Download Additional Exercises

1.
Calculate the future value of investing £5,000 at a compound interest rate of 5% per annum
for ten years. How much extra is earned on this investment by benefiting from compound
rather than simple interest?
Answer Guidelines
The formula for the future value from basic compound growth is: F t  F 0 ( 1 r ) . In this
example, F0 = £5,000, r = 0.05 and t is 10. The future value is therefore 8144.47.
The formula for the future value if simple interest is applied is F t  F 0 ( 1  tr ) giving a future
value of £7,500. Thus 644.47 extra interest is earned.
t
2.
An investor is trying to work out where they would make the most money. They have £2000
to invest for ten years. Using the concept of APR where possible, which of the following
schemes will make the investor richest?
(a) An annual compound rate of 8.5%
(b) A simple interest rate of return of 12%
(c) A quarterly compounded rate of interest of 9%
(d) A daily compounded rate of interest of 8.5%
Answer Guidelines
With some understanding, ranking of the alternatives will save computational effort. A daily
compounding rate of interest of 8.5% will be superior to an annual compounding rate of the
same value, therefore (a) need not be calculated.
To compare (c) and (d), the APR can be calculated rather than working out the future values
(which will, of course give the correct answers) so that the future value of the better rate can
be compared with the simple interest option.
 i c 
The formula for APR is APR   1    1 . For (c), i=0.09, c=4 and APR = 0.0931 or
 c  
9.31%. For (d), i=0.085, c= 365 and APR = 0.0887 or 8.87%. Hence (c) is better than (d).
The future value of (c), using 9.31% as the effective annual interest rate is, using
t
formula F t  F 0 ( 1 r ) would be £4871.12.
For the simple interest option (b), using the formula for future value, F t  F 0 ( 1  tr ) , the
future value would be£4,400.
The investor would therefore be richest with scheme (c).
3.
A suite of computers was purchased for the sum of £50,000 four years ago.
(a) If a reducing balance depreciation at the rate of 15% is used, what is the current
book value.
(b) The current replacement cost for the suite of computers if £40,000. How much would
need to have put into aside four years ago to save this amount of money? Assume a
rate of interest of 5%
Answer Guidelines
(a) The current book value of an asset is given by F t  F 0 ( 1 r )t where r is the rate of
4
depreciation. Hence the current book value in this case = £50 ,000 ( 10.15 ) =
£ 26100.31
(b) This question requires the present value to be calculated by rearranging the
t
compound interest rate formula, F t  F 0 ( 1 r ) , to take account of the parameters
which are known. In this case, the initial investment is required – F0 – rather than Ft.
Therefore F 0 
F t . F = 40,000 and r = 0.05, therefore F = £32,908.
t
0
t
( 1 r )
4.
A company wants to make a planned investment in machinery in five years time. How much
would be needed to be put into a sinking fund each year to generate £50,000 funds in five
years if interest rates are 6% per annum.
Answer Guidelines
This question is similar to an endowment mortgage calculation. After 4 years, with equal
payments and compound interest, the firm wants to have saved £50,000.
The future value of a saving scheme is given by FV = A + A(1+r) + A(1+r)2+….. A(1+r)t-1. In
this question, we know FV and r and want to identify A.
50,000 = A + A(1+0.06) + A(1+0.06)2 + A(1+0.06)3 + A(1+r)4
50,000 = A{+ A(1+0.06) + A(1+0.06)2 + A(1+0.06)3 + A(1+r)4
50,000 = A{1 + (1+0.06) + (1+0.06)2 + (1+0.06)3 + (1+0.06)4}
and
A
50 ,000
{1  (1  0.06)  (10.06)2  (10.06)3  (10.06)4 }
The denominator of this equation is a geometric series and so the sum can be calculated
using equation 13.4.3 i.e. S n  F *
( 1  Cn )
where C is the common factor.
(1C )
In this question C is (1 + 0.06) or 1.06. Therefore the sum for the five years,
S5,  1*
( 1  ( 1.06 )5 )
50 ,000
 £8865.25 .
 5.64 . Thus A 
5.64
( 1  ( 1.06 ))
5.
A parent decides to buy an annuity to pay for the costs of sending their child to University. It
is proposed to purchase this a full year before the student starts their course. It is assumed
that course fees will be £5000 per year and maintenance costs will be £6000 per year. The
proposed course is of 3 years duration and the annuity rate is 5%.
(a) What is the purchase price of an annuity which would guarantee the full £11,000 at
the beginning of each academic year
(b) What would be the difference if the parent only wanted to purchase an annuity to
cover the fees?
(c) How much extra would the parent save in (a) if the annuity rate changed to 8%
Answer Guidelines
(a) The calculation of an annuity is the same as for a repayment loan where the loan
value is the annuity purchase price and the annual payment the income received.
This is equation 13.5.4 in the book (page 408). For this example, A = 11,000, r = 5%
(0.05) and t = 3. The purchase price of an annuity would be £29,955.73.
(b) Using the same formula, the purchase price of the annuity would be £13,616.24.
(c) The purchase price of the annuity if the interest rate rose to 8% would be £28,348.07.
The saving would therefore be £1,607.66.
Note: Excel can be used to calculate these values. Care must be taken to ensure the
correct function is used and the correct parameters inserted. For this question the PV
function can be used and Excel help should be carefully consulted to identify how the
parameters should be entered.