Financial Mathematics

Objectives

After studying this Note, you will be able to:

discuss the importance of mathematics in financial calculations

describe the importance of time value of money

elucidate the concept of annuities and its uses

calculate the EMI of a loan

calculate the return on investment

explain the use of Excel as a tool for financial calculation

Introduction

Finance and mathematics are interrelated disciplines and share many common logic both at

the basic and advance level. Finance as a discipline is very quantitative in nature and use

mathematics as a tool to explore the inner depths since the use of quantitative techniques is

very deep. In this unit we will go through the various concepts of financial mathematics

related to time value of money, interest rates the effect of compounding and discounting,

effective, nominal and real rates of return and their impact on accumulation of the corpus.

TIME VALUE OF MONEY

That money has time value is the central concept in all the Mathematics, related to finance. A

dollar in hand today is worth more than a dollar in hand tomorrow is the basic premise behind

the investment decisions in the business world. The time value of the money comes on the

back of the assumption that the money is able to generate a positive rate of return. So in

essence we can say that all of us would like to have a cash inflow as early as possible and a

cash outflow as late as possible.

Let us take an example:

If you have 100/- and you invest it for a year and the rate of return is 10% per annum, what is

the amount you are going to have at the end of one year?

To calculate the above amount is not rocket science, we all are doing in since our middle

school using the formula of compound interest which says:

A=P(1+R)T

100

Where A is the final accumulated amount, P is the Principal, R is the rate of interest and T is

the time for which the amount is invested, so here in the case discussed P is 100/- R is 10%

and T is 1 year

The moment you apply the formula, you will get the A as 110.This is middle school

Mathematics. Now, in the terms of financial mathematics, we will say that Rs. 110 is the

future value of Rs. 100 at 10% rate of interest or in reverse we can say that Rs. 100 is the

present value of Rs. 110 receivable one year hence at 10% rate of return.

So we can see in the above illustration that to get to the future value we have to take the

present value and apply the formula for compound interest; the process is called

compounding and thus we can say that we find future value by compounding. The general

formula for compounding can be:

Future Value = Present Value (1+R)T

100

Where R and T are the rate and time period respectively.

Similarly, let us take another instance:

If you are going to get Rs. 112 one year down the line, the rate of return is 12%, what is the

current value of the same amount in your hand? In other words, we can frame the same

question as, ‘how much money should you invest today to get Rs. 112 at the end of the year if

the rate of return is 12%?’

How do you do it? If you had a good look at the last illustration, then you would be

compelled to use the formula, that was used in the last illustration in reverse, and you would

be correct, only the formula that would be used is A.

(1+R)

T

100

Here A is the amount in future, R is the rate of return and T is the time period. If we apply the

formula to the question given taking A=112, r=12% and T=1 year, then the answer is Rs.

100, i.e. 100 is the present value of 112 one year before. Or we can say: if you want to get

112 in a year’s time at 12% rate of return, then you will have to invest Rs. 100 today.

We have used a formula, which is reverse of compounding, to find the present value. This

process is known as discounting and the general formula for present value is:

Present Value= Future Value

(1+R)T

100

Here R and T are rate and time period respectively.

Now let us take a step forward, now that you have understood the concept of future value and

present value, let us take another example

If John is going to receive 100$ today and 100$ one year later and 100$ 2 years later and if

the interest rate is 10% per annum, how much money John will have at the end of 3 years?

The difference between the problem above and the problems we have faced in the earlier

questions is that there was a single value in the earlier problems and in these problems there

are many similar values happening at regular intervals of time.

The answer is simple, we need to calculate the value at the end of the period; that means, we

will need to calculate the future value, while the concept of the future value was simple in the

earlier example that we took, here there is a little difference.

The first 100$ will have an opportunity to compound for 3 years, while the second 100$ will

have the opportunity to compound for 2 years and the last 100$ which John has received will

have the opportunity to compound only for one year, so we will have to apply the formula

three times and add the result.

100(1+10)3

+100(1+10)2

+100(1+10)1

= 364.1

100 100 100

So we can say that the future value of the schedule of payment will be Rs. 364. In other

words, John is going to have Rs. 364.1 at the end of three years.

Then we move to the same thing in the context of present value. When there is present value

of multiple values, we have to calculate the present values separately and add all of them. For

example: if Kevin is going to receive Rs. 100 one year from now, Rs 100 two years from

now and Rs. 100 three years from now, what is the present value of this payment schedule in

the hand of Kevin today at the rate of 10%?

How do we calculate it? Simple, find the present values separately and add all of them.

This leads to 100 + (1+10)2

+ (1+10)

3 which result in 90.91+82.64+75.13=248.68.

100

100

Now this might look like a little more cumbersome and calculation oriented operation, which

it is if we look at it mathematically. While it is very difficult to find the short cuts or easier

methods, fortunately in this case we can go for the technical aids which can help us solve

these problems, the case in point being a financial calculator or financial functions of MS

Excel.

In this unit, we will be covering the financial functions of Microsoft excel in detail and we

will be looking at various complex financial problems which we will be able to solve easily

with the help of excel. We will be looking at the range of problems from calculating simple

EMIs of the loans to calculating the NPVs, IRRs and the payback periods of the projects. But

before we set ourselves in calculations with excel, we would look at some basic rules and the

tools which will help us use excel in correct and effective way.

Given below are the some basic concepts related to excel that we must keep in mind.

Time line

Time line is an imaginary line on which the periods and payments are marked. Given below

is a sample time line.

0 1 2 3 4 5 6 7 8 9 ……………………………n

Time 0 is today, Time n is the end of the periods, tick marks occur at the end of every period

or the beginning of the next period, the periods should be of equal duration, e.g. if the period

is weekly, then all the periods should be weekly only; and if the period is yearly, then all the

periods should be yearly only.

Before starting to solve any problem on financial mathematics using excel spreadsheet, we

should make time line first. After drawing the time line, we should mark the cash flows on

the ticks which occur at the end (or beginning of the period). Let us look at some of the

examples of the time lines, as used in the world of financial mathematics.

Time line depicting the payment of Rs. 100 two years hence

Time line depicting payment of Rs. 100 every year for next 3 years

Time Line for a payment of Rs 100 two years from now

Cash Flow Sign Convention

Before marking the time line with signs, we should be aware of the cash flow sign

convention. The convention simply says that if the money is coming to you, then you must

mark it positive; and if the money is going away, towards whatever cause, it should always be

marked as negative in the Time Line. This is very important, as not adhering to time

convention can lead to the very different results from what we would have been intending

and may result in a bigger mistake in our working environment. So, in simple words, we can

say that all the inflows of money will carry a positive sign and the money outflows will carry

a negative sign.

For example: If Reena has Rs 1 lakh today, which she is ready to invest, and starting from

today she is ready to invest 10000/- at the end of the year for next 5 years, this transaction

will be depicted in the timeline like this.

-100000 -10000 -10000 -10000 +FV Rate=10%

In this case, we can see that the money she has today is 100000 at time 0. So, it is the present

value. Since she is going to invest it, the money is going to go away from her presently;

therefore, we have taken it as –ve. Now after the end of the first, second and third periods,

she is going to invest Rs. 10000. Since the money is going to go away from her in those

times, we have taken a –ve sign on those cash flows. Now we come at the end of the time

line. Whatever amount she is going to get at the end of the period if the interest rate given

here is 10%, the amount is called Future Value. Since she is going to receive this amount,

there is an inflow and we have taken the amount as positive. Thus, you are going to observe

in most of the cases when we are calculating as per the investing situations, the present value

would be negative and the future value would be positive. So the money going out from our

hands, may it be for investing is taken as outflow and is inputted as negative in both time line

and excel sheet; and the money coming into our hands, may it be a loan, is taken as a positive

in both timeline and the formula in the excel sheet.

Time lines are very helpful in deciding which function of Excel should be employed to solve

the problem on hand based on its position on the time line, this would be very helpful in the

initial days with excel and it will prevent you from committing the normal mistakes that you

are prone to do in excel.

But before you start doing calculations in Excel, you must be familiar with the function tab of

excel; you have to go to functions and in the list of the functions of Excel, you will have to

select the category of the function which you would want to use.

The decision on which formula needs to be used depends on the position of the cash flow on

the number line.

1. If you have to find the value at the Time 0 (i.e. today), you have to use the formula of

present value (PV) and you are going to use PV function from the financial functions

of MS Excel.

2. If you have to find the value of the cash flow at the end of the time line, you have to

use the formula of future value (FV), and you will have to use the FV function in

Excel.

3. If you have both present value and future value but you need to calculate the equal

value at each tick then you need to calculate payment at each period (PMT), and you

will have to use the PMT function of the financial function of Excel.

4. If you have present value, future value and payment available, but you have to

calculate the number of ticks that will be there in the number line, then you will have

to calculate the number of periods (NPER) and you will have to use the NPER

function from financial functions of Excel.

5. If you have all the things mentioned in point one to four available with you and you

do not have rate, then obviously, you will have to calculate the rate (Rate). You will

have to use the rate function from financial functions of Excel.

6. In addition to this, the important formulae that you are going to use in excel are NPV,

IRR and Power.

a. NPV is used, when you have to calculate, the value today of a multiple payment

cash flow schedule, where the payment is not uniform, i.e. the amount in each

payment is not equal.

b. Similarly IRR is used, when there is a schedule of incoming and outgoings in

terms of payment and you have to find the resultant return on the inputs,

resulting from the outputs, that you are able to get.

Now, we have look at the major formulas in the excel sheet, which the time line is going to

help us to identify. The application of the formula on the timeline depends on the position of

the cash flow in the time line.

If we come to a conclusion after drawing the time line, that we have to find the value

at time 0, then we have to calculate the present value.

If we have to calculate the value at the end of the time period, then you have to

calculate the future value.

So, the next question comes to us, how do we calculate all of the mentioned functions?

For the point No.1, the formula that we are going to apply is called PV. The path to the

formula is as follows:

Home/functions/financial/PV.

And once you open the screen

Remember this sequence:

=XXX(Rate,NPER,PMT,PV,FV,Type)

To find FV ; FV will come in place of XXX

= FV(Rate, NPER, PMT, PV)

Rate stands for rate of return (clearly mentioned or expected)

NPER is number of periods (Number of periods of equal duration)

PMT is payment (equal payment in all the time periods)

FV is future value.

Type is a binary function which takes only two values 1 or 0.

If the payment is made at the start of the period, the value that needs to be taken is 1; if the

payment is made at the end of the period, then the value of the type function is 0. This is the

generic syntax of most of our formulae, since we can see that there are five fields out of

which, four would be known to you in most of the situations, and you will have to calculate

the fifth field, the value for which is not known.

CALCULATION OF ANNUITIES

The next concept that we come across while dealing with the problems of time value of

money, is annuity. Annuity is a regular payment, which one receives or gives according to a

schedule for a fixed period of time. If the payment is made at the beginning of the period,

then the annuity is called annuity due; if the payment is made at the end of the period, the

annuity is called regular annuity. While calculating in both the scenarios in MS Excel, the

difference will be in the field of type, which will be 1 in one case and 0 in the other case.

LOAN

AMORTIZATION

A loan is a typical borrowing arrangement, in which an amount is borrowed in lump sum and

it is returned in the equated installments, which are decided based on the tenure of the loan

along with interest rate at which the loan is secured. We can use the PMT function of Excel

to calculate the EMI of the loan.

Let us take an example: Pushpa borrowed 20 Lakk Rupees from ICICI bank to purchase a flat

in Bangalore, the rate of interest was 9 % and the tenure of the loan is 20 years. Calculate the

EMI that she will have to pay every month

Spreadsheet Solution

Use PMT function Rate =9%/12 NPER=240 Months PV= 20,00000

FV=0 Type =0

Doing this you will come to the conclusion that she will have to pay a monthly installment of

Rs. 17994/-

Now comes the second question, she wants to know, how much interest she is going to pay

over the currency of the loan.

Well, excel has two additional functions in the form of IPMT and PPMT, which can give you

interest or principal for specific number of installments, what we would ideally suggest that

you should be aware of loan amortization schedules, An amortization schedule is a typical

division of loan installment into interest and principal.

We will solve pushpa’s problem subsequently, let us first solve a problem of smaller

duration. Let us suppose we have to make an amortization schedule for a 3 year loan at 12%

for Rs. 20000.

The first step would be calculating the EMI, then we will divide the EMIs into interest and

principal for all 36 months.

After applying the PMT function as explained in the last example, we can say that the EMI

for this loan will be 664.16 per month. Now we have to create a schedule for this,

Follow the following steps and you will be able to make and amortization schedule for the

loan

1. Take a Spread sheet and mark a column as serial no. and number it from 1 through

36( as it is a 36 months loan)

2. Write the amount of EMI against each numbered cell( in this case 664.16, You can

write it in the first cell and then drag it for the rest of the cells)

3. Now in the next column, you will have to write the interest followed by the principal

part of the installment in the next column. This is how you can calculate the interest,

Since it is a reducing balances loan apply this formula in the relevant cell ,The

interest for the first month would be

20,000 x 12%= 200

12

S.No EMI Interest Principal Residual PR.

1 Rs. 664.29 200 Rs. 464.29 Rs. 19,535.71

2 Rs. 664.29 Rs. 195.36 Rs. 468.93 Rs. 19,066.78

3 Rs. 664.29 Rs. 190.67 Rs. 473.62 Rs. 18,593.17

4 Rs. 664.29 Rs. 185.93 Rs. 478.35 Rs. 18,114.81

5 Rs. 664.29 Rs. 181.15 Rs. 483.14 Rs. 17,631.67

6 Rs. 664.29 Rs. 176.32 Rs. 487.97 Rs. 17,143.70

7 Rs. 664.29 Rs. 171.44 Rs. 492.85 Rs. 16,650.86

8 Rs. 664.29 Rs. 166.51 Rs. 497.78 Rs. 16,153.08

9 Rs. 664.29 Rs. 161.53 Rs. 502.76 Rs. 15,650.32

10 Rs. 664.29 Rs. 156.50 Rs. 507.78 Rs. 15,142.54

11 Rs. 664.29 Rs. 151.43 Rs. 512.86 Rs. 14,629.68

12 Rs. 664.29 Rs. 146.30 Rs. 517.99 Rs. 14,111.69

13 Rs. 664.29 Rs. 141.12 Rs. 523.17 Rs. 13,588.52

14 Rs. 664.29 Rs. 135.89 Rs. 528.40 Rs. 13,060.12

15 Rs. 664.29 Rs. 130.60 Rs. 533.69 Rs. 12,526.43

16 Rs. 664.29 Rs. 125.26 Rs. 539.02 Rs. 11,987.41

17 Rs. 664.29 Rs. 119.87 Rs. 544.41 Rs. 11,443.00

18 Rs. 664.29 Rs. 114.43 Rs. 549.86 Rs. 10,893.14

19 Rs. 664.29 Rs. 108.93 Rs. 555.35 Rs. 10,337.79

20 Rs. 664.29 Rs. 103.38 Rs. 560.91 Rs. 9,776.88

21 Rs. 664.29 Rs. 97.77 Rs. 566.52 Rs. 9,210.36

22 Rs. 664.29 Rs. 92.10 Rs. 572.18 Rs. 8,638.18

23 Rs. 664.29 Rs. 86.38 Rs. 577.90 Rs. 8,060.28

24 Rs. 664.29 Rs. 80.60 Rs. 583.68 Rs. 7,476.59

25 Rs. 664.29 Rs. 74.77 Rs. 589.52 Rs. 6,887.07

26 Rs. 664.29 Rs. 68.87 Rs. 595.42 Rs. 6,291.66

27 Rs. 664.29 Rs. 62.92 Rs. 601.37 Rs. 5,690.29

In the

amortization chart above, we can see that, as the installment amount is constant over the

period, the amount of interest in the installment keeps on decreasing, while the amount of

principal in the installment keeps on increasing.

Thus if we close the loan, a little too prematurely, we will still giving up on majority of the

interest. Given below in the chart is the movement of interest and principal in a typical

amortization schedule.

INFLATION ADJUSTED INTEREST RATES

Inflation adjusted rate of return is a measure that accounts for the return periods inflation rate.

Inflation adjusted returns shows the return on an investment after taking away the effects of

inflation. To calculate Inflation Adjusted Return, use the following formula:

Inflation Adjusted Return = (1+Return)/(1+Inflation Rate)-1

Let us take an example:

28 Rs. 664.29 Rs. 56.90 Rs. 607.38 Rs. 5,082.90

29 Rs. 664.29 Rs. 50.83 Rs. 613.46 Rs. 4,469.45

30 Rs. 664.29 Rs. 44.69 Rs. 619.59 Rs. 3,849.86

31 Rs. 664.29 Rs. 38.50 Rs. 625.79 Rs. 3,224.07

32 Rs. 664.29 Rs. 32.24 Rs. 632.05 Rs. 2,592.02

33 Rs. 664.29 Rs. 25.92 Rs. 638.37 Rs. 1,953.66

34 Rs. 664.29 Rs. 19.54 Rs. 644.75 Rs. 1,308.91

35 Rs. 664.29 Rs. 13.09 Rs. 651.20 Rs. 657.71

36 Rs. 664.29 Rs. 6.58 Rs. 657.71 Rs. 0.00

A bond that pays interest annually yields a 8.05 percent rate of return. The inflation rate for

the same period is 4.15 percent. What is the real rate of return on this bond?

Solution: Real rate = 1+nominal rate/1+inflation rate-1*100 = 1.0805/1.0415-1*100= 3.74%

Summary

Financial Mathematics is a branch of mathematics, that help you understand the

functions and formulas of mathematics relevant to finance.

A dollar in hand today is worth more than a dollar in hand tomorrow is the basic

premise behind the investment decisions in the business world.

The formula of compound interest is: A=P(1+R)T

100

Future Value = Present Value (1+R)T

100

Present Value= Future Value

(1+R)T

100

Time line is an imaginary line on which the periods and payments are marked.

The cash flow sign convention states that if the money is coming to you, then you

must mark it positive; and if the money is going away, towards whatever cause, it

should always be marked as negative in the Time Line.

Annuity is a regular payment, which one receives or gives according to a schedule for

a fixed period of time.

A loan is a typical borrowing arrangement, in which an amount is borrowed in lump

sum and it is returned in the equated installments, which are decided based on the

tenure of the loan along with interest rate at which the loan is secured.

Inflation adjusted returns shows the return on an investment after taking away the

effects of inflation. Inflation Adjusted Return = (1+Return)/(1+Inflation Rate)-1

.1. Akbar expects to pay out the following in the next few years: End of Year 1 Rs.25,000 End of Year 2 Rs.22,000 End of Year 3 Rs.19,000 End of Year 4 Rs.16,000 End of Year 5 Rs.13,000 If Akbar desires to be able to pay out these amounts how much should he have now, assuming an annual rate of 9%? a) 75908 b) 78905 c) 79058 d) 80868 2. Geeta invests Rs. 2000 at the beginning of each month for 48 months. Her rate of return is 8% p.a. The investment’s value at the end of the said period will amount to_____________ A.1,10,000 B. 1,12,700 C. 1,15,000 D.1,11,500 3. Neeta wants to accumulate Rs.1,50,000 in three years time for a one month trip to the USA. Assuming she can get an 8% annual return on her investments, compounded quarterly, how much must she invest today in order to achieve her goal ? A. 115258 B. 116489 C. 117452 D. 118274 4 .A bank deposit of Rs. 25000 will earn an interest of Rs._____ at the end of one year, if it earns 10% p.a. compounded every month. A. 2599 B. 2617 C. 2745 D. 2799 5. A bank fixed deposit which promises to double your money in 8 years and 7 months is paying you interest @ __________% p.a. A. 7.55 B. 8.19 C. 8.41 D. 8.76 6. Gopinath has Rs.2,50,000 now in his account. If he is withdrawing Rs.25000 from this account at the end of each year, for how many years can this account last assuming an annual interest rate of 5% ? A.13.51 B.13.76 C.14.06 D.14.21 7. A Rs. 100 par value bond bearing a coupon rate of 10% will mature in 3 years. What is the value of the bond if the discount rate is 12%. A.94.20 B.94.70 C.95.20 D.95.70 8. Ajai and Kajal intend to go on an Alaskan cruise twelve years from now, which will cost Rs. 10,00,000 each. Assuming their total savings and investments is worth Rs.200,000 now and they can earn an annual interest of 12%, how much do they have to save at the beginning of each year to achieve this goal? A.44166 B.44661 C.45166 D.45661

9. Chetan borrowed Rs.5,000 five years ago. Assuming that he has to repay Rs.6,500 now, how much interest rate was he charged? A. 5.28 B. 5.39 C. 5.93 D. 5.82 10. Roger purchased an acre of land for Rs.8,00,000. He sells the land for X amount in 12.50 years. He estimated that the average annual rate of return on the antique was 20% compounded quarterly. Find “X”. A.9072890 B.9173920 C.9275680 D.9347820 11. Jacob purchased a house for Rs.19,00,000 for which he takes a loan of Rs. 10,00,000 to be repaid in 15 years at 8.25% per annum compounded monthly. How much is his approximate EMI? A.9500 B.9600 C.9700 D.9800 12. Sulakshana made an initial investment of Rs.20,000 in a Mutual Fund. If she subsequently invested Rs.14,000 at the end of each month into the same fund for the next 2 years, what is the value at the end of 5 years, of her investment assuming an annual return of 10% compounded monthly? A.501580 B.528180 C.532080 D.536780 13. Jigar has a Rs.1200 monthly loan payment, which is due to complete 5 years from now. Assuming that Jigar wants to repay the loan now in a lump-sum, what is the amount he should pay assuming annual discount rate of 10% compounded monthly? A.55498 B.56478 C.57878 D.58598 14. For the past 5 years, Leela has saved Rs. 45,000 at the end of each year in an account earning 7% interest compounded annually. She will continue to do so for another 5 years. What will be the balance of the account at the end of 7 years from now? A.676830 B.711830 C.726830 D.759830 15. Robert expects to receive an annual payment of Rs.2,50,000 from an inheritance fund at the beginning of the year for 8 years. What is the current worth of this fund assuming the fund is invested earning an annual rate of return 8.25% and the first payment will only be made 5 years from now? A.1028013 B.1036413 C.1048073 D.1049027 16.If Mohan has saved Rs.5000 each month end for past 6 years, what is the annual interest rate that he has earned from the account if the balance is now Rs.10,00,000, assuming monthly compounding? A.20 B.25 C.30 D.35 17. Jeff buys a used car paying$ 1500 down and $182.50 a month for 3 years A.What was the cash price of the car,if interest rate on the loan is j12= 18%?

B.What was the total interest on the loan? 18. Find the present value of an annuity of $380 at the end of each month for 3 years,at (a) J12=12% (b) j12-10.38% 19. An annuity with payments at the end of each month pays $200 for 2 years,then $300 for the Next year and then $400 for the following 2 years. Find the discounted value of these payments at j12=10%. 20.Jackie has made semiannual deposits of $500 for 5years into a saving fund paying interest at j2=6.1/4%. What semiannual deposits for the next 2 years will bring the fund up to $10,000? 21. A television set worth $780 may be purchased by paying $80 down and the balance in monthly installments for 2 years. Find the monthly installment if the dealer charges 15% compounded monthly, and the first installment is due in one month. 22. Mrs. Wong changes employers at age 46.She is given $8500 as her vested benefits in the company’s pension plan. She invests this money in a registered retirement savings plan paying j1=8% and leaves it there until her ultimate retirement at age 60.She plans on 25 annual withdrawals from this fund,the first on her 61st birthday. Find the size of these withdrawals. 23. A parcel of land,valued at $35,000, is sold for $15,000 down.The buyers agrees to pay the balance with interest j12=12% by paying $500monthly as long as necessary,the first payment due 2 years from now. Find the number of $500 payments needed and the size of the concluding payment one month after the last $500 payment. 24. A $5000 loan is repaid by 24 monthly payments of $175 each,followed by 24 monthly payments of of $160 each. What interest rate j1 is being charged? 25. You are offered a loan of $10,000 with no payments for 6 months,then $600 per month for 1 year, and $500 per month for the following year. What annual effective rate of interest does this loan charge? 26. A loan of $20,000 is to be repaid by annual payments of $4000 per annum for the first 5 years and payments of $4,500 per year thereafter for as long as necessary. Find the total number of payments and the amount of the smaller final payment made on year after the last regular payment. Assume an annual effective rate of 18%. 27. A store offers to sell a watch for $55 cash or $5 a month for 12 months. What nominal rate j12 is the store actually charging on the instalment plan, if the first payment is made immediately? 28..A “Fly by Night” Used Car Lot uses the following to illustrate their 12% finance plan on a car paid for over 3 years. Cost of the Car 12000.00 12% finance charge 4320.00 Total Cost 16320.00 Monthly payment=16320/36=$453.33