# Suggested Answers to Acquira Co.

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.

Correlations

Lehigh	1.000
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.

DB	Cons.

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.

I calculated:

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.

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