Pike Robert, Neal Bill. Corporate finance and investment: decisions and strategy
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Appendix B Solutions to selected questions 695
CHAPTER 6
1 The preference for IRR in practice is because:
(a) It is easier to understand (this is debatable).
(b) It is useful in ranking projects (although not always accurate).
(c) Lower-level managers do not need to know the discount rate. Where a risk-adjusted hurdle rate is used,
there may be considerable negotiation over the appropriate rate.
(d) Psychological. Managers prefer a percentage.
2 (a) years. The reciprocal is 25 per cent, which is the IRR for a project of infinite life.
(b) For a 20-year life, For an eight-year life,
(c) The annual cash flows are approximately the same for a long-lived project, the payback reciprocal is a
reasonable proxy for the IRR. The actual IRR is always something less than that given by the payback
reciprocal.
4 The two basic approaches in handling inflation are:
(a) forecast cash flows in money terms (i.e. including inflation) and discount at the money cost of capital; or
(b) forecast cash flows in current terms and discount at the real cost of capital.
The relationship between the two is given by:
where M is the money cost of capital, I is the inflation rate and P is the real cost of capital.
5 Bramhope Manufacturing
(i) (a) Additional investment: Additional annual (see below).
Therefore payback
(b) Time Cash flow (£) DF at 15% PV (£) DF at 17% PV (£)
0 (108,500) 1 (108,500) 1 (108,500)
1–8 24,300 4.48732 109,042 4.20716 102,234
8 20,500 0.32690 6,701 0.28478 5,838
7,243 (428)
(c) NPV AT
Workings: Existing project’s annual cash
Net project’s annual cash
Incremental cash
(ii) The project appears to offer a positive net present value and should be accepted. However, an NPV of
on a project costing is relatively small, and questions should be asked as to how sensitive
the key assumptions are to uncertainty. For example, is it realistic that the additional capacity can be sold
at the current price? Will there really be no increase in fixed overheads? If an advanced machine has been
developed after just two years, is an eight-year economic life optimistic?
£123,500£7,000
flow £94,300 £70,000 £24,300.
inflow £230,000 10.95 0.08 0.462 £94,300.
inflow £200,000 10.95 0.12 0.482 £70,000.
IRR approx. 17%.15% £7,243;
period £108,500>£24,300 4.5 years.
inflow £24,300£123,500 £15,000 £108,500.
11 P2
11 M2
1 I
IRR 19%.IRR 24%.
Payback four
CHAPTER 7
1 Capital budgeting involves the whole investment process from the original idea to the final post-audit.
The resource allocation process is the main vehicle by which business strategy can be implemented.
Investment decisions are not simply the result of applying some evaluation criterion. The investment decision
process is essentially one of search: search for ideas, search for information, search for alternatives and search
for decision criteria. The prosperity of a firm depends more on its ability to create profitable investment
opportunities than on its ability to appraise them.
Once a firm commits itself to a particular project, it must regularly and systematically monitor and control
the project through its various stages of implementation. Post-audit reviews – if properly designed – fulfil a
useful role in improving the quality of existing and future investment analysis and provide a means of initiat-
ing corrective action for existing projects.
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696 Appendix B Solutions to selected questions
2 Aims of post-audits:
(a) improve the quality of existing decisions and underlying assumptions
(b) improve the quality of future decisions
(c) basis for corrective action
3 (a) Total automation Partial automation
£ per unit £ per unit
Sales price 75 75
Variable cost (30) (55)
Contribution 45 20
Fixed cost (see note 1) (18) (5)
Net profit 27 15
Total cost per unit
Annual net profit
Break-even units (p/a) 4,000 2,500
Net present value:
Cash flow £ Cash flow £
Outlay
Inflows years 1–5
Scrap value year 5 –
Present value Present value
Outlay
Inflows at 3.27
Scrap value at 0.48 –
Net present value
Note 1
1250,0002
5 10,000
11,000,000 100,0002
5 10,000
404,000
519,500
48,000
654,0001,471,500
250,0001,000,000
££
100,000
200,000450,000
250,0001,000,000
£150,000£270,000
£60£48
(b) The total automation option produces a lower product cost. As the sales price and volume are identical
for both alternatives a higher annual profit is also reported by this alternative. Partial automation involves
lower fixed costs (depreciation), but variable costs are substantially greater. With the cost pattern above,
the total automation alternative results in a higher break-even point.
Remember that there is a higher initial capital cost with total automation. The accrual accounting
assumptions applied above acknowledge this with an annual charge for depreciation. The net present
value technique takes account of the capital cost in a different manner, charging the full cost immediate-
ly against discounted future cash flows. In this case, total automation generates a higher NPV but a high-
er initial outlay.
When the amount of the initial capital investment is considered, the option to undertake partial
automation looks attractive. It achieves a net present value of 80 per cent of total automation with only
one quarter of the capital expenditure. Partial automation shows a profitability index of 2.62, compared
with 1.52 for the total automation, which emphasises the attractiveness of the lower investment. The rec-
ommendation of the NPV technique must, however, be interpreted with care and considered in the light
of the overall company situation. Investment in total automation is the course of action which will max-
imise the wealth of the owners (the extra benefit exceeds the extra capital cost at 16 per cent). If the com-
pany is not in a capital-constrained situation then total automation should be pursued. If the company
has a number of projects, all competing for limited capital, then we would need to know the return from
the other investment opportunities and compare them with the incremental investment being directed to
total automation.
The data omit certain other factors which may be pertinent to the analysis, for example, quality differ-
ences, maximum machine capacities, maximum sales demand, the flexibility and interchangeability of
parts and equipment which may be beneficial.
For a lower risk investment with an attractive return, the company should opt for partial automation.
If they have extra capital available, electing for total automation of the product line will earn above their
extra cost of capital but nothing is known about other projects competing for this extra capital.
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Appendix B Solutions to selected questions 697
CHAPTER 8
1 While a capital project may have a high expected return, the risks involved may mean that there is the possi-
bility that the project will be unsuccessful – even to the extent of putting the whole business in jeopardy.
2 Project risk – variability in the projects’ cash flows; business risk – variability in operating earnings of the firm;
financial risk – risk resulting from the firm’s financing decisions, e.g. the level of borrowing.
3 Woodpulp project
Year CE NCF(£) ENCF(£) 10% DF PV(£)
1 0.90 8,000 7,200 0.90 6,480
2 0.85 7,000 5,959 0.83 4,938
3 0.80 7,000 5,600 0.75 4,200
4 0.75 5,000 3,750 0.68 2,550
5 0.70 5,000 3,500 0.62 2,170
6 0.65 5,000 3,250 0.56 1,820
7 0.60 5,000 3,000 0.51 1,530
PV 23,688
Accept the project. net cash flow.
4 Mystery Enterprises
Assuming a discount rate of 10 per cent and independent cash flows:
Coefficient of variation
£125
£71
1.76
SD
B
1£1022
2
11.12
2
1£1022
2
11.12
4
£125
NPV
£550
1.1
£450
11.12
2
£800 £71
£102Standard deviation
0.31500 4502
2
0.21600 4502
2
£10,500
0.21300 4502
2
0.31400 4502
2
Variance 1£2
0.213002 0.314002 0.315002 0.216002 £450Expected value Year 2 1£2
£102Standard deviation
0.31600 5502
2
0.21700 5502
2
£10,500
0.21400 5502
2
0.31500 5502
2
Variance 1£2
0.214002 0.315002 0.316002 0.217002 £550Expected value Year 112 1£2
ENCF ExpectedNPV £23,688 £13,000 £10,688.
CHAPTER 9
1
This portfolio has a risk only slightly below that of investment X, even though the correlation coefficient
is quite low. This suggests that the portfolio weighting involves too little of investment X if the intention is
to lower portfolio risk.
2866.25
29.4
23225 506.25 1354
2310.52
2
30
2
10.52
2
45
2
210.5210.5210.22130214524
Portfolio standard deviation 2310.52
2
s
x
2
10.52
2
s
2
y
210.5210.521r
xy
s
x
s
y
24
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698 Appendix B Solutions to selected questions
2 The appropriate formula is:
Proportion invested in asset
Also, remember that
(i) The in % in
(ii) The covariance is in % in
(iii) The covariance is 0; % in % in
(iv) The The weights are misleading as the answer is indeterminate under perfect
positive correlation.
3 (a) China:
Expected
Growth Probability IRR% Deviation Squared deviation Times probability
Rapid 0.3 50 25 625 187.50
Stable 0.4 25 – – –
Slow 0.3 0 625 187.50
Standard i.e. 19.4%
Scotland:
Expected
Growth Probability IRR% Deviation Squared deviation Times probability
Rapid 0.3 10 14.44 4.33
Stable 0.4 15 1.2 1.44 0.58
Slow 0.3 16 2.2 4.84 1.45
Standard
(b) Covariance calculation:
Expected portfolio return
Standard deviation
4 (a)
6% 3% 6% 15%
Eire: EV of IRR 10.3 20%2 10.3 10%2 10.4 15%2
210.1
3.2, i.e. 3.2%
23123.52 13.52 116.924
2310.25
2
19.4
2
2 10.75
2
2.5
2
2 12 0.25 0.75 4524
10.75 13.8%2 10.25 25%2 16.6%
Covariance 45
deviation 26.36
2.5%
Sum 6.36
3.8
value 10.3 10%2 10.4 15%2 10.3 16%2 13.8%
deviation 2375.0
19.4,
Sum 375.00
25
value 10.3 50%2 10.4 25%2 10.3 02 25%
covariance 21; %
B 99.5%.A 0.5%;
B 94%.A 6%;15; %
B 71%.A 29%;ovariance 36; %
cov
AB
r
AB
s
A
s
B
.
A
s
B
2
cov
AB
s
B
2
s
A
2
2cov
AB
Outcome (%) Deviation Sq’d Dev. p Sq’d
20 25 0.3 7.5
10 25 0.3 7.5
15 0 0 0.4 0
15 s 3.87Variance Total
5
5
Dev. p
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Appendix B Solutions to selected questions 699
3% 9% 8% 20%
Humberside: EV of IRR 10.3 10%2 10.3 30%2 10.4 20%2
CHAPTER 10
1 Expected
With more pessimistic expectations about market returns, this drops to:
2 (i)
(ii)
(iii)
(iv)
3 The intercept of the Security Market Line is the risk-free rate. It passes through the market portfolio which
has a Beta of 1.0. In diagrammatic terms:
19.3%ER
m
8.5%ER
j
1.71Beta
9%R
f
ER 5% 1.2338% 5% 4 8.7%
return 5% 1.23311.5% 5% 4 13%.
M
SML
Betab = 10
R
f
= 8%
ER
m
= 12%
Required
return
Outcome (%) Deviation Sq’d Dev. p Sq’d
10 100 0.3 30
30 100 0.3 30
20 0 0 0.4 0
60 s 7.75Variance Total
10
10
Dev. p
(b) (i) For a 50/50 split investment:
(ii) 75/25 split:
238.44 3.75 4
212.19 3.49, i.e. lower risk than either project
s 210.752
2
1152 10.252
2
1602
EV 10.75 15%2 10.25 20%2 11.25% 5% 16.25%
233.75 15 4
218.75 4.33
s 210.52
2
1152 10.52
2
1602 210.5210.5210213.87217.752
EV of IRR 10.5 15%2 10.5 20%2 17.5%
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700 Appendix B Solutions to selected questions
4
Comparing the expected returns with those currently achieved, A and C generate returns lower than
expected, which suggests they are overvalued. (B looks undervalued.)
5 (i) Projecting past returns into the future, the expected return on the whole market is 15% 7% 2 12%.
9.5% ER
C
5% 0.935% 4
11.5% ER
B
5% 1.335% 4
ER
A
5% 0.7310% 5% 4 8.5%
Required
return
8%0
M
CML
Risk
R
f
= 5%
ER
m
= 12%
(iii) For a 50/50 portfolio:
(iv) Bearing in mind there is no correlation between the risk-free rate and the market portfolio, the expres-
sion for portfolio risk is;
Alternatively, we could use the weighted average expression:
(v) As risk increases from 4% (the 50/50 portfolio) to 8% (the market portfolio), the return required increas-
es from 8.5% to 12%, suggesting a risk–return trade-off (assuming it is linear) of:
Thus, for every one percentage point increase in the portfolio standard deviation, the investor must
be compensated by an increase in portfolio return of 0.875%.
6 (a) Your graph should show every pair of observations lying along a straight line. This indicates perfect pos-
itive correlation between and The slope of the line, and hence the Beta, is
(b) Because all observations lie on the line of best fit, all variation in appears to be explained by variation
in i.e. there is no specific risk.
(c) of the market return. The mean return on the market Hence
Thus, systematic
You should expect to find the total risk is also 195.84 because there is no unique risk.
risk 10.82
2
13062 195.84.
s
2
m
15 22
2
1 10 22
2
p
16 22
2
306
portfolio 2.0.s
2
m
variance
R
m
,
R
j
0.8.R
m
.R
j
13.5%>4%2 0.875
10.5 8%2 10.5 02 4%
216
4, i.e. 4%
O
p
210.52
2
182
2
ER
p
10.5 12%2 10.5 5%2 8.5%
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Appendix B Solutions to selected questions 701
CHAPTER 11
1 The shares are ex-dividend, so the future required return is:
2 (i)
(ii)
(iii)
(iv)
3 Dividend growth is found by solving:
From tables, approx.
4 (i)
(ii)
(iii)
k
e
6% 0.6311% 6% 4 16% 3% 2 9%
Beta 10.8 0.752 0.6
k
e
6% 1.0311% 6% 4 16% 5% 2 11%
Beta 10.8 1.252 1.0
k
e
6% 0.8311% 6% 4 16% 4% 2 10%
0.042 0.158 0.20, i.e. 20%
Implied k
e
20p11.1582
£5.50
0.158
g 15.8%
11 g2
4
20.0p
11.0p
1.8182
11p 11 g2
4
20.0p
g 6.2%
k
e
9.3%
P
o
£3.64
D
o
£0.62
0.125, i.e. 12.5%
k
e
£0.8011.0452
£10.50
0.045 10.08 0.0452
5 Division Beta Market Premium % of assets
C 7% 0.7 8.00 12.6% 15
E 7% 1.1 8.00 15.8% 40
R 7% 0.8 8.00 13.4% 20
P 7% 0.6 8.00 11.8% 5
Total / 0.855 / 13.84% 100
k
e
R
f
6 (a) EV of
Outcome Deviation Squared deviation Prob. Sq’d
10% 16 0.6 9.6
20% 36 0.4 14.4
24.0 s 4.9Variance Total
6%
4%
dev. p
return 10.6 10%2 10.4 20%2 14%.
Overall required return 7% 0.85538% 4 17% 6.84%2 13.84%
10.105 0.44 0.16 0.152 0.855
Overall Beta 10.15 0.72 10.04 1.12 10.20 0.82 10.25 0.62
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702 Appendix B Solutions to selected questions
(b)
(c) The present
What is the risk of the market
Rearranging:
For Megacorp:
New Beta for Therefore, the new project lowers
Megacorp’s Beta.
8 (a) Since the asset Beta is a weighted average of the component segment Betas:
where of North, of West, of South.
Since North is 50 per cent more risky than South, and West is 25 per cent less risky than South, it follows that:
whence
(b) The asset Beta for East is:
(c) The asset Beta for Lancelot after the divestment and acquisition is again a weighted average of the com-
ponent asset Betas:
(d) If we evaluate projects in East on the assumption of all-equity financing, the cut-off rate is
R
f
b
E
1ER
m
R
f
2 10% 1.12115%2 26.8%
0.56 0.1875 0.375 1.12
¢
1
2
1.12≤ ¢
1
4
0.75≤ ¢
1
4
1.5≤
b
A
¢
1
2
b
E
≤ ¢
1
4
b
W
≤ ¢
1
4
b
N
≤
1.0 1.4
1.6
2.0
1 1.12 1.12
b
S
Revenue sensitivity factor Operational gearing factor
b
E
b
S
Relative risk factor
1b
E
2
b
S
1.00, b
N
1.50, b
W
0.75.
11.5b
S
2
4
10.75b
S
2
4
b
S
2
1.06
b
S
bb
W
bb
N
b
b
A
¢
1
4
b
N
≤ ¢
1
4
b
W
≤ ¢
1
2
b
S
≤ 1.06
Megacorp 10.8 1.22 10.2 0.052 0.95.
Project Beta
10.1214.92
9.33
0.49
9.33
0.05
s
j
114210.82
1.2
9.33
s
m
s
j
r
jm
Beta
j
1s
m
2?
Beta of project
cov
jm
s
m
2
r
jm
s
j
s
m
s
m
2
r
jm
s
j
s
m
Beta 1.20.
S.d. 2310.82
2
1142
2
10.22
2
14.92
2
210.8210.2210.362114214.924
ER 10.8 30%2 10.2 14%2 24% 2.8% 26.8%
proportion 20% s.d. 4.9% Erewhon ER
14%
proportion 80% s.d. 14% Megacorp ER 30%
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Appendix B Solutions to selected questions 703
(e) There are numerous problems involved in obtaining tailor-made project discount rates. First, given that
we may have to take the Beta from a surrogate company, ungearing the Beta of its shares requires an accu-
rate assessment of the values of the equity and debt. If the company is quoted, the market valuation of
equity can be taken, but not all corporate debt is quoted. Thus the debt:equity ratio used will often be a
mixture of market and book values, even for quoted companies. Second, decomposing the ungeared Beta
into segmental asset Betas strictly requires weighted average calculations based on market values. Since
corporate segments are not generally quoted on stock markets, book values invariably have to be used.
Third, to measure project Betas requires consideration of whether the project is of a different order of risk
from ‘typical’ projects in the division. If so, the revenue sensitivity factor requires estimation, mainly
based on guesswork for unique projects. In addition, the project gearing factor must also be estimated.
Fourth, the general problems relating to specification of the risk-free return and the risk premium on the
market portfolio still have to be addressed.
CHAPTER 12
1 Companies issue a type of call option when issuing share warrants (giving the holder the right to buy shares
at a fixed price) and convertible loan stock (giving the holder the right to exchange the loan for a fixed num-
ber of shares).
2 Gaymore plc
Traded options give the holder the right, but not the obligation, to buy (a call option) or sell (a put option) a
quantity of shares at a fixed price on an exercise date in the future. They are usually in contracts of 1,000 shares
and for three, six or nine months.
Holders of a put option in Gaymore plc have the right to sell shares in April at 500p. For this right they cur-
rently have to pay a premium of 47p, or on a contract of 1,000 shares.
If the share price falls below 453p (i.e. 500p–47p), the shares become profitable and the holder is ‘in the
money’. So if they fall to 450p, the investor can buy shares at this price and exercise his or her put option to
sell shares for 500p, a profit of 50p per share which, after the initial cost of the option, gives a net profit of 3p
per share or on the contract.
If the share price moves up to 510p by April, the option becomes worthless, and the investor loses his or
her 47p premium.
Options such as this one can be used either to speculate or to hedge on share price changes for a relative-
ly low premium.
3 The terms for an exchange traded option are standardised whereas the terms for traditional options can vary
from contract to contract. It is the development of standardised options that has facilitated the trading of these
instruments.
6 The put – call parity relationship is:
From this:
where
The value of the call option is 30p.
7 The put – call parity relationship is:
E>11 R
f
2 30 5 19 16p
S P C E>11 R
f
2
C 25 10 15 30
S 25, P 15, E>11 R
f
2 10.
C S 1E>11 R
f
2 P
S P C E>11 R
f
2
£30
£470
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704 Appendix B Solutions to selected questions
8 Inputs:
Valuation equations:
Calculations:
The estimated value of the call is 2.12p. This implies that if the call is bought at this price the share price
would have to increase from 38.00p to 40.12p by the end of the six-month period for the investor to break
even.
9 Put–call parity
10 (a) Intrinsic value: The intrinsic value of an option is the difference between the value of the underlying
asset and the exercise price of the option. September 800 call is in the money while the April put is out
of the money.
Time value: Options are more valuable the greater is the likelihood of a significant change in the value
of the underlying asset. If an option expires tomorrow we can certainly say that there is a smaller likeli-
hood of a significant change than if the option expires in six months. Quite simply, the longer the time
to expiration, the more opportunity there is for the price to move and therefore at March the September
option is more valuable than the April option.
(b) Short straddle (written straddle)
E.g. June 800 written premiums 53.5 20 73.5pcall put:
S P C Ke
rt
1 420.5 38.5 50.5 420e
0.5r
1 0.9726 e
0.5r
1 0.5r 0.02776
1 r 0.0555 5.55%
2.12
19.93 17.81
C
1
38.00 0.5245 40.00e
0.10 0.5
0.4682
N1d
2
2 N10.07982 0.4682
N1d
1
2 N10.06152 0.5245
0.07986
30.05129 0.044>30.2 0.70714
d
2
3ln 38>40 10.10 0.20
2
>220.54>0.220.5
0.06156
30.05129 0.064>30.2 0.70714
d
1
3ln 38>40 10.10 0.20
2
>220.54>0.2>20.5
d
2
ln1S
1
>X2 1r VAR1R2>22t
SD1r22t
d
1
SD1r22t
d
1
ln1S
1
>X2 1r VAR1R2>22t
SD1r22t
C
t
S
t
N1d
1
2 Xe
rT
N1d
2
2
SD1R2 0.20
T 0.5
r
f
0
.
10
S
T
38p
X 40p
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