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1
END OF CHAPTER EXERCISES - ANSWERS
Chapter 14 : Stock Valuation and the EMH
Q1
Does the dividend discount model ignore the mass of investors who have bought their
shares with the intention of selling them in, say, 3 years time?
A1
The dividend discount model implies that the ‘fair’ price today of a share you intend to sell in 3
years time is :
[1]
P=
D3
P3
D1
D2
+
+
+
2
3
(1 + R) (1 + R)
(1 + R)
(1 + R) 3
where we have assumed the discount rate is constant. But you need to forecast what

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1
END OF CHAPTER EXERCISES - ANSWERS
Chapter 14 : Stock Valuation and the EMH
Q1 Does the dividend discount model ignore the mass of investors who have bought their shares with the intention of selling them in, say, 3 years time? A1 The dividend discount model implies that the ‘fair’ price today of a share you intend to sell in 3 years time is : [1]
3333221
)1()1()1(
)1(
RP R D R D R DP
+++++++=
where we have assumed the discount rate is constant. But you need to forecast what you think P
3
will be. The logical way to do this is to note that : [2]
....)1()1(
2543
++++=
R D R DP
But when we substitute equation [2] into equation [1] we obtain the result that the fair price depends on all future dividends, that is, the dividend discount model. Therefore the dividend discount model implicitly includes the behaviour of those (rational) investors who want to sell the stock in 3 years time. Q2 What practical use is there in knowing the beta of your stock portfolio? A2 Portfolio beta
β
p
is a weighted average of the individual betas of the stocks in your portfolio :
β
p
=
Σ
w
i
β
i
where w
i
= proportion of the total dollar value of your portfolio held in asset-i. The portfolio beta is a measure of the responsiveness of your portfolio return to changes in the market portfolio: ER
p
– r =
β
p
(ER
m
– r)
Hence, if you feel you are overexposed to changes in the market return, you can alter the composition of your stock portfolio to achieve a lower desired value of
β
p
Q3 Why might stock prices be highly volatile even though all investors act in a perfectly rational way? A3 The Gordon growth model is useful here. In an efficient market the price of a stock is given by :
© K. Cuthbertson and D. Nitzsche (2008)
‘Investments’
, J. Wiley, 2
nd
edition
2
g R DP
−=
1
where D
1
= expected dividends in year-1, R = required (risk adjusted) return on this stock, g = growth rate of dividends. Suppose D
1
= 10, R = 0.10 and g = 0.05 then P = 200. Now suppose there is an upturn in the economy so investors alter their view of future dividend growth by a mere 1% p.a. so that g now equals 0.06. The new stock price is P = 10/0.04 = 250. The change in the stock price is a relatively large rise of 25%. Similarly, changes in investors’ views about inflation or the risk premium (i.e. the stock’s beta) can lead to a change in R and a change in the stock price. Hence large changes in stock prices are likely to occur when investors are rational and forward looking. Q4 Why is it better to short sell an overvalued stock and simultaneously buy a different undervalued stock, rather than simply just buying the undervalued stock ? A4 You buy an undervalued stock when its quoted price P
A
is below the fair value V
A
(e.g. calculated using the dividend discount model), expecting the market price to rise towards its fair value. However, if economy wide “bad news” about dividends arrives, this will lead to a reappraisal of fair value which may now be below P
A
and hence you are now holding an overpriced stock. Suppose however that there had been another stock-B available which was initially overpriced, that is P
B
> V
B
then you would short-sell this stock in the hope its price falls. The arrival of economy wide ‘bad news’ will further depress V
B
, so you would make even more money when P
B
falls to the new lower value for V
B
. Hence the ‘extra’ gains on stock-B partly compensate for possible losses on stock-A, after the arrival of the economy wide ‘bad news’ (e.g. slower growth in output). Q5 The dividends of company-X are expected to grow at the constant rate of 5% p.a. The last dividend pay-out was $1.80 per share. The risk adjusted (required) rate of return is ER = 11% p.a. The current market price of the share is $ 35. Should you purchase the share? A5 Gordon Growth Model: Fair (“Correct”) Value of share V = D
0
(1+g)
/ (R-g) = 1.80 (1.05) / (0.11-0.05) = $31.50 The current quoted market price is $35 (i.e. it’s overvalued). Since you expect the actual price to fall towards the fair value, you would not buy the share. If anything you would short sell the share. Q6 The past performance of “BubbleStock” is:
Cents 2008 2009 2010 EPS (Earnings per share)
10 12 15
Dividends per share
4 4.8 6 “BubbleStock” is expected to produce earnings per share in 2011 of 20 cents and in 2012 of 26 cents. Earnings growth thereafter is expected to be 10%. (a) What is the pay-out ratio? If the rate of return on “BubbleStock” required by investors is 14%, what is the fair price for the share at the end of 2010?
© K. Cuthbertson and D. Nitzsche (2008)
‘Investments’
, J. Wiley, 2
nd
edition
3
(b.) In the past 5 years “BubbleStock” shares have provided a 10%, 12%, 3%, 6% and 8% return. Estimate the expected return and standard deviation of “BubbleStock”. (c.) The expected return on the market portfolio is 7%, the standard deviation of the market portfolio is 6%, correlation of “BubbleStock” with the market return is 0.7 and the risk-free rate is 5%. If the current market price of “BubbleStock” is P = 235, is it a good buy (or should we just say “goodbye” to the stock)? A6 (a.) 40% of earnings is being paid out as dividends (e.g. 4/10, 4.8/12, 6/15). Hence we can expect D(2011) = 0.4(20 cents) = 8 cents. D(2012) = 0.4(26 cents) = 10.4 cents PV of first two dividend payments V
2
= 8 / 1.14 + 10.4 / 1.14
2
= $15.02 Value
at the end 2012
of dividends from 2013 onwards V(2012) = 10.4 (1.1) / (0.14-0.10) = 286 Discounting back to end 2010 V
*
= 286 / (1.14)
2
= $220.07 Hence fair value V = $15.02 + $220.07 = $235.10 The fair price for “BubbleStock” stock (consistent with a required return of 14%) is
$ 235.10
(b.) Mean (R
Bubble
) = (1/5) (10 + 12 + 3 + 6 + 8) = 7.8 Var(R
Bubble
) = (1/5) ((10-7.8)
2
+ (12-7.8)
2
+ (3-7.8)
2
+ (6-7.8)
2
+ (8-7.8)
2
) = (1/5) (4.84 + 17.64 + 23.04 + 3.24 + 0.04) = 9.76
σ
(R
Bubble
) = 3.1241 (c.) r = 5% E(R
m
) = 7%
σ
m
= SD(R
m
) = 6%
σ
(R
Bubble
) = 3.12141% (from b.)
ρ
= Corr(R
Bubble
, R
m
) = 0.7 Cov(R
Bubble
, R
m
) =
ρ
σ
m
σ
( R
Bubble
)
t
= (0.7) (6) (3.1241)
= 13.1212
β
Bubble
Cov (R
Bubble
, R
m
) /
σ
m
2
= 13.1212 / 6
2
= 0.3645
Knowing Bubble’s beta, we can use the SML to calculate it required return: E(R
Bubblet
) = r + (ER
m
- r)
β
Bubblet
= 5% + (7% – 5%) 0.3645 = 5.73% With a discount rate of 14% Internet has a fair price of $235.10 - see (a). Using the “correct risk adjusted discount rate of 5.73 % given by the SML would give a fair value much greater than $235.10. Hence the quoted price of $235 is below the fair value and you should buy the stock. Alternatively, you can note that at a price of $235.10 the IRR on the stock is 14% - see (a). However, the required (risk adjusted) rate of return is only 5.73%, hence invest in the stock because its current risk adjusted IRR > required rate of return, of 5.73% given by the SML. Q7 A firm is expected to pay dividends of 20p at the end of the year t=1. Dividends are then expected to grow at 5%. The (risk adjusted) required rate of return for this firm is
© K. Cuthbertson and D. Nitzsche (2008)
‘Investments’
, J. Wiley, 2
nd
edition
4
11%. What would you expect its current market price to be? If the dividend payout ratio is 60% what would you expect the price earnings ratio to be? A7 Using the Gordon growth model: P = D
1
/(R-g) = 20/(0.11-0.05) = 333.3p D
1
= pE
1
= 0.6E
1
Hence : E
1
= D/0.6 = £33.3 Thus P/E
1
= 333.3/33.3 = 10. Q8 The dividends of company-X are expected to grow at the constant rate of 5% p.a. from now on. The last dividend has just been paid and was $1.80 per share. (a.) Given the business risk of company-X investors require an average rate of return on the stock of ER = 11% p.a. Show that the “fair value” of this share is $31.50. The current market price of the share is $28. As a speculator should you buy or sell this share? What are the risks involved in your strategy? (b.) What would you have done if the market price had been $36 (rather than $28)? (c.) Suppose share-A has a market price of $28 and another share-B (of a firm in the same industry sector) has a market price of $36 and both have the same ‘fair value’ calculated in (a). They are both mispriced because other traders have not yet correctly recognised the ‘true’ profit potential of each of these firms. How can you speculate on these two shares while largely insulating your strategy from any general fall in all shares (i.e. the whole market – or ‘market risk’)? (d.) To simplify the calculations, assume that general economic conditions deteriorate (because of a rise in interest rates by the Central Bank) and both the market price of A and B fall by $1, which simply reflects the fall in ‘fair value’ (by $1) of both shares (i.e. the rise in interest rates has raised the discount factor for both firms and hence the fair value falls by $1). What happens if you have to ‘close out’ your positions in A and B
immediately after
this general fall in prices (and hence cannot wait for any miss-pricing to be corrected) ? A8
(a.) Gordon Growth Model: Fair (“Correct”) Value of share V = D
0
(1+g)
/ (R-g) = 1.80 (1.05) / (0.11-0.05) = $31.50 If P = $28 and V = $31.50 then the share is undervalued by $3.5 (about 11%). You should buy the share today at P = $28. If V remains unchanged over say the next week, then as more traders notice the share is
undervalued,
they too will purchase the share. This will push the share price up towards its ‘fair value of $31.5 and if you ‘got in fast’ you might then close out your position by selling at a price close to $31.5 by the end of the week. The danger is that the company announces some ‘bad news’ over the next week (e.g. “profits this year will be lower than previously forecast because we have not had our option accepted by MGM on our new film
Harry Potter and the Money Making Machine
and the ‘pilot’ cost a lot to make“). This is known as idiosyncratic or
specific risk
– as it affects only this firm (or at most a small group of film/movie companies). This would immediately lower forecasts of future dividends for this firm and hence V might fall to $26 and now you are left currently holding an overvalued share. (For an example of market risk – see below) (b.) If the current quoted market price is P = $36 and V = $31.50 then the share is overvalued. You expect the actual price to fall towards the fair value. Hence, you would short sell the share, in the hope that if your calculation is correct, you will be able to buy it back at a lower price in the future and return the ‘srcinal’ share to your broker. (c.) To minimise risk you should (short) sell a share which is overvalued (i.e. P > V) and buy another which is undervalued (i.e. P < V). Then, if there is a change in the overall
© K. Cuthbertson and D. Nitzsche (2008)
‘Investments’
, J. Wiley, 2
nd
edition

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