The time value of money For civil 7th, computer 7th and civil 4th

Concept of the Time Value of Money (TVM)

The Time Value of Money (TVM) is a fundamental financial principle saying that the money available today is more

valuable than the same money to be received in the future due to its potential earning capacity.

Why Does Money Have Time Value?

a. Interest earnings – Money today can be invested to earn interest or returns.
b. Inflation – Prices rise over time, reducing future purchasing power. Purchasing power of money goes
on decreasing.
c. Risk & Uncertainty – Future cash flows are uncertain; having money now is safer.
d. Consumption Preference – People prefer immediate consumption over delayed fulfilment.
The time value of money is the concept to learn and understand the value of cash flows ie inflow or out flow occurs at
different point in time. Some writers like to call mathematics of finance to time value of money. It has great importance
in financial decision making. It's because the effects of financial decision remain for long years. For example, the benefit
of investment made in the fixed assets remains up to the life period of that assets. The interest, dividend and so on
should be paid in future on the funds raided form long term sources. The financial decision is, therefore, made by
comparing cash outflows or expenditure and cash inflows or revenues.

Importance of time value of money

Money has interest cost due to various reasons like inflation, different risk and time preferences. Most financial decision,
personal as well as all business involve time value of money considerations. The important of the time value of money
could be stated as follows.
a. Financial decision: it is essential to know the concept of time value of money to make appropriate financial decision.
The more latter the expected benefit is received, the less is its present value. On account of this, the concept of time
value of money is considered a powerful tool of evaluation the future returns.
b. Valuation of securities and other assets: the concept of time value of money facilitates the managers and individual
investors for valuation of securities and otter assets before they go for making investment decision about different
securities and assets. For this purpose, different security valuation models have been developed and used in time
value of money framework.
c. The cost of capital: a company needs the capital for long term investments. For this it is necessary to calculate the cost
of capital. For the calculation of cost of capital, the models of time value of money are used.
d. Working capital management: the concept of time value of money is also important for working capital management.
This is used in evaluating proposed credit policies and the firm’s efficiency in managing current assets and liabilities.
e. Lease analysis: a company can lease or purchase the fixed assets. For the analysis of lease versus purchase decisions
the concept of time value of money is used. It helps to calculate the present value of cost of leasing and cost of
buying. On the basis of the present values of these two alternatives, the company can decide on appropriate sources
of financing.

1
 Techniques of computing time value of money

For the meaningful comparison of cash flows at different periods of time, they should be converted into a same point
of time or should be expressed in common denominator. There are two techniques or methods of making such
transformation. They are future value analysis or compounding and present value analysis or discounting technique.

1. Future value analysis or compounding technique

The value of a payment or serried of payments at a given time in future, when compounded at an appropriate interest
rate is called future value. The future value is a total of the sum of present value and interest. The process of going
from today's value ie present values to future values is known as compounding.
This section deals with the process of computing future or compounded value. Generally, the future value at the end of
specific period n is calculated as

FVn= PV (1 + i) n
Using table
FVn= PV (FVIFi%,n years)
Where
FVn = future value at the end of n period (year)
PV = Present value
i= interest rate
n =numbers of period (years)
FVIFi%,n years = future value interest factor for i (interest) and n (period)

Example
Suppose Rs 100 deposited in bank account. The bank provides 5 percent per annum compound interest. How much
rupees will be after five years?
Solution
Given
PV = Rs 100
i= 5% = 0.05
n= 5 yea r
We have
n
FVn= PV (1 + i)
5
= 100(1+0.05) = Rs 127.63
Alternatively
We can use also table
FVn= PV (FVIFi,%nyears )
FVn= 100(FVIF5%5years)
= 100x 12.763 = Rs. 127.63
1. Present value analysis or discounting technique
The value today of a future payment or series of payments discounted at the appropriate discount rate is called present
value. In other words, present value is the value today of a future cash flow or series of cash flow. The process of finding
present value is called discounting.
In this section an attempt has been make to explain the process of discounting or calculating the present value.
Generally, the present value at the end of specific period n is calculated as

2
 FVn
-n
PV = ------------ or PV =FV (1 + i)
(1 + i) n
Using table
PV= FV (PVIFi%nyears )
Where
PVIFi%nyears = present value interest factor for i (interest) and n (periods)
The process of finding out present value is just opposite to compounding. The process converting future amount to
present value is known as discounting. It is because the present value is always less than future amount. The rate of
interest used for this purpose is called discount rate.
The present value depends on the rate at which future rupee is discounted. If discounted at 4%, the present value of Rs
100 to be received after 5 year becomes RS 82.19. The present value becomes Rs 78.35 at 5% and Rs 74.73 at 6%.
Example
What is the present value of money Rs 1000 receivable 6 years hence if the rate of discount is 10 percent?
Solution
Given
FV = Rs 1000
i = 10% = 0.10
n = 6 year
PV =?
We have
FVn
PV = ------------
(1 + i) n
= 1000/ (1+0.10)6 = RS 564.50
Alternatively
Using table
PV= FV (PVIFi%nyears)
PV = 1000(PVIF10%6years)
PV = 1000x 0.564.5 = Rs. 564.50

Example: Ram needs Rs 250000 at the end of fifth year. How much money does he need to deposit now if his bank
provides him 9% interest rate .

Calculation of number of years

Example: how much will it take to double your investment if bank provides 10% annual interest? Few times asked in PU
Calculation of rate
Example: Nabil Bank offers to its clients that if anyone deposit Rs 200000 now in a saving account then he or she will
receive Rs 350000 at the end of the 6th year. Calculate the interest rate provided by the bank.

Annuities
An annuity is a stream of constant cash flows (payment or receipt) occurring at regular intervals of time. In other words,
an annuity is a series of equal amount of payment or receipt at fixed intervals for a specific number of periods. Example
of an annuity is the premium payment of a life insurance policy, pension payment, royalties etc, these cash flows are
usually annual but can occur at other intervals, such as monthly (rent, car payments etc.)
The following things are needed for annuity

3
 a. Same amount should be either deposited or receipt
b. Every year it should be done at the end of the year.
In an annuity, series of equal payments or receipts also called even series of payment or receipts. In business such series
of payments are called even cash flows. An annuity can be an ordinary or annuity due.
- Ordinary annuity: when the cash flows occur at the end of each period, the annuity is called an ordinary annuity. It is
also known as deferred annuity. In other words, if the series of payments or receipts are made at the end of each period,
it is known as ordinary annuity.
- Annuity due: when the cash flows occur at the beginning of each period, the annuity is called an annuity due. In other
words, if the series of payments or receipts are made at the beginning of each period, the annuity is known as an
annuity due.

Future value of an ordinary annuity

An ordinary annuity consists of a series of equal payments made at the end of each period. In this regard, future value of
an annuity is the total sum of an equal amount which is invested or paid each period and retained until the end of the
period. This can be expressed algebraically as follow.
FVAn = PMT {(1+i) n-1}/i
Using table
FVAn = PMT (FVIFAi%,nyear)
Where
PMT = periodic payment
i =interest rate
n = timing of annuity
Where
FVIFAi%,nyear = future value interest factor for an annuity of n periods at i interest

F= A [
( 1+i )N −¿ 1
i ]
Example
Suppose you have decided to deposit Rs 30000 per year in a saving account of Nepal Bank LTD. What will be the
accumulated amount in your account at the end of 30 years, if the interest rate is 9 percent?
Solution
Given
Regular deposit (Annuity) = Rs 30000
Interest rate i = 9%
Period or year n = 30 years
We have
The accumulated sum will be
FVAn = PMT {(1+i) n-1}/i
= 30000{(1+.09)30-1)}/0.09
= 30000x 136.308
= Rs. 4089240
Alternatively
By using table
FVAn = PMT (FVIFAi%,nyear)
= Rs 30000x (FVIFAi%,n year)
= 30000x 136.308
= Rs 45089240

4
Future value of annuity due
If the cash flows occur at the beginning of each period, it is known as annuity due. Since the cash flows of an annuity due
occur one period earlier in comparison to the cash flows on an ordinary annuity, the following relationship holds:
FVAn Due

F= A [ i ]
( 1+i )N −¿ 1
(1+i)
By using table
FVAn Due= PMT (FVIFAi%, n year) (1+i)
Example
What will be the future value of a 10 years annuity due, with 7 percent interest rate and Rs 10000 yearly deposit?
We have
FVAn Due = PMT (FVIFAi, n) (1+i)
= 10000x (FVIFAi, n)(1.07)
= 100000x 13.816x1.07
= RS 147831.20
Example: Calculate the future worth of Rs 50000 uniform continuous cash flow from the beginning of each 5 years
deposit at 10%
Present value of an ordinary annuity
Pension funds, insurance obligations, and interest received form bonds all involve annuities. To value them, we need to
know the present value of each. It can be found by using following equation
P= A ¿

By using table
PVA n = PMT (PVIFAi%, n year )
Where, PVIFAi%, n year = present value interest factor for an annuity of n periods at i discount rate.
Example
Suppose that we are given alternative of 3-year annuity having Rs 1000 per year or the lump sum payment today. If we
do not need money for coming there years and take lump sum, we deposit in saving account of the bank yielding 4
percent rate of interest. How much should be lump sum payment to make equal to annuity?
Solution
By using the table
We have
PVA n = PMT (PVIFAi, n)
= 10000(PVIFAi, n)
= 10000x 2.7751
= RS 2775.10
Example: Mr. Hari needs Rs 50000 every year for study for 5 years. His father is thinking to deposit a lumpsum now to
meet this requirement. How much money does he need to deposit now if interest rate is 7%?
Present value of annuity due
If equal series of return occur at the beginning of each period, that is to be considered as the present value of annuity
due. With the present value of annuity due, we simply receive each cash flow 1 year earlier that is we receive it at the
beginning of each year, rather than at the end of each year. Thus, since each cash flow is received 1 year earlier, it is
discounted back for one less period. To determine the present value of an annuity due, we merely need to find the
present value of an ordinary annuity and multiply that by (i+1), which in effect cancels out 1 year’s discounting. Simply

5
 PVA n Due P= A ¿ (1+i)
=

By using table
PVA n due= PMT (PVIFAi%, n year ) (1+i)
Example
Suppose your mother wished to deposit enough money now to meet your college fees for the next three years of Rs
6000 per year. What is the present value of this annuity if the rate of interest is 5% and each payment is made at the
beginning of the year?
Solution
We have
PVA n due= PMT (PVIFAi, n) (1+i)
= 6000x (PVIFAi, n) (1.05)
= 60000x2.7232x1.05
= RS 17156.16
Finding out interest rate or discount rate
Discount rate is one of the most important factors in time value of money concept. So it should be given to calculate
present value or future value. If not given it has to be found. It can be clear with the help of following illustration.

Example: A bank offers to lend you Rs 2000, if you sign a note to repay Rs 3221 at the end of five year. What rate of
interest are you paying?
Solution
Given
PV = Rs =2000
FVn= Rs 3221
n = 5 years
We have
n
FVn= PV (1 + i)

So,
3221 = 2000(1+ i)5
(1+ i)5= 3221/2000
(1+ i)5 = 1.6105
1+i = (1.6105)1/5
i = 0.10 = 10%
Example
A finance company advertises that it will pay a lump sum of Rs 8000 at the end of 6 years to investors who deposit
annually Rs 1000 for 6 years. What interest rate is implicit in this offer?
Solution
The interest rate may be calculated in two steps
a. Find the FVIFA i 6years for this contract as follows
Rs 8000 = Rs 1000(FVIFAi, 6years)
FVIFAi, 6years = 8000/1000
=8
b. Look at the FVIFAi, n table and read the row corresponding to 6 years until you find a value close to 8. Doing so, we find
that
FVIFAi12% 6 is 8.1152

6
So, we conclude that the interest rate is slightly below 12 percent.
Finding out the number of periods
Number of periods is also another important and essential factor in time value of money. It has to be found out.
Following illustration helps to know the way to find out missing number of periods.
Example: Bimala Gurung was student at the university in 2007; she borrowed Rs 12000 in student loans at an annual
interest rate of 9%. If she repays Rs 1488 per year, how long, to the nearest year, will it take her to repay the loan?
Solution
Given
Borrowed amount PV = Rs 12000
Interest rate i = 9%
Annual installment PMT = Rs 1488 per year

Time to repay the loan nearest to year n =?

We have
PVA n = PMT (PVIFAi, n)
Or Rs 12000= 1488(PVIFA9%, n)
PVIFA9%, n = 8.0645
According to the PVIFA table, the value 8.0645 is laying at 9% in 15 years. Therefore, 15 years will take to repay the loan
to Bimala Gurung.
Alternatively
We can also find out the number of year by using the formula
We have

PVA n = PMT {1- 1/ (1+I) n}/i

12000= 1488[1-1/(1+.09)n]0.09
12000/14878 =[ 1-1/(1.09)n]/0.09
8.0645=[ 1-1/(1.09)n]/0.09
8.0645x0.09 = [ 1-1/(1.09)n]
0.7258 = =[ 1-1/(1.09)n]
1/(1.09)n = 1-0.7258
1/(1.09)n = 0.2742
(1.09)n = 1/ 0.2742
(1.09)n = 3.6469
Taking long in both side
n log 1.09= log 3.6469
n 0.3742 = 0.5619
n = 0.56419/0.3742
n = 15.077 years
Or we can say 15 years
Example
You want to take up a trip to the moon which costs Rs 1000000; the cost is expected to remain unchanged in nominal
terms. You can save annual Rs 50000 to fulfill your desire. How long will you have to wait if your savings earn an interest
of 12 percent?
Uneven cash flow stream
As we know the amount of periodical payments in annuity remains constant. Such constant amounts of periodical
payments are called even series of cash flows and varying amount of periodical payments is called uneven cash flows. In
business situation, constant cash flows do not exist, always. Financial decisions are also concerned with uneven cash
7
flows. For example, business projects do not normally provide constant or even cash flows; dividend on common stock
may vary from year to another. But the interest on bonds and debentures remain constant until their maturity.
The computation of the present value of uneven series of future cash flows differs with constant cash flows. Therefore, it
is necessary to understand for financial analysis dealing with uneven cash flows.
Present value of an uneven cash flow steam
The present value steam of uneven steam of future income is found out as the sum of present values of individual
components of present value steam.
Alternatively
Year Cash stream pv@ …% Present value
1 Xxx Xxx Xxx
2 Xxx Xxx Xxx
3 Xxx Xxx Xxx
n Xxx xxx xxx
Total present value xxx
Example
Find the present values of the following cash flow steams. The appropriate discount rate is 6%.
Year cash stream
1 100
2 200
3 200
4 200
5 200
6 0
7 1000
What is the value of steam at a0 6 percent discount rate?
Year Cash stream pv@ 6% Present value
1 100 .9434 99.34
2 200 .8900 178
3 200 .8396 167.92
4 200 .7921 158.42
5 200 .7473 149.46
6 0 .7050 0
7 1000 .6651 665.10
Total present value Rs 1413.24
Future value of uneven cash flow steam
The future value of an uneven cash flow stream is also known as terminal value. It can be found by compounding each
cash flow to the end of the steam and then summing the future value.
The future value of uneven cash flow stream may be calculated with the help of the following formula.
Where
n = total number of years
Alternatively
Year Cash stream pv@ …% Present value
1 Xxx Xxx Xxx
2 Xxx Xxx Xxx
3 Xxx Xxx Xxx
4 Xxx Xxx Xxx
n Xxx xxx xxx
Total future value xxx

8
Example
Find the future values of the following cash flow stream with discount at 7 percent.
Year 1 2 3 4 5
Cash flow 500 300 200 700 900
Calculation of future value at 7%
Solution

Year Cash stream fv@7% Future value

1 500(four year) 1.3108
2 300(three year) 1.2250
3 200(two year) 1.1449
4 700(one year) 1.07
5 900(zero year) 1
Total future value

Interest calculation
Interest, charge for the use of credit or money, usually figured as a percentage of the principal and computed ann
ually. Interest depends upon the total amount of money and the length of time over which it is borrowed. Normally, it is
calculated as annual rate and interest payments are made by borrowers to lenders. There are two methods on interest
calculation.
i. Simple interest
ii. Compound interest
Simple interest

Interest computed only on the principal and (unlike compound interest) not on
principal plus interest earned or incurred in the previous period(s).
When the total amount of interest earned is directly proportional to the initial principal amount, then that is called
simple interest. Simple interest is not used in commercial practice at this modern time.
Simple interest is calculation form the following formula
I=PNr
Where, I is amount of simple interest, N is number years, and r is the rate of interest.
i. Compound interest
Interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan.
Compound interest can be thought of as “interest on interest,” and will make a deposit or loan grow at a faster rate than
simple interest, which is interest calculated only on the principal amount. The rate at which compound interest accrues
depends on the frequency of compounding; the higher the number of compounding periods, the greater the compound
interest. Thus, the amount of compound interest accrued on $100 compounded at 10% annually will be lower than that
on $100 compounded at 5% semi-annually over the same time period. Compound interest is also known as
compounding.

Whenever the interest charge for any interest period (year) is based on the principal loan amount plus any accumulated
interest charges up to the beginning of that period, the interest is said to be compounded. It is mostly used in practice.

If interest earned or charged in each period is calculated based on the total amount, including original principal plus the
accumulated interest that has been left in the account, at the end of the previous period than it is called compound
interest
following notation is utilized for compounding interest calculations.
I= effective interest rate per interest period]
9
N=number of compounding periods
P=Present sum of money
F= future sum of money
A= end of period cash flows in a uniform series
Nominal interest rate and Effective interest rate

If a financial institution uses a unit of time less than one year in length such as a month, a quarter etc, the institution
usually quotes the interest rate on the annual basis. In this situation, the basic annual rate of interest is known as the
normal interest rate and it is represented by r. hence, nominal interest rate may be stated for any time less than one
year such as, 6 months, quarter, month, week, day etc

For example, if the interest rate is 12% per year and compound semiannually ( or if the interest rate is 6% per interest
period and the interest period is six months) in this situation, nominal interest is 12% i.e. r= 12%. But the actual rate of
interest would be greater than 12% because of the compounding twice during the year.

The actual rate of interest earned or paid during one year is known as effective rate. It is also expressed on the annual
basis.

= (1+0.12/2)2-1
= 0.1236 = 12.36 %
Example: a bank charges 1% interest per month on car loans. What is APR (annual percentage rate) and what is EAR?
Annual percentage rate, r= 1%x12= 12%
Effective annual rate EAR=
Where nominal interest rate, r= 12% per year
Number of compounding periods in a year, M= 12 times (compounding is monthly and there are 12 months in a year)
Example
You lend Rs 100 for three years at 10% interest compounding annually. How much would your earn interest and get the
end of the three years.
Solution
Given p= Rs 100, n= 3 years, i= 10% per year, F=?
We have: F= p (1+i) n
Simple interest method
F=pnr+p= 100x.1x3+100=130
Compound interest method
F= p (1+i) n
F= 100(1+0.1)³ = Rs 133.10
What will be the maturity amount of Rs 15000 after five years for nominal rate of interest is 9% per year, when
compounded yearly and quarterly?

Example: Suppose you borrow Rs 8000 from a bank, that charges interest at a rate of 10% compounded annually. How
much would you owe at the end of the year 4.

Example. Suppose you have invested Rs 1000 at present. How long does it take for your investment to double if interest
rate is 8% compounded annually?

Example: A man wants to have Rs 1000000 for his daughter's education after five years. How much should he deposit
now at saving account to invest for that purpose, if interest rate is 9% compounded annually,
10
Example: Ram deposit Rs 5000 now in a bank which gives 10 % interest per year, he draws Rs 2000 at the end of 3rd
year. What will be the remaining amount at the end of 5th year?
Example: Find the effective interest rate when the nominal rate of interest is 18% per year and compounding is
Monthly, daily, hourly
A person deposits Rs 4000 at a nominal interest rate of 20% for 5 years. Find the maturity of the deposit when the
interest is compounding quarterly.
Interests formulas relating a uniform series to its present and future sum
Uniform series (equal payment)
Uniform series shows equal cash flow amount (payment or receipts) occurring at the end of each equal periods. Such
uniform series is often called an annuity and it is denoted by A. commercial installments, rental payments, insurance
payments etc are some examples of uniform series.

If an amount A occurs at the end of each period for N periods at i% interest rate per period, the future sum at the end of
nth period can be written as,

F= A [ ( 1+i )n−¿ 1
i ]
Problem, if you deposit Rs 6000 each end of each year, how much money would be accumulated to your account at the
end of 10 years? When I = 7% compounded annually.
Solutions:
Given
A= 6000, N=10 years, I =7% per year and F=? By using uniform series compound amount factor,
F=A(F/A, 7%, 10)
Rs 82898.68
Finding P when A is given

P= A ¿

Example: how much money should you deposit now in your bank account so that you may have five ends of year
withdrawals of Rs 500 each? If i= 12% compounding annually.
Given
A=Rs 500, N= 5 years, I =12%and P=?
Using uniform series present value factor
P=A (P/A, 12%
Answer 1802.38
Semiannual and other compounding period s
Until now, we have assumed that the compounding period is always annual, however, it need not be as evidenced by
saving and loan associations and commercial banks that compounded on semiannual, quarterly, and some cased
monthly, weekly and a daily basis. It may occur more frequently than once a year.
Semiannual compounding: it involves two compounding periods within the year. Instead of the stated interest rate
being paid once a year, one half of the stated interest rate is paid twice a year.
Quarterly compounding: it involves four compounding periods within the year. One fourth of the stated interest rate is
paid four times a year.
Monthly compounding: it involves twelve compounding periods within a year. One twelfth of the stated interest rate is
paid twelve times a year.
Weekly compounding: it involves 52 compounding periods within a year. 1/ 52 nd of the stated interest rate is paid 52
times a year

11
Daily compounding: it involves 365 compounding periods within a year. 1/365th of the stated interest rate is paid 365
times a year.
In above compounding formulate for calculating present value and future value remains same as before. Just use the
following tricks
Compounding Multiply nxm Divide i/m
Semi annual nx2 i by 2
Quarterly nx4 i by 4
Monthly nx12 i by 12
Weekly nx52 i by 52
Daily nx365 i by 365
Continuous Compounding 1+i=e ko power r
Sometimes compounding may be done continuously. It is called continuous compounding. In the extreme case, interest
can be compounded continuously. Continuous compounding involves compounding over every microsecond, the
smallest time period imaginable.
The formula for future value while interest compounded continuously 1+i=er
FV (continuous compounding) PV (e = rn
)
Where
e= 2.7183(base of natural logarithm)
i= annual interest rate
n= number of yeas
Pv = F/ern

For annuities : PV = A
[ ( e rn−1 )
e rn ( e r −1 ) ]
fV = A
[ ( ern−1 )
( e r−1 ) ]
Example: calculate the future worth of Rs 50000 uniform continuous cash flow at the end of each five years deposited at
10% compounded continuously.
Solution: given
A= Rs 50000 continuous cash flow
r = 10% compounded continuously
n= 5 years
Fv = ?
EMI
Easy Monthly Installment/ Equal Monthly Installment or Equated Monthly Installment
In above examples, we assumed that loan is to be repaid in equal annual payment. But in practice, most loans require
monthly or quarterly or any other periodic payments. In case of such periodic payments, at first we should compute
periodic rate. (APR/number of payments in a year) and number of periods (number of years x number of compounding
periods in a year) If installment is paid monthly, we can compute monthly payment, also called equal monthly
installment EMI using following way
AMOUNT OF LOAN
EMI =------------------------------
PVIFAi/m, nxm
Where, m is number of times installment paid in a year. If monthly installment is paid then m will be 12
If formula is needed then, multiply n and m (mxn) divide i with m (i/m)

12
 • Suppose a loan of Rs 100000 for five years. Bank charges 12 % interest. Loan is repayable in monthly
installments. Compute EMI.
Solution
Given
Loan amount = 100000
Interest rate i = 12%
Number of year n = 5 years
Payments in a year m = 12 times
We have
AMOUNT OF LOAN
EMI =------------------------------
PVIFAi/m, nxm
100000
EMI =------------------------------
PVIFA12%/12, 5x12
= 100000/44.9550= Rs. 2224.45
Gradient
1. Interest calculation for uniform Gradient / Arithmetic Gradient
An arithmetical progression of cash flow service which begins to increase or decrease from the end of second
period onwards is called linear gradient series and it is generally known as gradient. Some economic analysis
problems involve receipts expenses that are projected to increase or decrease by a uniform amount for each
period.

F= A [ i ] [i i ]
( 1+i )N −¿ 1 G ( 1+i )N −¿ 1 NG
+ −
i
P= A ¿

2. Geometric Gradient Series

Some engineering economic problems relating to construction costs involve cash flows that increase or
decrease over time not y constant amount but by a constant percentage is said to be a geometric
gradient. It is also called compound growth cash flow.

A geometric gradient refers to a series of cash flows that increase or decrease by a constant
percentage (g) each period.

 A1= First cash flow

 r = Discount rate
 g = Growth rate per period
 n = Number of periods
Case 2: When r=g

13
Example: A company expects to receive a series of maintenance payments starting at Rs
5,000 in Year 1, growing at 4% per year for 10 years. If the discount rate is 7%, what is
the present value of these cash flows?

Example:
Ramesh, a Civil Engineer is planning to deposit a total of 20% of his salary which is 250000 per year
now each year in mutual fund. He expects 7% salary increase each year for 15 years. If the mutual fund
will give average 10% annual return, what will the sum amount at the end of 15 years? And if salary
increase by Rs 25000 per year, what will be the amount?
Case one

Year Total salary increase@ 7% per year 20% deposit

0 250000 50000
1 267500 53500
2 286225 57245

15 689758 137951
Second case

Year Total salary increases by Rs 25000 per year 20% deposit

0 250000 50000
1 275000 55000

15 625000 125000
1. A woman desires to have Rs 100000 in her retirement saving plan after working 20 years. She will accomplish it by
depositing money end of each year that earns 7% per year. How much must she deposit each year?

14
2. Your father wants to have Rs 2000000 for the study of your sister after the period of 5 years. How much money does
he has to deposit each year in saving account that earns 8% interest annually?
3. Mr. Adhikari wants to have Rs 3000000 for e studies of his son after the period of 15 years. How much rupees does
he has to deposit end of each year for 10 continuous years in a saving account that earns 16% interest annually?
4. What uniform payment at the end of 8 successive years is needed to equivalent Rs 1000 at the beginning of the first
year with 10% rate of interest?
5. Suppose one has present amount of Rs 1000 and it is desired to determine what equivalent uniform end of year
payments could be obtained from it for 10 years. If interest is 20% compounded continuously.
6. A person needs Rs 1200000 immediately to pay a new vehicle and he borrows it from a bank. He wants to repay that
loan semiannually with equal installment basis over the next eight years. What amount will he pay for each
installment if interest rate is being charged 7% compounding continuously? Hints: divide i by 2 ie 3.5%, use 16
periods and calculate A with the help of continuously compounding formula for Pv.
7. What will be the future worth at the end of 5 years of a uniform continuous cash flow of Rs 500 with interest is
compounded continuously at the nominal annual rate of 8%?
8. Calculate the future worth of Rs 50000 uniform continuous cash flow from the beginning of each 5 years deposit at
10% compounded continuously.
9. A person is planning for his retired life and has 10 more years of service. He would like to deposit 30% of his salary,
which is Rs 5000 at the end of first year and thereafter he wishes to deposit the amount with an annual increase of Rs
1000 for the next 9 years with an interest of 15%. Find the total amount at the end of 10th year.
10. Assume that it is now January 1, 1996. On January 1, 1997, you will deposit Rs 1000 onto a saving account that
pays 8 percent.
i. If the bank compounds interest annually, how much will you have in your account on January 1, 2000?
ii. What would your January 1, 2000 balance be if the bank used quarterly compounding rather than
annual compounding?
iii. Suppose you deposited the Rs 1000 in 4 payments of Rs 250 each on January 1 of January 1997, 1998,
1999 and 2000, how much would you have in your account on January 1, 2000 based on 8 percent
annual compounding?
11. How many deposits of Rs. 25000 each should Dr. Thakur make each month so that the final accumulated amount
will be Rs 1000000 if the bank interest rate is 12% per year? Hints first calculate the monthly effective rate,,,
(1+i)1/m-1 , use the effective rate and calculate number of period.
12. You wish to study your son in medical college after 20 years. Recently, government has fixed total 35 lakhs to
complete MBBS studies. How much you need to deposit on each year to meet your desire if bank providing 10%
interest rate per year for your fixed account. ( PU 2017 spring )
13. Ram invested at high yield account aimed to get the double of his investment at the end of 10 years. Compute the
effective interest rate he received on the account. ( PU 2017 spring )
14. A man aged of 30 years now had borrowed Rs 400000 from a bank for his further studies at the age of 20 years. The
bank was charged interest at 10% per year compounded quarterly. He wishes to pay that loan from last 10
semiannual way with equal installment basis and now he has just cleared the loan. What amount did he pay in each
installment? ( more than 3 times asked in PU exam)
Hints: calculate quarterly interest rate iq = 10%/4 = 2.5%
Change it into effective semiannually rate
I semi = (1+iq)2-1 = 5.06%

15
 Compounding the initial amount ie 400000 to the final year use effective semiannual rate and 20 periods answer is
Rs 1073515
Find out A using FVIFA effective semiannual rate and 10 periods answer is Rs 85110.62
15. What will be the amount at the end of 8 years if you deposit Rs 15000 per month for five years continuously if
nominal interest rate is 12% compounded quarterly? Pu 2024 fall old course

Given: monthly deposit = 15000, deposit period 5 years , 60 months, total investment period 8 years, , nominal interest
rate 12%, compounding frequently quarterly 4 times in a year, every three months.
Setp 1 convert the nominal rate to effective quarterly rate
The nominal annual rate is 12%, so the quarterly rate is 12%/4 =3%= 0.03
Step 2: calculate the future value of monthly deposits over 5 years
Since deposits are made monthly but compounding is quarterly, we adjust the monthly deposit to align with the
quarterly compounding
Number of quarters in five years = 5x4 = 20 quarters
Total deposit per quarter 3x15000 = 45000, since
Future value of an ordinary annuity
Fv = A x Fvifa3%20 period
= 45000x26.87= 1209150
Step 3, let the investment grow for the remaining three years
Now, the cumulated mount after 5 years grow for another 3 years (12 quarters at the same quarterly rate)
Future value of a single sum Fv= Pv x Pvif3%12 years
1209150x1.4258= 1723000
16. A man is planning to retire in 25 years. He wishes to deposit regular money every 3 months until he retires so that
he will receive annual payments of Rs 450000 after the first year of his retirement for the next 10 years. How much
must he deposit if the interest rate is 8%, compounded monthly?

Solution: step one

Let us find out how much money is needed at retirement to fund the 10 annual payments of Rs 450000
This is an ordinary annuity, where
Annual payment A = Rs 450000
Number of payments n = 10 years
Interest rate = 8%, compounded monthly,
First, we need the effective monthly interest rate, since the compounding is monthly but payments are made annual
Calculation of monthly effective interest rate
The nominal annual rate i= 8% = .08
Compounding periods in a year m = 12
Monthly effective rate = (1+i/m) m-1
= 8.3%
Now calculate the total amount required in 25th year so that he can receive Rs 450000 every year for 10 years using 8.3%
interest rate
Pv in the 25th year considering 10 years and 8.3% interest rate.
Pv=?
A= 450000
n = 10 years
I =8.3%
The value will be 2974860

16
Step 2, calculation of quarterly deposits needed to reach the retirement fund
Now we need to find out how much he should deposit every quarter for 25 years to accumulate Rs 2974860
This is a future value of annuity problem, where
Fv = Rs 2974860
Number of years = 25 years
Deposits are every three months, it means quarterly,
Total number of deposit N = 25x4= 100 periods
To calculate the amount required we need to calculate quarterly effective rate
Nominal interest rate i = 8%
Compounding period in a year = 12 times
Monthly interest rate im = 8/12 = 0.0066667
Effective quarterly rate iq
(1+monthly rate)3-1 = 2.01%
Now we calculate amount required for every three month till 25 years,,, in 25 years there will be 100 times 3 months
2974860= use 100 periods, 2.01 interest rate than calculate A using fv annuity formula…
The answer will be 9468.14
Goal: Re450,000/year for 10 years after retirement.

Time until retirement: 25 years.

Deposit frequency: Every 3 months (quarterly).

Interest rate: 8% per year, compounded monthly.

Required quarterly deposit: ≈ Rs 9,468.ceive Rs 450,000/year for 10 years after retirement.

Time until retirement: 25 years. Deposit

Time until retirement: 25 years.

Deposit frequency: Every 3 months (quarterly).

Interest rate: 8% per year, compounded monthly.

Required quarterly deposit: ≈ Rs 9,468. Rs 9,468. Goal: Receive Rs 450,000/year for 10

years after retirement. Time until retirement: 25 years. Deposit frequency: Every 3 months

17
(quarterly). Interest rate: 8% per year, compounded monthly. Required quarterly deposit: ≈
Rs 9,468.

18