Revision exercise 1 solutions kho tài liệu bách khoa

Finance – Revision Exercise 1 - solutions

Finance – Revision Exercise 1 - Solutions
In the solutions:
PV
=
Present value
PVIF =
Present Value interest factor (Table A.1)
PVA =
Present value of an annuity
PVAIF =
Present value of annuity interest factor (Table A.2)
FV
=
Future value
FVIF =
Future Value interest factor (Table A.3)
FVAIF =
Future Value interest factor (Table A.4)
CF
=
Cash Flow
m
=
number of compounding periods per year
n
=
number of years
r
=

interest rate

1.
PV =

What is the present value of £26,250 to be received at the end of 7
years, if the relevant interest rate is 12%?
FV
(1 + r)n

Or, using the tables
PV =

FV * PVIFr,n (Table A.1)

PV =

26250 * 0.4523

2.
FV =

=

£11,873

What is the future value of £1200 if it earns 14% compounded
annually for 15 years?
PV * (1 + r)n

Or, using the tables
FV =

PV * FVIFr,n (Table A.3)

FV =

1200 * 7.1379

3.
APR

=

£8,565

What is the Annual Percentage Rate (APR, – or effective rate in USA)
if a 10% annual rate is compounded monthly?
=

(1 + (r/m))m

=

(1 + (0.1/12))12

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Finance – Revision Exercise 1 - solutions

4.
PVA

5.
FVA

6.

=

1.104713

=

10.47%

What is the present value of a stream of cash flows of £500 per annum
for 5 years at 8% interest rate?
=

CF * PVAIFr,n (Table A.2)

=

500 * 3.9927

=

£1996

What is the future value of a stream of cash flows of £10,000 per
annum for 8 years at 11% interest?
=

CF * FVAIFr,n (Table A.4)

=

10000 * 11.859

=

£118,590

What is the return offered if you can invest £1200 now for 6 years and
obtain £1900 at the end of that time?

FV

=

PV * FVIFr,n (Table A.3)

FVIFr,n

=

FV
PV

=

1900
1200

=

1.5833

Look up table A.3, 6 years and across until you get to the closest number to 1.583,
then read up for the interest rate = 8%.
Using the calculator it will be 1.58331/6 = 1.0795979
7.
(1 + R)

= 7.96%

If a company’s nominal cost of capital is 12% and inflation is 3%,
what is the real cost of capital for the company?
=

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(1 + N)
(1 + I)

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Finance – Revision Exercise 1 - solutions

(1 + R)
R
8.

=

1.12
1.03

=

1.087378

=

8.738%

What is the nominal return if the real return is 4% and the inflation
rate is 3%?

(1 + N)

(1 + N)
N

=

(1 +R) * (1 + I)

=

1.04 * 1.03

=

1.0712

=

7.12%

What would be the annualised return over two years, if in year two
the real return remained the same, but the inflation rate rose to 5%?
(1 + N)

=

(1 +R) * (1 + I)

=

1.04 * 1.05

=

1.092

Two year return = (1.0712) * (1.092)
(1 + N)2
=
1.16975
this is the 2 year compound growth factor, we
want the annualised return, so we need to take this return to the root of the number of
years;
=

1.169750.5

=

1.08155

Subtract 1 to get the interest rate = 8.155%

9.

If a bond has a coupon of 7.5% and a nominal value of £100, and is
priced at £110.22 in the market, what is the interest yield?
Interest yield (or running yield)

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=

Coupon Payment £
Market price

=

7.5
110.22

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Finance – Revision Exercise 1 - solutions

=

0.068

=

6.8%

10.
If you buy shares in BAe Systems today for 462p, sell them in three
months time for 488p and receive a quarterly dividend of 6p, what is your
holding period return?
What is this return annualised?
Holding period return =

Div. + selling price – buying price
Buying price

=

6 + 488 – 462
462

=

0.069264

=

6.9264%

To annualise, (1 + r)m where m = number of periods in a year

annualised return

=

(1.069264)4

=

1.30719

=

30.719%

11. There is a 20% probability of getting a return of –8% (minus 8%), a 25%
chance of getting a return of 2%, a 35% chance of getting a return of 9%

and a return of 15% has a probability of 20%. What is the expected
return and standard deviation of this investment?
rtn
-0.08
0.02
0.09
0.15

prob
0.2
0.25
0.35
0.2
exp rtn =

rtn * prob rtn - E( r ) rtn - E( r )^2 p* (rtn - E( r)^2
-0.016
-0.1305
0.0170303 0.00340605
0.005
-0.0305
0.0009303 0.00023256
0.0315
0.0395
0.0015603 0.00054609
0.03
0.0995
0.0099003 0.00198005
0.0505
Variance =

0.00616475
Std dev =
0.07851592

Expected return = 5.05%
Standard deviation =
7.851%

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Finance – Revision Exercise 1 - solutions

12. What will a sum of £7000 at 8% compounded semi-annually be worth in
10 years time?
FV =

PV*(1 + (r/m))mn

Where m = number of times compounded per year and n = number of years.
FV =

7000 * (1 + (0.08/2))2*10

=

7000 * (1.04)20

=

7000 * 2.1911

=

£15,337.86

13. What will a sum of £12,000 at 12% compounded monthly be worth at the
end of 2 years?
FV =

PV * FVIFr,n

(Table A.3)

=

12,000 * FVIF0.01,24

=

12000 * 1.2697

=

£15,236

or,
FV =

=

PV * (1 + (r/m))mn
PV * (1 + (0.12/12))12*2

14. If you were offered £10,000 four years from today, or £1000 a year for ten
years and the return you can earn on your investments is 7%, which
would you prefer?
PV

PVA

=

FV * PVIF0.07,4

=

10000 * 0.7629

=

£7,629

=

CF * PVAIF0.07,10

=

1000 * 7.0236

=

£7,023.60

You would prefer £10,000 in four years time.

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Finance – Revision Exercise 1 - solutions

15. How long does it take £14,500 to grow into £23,352 at 10% interest?
FVIFr,n

=

FV
PV

=

23352
14500

=

1.6105

Look up future value factors table (A.3), down the 10% column until you come to
1.6105 and read across for the number of years = 5.
16. A company is expecting £4000 a year from a contract for the next 8 years
(first cash flow at the end of year 1). What is the present value of this
contract if the interest rate is 7%?
PVA

=

CF * PVAIFr,n (Table A.2)

=

4000 * 5.9713

=

£23,885

17. A company wants to redeem a bond at the end of five years time. The
amount outstanding is £250m. If the company can earn 7% on its cash,
how much must it save each year to have the £250m in 5 years time to pay
off the bond?
FVA

=

CF * FVAIFr,n (Table A.4)

CF

=

FVA
FVAIFr,n

=

250
5.7507

=

£43.473m per year

18.
If you can borrow at 9% compounded daily, what is your effective rate of
interest (your true borrowing rate that reflects the daily compounding?
The true borrowing rate will be higher than the annual quoted rate. The calculation is;
(1 + (i/m))m – 1,

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Finance – Revision Exercise 1 - solutions

where i is the annual interest rate and m is the number of compounding periods per
year.
Here we have;
(1 + (0.09/365))365 -1
= (1 + 0.000246575)365 – 1
= 1.09416 – 1
= 9.416%
If you borrowed £15,000 for 5 years, how much would you pay back?
You will use the same formula, but you don’t need to subtract the 1 at the end. This
will give you a compound growth factor that you can multiply your cash sum by. The
formula becomes;
CF * (1 + (i/m))mn where n is the number of years.
£15,000 * (1 + (0.09/365)365*5
=

£15,000 * (1.000246575)1825

=

£15,000 * 1.568225

=

£23,523

You borrow £15,000 over 5 years, at an interest rate of 9% compounded daily, which
means that you will repay £23,523 over the life of the loan. This compares to
repaying £23,079 if the interest was compounded annually (£15k * 1.095).

19.

If you were charged 3% interest per month, how much would you pay
back over the life of a £5,000, two year loan?
Using the framework from Q.18, we have;
CF * (1 + (i/m))mn where n is the number of years.
= £5000 * (1 + 0.03)12*2
= £5000 * (2.03279)
= £10,163.97
Here, we don’t need to calculate the i/m rate (the per period interest rate), because it is
already given in the question (3% per month – it is not expressed as the annual rate

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Finance – Revision Exercise 1 - solutions

here). We are not working out the true annual interest rate, just the amount we need to
repay. The true annual interest rate is 42.576%, which sounds a lot worse than 3% per
month.
20. You have twins that are five years old just now. You estimate that you will
need a fund of £80,000 at the end of six years to be able to fund their
schooling. A financial product is offering a saving rate of return of 6%
per annum.
i)

How much do you have to deposit now to grow to £80,000 over the six
years?
PV =

FV * PVIF0.06,6 (Table A.1)

=

80000 * 0.7050

=

£56,400

or,
PV =
ii)

FV
(1 + 0.06)6

How much would you have to save each year for it to grow to £80,000
at the end of six years?
FVA

=

CF * FVAIFr,n

CF

=

FVA

FVAIF0.06,6

=

80000
6.9753

=

£11,469 needs to be saved each year.

21.
Assume HSBC has a required rate of return of 8%, it has earnings per
share (eps) for fiscal year 2009 of 22.32p. How long will it take for the eps to
double at that required rate of return?
FVIFr,n here is 2.0 (eps doubling), so we just need to look up Table A.3 and 8%,
read down for 2.0 and read off the year = 9 years. Compound growth of 8%
annually will double the value of money in 9 years.
22.
Tesco had a total capital expenditure in 1998 of £841m, in 2002 this was
£2027m, in 2006 it was £2.8bn, in 2009 it was £3.1bn and in 2011 it was £3.7bn.

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Finance – Revision Exercise 1 - solutions

What was the annualised growth rate in capex at Tesco from ’98-02, 02-06, 06-09,

09-11, and for the entire period?
For 1998 – 2002, the growth was;
FV =

PV * FVIFr, n

2027 =

841 * FVIFr,4

FVIFr,4 =

2027
841
=

2.4102

To find out the annualised growth rate, take to the power of 1 over the number of
periods, = 1/4 = 0.25
=

2.41020.25

=

1.24599

=

24.599% compound growth per annum

Tesco

1998
2002
2006
2009
2011

Capex
841
2027
2800
3100
3700

eps dividend
8.64
3.87
11.86
5.60
20.2
8.63
27.5
11.96
33.1
14.96

Tesco growth

rates

1998
2002
2006
2009
2011

whole period
growth:

Capex
841
2027
2800
3100
3700

growth
24.60%
8.41%
3.45%
9.25%
12.07%

Eps
8.64
11.86
20.20
27.50

33.10

growth
8.24%
14.24%
10.83%
9.71%
10.88%

dividend
3.87
5.60
8.63
11.96
14.96

growth
9.68%
11.42%
11.49%
11.84%
10.96%

Dividend and eps growth have picked up considerably in the second period.
Tesco continue to invest heavily in capex to drive the business forward (capex

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Finance – Revision Exercise 1 - solutions

continues to grow although at a slower pace). The rapid growth in capex in the
early period has delivered faster earnings and dividend growth in the second
period.
In the more recent times (2006-2009), capex growth has slowed to only 3.45% per
annum, this may already be slowing the growth of eps (down from 14.24% growth
to 10.83% growth), which ultimately will affect the ability of the company to
grow dividends at the current rate. Through to the 2011 period capex has picked
up again (international investments), but in early 2012, Tesco warned of a poor
UK trading performance, which sent the shares down by almost 20%. Results out
in April 2012.
The dividend payout ratio (div/eps) fluctuates around 45%. It looks like the firm is
very good at generating strong returns from its investment.
23.
Tesco’s eps in 1998 was 8.64p, in 2002 it was 11.86, in 2006 it was 20.20p,
in 2009 it was 27.50p and in 2011 it was 33.1p. What was the growth rate in eps
from ’98-02, 02-06, 06-09, 09-11, and for the entire period?
Same process as Q.22, see table for answer.
24.
Tesco’s dividend was 3.87p in 1998 and in 2002 it was 5.60p, in 2006 it was
8.63p, in 2009 it was 11.96p, and in 2011 it was 14.46p. What was the dividend
growth rate from ’98-02, 02-06, 06-09, 09-11, and for the entire period?
Same process as Q.22 , see table for answer.
25. What would you pay for an asset that produces a level cash stream of
£50,000 every six months for the next 10 years, if your opportunity cost of
capital is 8% (annual rate), compounded per period?
PVA

=

CF * PVAIFr,n (Table A.2)

=

50000 * PVAIF4,20

=

50000 * 13.590

=

£679,500

or, working out the 20 period annuity factor using the formula (the method you
would use if the interest rate was not a whole number);
PVA

=

1-

[1/(1 + r)n]
r

(formula from the top of Table A.2)

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Finance – Revision Exercise 1 - solutions

=

1 – [1/(1 + 0.04)20]
0.04

=

£679,516

26. A company has borrowed £100,000 to buy new machinery. It has to repay
the loan in equal instalments over a 5 year period. If the borrowing rate
is 12 %, what is the annual repayment?
PVA

=

CF * PVAIFr,n (Table A.2)

100000

=

CF * 3.6048

CF

=

100000
3.6048

=

£27,740.80

27. You run a small business and you buy a fleet of four delivery vans. You
arrange the deal through a finance company. They lend you £120,000, but
you have to repay the loan in equal repayments over the next 30 months.
The payments are £4649.72 per month, what is the annualised interest
rate being charged?
PVA

=

CF * PVAIFr,n (Table A.2)

120000

=

4649.72 * PVAIFr,30

PVAIFr,30 =
=

120000
4649.72
25.808

Look up table A.2, 30 periods until you get to 25.808, = 1% per month.
Annualised:
=

(1.01)12
12.6825%

28. Bryan Griggs is a world renowned corporate guru and has signed a £34m
contract with Mckenzies the giant Scottish consulting firm. The contract
will entail Griggs being paid £2m per annum for three years and then
£5m per annum for the next 4 years, and assuming he makes it to the end
of the contract, he will be paid a bonus of £8m. If the opportunity cost of
capital is 8%, what is the present value of the contract?
PVA

£2m * PVAIF0.08,3

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=

£5.1542m

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Finance – Revision Exercise 1 - solutions

PVA

£5m *( PVAIF0.08,7 - PVAIF0.08,3)
= £5m * 2.6293
=
£13.1465m

PV

£8m * PVIF0.08,7

=

£4.668m
£22.9687m

29. A rival company in your sector is offering you a joint venture deal – if you
pay £200,000 to the other company today, it will pay you a cash flow of
£30,000 per annum at the end of year 5 and then for a further 12 years. If
your cost of capital is 10%, would you sign the contract?
There is a 13 year annuity at 10% = 7.1034 (Table A.2), multiply £30,000 cash
flow by this = £213.1. The cash flow starts from the end of year 5, the annuity
will bring the value back to the start of year 5 so we use the discount rate for the
end of year 4 = 0.683, multiply the 213.1 by this = £145,550 – don’t accept the
contract.
Or (17 year annuity – 4 year annuity) * 30,000 = £145,550
30. Your company has the following four options:

-

sell the division today for £10m cash
accept an earn out with four fixed payments of £3.25m each, with
the first payment today
accept royalties of £1m per annum for the next 20 years with the
first payment today
accept £5m now and £10m in 6 years time.

The relevant interest rate is 8%.

1
2
3
4

=
=
=
=

£10m
£3.25 * PVAIF0.08,4*1.08
(PVAIF0.08,20)*1.08
£5m + (PVIF0.08,6 * £10m)

=
=
=
=

£10m
£11.625m
£10.6035m
£11.302m

Choose the second option
31. Your company can buy a machine for £350,000 that will generate cash
savings of £70,000 per annum for the next 6 years as a result of lower
labour costs. If the opportunity cost of capital is 9%, should you buy the
machine?
PVA

=

- £350000 =

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CF * PVAIF0.09,6 (Table A.2)
70000 * 4.4859

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Finance – Revision Exercise 1 - solutions

-£350000 =

£314,013

NPV

-£35,987

=

Don’t buy the machine.
32. You have gone on ‘Who wants to be a millionaire anyway’ and won the
star prize of £1m, but find out that your prize will be paid in equal annual
instalments of £100,000 starting today. You were so confident of winning
that you have already spent £500,000.
You are contacted by a financial company who are willing to buy your stream of
cash flows off you for a single payment today. Your opportunity cost of capital is
8%. They offer you £700,000, do you accept or reject?
A fair price today for the prize would be:
PVA
A.2)

=

CF * PVAIF0.08,10 * (1 + r) [as the annuity starts today]

=

100000 * 6.7101 * 1.08

=

£724,690.80

(Table

You should be able to sell the prize for that amount today, so you would reject the
offer of £700,000.

33. What is a convertible bond and what are the advantages and
disadvantages of it?
It is a bond (usually with a lower coupon than a straight bond would have) which
is convertible into shares at some point in the future. It is a means of raising
finance for companies, at the current time companies are raising cash this way
because they don’t want to sell shares at low current valuations. If the convertible
goes well the shares will rise above the conversion price, and the bondholders will
convert into equity. This will save the company having to find the cash to redeem
the bond.
The downside is that there are now more shareholders – the original shareholders
have been diluted.
The convertible is a way of resolving some of the agency problems. If the
managers enter a risky project and it pays off, the shareholders would get all the
benefits – with the convertible the project is still undertaken, but part of the upside
will be shared with the bondholders.

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Finance – Revision Exercise 1 - solutions

34.

What are the agency problems between shareholders and managers and
how are they resolved? How effective are the resolution techniques you have
identified?
The dangers that managers expropriate too much of the company’s wealth for their
own use. See the black class folder cuttings from FT and Business Week focussing on
the problems that stock options have caused. They are a useful tool but they are being
misused by managers.
Agency problems can range from J. Messier at Vivendi getting the company to buy
him a $17m apartment for him to live in New York, to the rewarding of managers to
the tune of hundreds of millions of dollars for a miserable performance. The banks
took huge risks to generate higher returns that would result in higher bonuses for
themselves – at the expense of the shareholders (and taxpayers).
Managers and shareholders should have their interests aligned so they are moving
towards the same goal. Executive options were supposed to achieve that. The
institutional shareholders need to be stronger in the face of managers. In the UK for
more and more companies, the directors are obliged to buy a significant number of
shares. This means they are more aligned with ordinary shareholders. When the
shares go down, they will feel the pain in the same way that shareholders do. This
wasn’t the case with options.
35.
What justification is there for shareholder wealth maximisation being the
main corporate aim, why not any of the other common goals?
It is the shareholders who own the company, they have put the money in, the
customers haven’t, the community hasn’t, the government hasn’t. The
shareholders are risking their capital in the company and the company has to
operate on the basis that they aim to maximise the shareholder’s return. If the
managers don’t do it – get someone who will, or the company may be taken over.
That doesn’t mean the company puts shareholder wealth absolutely above
everything else. Companies can suffer if they get a bad reputation in the wider
marketplace. Consumers are quick to castigate companies for behaviour that is

lacking in any thought for the wider community. So, in recent years Nike have
been lambasted for having no factories in the US, but continuing to charge huge
price for their products. BP is accused of cutting back on safety standards. Other
companies are caught up in bribery and corruption scandals.
Companies want to be perceived as good corporate citizens and will work to try
and maintain that image. If this means that some extra costs are incurred then so
be it, this is a price the companies are willing to pay, not to suffer in the court of
public opinion. So this means that shareholder wealth maximisation will be
followed but with a softer face.

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Finance – Revision Exercise 1 - solutions

36. How would you describe the key differences between shareholders in a
company and bondholders in a company?
Shareholders can only lose what they put in, they can’t be pursued for the debts of
the company. They have a vote in the running of the company. They are last in
line to receive cash from the company and will usually get nothing if the
bondholders put the company into liquidation. The shareholder’s return will be in
the form of a dividend (not guaranteed) and a capital gain (risky – they have to
realise the gain).
The bondholders do not have any say in the running of the company – they have
no vote. They receive a fixed annual return and their capital back at the end of the
bond’s life.
37. What is the usual order of preference for financing a company and why is
this the case?

Retained earnings, then debt finance, then equity finance.
Retained earnings is top because the finance is available for use without the
scrutiny of the financial markets. It is ‘internal’ finance.
Debt and equity finance is ‘external’ finance as it has to be raised on the capital
markets and the company will face scrutiny because they have to publish a
prospectus detailing the use of the cash. Debt is preferred to equity because there
is a tax break on the interest payments – debt is cheaper. The markets will usually
look at a company that is selling equity as having problems with the debt markets
and this is something of a last resort (the equity market).
38.
How far do listed company responsibilities go? Any further than the
shareholders?
How ethical does a company need to be? Nike used to manufacture all their shoes
outside the US in very cheap labour factories. Consumer pressure (rather than
shareholder pressure) forced the company to shift some production to the US. Does
the company owe the extended community anything? – No, why should shareholders
cash be spent in this way if it doesn’t have to. – Yes, it is an investment by the
company in being recognised as a ‘good corporate citizen’ which will be rewarded by
loyalty and good feelings from customers and other stakeholders. Tesco is a giant
corporation making £2 billion in profits but it is still in touch with its customers (and a
larger supporter of schools and local charities) and is valued and respected. [The big
banks in the UK are perceived as not making a similar effort and don’t instil the same
trust in their customers.]
39 .
How should chief executives be rewarded? Is Larry Ellison worth $700m
a year? Should Stan O’Neal have been given $160m for what he did at Merrill
Lynch? Should the average remuneration package of a top S&P 500 or FTSE
100 CEO run into the tens of millions of dollars per year? What risks are they
taking?

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Finance – Revision Exercise 1 - solutions

BP in January 2005 announced that it was stopping rewarding its staff with share
options, instead shares would form part of their compensation. Shares give an
ownership in the company, options do not. Executives can simply exercise the
options and sell at the same time, pocketing any gain. If options are ‘under the water’
ie, the share price is well below the option exercise price, then they may not be much
use in motivating the executives. Shares will have a value, even if there has been a
fall.
As far as the level of remuneration is concerned that is up to the shareholders. But
they should devise a system that rewards good performance with increases in
remuneration and poor performance with reductions in remuneration. Too many
bosses remuneration goes up whether the company does well or does poorly. Poor
performance should not be rewarded as it has been at a number of banks in the past
year. There is a mismatch between risk and reward for the executives, they have all
the rewards and the shareholders have all the risks.

zfm1573208547.doc 2010

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