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TUTORIAL – 2A
Financial Mathematics
This chapter deals with problems related to investing money or capital in a business venture. When
an individual or a company makes an investment in a business, a return in the form of a profit,
dividend or interest is expected. Before making an investment, the investor would like to know the
return he would receive from such an investment and in what time and whether at regular intervals or
at once at the end of a said period.
In any investment decision, time is an important factor. The longer an investment continues, the
greater will be the return required to the investor.
Let us deal with the all-important form of return known as “interest”. When an amount of money is
invested over a number of years, the interest earned can be dealt with in two ways:
1. Simple interest
2. Compound interest
SIMPLE INTEREST
Interest is the profit return on investment. If money is invested then interest is paid to the investor. If
money is borrowed then the person who borrows the money will have to pay interest to the lender.
The money which is invested or borrowed is called the principal. Simple interest is interest earned in
equal amounts for fixed periods and it is a given proportion of the principal. With simple interest the
principal always stays the same no matter how long the investment lasts.
If a sum of money is invested for a given period of time, then the amount of simple interest which
accrues depends upon the period of time, the interest rate and the amount invested.
To calculate the simple interest the following formula could be used.
I = P * n* r
Where I – Simple interest
P – Principal
n – Period
r – Rate of interest (a proportion)
ICASL - Business School Programme
Quantitative Techniques for Business (Module 3)
2
Alternative Formula
A= P (1+ nr)
Where P = the original sum invested
r = the interest rate (expressed as proportion, so 10% = 0.1)
n = the number of periods (normally years)
A = the accrued amount after n periods, consisting of the original
capital (P) plus interest earned.
Worked Example- 1:
Rs. 700,000 is invested at 7% per annum. How long will it take for the investment to reach
Rs. 798,000?
Solution -1:
The interest element = Rs. 798,000 – Rs. 700,000 = Rs. 98,000
We therefore have P = 700,000, I = 98,000 and r = 0.07
98,000 = 700,000 * n * 0.07
n =
= 2 years
Or alternatively A = 798,000, P = 700,000 and r = 0.07
A= P (1+ nr)
798,000 = 700,000 [1 + 0.07n] 1.14 = 1 - 0.07n n = 0.14/0.07 = 2 years
Worked Example- 2:
Which receives more interest per annum? Rs. 50,000 invested @ 8% per annum or Rs.
60,000 invested @ 7% per annum. What is the annual difference?
Solution -2:
Option -1: P = 50,000, n = 1 and r = 0.08
Interest per annum = 50,000 * 1* 0.08 = 4,000
Option -2: P = 60,000, n = 1 and r = 0.07
Interest per annum = 60,000 * 1* 0.07 = 4,200
Annual difference = Rs200
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COMPOUND INTEREST
This is different from simple interest in that the interest earned is added to the principal which
also attracts interest. If money is invested at compound interest, the interest due at the end of
each period is added to the principal for the next period.
For example, if Rs200,000 is invested to earn 10% interest, after one year the original
principal plus interest will amount to Rs220,000, which will be the principal for the second
year, after two years the principal will become Rs242,000, after three years the total
investment will be Rs266,620 and so on.
Original investment 200,000
Interest in the first year (10%) 20,000
Total investment at the end of one year 220,000
Interest in the second year (10%) 22,000
Total investment at the end of two years 242,000
Interest in the third year (10%) 24,200
Total investment at the end of three years 266,200
The basic formula for compound interest is
A = P (1 + r)n
Where P = the original sum invested r = the interest rate, expressed as proportion, (so 12% = 0.12) n = the number of periods A = the accrued amount after n periods. Worked Example -3:
Rs. 2,500 invested on 1 January 1995 had grown to be worth Rs. 61,482 on 31 December
2009. The equivalent annual compound growth rate (to one decimal place) is
A) 23.8% B) 22.6% C) 22.4% D) 24.2%
Solution -3:
A = Rs61,482 P = Rs2,500 n = 15 years r = ?
A =
61,482 = 2,500 *
= {1 + r}
1.238 = 1 + r r = 0.238 (3dp) r = 23.8 %
Correct answer - (A)
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Worked Example - 4:
An investment quadruples in value in eight years. The annual percentage compound interest
growth rate is closest to
A) 15 B) 19 C) 22 D) 37
Solution:
Let the initial investment be Rs. M, and then M becomes 4M in 8 years.
P = M, A = 4M n = 8 r =?
A =
4M = M *
= {1 + r}
= 1 + r 1 + r = 1.189 r = 0.189 r = 19% (approx.)
Correct answer - (B)
WITHDRAWALS OF CAPITAL OR INTEREST
If an investor takes money out of an investment, it will cease to earn interest. For example, if an
investor puts Rs800,000 into a bank deposit account which pays interest at 10% per annum, and
makes no withdrawals except at the end of year 2 and 3, when he takes out Rs500,000, and
Rs400,000 respectively, what would be the balance in his account after four years?
Rs
Original investment 800,000
Interest in year 1 (10%) 80,000
Investment at the end of year 1 880,000
Interest in year 2 (10%) 88,000
Investment at the end of year 2 968,000
Less withdrawal 500,000
Net investment at the start of year 3 468,000
Interest in year 3 (10%) 46,800
Investment at the end of year 3 514,800
Less withdrawal 400,000
Net investment at the start of year 4 114,800
Interest in year 4 (10%) 11,480
Investment at the end of year 4 126,280
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This can be shown using a table given below: Amount available at the beginning of the year (Rs)
Interest receivable (Rs)
Withdrawals(Rs) Amount available at the end of year (Rs)
800,000 80,000 - 880,000 880,000 88,000 500,000 468,000 468,000 46,800 400,000 114,800 114,800 11,480 - 126,280
CHANGES IN THE RATE OF INTEREST
If the rate interest changes during the period of an investment, the compounding formula must be
amended slightly, as follows.
A = P (1 + r1)x (1+r2) n-x
Where r1 = the initial rate of interest x = the number of years in which the interest rate r1 applies r2 = the next rate of interest n - x = the (balancing) number of years in which the interest rate r2 applies.
Worked Example - 5:
An investor places Rs80,000 into an investment for 10 years. The compound rate of interest earned is
8% for the first 4 years and 12% for the last 6 years. At the end of the 10 years the investment is
approximately worth
A) Rs224,680 B) Rs214,830 C) Rs246,730 D Rs268,120
Solution -5:
A = 80,000
A = 80,000 * 1.084 * 1.12
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A = Rs. 214,830 (approx.)
Correct answer - (B)
REGULAR INVESTMENTS
An investor may be encouraged to add to his investment from time to time when he finds that it is a
profitable venture. In this chapter, the problems encountered are based on uniform time intervals.
That is, it is assumed that the deposits are of equal amounts and made at regular intervals.
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A person invests Rs. 100,000 now, and a further Rs. 100,000 each year for three more years. How
much would the total investment be worth after 3 years if interest is earned at the rate of 10% per
annum?
In problems such as this, we call now ‘year 0’, the time one year from now ‘year 1 and so on.
Year 0 The first year’s investment will be Rs. 100,000 (1.10)4 = 146,410
1 The second year’s investment will be Rs. 100,000 (1.10)3 = 133,100
2 The third year’s investment will be Rs. 100,000 (1.10)2 = 121,000
3 The fourth year’s investment will be Rs. 100,000 (1.10) = 110,000
510,510
The amount available of a regular investment at the end of a given period can be calculated using the
equations given below:
Case - I: (For investment made at the end of the year)
If a fixed amount (Rs. A) is invested at the end of each year for a given period (n years) at a given
rate of interest (r), the amount available at the end of n years (Rs. S) can be expressed as
Where r is the rate of interest expressed as proportion and R = 1 + r
Case - II: (For investment made at the beginning of the year)
If a fixed amount (Rs A) is invested at the beginning of each year for a given period (n years) at a
given rate of interest (r), the amount available at the end of n years ( Rs. S) can be expressed as
Where r is the rate of interest expressed as a proportion and R = 1 + r
Worked Example - 6:
Mr. Perera invested 12 annual payments of Rs2000 into an investment fund earning a compound
interest of 6% p.a. If the first payment was at year zero calculate the value of the fund at year 12.
A) Rs. 38,140 B) Rs. 53,840 C) Rs. 33,140 D) Rs. 35,760
Solution- 6:
S = Rs. 35,760
Correct answer - (D)
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Worked Example - 7:
At the end of each year a company sets aside Rs. 100,000 out of its profits to form a reserve fund. This is
invested at 10% p.a. compound interest. If the first deposit will be made in one year’s time what will be
the value of the fund after four years?
A) Rs. 356,100 B) Rs. 510,000 C) Rs. 464,100 D) Rs. 484,800
Solution- 7:
S = Rs. 464,100
Correct answer - (C)
AMORTIZATION SCHEDULE (For an agreed amount of repayments)
An Amortization Schedule is a statement which shows the outstanding amount of a loan period by
period. Our syllabus deals with the loans and mortgages which involve fixed repayments. The amount
of repayment can be either calculated in which case every repayment from the first to the final is the
same and it covers the principal amount and the interest, or as agreed by the two parties in which
case every payment is the same except for the final payment which is the balance due on the loan at
the end. And the final repayment would be smaller than the other repayments. The calculation of
repayment will be discussed in annuities in detail and here we consider the latter.
Example:
A customer of your firm has purchased a computer costing Rs. 160,000. The customer has paid a Rs.
60,000 deposit and has agreed to pay off the rest of the purchase price by instalments of Rs. 35,000
per year payable at the end of each year. Interest is charged on the outstanding balance at 17% per
year.
a) Draw up a schedule of the payments until the debt is paid off, round your interest calculations to
the nearest Re.
b) How many full payments of Rs. 35,000 are made?
c) What is the value of the final payment
d) How much is paid in total for the computer?
Solution
Year Amount outstanding at
the beginning
Interest
payable
Repayment Amount outstanding at the
end
1 100,000 17,000 (35,000) 82,000
2 82,000 13,940 (35,000) 60,940
3 60,940 10,360 (35,000) 36,300
4 36,300 6,171 (35,000) 7,471
5 7,471 1,270 (8,741) NIL
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a) 4 full payments of Rs. 35,000 have been made.
b) Final payment would be Rs8,741
c) Total amount paid for the computer = Rs. 60,000 + 4 x (Rs. 35,000) + Rs. 8,741
= Rs. 208,741
Practice Questions
1. Sarah invested Rs. 5,000 in a bank deposit account, which pays interest of 9% per annum, added to
the account at the end of the year. She made one withdrawal of Rs1,500 at the end of 3 years. What
was the account (to the nearest Re) at the end of 5 years
A) Rs. 9,511 B) Rs. 5,992 C) Rs. 5,119 D) Rs. 5,911
2. After 15 years an investment of Rs. 60,000 has grown to Rs. 674,000. What annual rate (to one
decimal place) of compound interest has been applied?
A) 17.5% B) 19.2% C) 20.5% D) 17.8%
3. Perera invests Rs700 on 1 January each year starting in 2010. Compound interest of 10% has been
credited on 31 December each year. To the nearest Re, the credit of his investment on 31 December
2019 will be
A) Rs. 12,972 B) Rs. 11,156 C) Rs. 10,456 D) Rs. 2,272
4. An item of equipment currently costs Rs. 4,000,000. The rate of inflation for 3 years is expected to be
8% per annum then 10% per annum for the following 2 years. The price of equipment is expected to
increase in line with the inflation. The price (to the nearest Rs000) after 5 years will be
A) 6,907,000 B) 6,156,000 C) 6,097,000 D) 7,906,000
DISCOUNTING
Discounting is the reverse of compounding. As defined in the previous chapter, compounding
can be stated as “if we invest rupees P now for n years at the rate of r per annum, we should obtain
P (1+r)n in n years’ time”. Discounting is, therefore can be stated as “ if we wish to have rupees A
in “n” years’ time, how much we need to invest now (at year 0) at an interest rate of r in order to obtain
the required sum of money in the future”.
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The formula for discounting is
P = A (DCF) Where DCF =
A is the sum to be received after ‘n’ time periods
P is the present value of that sum
r is the rate of return, expressed as a proportion
n is the number of time periods (usually years)
* The rate “r” is sometimes called a cost of capital.
Note: For given rate of discount and period of time, the DCF value can be obtained either by using the
formula or obtained from the DCF Table.
INVESTMENT APPRAISAL OR PROJECT EVALUATION
A project should be evaluated before it is undertaken. That is, the expenses to be incurred by the
project and the return from the project are studied carefully and then whether the project makes a
profit or not, is analysed. This analysis is known as “project evaluation”.
The expenses incurred for the project at various points of time are known as cash outflows and the
return from the project when considered in terms of cash are known as cash inflows.
The cash outflows and inflows are converted to one point of time (year – 0) since the rupee value now
is not the same as in the future. They are usually converted to present values (or today’s rupee
value). The difference between the two is calculated. This is known as “Net Present Value”, denoted
by NPV.
Interpretation of NPV
1. If the NPV of a project is positive, the project is in profits and hence it is considered to be acceptable.
2. If the NPV of a project is negative, the project makes a loss and hence it is considered to be
unacceptable.
Note: If the present value of all cash inflows equals that of the cash outflows then the project makes
neither profits nor losses. This position is known as “break-even” situation.
Example:
A company purchased a machine now for Rs. 1,000,000. The accountant of the firm estimates that
the machine would contribute Rs. 250,000 per annum to profits for next five years, after which point of
time it can be disposed of, for Rs. 50,000. Determine the NPV of the machine if the rate of discount is
10% per annum.
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You may assume, for ease of calculations that all inflows occur at year-ends.
The NPV calculations are usually shown in tabular form as shown below:
Year Cash flows Discount factor Present value
Rs Obtained from
table A
Rs
0 –1,000,000 1.000 –1,000,000
1 250,000 0.952 238,000
2 250,000 0.907 226,750
3 250,000 0.864 216,000
4 250,000 0.823 205,750
5 300,000 0.784 235,200
NPV = 121,700
THE INTERNAL RATE OF RETURN METHOD
The internal rate of return method (IRR) of evaluating a project is an alternative to the net present
value method. With the NPV method, the net present value of a project at a given rate of discount is
calculated to see whether the project is viable or not. If the NPV is zero, then it indicates that the
return from the project is equal to the rate used for discounting.
The internal rate of return of a project is the rate of interest at which the NPV of the project is zero. It
is the rate at which a project makes neither profits nor losses. It is obtained generally by a trial and
error method as follows:
1. Determine a discount rate at which NPV is small and positive;
2. Determine another (larger) discount rate at which NPV is small and negative;
There is no precise formula for calculating the IRR of a given project. However, it can be calculated
(using a linear interpolation technique) either
a) by formula
b) graphically
Estimation of IRR by formula
The exact formula equivalent of the graphical linear interpolation method is given below:
If a project makes NPVs of N1 and N2 at discount rates of r1 and r2 respectively, the IRR of
the project would be
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IRR =
Or
IRR =
Where N1 is the NPV at the rate of r1%
N2 is the NPV at the rate of r2%
Graphical estimation of IRR
In order to estimate the IRR of a project graphically:
- scale the vertical axis to include both NPVs
- scale the horizontal axis to include both discount rates
- plot the two points on the graph and join them with a straight line
- identify the estimate of the IRR where this line cuts the horizontal axis
Worked Example - 1:
A client of yours has asked for your advice to choose one of the two contracts awarded to him. The
first contract generates net cash inflows of an initial Rs10 million at the start of the contract and then
Rs1 million at the end of each of the 4 years of the contract. The second contract also lasts for four
years and generates net cash inflows of Rs4 million at the end of each year. The client’s current rate
of interest is 8% per annum.
a) Which contract would you advise the company to choose?
b) The rate of interest is forecast to fall to 5% before either of the contracts is due to start. How would
this affect your advice?
Solution - 1
Cash flows with Contract - 1
Year Net cash inflows (Rs million) DCF (8%) Present value
0 10 1.000 10.000
1 1 0.926 0.926
2 1 0.857 0.857
3 1 0.794 0.794
4 1 0.735 0.735
NPV 13.312
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Note: Since the cash flows from year 1 to year 4 are the same the NPV could be obtained using the
shortcut method discussed below:
NPV = Initial cash inflow now + annual cash inflow × Cum DCF (8%, 4 years)
(The cumulative DCF could be obtained from Table – B of Mathematical Tables of ICASL)
NPV = 10 + (1× 3.312) = Rs. 13.312 million
Similarly the NPV of Contract -2 would be given as follows
NPV = Annual cash inflow × Cum DCF (8%, 4 years)
NPV = 4 × 3.312 = Rs. 13.248 million
Hence, Contract-1 would be more profitable.
If the rate of interest changes only the DCF will get changed cash flows remain the same.
At 5% rate of discount
NPV of Contract – 1
= Initial cash inflow now + annual cash inflow × Cum DCF (5%, 4 years)
NPV = 10 + (1× 3.546) = Rs13.546 million
NPV of Contract -2 would be
= Annual cash inflow × Cum DCF (5%, 4 years)
NPV = 4 × 3.546 = Rs. 14.184 million
Hence the decision would be reversed.
Worked Example - 2:
The Finance Division of a company has been asked to evaluate the following proposals for the
maintenance of a new Central Air Conditioning System with a life of six years.
Proposal – 1:
The supplier of the air conditioner will make a charge of Rs. 180,000 per year on a six year contract.
Proposal – 2:
The company will carry out its own maintenance estimated at Rs. 100,000 per annum now, rising at
10% per annum with a major overhaul at the end of year 4 costing an additional Rs. 300,000.
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The discount rate is 12%, and all payments are assumed to be made at year ends.
a) Calculate the maintenance cost for each year, if the company provides its own maintenance
b) Calculate the present value of the cost of maintenance, if the company carries out its own
maintenance
c) Calculate the present value of the supplier’s maintenance contract
d) Recommend, with reasons, which proposal should be adopted
Solution - 2:
If the company provides its own maintenance, the present value of cost maintenance would be as
follows:
Year Maintenance cost DCF (12%) Present Value
1 (110,000) = (110,000) 0.893 (98,230)
2 (121,000) = (121,000) 0.797 (96,437)
3 (133,100) = (133,100) 0.712 (94,767)
4 (146,410) + (300,000) = (446,410) 0.636 (283,917)
5 (161,051) = (161,051) 0.567 (91,316)
6 (177,156) = (177,156) 0.507 (89,818)
Present value of cost of maintenance (754,485)
If the supplier provides maintenance, the present value of cost of maintenance would be
Year Maintenance cost
1 (180,000)
2 (180,000)
3 (180,000)
4 (180,000)
5 (180,000)
6 (180,000)
Since the cash flows are the same the present value of the maintenance could be obtained using the
cumulative DCF table (Table – B of Mathematical Tables of ICASL)
Present value = 180,000 × Cum DCF (12%, 6 years) = 180,000 × 4.111 = Rs. 739,980
If we compare the two options given above, Proposal - 1 (supplier’s maintenance contract) is more
economical and hence it is recommended.
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ANNUITIES
An annuity is a sequence of fixed equal payments (or receipts) made over uniform time intervals
without any interruption. It is an agreement whereby a person pays (or receives) a fixed amount at the
end (or beginning) of each period.
Examples:
- Weekly wages, - Monthly salaries, - Pension scheme, - Insurance premiums, - House-purchase mortgage payments, - Hire-purchase payments
Annuities may be paid
- at the end of payment intervals
This is the most common form of annuity. It is known as an ordinary annuity. This is
where the amounts are payable (or receivable) in arrears.
Or
- at the beginning of payment intervals
This form of annuity can be observed in certain types of investment or insurance
premiums where it will not be deemed to have started until the first deposit or
payment has been made.
PV of an annuity
Case – 1: An ordinary annuity
Consider an ordinary annuity of Rs. A, payable at the end of each year, for n years. If the rate of
interest is r%, the present value (or cost) of the annuity would be as follows:
An ordinary annuity starts from year -1 and it goes on till year - n
Year Cash flows
1 A
2 A
3 A
- -
- -
- -
n A
15
To calculate the present value of a fixed annual cash flow, we can multiply the annual cash
flows by the sum of the DCF factors for the relevant years. This total factor could either be
obtained from the Table
PV = annual amount × Cumulative DCF (r%, n years)
Where Cum. DCF =
Worked Example -3:
The present value of a 5-year annuity receivable which begins in one year’s time at 5% per annum
compound interest is Rs. 60,000. The annual amount of the annuity, to the nearest Re, is
A) Rs. 12, 000 B) Rs. 13, 860 C) Rs. 25, 976 D) Rs. 300, 000
Solution - 3
PV = annual amount × Cum DCF (5%, 5)
60,000 = A × 4.329
A = 60,000/ 4.329 = Rs. 13,860
Correct answer - (B)
Case – 2: A due annuity
Consider a due annuity of Rs. A, payable at the beginning of each year, for n years. If the rate of
interest is r%, the present value (or cost) of the annuity would be as follows:
A due annuity starts from year -0 and it goes on till year (n-1)
Year Cash flows
0 A
1 A
2 A
- -
- -
n - 1 A
PV = amount at year 0 + annual amount × Cumulative DCF (r%, n-1 years)
PV = annual amount × [1 + Cumulative DCF (r%, n-1 years)]
16
Worked Example -4:
A farmer is to lease a field for 6 years at an annual rent of Rs. 50,000, the rentals being paid at the
beginning of each year. What is the present value of the lease at 7%?
A) Rs. 190,000 B) Rs. 200,000 C) Rs. 238,300 D) Rs. 255,000
Solution – 4:
PV = annual amount × [1 + Cumulative DCF (r%, n-1 years)]
PV = 50,000 × [1 + Cum DCF (7%, 5)]
PV = 50,000 × [1 + 4.100] PV = Rs. 255,000
Correct answer - (D)
PERPETUITY
A Perpetuity is the same as an annuity except that payments go on forever. The present value (or
cost) of an annuity for every year in perpetuity can be expressed as
PV =
Where r is the cost of capital as a proportion
Worked Example -5:
AB Ltd wants to undertake a project which costs Rs. 2 million now and generates an annual cash flow
of Rs. 250,000per annum for every year in perpetuity. If the rate of discount is 12% is the project
viable?
Solution – 5:
Year Cash flows DCF (12%) PV
0 (2,000,000) 1.000 2,000,000
1 - ∞ 250,000 1 0.12 = 8.3333 2,083,333 (approx.)
NPV = 83,333
Hence, the project is viable at 12% rate of discount.
17
MORTGAGES AND LOANS
Though the two terms, mortgage and loan are treated alike for calculation purposes, there is a
difference between the two in commerce and business. A mortgage is a method of using property as
security for the payment of debt. That is, it is a type of loan that is secured with real estate or personal
property.
If an amount of money is borrowed over a period of time, one way of amortizing the debt is by paying
a fixed amount at uniform time intervals. This fixed amount includes both repayment of capital and
interest. Generally the bank mortgages or loans are of this type.
There are several ways of calculating the amount of repayments of a mortgage; the following methods
are the ones which could be understood easily. To explain the methods a simple example is shown
below:
Example:
A company obtains a loan of Rs50,000 at 6% interest per annum repayable in equal annual
instalments at every year-end over the next 5 years. Calculate the annual payment necessary to
amortize the debt and prepare the amortization schedule?
Method – 1: Using the DCF Table
Year Amount borrowed Amount settled DCF (6%) Present Value
Loan Repayments
0 50,000 - 1.000 50,000
1 A
2 A
3 A Cum DCF
4 A = 4.212 4.212 A
5 A
--------- -----------
50,000 4.212 A
===== ======
4.212 A = 50,000
A = Rs. 11,871 (approx.)
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Method – 2: Using a formula.
A =
Where A – Repayment amount L – Loan amount n – Period
r – Rate of interest (a proportion) and R = 1 + r
A =
A = Rs. 11,870 (approx.)
Note: A difference of Re1 is due to rounding off.
Amortization Schedule
Year Amount outstanding
at the beginning
Interest
payable
Repayment Amount outstanding at
the end
1 50,000 3,000 11,870 41,130
2 41,130 2,468 11,870 31,728
3 31,728 1,904 11,870 21,762
4 21,762 1,306 11,870 11,198
5 11,198 672 11,870 NIL
Worked Example -6:
A mortgage of Rs. 100,000 is arranged now for 5 years at a rate of interest of 11%. Interest is
compounded on the balance outstanding at the end of each year. The loan is to be repaid by 5 annual
instalments, the first being due after the end of one complete year.
a) Find the gross annual instalments
b) Find the amount outstanding after two complete years
c) If the rate of interest changes to 13% after two complete years, find the revised annual
instalments.
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Solution - 6
(a) A =
A = Rs. 27,057 (approx.)
(b)
Year Amount outstanding
at the beginning
Interest
payable
Repayment Amount outstanding at
the end
1 100,000 11,000 27,057 83,943
2 83,943 9,234 27,057 66,120
(c) A =
A = Rs. 28,003 (approx.)
1. The present value of Rs. 50,000 receivable 3 years from now, assuming a rate of discount of 9%, is
A) Rs. 38,600 B) Rs. 42,000 C) Rs. 63,500 D) Rs. 64,935
2. An investment has a net present value of Rs. 4,000 at 10% and one of – Rs. 2,000 at 15%. What
is the approximate IRR?
3. A job carries a monthly salary of Rs. 10,000, payable in arrears. The net present value of next year’s
salary, assuming an annual rate of interest of 12% is
A) Rs. 89, 000 B) Rs. 94,700 C) Rs. 103,700 D) Rs. 112,460
4. The annual rent of a building is Rs. 120,000 payable in advance at the beginning of each year. At an
interest rate of 14%, the present value of the rental payments is Rs. 531,960. The length of the lease
is
A) 3 years B) 4 years C) 5 years D) 6 years
5. A fixed-interest Rs. 200,000 mortgage with annual interest compounded at 6% each year, is to be
repaid by 15 equal annual payments. The annual repayment will be closest to
A) Rs. 14,133 B) Rs. 20,593 C) Rs. 31,954 D) Rs. 83,400
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6. A company needs to have a balance of Rs. 500,000 in exactly three years from now. It plans to
achieve this by putting 12 equal quarterly sums into a Fund. The first sum will be deposited in
three months from now. The fund attracts compound interest of 2.5% each quarter.
a) Calculate the size of the quarterly sum required for the Fund.
b) Demonstrate simply why your answer is reasonable.
7. A small company is faced with 3 options.
OPTION -1: To trade in the existing computer for Rs. 25,000 and buy a new computer priced at
Rs. 95,000, paying the difference of Rs. 70,000.
OPTION -2: To upgrade the existing computer at a cost of Rs. 30,000.
OPTION -3: To continue with the existing computer as at present for another four years.
OPTION Initial cost (Rs.) Market Value at the end
of 4 years (Rs.)
Annual maintenance & repair cost
payable in advance (Rs.).
1 70,000 25,000 15,000
2 30,000 12,000 20,000
3 0 0 28,000
a) Determine the most economical option based on net present value criterion using a discount rate of 12% per annum.
b) What other non-financial factors should be taken into consideration before making a decision?
8. S Ltd has developed a vehicle security device and is considering manufacturing marketing the
new product. The project will require a Rs20 million investment.
The following estimates of costs and revenues for the product over the 5 years have been made.
Sales forecast:
Year Quantities sold (units sold) Selling price (Rs per unit)
1 5,000 2,500
2 15,000 2,300
3 22,000 2,000
4 15,000 2,000
5 5,000 2,000
New plant and machinery will be purchased at a cost of Rs20 million; this will have a resale
value of Rs1.5million at the end of 5 years.
Labour cost will be Rs400 per unit in year 1, rising by Rs20 per unit in each succeeding year.
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Material costs will be Rs800 per unit for the first two years of production rising by 10% in year
3 and by a further Rs60 in each of years 4 and 5.
Other variable costs will be Rs100 per unit and are expected to remain at that level for the
duration of the project.
Fixed costs of the production will be Rs2 million for the first two years rising by 10% in year 3
and by a further 5% in each of years 4 and 5.
The cost of capital to the company is 12%.
Calculate the net present value of the project and comment on whether the investment should be
initiated?
9. A Rs100,000 mortgage is arranged now for 5 years at a rate of interest of 11%. Interest is
compounded on the balance outstanding at the end of each year. The loan is to be repaid by 5
annual instalments, the first being due after the end of one complete year.
1) Find the gross annual instalments
2) Find the amount outstanding after two complete years
3) If the rate of interest changes to 13% after two complete years, find the revised annual
instalments.
10. An oil well is currently producing annual (year-end) cash flows of Rs50 million. The best geological
evidence suggests that the well has reserves that will last for another 10 years, at the present rate
of extraction. A special pump could be installed, at a cost of Rs75 million, that could double the
rate of extraction but halve the life of the well. After the well had been exhausted, this special
pump could be sold for Rs1 million. The immediate introduction of the special pump is now being
considered. Last year’s earnings have just been distributed and the existing equipment has no
resale value.
You are required
I. to tabulate the annual effect on earnings over the next 10 years of introducing the special
pump;
II. to compare the net present value of the options if the cost of capital to the company is 8%
p.a.;
III. to find whether the pump should be installed if the cost of capital were to be 12%.
