The time value of money concept
✓ also called the time preference for money
✓ value of money is time dependent
✓ value of a unit of money is different in different time periods
✓ one dollar today is worth more than one dollar tomorrow!!!
➢ value of a sum of money received today is > its value received after some time
➢ conversely, a sum of money received in future is < valuable than it is today
➢ shows that, all things being equal, it is better to have money now than later
✓ Why does one dollar today worth more than one dollar tomorrow?
✓ Why?
➢ consumption preference – a strong preference for immediate rather than delayed
consumption
➢ inflation – erodes purchasing power of money (value decreases with time)
➢ uncertainty (risk) – default risk (‘a bird in the hand’)
➢ investment opportunities – cash received today can be invested to earn income
✓ interest
➢ compensation/cost of paying money in the future
➢ simple interest – payable only on the principal amount
➢ compound interest – payable on both principal & interest earned but not paid
➢ compounding period – number of times interest is accrued/payable in a year
❖ e.g., compounded annually, semiannually, quarterly, monthly, daily, etc.
✓ time line – a time frame to visualize issues related to the concept of time value of
money
✓ symbols
➢ P = also called principal/amount of money loaned/borrowed now/today
➢ y = number of years P will be outstanding
➢ c = number of times interest is accrued/payable in a year (frequency)
➢ r = interest rate per period = annual interest rate/c
➢ n = number of periods P will be outstanding = (y) x (c)
➢ I = amount of interest per n
a) simple I = (P) x (r) x (n)
b) compound = P x (1+r)n
➢ M = amount to be collected/paid after n periods = P + I
Note that r & n should be expressed in the same time measure (e.g., month)
✓ Time line
Period (e.g., y)
0
Cash
P
1
2
3
4
M
Examples-1
On January 1, 2022, Mr. Ali deposited Birr100 in a bank saving account that pays 12%
interest compounded monthly. He kept the money for 2 years.
a) How much is the principal?
b) How much is interest rate per year?
c) How frequent does the bank pay interest per year?
d) How much is interest rate per month?
e) For how many periods will the money stay at bank?
f) Calculate simple interest for 2 years.
g) Calculate compound interest for 2 years.
h) How much money will Ali get at the end of the second year of deposit?

Periods (e.g., y)
0
P=1,000
4
FVp=?
Example-2
On January 1, 2020 MCC Co deposited Br. 10,000 cash in a bank account that pays
12% interest. Determine the sum to be accumulated in the account after 2 years if
interest is compounded a) annually? b) quarterly? c) monthly? d) continuously?
Answer:
P = 10,000
c=4
y=2
n=2*4=8
r = 12%/4 = 3%
FVp=?

In example-2 above, how long it will take MCC to double its Br 10,000 deposit with
12% annual interest rate and interest rate paid a) annually? b) quarterly?

Periods (e.g., y)
0
PVp=?
4
FV=1,000
Example-3
What is the value today of Br. 10,000 to be received after 2 years if the market interest
rate is 12% compounded a) annually? b) quarterly? c) monthly?
Answer:
M = 10,000
c=4
y=2
n=2*4=8
r = 12%/4 = 3%
PVp=?
✓ Annuities – a finite series of equal cash receipts/payments at equal intervals of time
with fixed interest (discount) rate
➢ fixed equal payment per period (R)
➢ fixed equal time interval between payments i.e., equal periods (n)
➢ fixed equal interest rate per period (r)
✓ Two types
➢ Ordinary annuity – due at the end of each period
Periods (e.g., y)
0
Annuities (Rs)
1
2
3
4
100
100
100
100
4
➢ Annuity due – due at the beginning of each period
Periods (e.g., y)
0
1
2
3
Annuities (Rs)
100
100
100
100
✓ Definition
➢ a sum to which a series of end of period deposits (R) will grow
0
1
2
3
4
100
100
100
100
FVAo=?
Example-4
Ali plans to deposit Br. 500 at the end of each of the next four years in a bank account
that pays 12% interest compounded annually. How much will be the balance of the
account at the end of year-4?
Answer:
R = 500
c=1
y=4
n=1*4=4
r = 12%/1 = 12%
FVAo
✓ Definition
➢ a sum that must be invested now to guarantee a desired series of payments (R) at
the end of each future period
0
1
2
3
4
100
PVAo=?
100
100
100
Example-5
On January 1, 2020 BAC Co purchased equipment by signing a note promising to pay
Br. 10,000 at the end of each of the next five years starting December 31, 2020. What is
the cost of the equipment today if the market interest rate is 10% compounded annually?
Answer:
R = 1,000
c=1
y=5
n=1*5=5
r = 10%/1 = 10%
PVAo
✓ Definition
➢ a sum to which a series of beginning of period deposits (R) will grow
0
1
2
3
4
100
100
100
100
FVAd=?
Example-6
Ali plans to deposit Br. 500 at the beginning of each of the next four years in a bank
account that pays 12% interest compounded annually. How much will be the balance of
the account at the end of year-4?
Answer:
R = 500
c=1
y=4
n=1*4=4
r = 12%/1 = 12%
FVAd
✓ Definition
➢ a sum that must be invested now to guarantee a desired series of payments (R) at
the beginning of each future period
0
1
2
3
100
PVAd=?
100
100
100
4
Example-7
On January 1, 2020 BAC Co purchased equipment by signing a note promising to pay
Br. 10,000 at the beginning of each of the next five years starting December 31, 2020.
What is the cost of the equipment today if the market interest rate is 10% compounded
annually?


The present value of an annuity that pays A dollars at the end of each of the next ‘t’ years,
assuming a constant interest rate ‘r’ compounded continuously, is:

To calculate the present value of an annuity which pays A dollars a year forever,
compounded continuously, is:
Example-8
An investment offers a perpetual cash flow of Br. 1,000 every year. The required rate of
return on similar investment is 8%.
a) What is the value today of the investment if interest is compounded annually?
b) What about future value of the investment?

Assuming a 10 percent interest rate compounded continuously, what is
the present value of annuity that pays $500 a year a) for the next five
years, b) forever?

Example-9
BAC Co purchased a multipurpose equipment promising to pay Br. 10,000 at end of
each of the next five years starting December 31, 2020. The company has also agreed to
increase the payment by 5% each year. How much is the cost of the equipment if the
market interest rate is 8% compounded annually?
Example-10
BAC Co is planning a fund for purchase of multipurpose equipment. It will deposit Br.
10,000 at the end of each of the next five years starting December 31, 2020. The
company plans to increase each yearly deposit by 5% to accommodate for price increase
over the coming five years. How much will be the balance of the fund at the end of the
5th year if the market interest rate is 8% compounded annually?
Example-11
BAC Co purchased a multipurpose equipment promising to pay Br. 10,000 at end of
each year starting December 31, 2020 for indefinite period. The company has also
agreed to increase the payment by 5% each year. How much is the cost of the equipment
if the market interest rate is 8% compounded annually?

Example-12
BAC Co received the following interest offers from three different banks. Which of
these is the best if you are thinking of opening a savings account? Which of these is best
if they represent loan rates?
Bank
A
B
C
Nominal interest
11% compounded daily
11.5% compounded quarterly
12% compounded annually
EAR
11.63%
12.01%
12.00%
4. Finding R (regular payment)
Example-13
BAC Co is planning to accumulate Br.100,000 over the coming five years to purchase
multipurpose equipment. Determine the amount the company must accumulate at the
end of each year if the market interest rate is 8% compounded annually.
Answer:
FVAo=100,000
c=1
r=8%/1=8%
n=5years *1=5
R=?
5. Finding n (number of regular payments)
Example-14
BAC Co is planning to accumulate Br.100,000 to purchase a multipurpose equipment.
Determine number of yearly deposits of Br.17,046 which the company must make at the
end of each year if the market interest rate is 8% compounded annually.
Locate 5.8665 in the FVAo table under r=8% and its corresponding n value to the left
6. Finding r (interest rate per period)
Example-15
BAC Co purchased an equipment for Br.100,000 on credit. It promised to pay Br.15,600
at the end of each of the coming 10 years. Determine the interest rate per period, if it is
compounded annually.
Locate 6.4103 in the PVAo table under n=10 and its corresponding r value at the top

Example-16
An investment will pay Br.1,000 at the end of each of the next 3 years, Br.2,000 at the
end of Year 4, Br.3,000 at the end of Year 5, and Br.5,000 at the end of Year 6. If other
investments of equal risk earn 8% annually, what is this investment’s present value? Its
future value?
8. Deferred annuity
✓ Definition
➢ an annuity where payments (R) are delayed until a certain period has elapsed
➢ usually it has two stages, namely, accumulation and payment stages
Start of deferred
annuity
Single payment (P)
End of deferred
annuity
Deferred period
Accumulated stage
FV of P
nR
R payment stage
FV = 0
Example-17
ABC Co wants to invest an amount of money today (January 1, 2000) such that its
employees can receive Br.5,000 at the end of every month for 10 years when they retire
from work. If the initial investment can earn 9% compounded annually until December
31, 2010 and then 6% compounded interest when the fund starts paying out on January
31, 2011. How much money must the company invest today?
Given:
5.
PVAo = FV of single sum
= 5,000[(1-(1/((1+0.005)^120))]/0.005
Ordinary annuity year=10
= 450,367.27
Single sum year = 11
FVp = 450,367.27
Annual interest rate – single sum = 9%
PVp = 450,367.27/[(1+0.09)^11]
Annual interest rate – ordinary annuity = 6%
= 174,532.11
6.
Compounding period – single sum = 1
7.
Compounding period – ordinary annuity = 12
8.
r – single sum = 9%/1=9%
9.
r – ordinary annuity = 6%/12= ½ %
1.
2.
3.
4.
R=5,000
10. n – single sum = 11*1=11
11. n – ordinary annuity = 10*12=120
12. P=?
1.
If the interest rate is 10% how much money would one need to receive now to be equivalent to
$1 million received two years from now if:
2.
a)
Interest is compounded annually?
b)
Interest is compounded semiannually?
c)
Interest is compounded monthly?
d)
Interest is compounded continuously?
Suppose that the interest rate (r) is such that the present value of receiving $V2 in t2 years from
now is the same as the present value of receiving $V1 in t1 years from now, t2 > t1. Assume
that interest is compounded annually.
a.
Show that V2 > V1.
b.
Show that the present value of receiving $V2, (t2 + k) years from now is also equal to the
present value of receiving $Vl, (t1+ k) years from now for any value of k. (That is, it is the
absolute difference between time periods that matter.)