Yale School of Management
1) Calculate the historical quarterly mean, and standard deviation for each stock and the market. Convert these numbers to annualized values by multiplying the quarterly mean returns by four, and the quarterly standard deviations by the square root of four.
Quarterly Lehigh DB Cons. S&P
mean 0.001 0.027 0.156 0.030 std 0.117 0.069 0.150 0.033 S&P beta 1.702 0.288 3.078 1.000 Annualized Lehigh DB Cons. S&P
mean 0.003 0.107 0.623 0.118 std 0.235 0.138 0.300 0.066
2) Calculate the correlations among each of the four series, based upon quarterly data.
DB 0.421 1.000
Cons. 0.364 0.203 1.000
S&P 0.476 0.137 0.673 1.000
Lehigh DB Cons. S&P
3) Estimate the beta of each of the three companies with respect to the S&P 500, based upon quarterly data.
There are two ways to approach this. One is to calculate the covariance of the company return with the S&P and then divide by the S&P variance, and the other way is to use the regressions package in advanced math tools functions provided by Excel or other spreadsheets. These numbers are estimated by regressing the column of company returns on the column of S&P 500 returns.
Quarterly Lehigh DB Cons. S&P
Beta 1.702 0.288 3.078 1.000
4) Assume that the current riskless rate is 5.5%, and that the equity risk premium is 8%. Calculate the expected return for each of the three companies, based upon the Capital Asset Pricing Model.
Recall that the CAPM model is:
E[Ri] = Riskless Rate + ß*(Equity Risk Premium) The riskless rate and the equity risk premium are given, and you calculated the betas above.
Annual Lehigh DB Cons. S&P
CAPM ret. 0.191 0.078 0.301 0.135
5) Compare the actual returns over the period to the expected returns based upon the CAPM. Assume the t-bill rate was unchanged over the period. The difference between the actual return and the CAPM expected return is called Jensen's alpha. Are the alphas for each firm positive or negative? Is this a violation of the CAPM? Why or why not?
Jensen's Alpha for asset i is calculated as:
Alphai = Ri - [Rf + ßi (Rm - Rf)]
Where Rm is equal to the actual market return over the period, NOT the expected market return over the period. Thus, the alpha for Lehigh is:
Lehigh Alpha = Lehigh Return - (Tbill Return + ßLehigh (S&P Return - Tbill Return)
Lehigh Alpha = 0.003 - (.055 + 1.702(.118 - .055)) = -0.159
Annual Lehigh DB Cons. S&P
Alpha -0.159 0.034 0.373 0.000
6) Assume there are no taxes. Estimate the asset beta of each of the three firms. Recall that, under the assumption that the beta of debt is zero,
ßAsset = Wequity * ßEquity
ßAsset = (E/E+D) * ßEquity
we are given:
Info. Lehigh DB Cons.
Debt/Equity 0.000 0.200 1.000 Equity 4.000 1.000 1.000 Debt 0.000 0.200 1.000 E/(E+D) 1.000 0.830 0.500 Wa calculated: BetaAsset 1.702 0.239 1.539
Here the case explains that DB and Consolidated are "$1 Billion companies." This means that they have an outstanding equity value (Share Price * # shares outstanding) of $1 Billion. If you make the alternative assumption that $1 Billion refers to the value of the debt + the value of the equity, that is acceptable. It won't change percentages. For example, if you assumed that $1 Billion for DB was the sum of the debt + equity, how does this change things? For Lehigh, assume D/E = 20% and D+E = $1 B. First, solve for D: D= .2E. Substitute this into D+E = $1B: .2E + E = $1B, so 1.2E = $1B, so E = 1/1.2, or .833. In other words, it makes no difference!
7 & 8) Assume Lehigh merges with DB & Assume Lehigh merges with Consolidated
What would be the combined standard deviation of the new firm?
Use the formula for the standard deviation of a portfolio, assuming the weight on Lehigh is .8 and the weight on Consolidated is .2
What would be the beta of the new firm?
ßportfolio = W1 ß1 + W2 ß2
What would be the new expected return of the new firm?
E[Rportfolio] = W1 E[R1 ]+ W2 E[R2]
What is the expected Sharpe ratio of the new firm?
Sharpeexpected = (E[Rportfolio] - Rf)/ std(portfolio)
Lehigh + DB Lehigh + Cons.
WLehigh 0.800 0.800
std(port) 0.201 0.109
beta(port) 1.419 1.977
E[Rportfolio] 0.169 0.213
Expected Sharpe 0.56 1.46
9) Assume that a single-stage discounted cash flow model under certainty fairly represents the relationship between current price, earnings, discount rates and growth for each firm. Using analysts' forecasts of the growth rate in earnings, gi, compared the discount rates implied by the P/E ratios for DB and Consolidated to those implied by the CAPM. Does either firm plot above or below the security market line?
Use the formula for a perpetuity, with a constant growth rate in earnings, and solve for the discount rate implicit in the P/E ratio & earnings forecast:
Pt = Et+1/(rddm - g). This implies: rddm = [Et+1 / Pt ] + g. The security plots above the SML if
rddm - rCAPM. You may call this the "expected alpha" of investing in the security, if you believe that the constant growth rate perpetuity model holds true.
forecast g 0.15 0.20
P/E 20.00 30.00
R ddm 0.20 0.23
exp. alpha 0.12 -0.07
10) What inferences do you draw from your analysis? In particular, which merger increases the probability of the new firm achieving a return in excess of treasury bills?
For this, use the expected Sharpe ratio calculated above. Lehigh alone has an expected Sharpe ratio of .58. A merger with DB does not increase this ratio. A merger with Consolidated will increase this to 1.46, which increases the probability of exceeding T-Bills.
Is either firm a clear bargain?
If the CAPM holds exactly, neither firm can be bargain, however, below, we consider the valuation given by the CAPM discount rate.
What recommendation will you make to Stacey regarding the effect of either acquisition upon Lehigh?
Since the expected alpha of one firm is negative, and one is positive, you could recommend a merger with DB. On the other hand, you could argue that the market is efficient, and that there is no benefit to merging the firms at all.
Assuming the dividend growth model is approximately correct, and the CAPM discount rate is the appropriate one, what price would you suggest that Stacey pay for DB? What price would you suggest Lehigh pay for pay for Consolidated?
This is a bit tricky, because it depends upon the reliability of the single-stage discounted cash flow model used in question 8. Note that, when you use the CAPM in the dividend-discount model, you get some strange results, that is you calculate: P = Et+1/(rCAPM - g). You can figure the earnings from the P/E ratio of the firm, and the value of the firm's equity. For DB, with a P/E ratio of 20, that means the E/P = .05, and the E = .05*$1Billion, or $50 million year.
DB price -0.69
Consolidated Price 0.33
It says that the price of DB is negative, a nonsensical answer. You thus infer that the dividend discount model is wrong, the growth expectations are incorrect, or that the CAPM discount rate is much too low.