Project #18687 - portfolio anaysis

4.7 Capital Asset Pricing Model and Its Applications to Capital Budgeting

We start by constructing the Capital Market Line (CML) in a financial market with one risk-free asset B and two risky assets S1 and S2. The return on the risk- free asset B equals Rf and we denote by Ri, i = 1,2, the return on the risky asset Si, i = 1,2. For i = 1,2, let μi denote the expected return on the risky asset Si, σi2 its variance and ρ the correlation between the two assets. In other

words, μi = E(Ri), σi2 = var(Ri) and ρ = cov(Ri,Rj). Consider an individual σ1 σ2

Project Project’s “beta” β Project’s expected cash-flows next year

  1. I1  1.8 $125

  2. I2  1.8 $115

  3. I3  1.8 $110

26 CHAPTER4. PORTFOLIOANALYSIS investor who optimizes her portfolio P according to the mean-variance criterion

and denote by wi, i = 1, 2, her portfolio weights on the two risky assets.

  1. What are the expected value and the variance of the return RP on the in- vestor’sportfolioP?(hint:computetheportfolio’sexpectedreturnμP and varianceσP2 asfunctionsofRf,wi,μi,σiandρ(i=1,2))

  2. Assume that no short sales on the risky assets S1 and S2 are allowed. What is the minimum variance frontier of portfolios composed of the two risky assets S1 and S2 only? Show that this minimum variance frontier can be represented by a hyperbola in a (σ, μ) space. Given that no short sales are allowed, which parts of the previous hyperbola correspond to feasible portfolios? (hint: compute the variance σP2 of a portfolio composed of the risky assets only, as function of its expected return μP and of μi, σi and ρ (i = 1, 2))

  3. Intheaboveframework,assumethatμ1 =2,μ2 =1,σ1 =2andσ1 =1. Plot the efficient sets (i.e. the minimum variance frontiers) corresponding to the cases ρ = 1, ρ = 1/2, ρ = 0 and ρ = 1 (use the same graph for the four cases). What happens in the limiting cases ρ = 1 and ρ = 1?

  4. We now assume that ρ = 1/2 and that the risk-free interest rate Rf equals 1/2. Derive the equation of the Capital Market Line. What is the com- position of the investor’s optimal portfolio? Compute the expected return μt and the variance σt2 of the tangency portfolio t. (hint: use the two fund separation result)

    We now assume that there is some large number K of individual investors in the economy, each choosing their optimal portfolio according to the mean- variance optimality criterion. We start by generalizing the previous analysis to the case where there are N risky assets Si , i = 1, ..., N , and one risk-free asset B in the market. Let R denote the N-vector of returns on the risky assets, R = (R1, ..., RN ), E its expectation and Σ its variance-covariance matrix,

? E(R1) ? ? var(R1) cov(R1, R2) · · · cov(R1, RN ) ? .cov(R ,R ) var(R ) ··· cov(R ,R )

?.??12 2 2N? E=?.?andΣ=?.....?.

? .? ? . . . . ? E(RN) cov(R1,RN) cov(R2,RN) ··· var(RN)

We further assume that E(Rj) ?= E(Ri) if i ?= j and that Σ is positive definite.

4.7. CAPITALASSETPRICINGMODELANDITSAPPLICATIONSTOCAPITALBUDGETING2

  1. What is the composition of the tangency portfolio t, as a function of the vector of expected returns E, risk-free rate Rf , and the variance-covariance matrix Σ? (hint: compute the tangency portfolio weights on individual as- sets)

  2. At equilibrium, we know that the tangency portfolio t and the market port- folio m are equivalent. Let Rm denote the return on the market portfolio. Use the previous result to compute the expected return E(Rm) and the vari- ance var(Rm) of the market portfolio. Show that the Capital Asset Pricing Model formula holds. (hint: compute cov(Ri, Rm))

  3. Assume that the risk-free rate is 2%, and that the market-risk premium, E(Rm)Rf , is 8%. Plot the Security Market Line (SML). What can we say about portfolios of assets whose expected return is above/below the SML?

    We now show that the Capital Asset Pricing Model (CAPM) formula can also be used for capital budgeting. Consider an all-equity company A which faces a project I. Today’s cost of this project is C0 and it will generate a risky cash-flow C1 in one year form now. Let SI be a financial asset with risk comparable to those of the project I, and βI the “beta” on the project I. As previously, we denote by Rf the risk-free return and by Rm the return on the market portfolio.

  4. When the cash-flows generated by a project are known with certainty, the Net Present Value formula tells us they should be discounted at the risk- free rate Rf . This simple capital-budgeting rule can be generalized to risky cash-flows in which case the discount rate of a project should be equal to the expected return on a financial asset of comparable risk. Under CAPM, what is the NPV of the project I?

  5. Suppose that the “beta” of the company A equals 1.8, i.e. βA = 1.8. Be- cause the new project I is similar to the firm’s existing ones we assume that the “beta” on this new project is equal to the company’s “beta”, i.e. βI = βA. Further, we still suppose that the risk-free rate is 2%, and that the market-risk premium, E(Rm) Rf , is 8%. What is the discount rate on the project I?

  6. The company A is evaluating the following three projects:

 

28 CHAPTER4. PORTFOLIOANALYSIS

Each project initially costs $100. All projects are assumed to have the same risk as the firm A as a whole. For every Ik , k = 1, 2, 3, compute the project’s internal rate of return R ?I,k and the project’s NPV. Should those projects be accepted or rejected?

  1. We now use the Security Market Line (SML) to estimate the risk-adjusted discount rate for risky projects. Consider the relationship between the project’s internal rate of return R ?I and the company’s “beta” βI : let the SML repre- sent the set of projects with zero NPV. Construct the SML in the (βI , R ?I ) diagram. Illustrate the projects I1, I2 and I3 in the above diagram. What are the accept/reject regions?

    In the previous analysis, we have assumed that the company A was all- equity. Suppose that the firm now uses both debt and equity to finance its investments. We now compute what the discount rate of a project should be given that the company has both debt and equity.

  2. Assume that the company A can borrow at some rate R. Knowing that the interest on debt is tax-deductible and that the corporate taxation rate is TC , what is the effective interest rate rB on debt as a function of R and TC? Assume that the interest rate paid by A is 10% and that the corporate tax rate is 34%. What is in this case the value of rB ?

  3. As shown previously, the return on equity of the company A, denoted rS, can be computed by using the Security Market Line (SML). Knowing that therisk-freerateisRf =2%,themarketpremiumE(Rm)Rf is8%and that the company’s “beta” is 1.8, compute rS .

  4. In cases where the company has both debt and equity, it evaluates projects by using a discount rate r which is obtained as the weighted average of rB and rS. The weights used in the computation of r correspond to the proportion of debt and equity the company has. If the company has 40% debt and 60% equity, what is the value of r?

  5. Assume that the company still faces the same projects I1, I2 and I3, with cash-flows as defined in 3.3. Should those projects now be accepted of rejected?

Project Project’s “beta” β Project’s expected cash-flows next year

  1. I1  1.8 $125

  2. I2  1.8 $115

  3. I3  1.8 $110

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