iπ i denotes the total

4 Single Index Model 3 1. Return now to the linear model: R i = a i + β i R m 2. a i is the component of R i that is independent of the market 3. β i is a constant that determines how R i varies relative to changes in the market return R m 4. Because a i is random, we may write it in the form a i = α i + ɛ i where α i = E (a i ) and E (ɛ i ) = 0 5. Also assume that ɛ i are uncorrelated with the other ɛ j and also uncorrelated with R m

5 Single Index Model 4 1. We see that our linear model may be written as R i = α i + β i R m + ɛ i with cov(ɛ i, ɛ j ) = 0 when i j and cov(ɛ i, R m ) = If we let r i = E(R i ), r m = E(R m ), σ 2 i = V(R i ) and σ 2 ɛi = V(ɛ i ) and σ ij = cov(r i, R j ) = β i β j σ 2 m r i = α i + β i r m σ 2 i = β 2 i σ 2 m + σ 2 ɛi

6 Single Index Model 5 1. β i σ m is called the systematic risk. 2. σ ɛi is called the non-systematic risk. 3. In a well-diversified portfolio the non-systematic risk can be made negligible compared to the systematic risk. 4. Eg, for a portfolio of risky assets with weights x i we obtain µ P = α P + β P r m ; α P = i x i α i ; β P = i x i β i and σ 2 P = β2 P σ2 m + i x 2 i σ 2 ɛi

7 Single Index Model 6 1. When the portfolio is well-diversified we have approximately x i 1/n for each i and the portfolio variance becomes σ 2 P β2 P σ2 m + 1 n σ2 ɛ; σ 2 ɛ = 1 n i σ 2 ɛi 2. for large values of n, 1 n σ2 ɛ 0 so it may be ignored. 3. We see that for a well-diversified portfolio σ P β P σ m

8 Single Index Model 8 [BLACKBOARD: estimation of betas]

9 Adding a Riskless Asset 1 1. CAPM is an extension of the Portfolio Theory we discussed earlier. 2. The key feature of CAPM is that it assumes the existence of a riskless asset in the market. 3. The presence of the riskless asset leads to a degenerate efficient frontier. It becomes a straight line instead of a hyperbola in the µσ-plane. It is called the Capital Market Line.

10 Adding a Riskless Asset 2 1. We consider a portfolio of n risky assets S 1, S 2,..., S n and one riskless asset S Assume that the return on S 0 is r S 0 is riskless, so E(R 0 ) = R 0 = r 0 V(R 0 ) = 0 4. Also cov(r 0, R i ) = 0 i = 0, 1, 2,..., n

11 Adding a Riskless Asset 3 The covariance matrix for the corresponding (n + 1)-asset portfolio has structure s Ŝ = s 1n s n1... s nn

12 Adding a Riskless Asset 4 1. Obviously Ŝ does not have an inverse. Hence we cannot apply the portfolio analysis of the previous section. Therefore it is degenerate. But a solution is still available. 2. Let x, S, e and r denote the same quantities defined in the previous Chapter for the case of n-risky assets. 3. We will use the notation µ, σ to denote quantities which include the riskless asset. 4. Let x 0 denote the amount allocated to the riskless asset. Then The budget constraint becomes x ê = x 0 + x e = 1 so x 0 = 1 x e

13 Adding a Riskless Asset 5 1. The return on the portfolio can be expressed as µ = x r. Equivalently µ = r 0 + x r and r = r r 0 e 2. Finally, the presence of the riskless asset cannot affect the portfolio variance and so σ 2 = x Ŝ x = x Sx = σ 2

14 Adding a Riskless Asset 6 1. The EF for the (n + 1)-asset portfolio is therefore the solution to the unconstrained optimisation problem: min Z(x) = t µ σ2 = t(r 0 + x r) x Sx 2. NOTE: the budget constraint is automatically satisfied (since we have removed the dependence of the problem on x 0.) Therefore, we minimise Z with respect to the risky allocations x only. 3. The matrix S is the corresponding covariance matrix of the risky assets and it is non-singular.

15 Adding a Riskless Asset 7 1. As previously, the critical line (which maps to the EF in the µ σ-plane) is given parametrically by a vector equation x = x(t) for t For the minimum we must have Z x = Sx tr = 0

16 Adding a Riskless Asset 8 Then, since S is non-singular x = t(s 1 r) is the required solution, giving µ = r 0 + t(r S 1 r) = r 0 + ct σ = t 2 (r S 1 r) = ct 2 = ( µ r 0 )t.

17 Adding a Riskless Asset 9 1. The common term c = r S 1 r can be evaluated in terms of known quantities as c = r S 1 r = (r r 0 e) S 1 (r r 0 e) = ar0 2 2br 0 + c, ( ( a = d r 0 b ) ) d a a where a, b, c are as given in Chapter 7 = dσ 2 0,

18 Adding a Riskless Asset 10 σ 2 0 is the portfolio variance on the EF for the risky assets only, corresponding to a portfolio return of r 0, i.e. σ 2 0 = ar 2 0 2br 0 + c d = a d (r 0 b a )2 + 1 a

19 Adding a Riskless Asset 11 The Critical Line 1. From the previous x = t(s 1 r). 2. This the vector equation of a line in n-dimensional x space. 3. It passes through the origin x = 0 at t = 0.

20 Adding a Riskless Asset 12 The EF (Capital Market Line) Using the above we find σ 2 = dσ 2 0 t2 = ( µ r 0 ) 2 /dσ 2 0 and since d > 0 we obtain σ = µ r 0 σ 0 d This straight line in the µ σ-plane is the degenerate form of the efficient frontier, when one of the assets is riskless. In this context, it is called the Capital Market Line or CML for short.

21 Adding a Riskless Asset 13 See notes, p. 112

22 CAPM 1 1. Let x be the allocation vector for any feasible portfolio (but not necessarily efficient). 2. Let x m denote the allocation vector for the market portfolio M (containing only risky assets). cov(r, R m ) = x Sx m = t m x r = t m ( µ r 0 ) ( µ r0 ) = σm 2 µ m r 0 3. where µ = E(R) and µ m = E(R m ) and we used that σm 2 = t m ( µ m r 0 ))

23 CAPM 2 We therefore get µ r 0 = β( µ m r 0 ) or E(R) = r 0 + β(e(r m ) r 0 ) (1) where β = cov(r, R m) σ 2 m This pair of equations implies that the expected return on any asset (or portfolio of assets) is a linear combination of the expected returns on the market portfolio and the riskless asset. (2)

24 CAPM 3 1. Important to realise that the asset (or portfolio) under consideration need not lie on the CML, 2. The parameter β (or beta) is a measure of the risk of the asset (or portfolio) relative to the market portfolio. 3. A high β (β > 1) asset has higher risk than the market portfolio and has a higher expected return.

25 CAPM 4 4. A low β (β < 1) asset has lower risk than the market portfolio and is penalised by lower expected return. 5. CAPM also infers a linear return between µ and β, that is between the expected return and the associated risk of an asset (or portfolio). The corresponding line in the µβ-plane is called the Security Market Line aka SML

26 The One-Fund Theorem 1 The One-Fund Theorem There exists a single fund M of risky assets such that any efficient portfolio can be constructed as a combination of the fund M and the risk-free asset P 0.

27 The One-Fund Theorem 2 1. In equilibrium, all investors will select their portfolios on the CML since this is the new EF in the presence of a riskless asset. 2. But we already know two points on the CML: P 0, the riskless asset and M the market portfolio of risky assets. These two point uniquely determine the CML.

28 The One-Fund Theorem 2 3. Indeed, if r m denotes the expected return on the market portfolio we have µ = x 0 r 0 + (1 x 0 )r m (3) σ = (1 x 0 )σ m, (4) equivalently, on eliminating x 0, µ r 0 σ = r m r 0 σ m

29 The One-Fund Theorem 3 This last result states the important result: The price of risk of any asset or portfolio on the CML is equal to the price of risk of the market portfolio. All investors will choose portfolios which are equivalent to some proportion x 0 of the riskless asset P 0 and a proportion (1 x 0 ) of the market portfolio M. The relative proportions of the risky assets contained in M will be the same for all investors, since these quantities are independent of investor preferences. of investor preferences

30 The One-Fund Theorem 4 1. Efficient portfolios between P0 and M are called lending portfolios, since they can be obtained by purchasing the riskless asset (i.e. lending) and the market portfolio. 2. Efficient portfolios between M and infinity are called borrowing portfolios, since these can be obtained by short-selling the riskless-asset asset (i.e. borrowing) and investing the proceeds in the market portfolio.

31 The One-Fund Theorem 5 What is the market portfolio M??? Since all investors should purchase the same fund of risky securities, a little reflection indicates that it might be the portfolio of risky assets, with weights proportional to their market capitilization. That is, the weight of a risky asset in the market portfolio is equal to the proportion of that assset s total capital value to the total market value. Thus, in equilibrium, M should be precisely the same market portfolio considered in Section 1 in the lecture notes.

32 The Characteristic Line 1 From Equations (1) and (2), the CML is consistent with the regression line connecting the random variables R and R m. This regression line is called the Characteristic Line in the context of CAPM and can be written R = r 0 + β(r m r 0 ) + ɛ, where ɛ is a zero mean random variable or residual.

33 The Characteristic Line 2 In this presentation of the CAPM, it is evident that the return on a asset (or portfolio) contains 2 sources of risk: 1. through the value of the parameter β; and 2. through parameter ɛ. The β of an asset determines the systematic or non-diversifiable risk. This is the risk related to the covariance of the asset and the market portfolio. The second component, ɛ, determines the non-systematic or diversifiable risk, because its effects may be virtually eliminated through portfolio diversification.

34 The Pricing Model1 An asset with intial price, X 0 and whose price at time T in the future X T is unknown. A number of models for pricing risky such a risky asset include the Principle of Expected Return and the Principle of Expected Utility. The CAPM provides another pricing model BLACKBOARD: Do derivation

35 The Pricing Model 2 We have deduced that X 0 = 1 ( E(X T ) cov(x ) T, R M ) 1 + r 0 σm 2 (r m r 0 ), which is the pricing formulation of CAPM for a risky asset. 1. The 1st term (1 + r 0 ) 1 E(X T ) is simply the discounted expected future payoff of the asset, corresponding to the price determined by the Principle of Expected Return 2. The 2nd term is a risk premium dependent on the asset s covariance with the market and the excess return of the market over the risk-free rate.

36 The Pricing Model 3 3. The formula is linear in X T. This ensures that it is consistent with arbitrage free pricing. EG if 2 assets have random pay-outs X T and Y T at time T in the future, then the combination P T = ax T + by t must have price today equal to P 0 = ax 0 + by 0. If this wasnt the case we could ensure an arbitrage profit by purchasing the 2 asset portfolio if P 0 < ax 0 + by 0 or selling the portfolio if P 0 > ax 0 + by 0.