Time Value of Money
LECTURER: ISAAC OFOEDA
Chapter Objectives
• Understand what gives money its time value.
• Explain the methods of calculating present
and future values.
• Highlight the use of present value technique
(discounting) in financial decisions.
• Introduce the concept of internal rate of
return.
Time Preference for Money
• Time preference for money is an individual’s
preference for possession of a given amount
of money now, rather than the same amount
at some future time.
• Four reasons may be attributed to the
individual’s time preference for money:
–
–
–
–
risk
preference for consumption
investment opportunities
inflation
Required Rate of Return
• The time preference for money is generally
expressed by an interest rate. This rate will be
positive even in the absence of any risk. It may
be therefore called the risk-free rate.
• An investor requires compensation for
assuming risk, which is called risk premium.
• The investor’s required rate of return is:
Risk-free rate + Risk premium.
Time Value Adjustment
• Two most common methods of adjusting cash
flows for time value of money:
– Compounding—the process of calculating future values of
cash flows and
– Discounting—the process of calculating present values of
cash flows.
Future Value
• Compounding is the process of finding the future
values of cash flows by applying the concept of
compound interest.
• Compound interest is the interest that is received
on the original amount (principal) as well as on
any interest earned but not withdrawn during
earlier periods.
• Simple interest is the interest that is calculated
only on the original amount (principal), and thus,
no compounding of interest takes place.
Future Value
Lump sum payment
• The general form of equation for calculating
the future value of a lump sum after n periods
may, therefore, be written as follows:
Fn  P (1  i ) n
• The term (1 + i)n is the compound value
factor (CVF) of a lump sum of Re 1, and it
always has a value greater than 1 for positive i,
indicating that CVF increases as i and n
increase. F =P  CVF
n
n,i
Example
• If you deposited GHȼ55,650 in a bank, which
was paying a 12 per cent rate of interest on a
ten-year time deposit, how much would the
deposit grow at the end of ten years?
Future Value of an Annuity
• Annuity is a fixed payment (or receipt) each
year for a specified number of years. If you
rent a flat and promise to make a series of
payments over an agreed period,
you
have
 (1  i ) n  1 
created an annuity. Fn  A 

i


• The term within brackets is the compound
value factor for an annuity of Re 1, which we
shall refer as CVFA.
Fn =A CVFA n, i
Future Value of an Annuity
• Annuity is a fixed payment (or receipt) each
year for a specified number of years. If you
rent a flat and promise to make a series of
payments over an agreed period, you have
created an annuity
 (1  i ) n  1 
Fn  A 

i


• This is used in the case of ordinary annuity (payment in
arrears)
• The term within brackets is the compound
value factor for an annuity of Re 1, which we
shall refer as CVFA Fn =A CVFA n, i
Future Value of an Annuity
• Suppose that a firm deposits GHȼ 5,000 at the
end of each year for four years at 6 per cent
rate of interest. How much would this annuity
accumulate at the end of the fourth year? We
first find CVFA which is 4.3746. If we multiply
4.375 by GHȼ5,000, we obtain a compound
value of GHȼ 21,875:
F4  5,000(CVFA 4, 0.06 )  5,000  4.3746  Rs 21,873
Future Value of an Annuity
• If the payments are made at the beginning of
each period, then we have an annuity due.
Rental lease payments, life insurance
premiums, and lottery payoffs (if you are lucky
enough to win one) are examples of annuities
due.
𝟏+𝒊 𝒏−𝟏
𝑭𝑽 = 𝑪𝑭
𝟏+𝒊
𝒊
Example
• A customer has instructed his bankers to transfer
GHȼ400 from his current account to his savings
account at the end of each year for 4 years.
Interest rate on savings is 10% per annum. What
is the size of the savings account at the end of the
4th year.
• Assuming the customer’s instruction to the bank
was that the first transfer should be made now,
followed by the same amount of GHȼ400 for
three more years. What is the size of the account
after 4 years.
Future Value of an Uneven Periodic
Sum
• Investments made by of a firm do not
frequently yield constant periodic cash flows
(annuity). In most instances the firm receives
a stream of uneven cash flows. Thus the
future value factors for an annuity cannot be
used. The procedure is to calculate the future
value of each cash flow and aggregate all
present values.
𝑭𝑽 = 𝑪𝑭𝟏 𝟏 + 𝒊 𝒏−𝟏 + 𝑪𝑭𝟐 𝟏 + 𝒊 𝒏−𝟐 + 𝑪𝑭𝟑 𝟏 + 𝒊 𝒏−𝟑
Example
• Dosky expects to receive GHȼ300, GHȼ500,
GHȼ1000 and GHȼ1500 in year1,2,3 and 4
respectively. What is the value of these cash
flows at the end of the fourth year?
Application of the “f” Rule
• The above discussion assumes that payments
are made once a year. However, in real life
situation, payments may be made daily,
weekly, monthly, quarterly or even semiannually. In this case, we apply the f rule.
That is, wherever we see r we divide it by f
and wherever we see n we multiply by f.
Sinking Fund
• Sinking fund is a fund, which is created out of
fixed payments each period to accumulate to
a future sum after a specified period. For
example, companies generally create sinking
funds to retire bonds (debentures) on
maturity.
• The annuity formulas may be used in
computing the periodic payments for both
ordinary annuity or annuity due.
Example
• Cleanborn Ltd has bought an asset with a life
span of 4 years. At the end of the 4 years,
replacement of the asset will cost GH¢12,000.
In this direction, the company has decided to
provide for the future commitment by setting
up a sinking fund account into which equal
annual investment will be made at the end of
each year. Interest rate on the investment will
be 12% per annum.
Example Cont’d
• Required:
a. Calculate the annual instalments
b. Draw up the sinking fund schedule to show
the growth fund
c. Assuming the first payments will be made
now and 12 months thereafter, what are the
annual payments?
d. Draw up the sinking fund schedule under (c)
Solution-----Arrears
• Finding the periodic payment
𝟏. 𝟏𝟐 𝟒 − 𝟏
𝟏𝟐, 𝟎𝟎𝟎 = 𝑪𝑭
= 𝟐𝟓𝟏𝟎
𝟎. 𝟏𝟐
Sinking Fund Schedule
Year
Bal b/f
Payments
Interest
Bal c/f
1
0
2510.0
0
2510.0
2
2510.0
2510.0
301.2
5321.2
3
5321.2
2510.0
638.5
8469.7
4
8469.7
2510.0
1016.4
11996.1
Solution-----Advance
• Finding the periodic payment
Sinking Fund Schedule
Year
Bal b/f
Payments
Interest
Bal c/f
1
0
2241.8
269.0
2510.8
2
2510.8
2241.8
570.3
5323.0
3
5321.2
2241.8
907.6
8470.6
4
8469.7
2241.8
1285.4
11996.9
Changes in Interest Rates
• If the interest rates changes during the period
of an investment, the compounding formula
must be amended as follows;
Present Value
• Present value of a future cash flow (inflow or
outflow) is the amount of current cash that is
of equivalent value to the decision-maker.
• Discounting is the process of determining
present value of a series of future cash flows.
• The interest rate used for discounting cash
flows is also called the discount rate.
Present Value of a Single Cash Flow
• The following general formula can be
employed to calculate the present value of a
lump sum to be received after some future
Fn
n


P


F
(1

i
)
n 
periods:
n
(1  i )
• The term in parentheses is the discount factor
or present value factor (PVF), and it is always
less than 1.0 for positive i, indicating that a
future amount has a smaller present value.
PV  Fn  PVFn ,i
Example
• Suppose that an investor wants to find out the
present value of GH¢50,000 to be received
after 15 years. Her interest rate is 9 per cent.
First, we will find out the present value factor,
which is 0.275. Multiplying 0.275 by
GH¢50,000, we obtain GH¢ 13,750 as the
present value:
PV = 50,000  PVF15, 0.09 = 50,000  0.275 = Rs 13,750
Present Value of an Uneven Periodic
Sum
• Investments made by of a firm do not
frequently yield constant periodic cash flows
(annuity). In most instances the firm receives
a stream of uneven cash flows. Thus the
present value factors for an annuity cannot be
used. The procedure is to calculate the
present value of each cash flow and aggregate
all present values. 𝑪𝑭
𝑪𝑭
𝑪𝑭
𝑪𝑭
𝑷𝑽 =
𝟏
𝟏+𝒊
+
𝟏
𝟐
𝟏+𝒊
+
𝟐
𝟑
𝟏+𝒊
+⋯
𝟑
𝒏
𝟏+𝒊
𝒏
Present Value of an Uneven Periodic
Sum
• Your company invested in a project which is
expected to generate GH¢50,000 in year1,
GH¢70,000 in year2 and GH¢85,000 in year3.
If the cost of capital for the company is 15%,
what will be the value of the future cash flows
today.
Present Value of an Annuity
• The computation of the present value of an
annuity can be written in the following
1
general form:
1 
P  A 

n
 i i 1  i  
• The term within parentheses is the present
value factor of an annuity of GH¢ 1, which we
would call PVFA, and it is a sum of singlepayment present value factors.
P = A × PVAFn, i
Present Value of an Annuity
• If the payments are made at the beginning of
each period, then we have an annuity due.
Capital Recovery and Loan
Amortisation
• Capital recovery is the annuity of an
investment made today for a specified period
of time at a given rate of interest. Capital
recovery factor helps in the preparation of a
loan amortisation (loan repayment) schedule.
 1 
A= P

PVAF
n ,i 

A = P × CRFn,i
The reciprocal of the present value annuity factor is called the capital
recovery factor (CRF).
Example
• You have applied to your bankers for a loan of
GH¢30,000 to complete your dream house for
deductions to be made over 3 years equal
annual instalments. Your bankers, however,
maintained that your 40% annual salary which
amounts to GH¢12,000 cannot meet both the
principal and interest payment. It is the bank’s
policy to maintain a debt service ratio of 40%.
Interest rate charged by the bank is 18% per
annum.
Example
• Required:
a. Calculate the size of the loan you qualify for.
b. Prepare amortization table to show how the
loan will be liquidated
Solution
• Value of the loan
Annual Payment = GH¢12,000
Solution
Year
1
2
3
Amortization Schedule
Bal b/f Interest Instalment Principal payment
26091.27 4696.4
12000.0
7303.6
18787.7 3381.8
12000.0
8618.2
10169.5 1830.5
12000.0
10169.5
Bal c/f
18787.70
10169.48
-0.01
Example
• Mr Zidi borrowed GH¢20,000 to renovate his
house. He will pay it back in equal annual
payments, which begins today and every 12
months for 4 years. Interest rate on the loan is
8% per annum.
a. Calculate the annual loan payments
b. Show the amortization table
Solution
• Finding annual payment
Solution
Year
0
1
2
3
4
Bal b/f
20000.00
15361.9
11952.7
8270.8397
4294.3969
Amortization Schedule
Interest Instalment Principal payment
0.0
4638.11
4638.1
1229.0
4638.11
3409.2
956.2
4638.11
3681.9
661.7
4638.11
3976.4
343.6
4638.11
4294.6
Bal c/f
15361.89
11952.73
8270.84
4294.40
-0.16
Present Value of Perpetuity
• Perpetuity is an annuity that occurs
indefinitely. Perpetuities are not very common
in financial decision-making:
Present value of a perpetuity 
Perpetuity
Interest rate
Present Value of Growing Annuities
• The present value of a constantly growing
annuity is given below:
A  1 g 
P=
1  

i  g   1  i 
n



Present value of a constantly growing perpetuity is given by a
simple formula as follows:
A
P=
i–g
Finding Interest Rates
𝒊=
𝑭𝑽 𝒏
𝑷𝑽
-1
Your brother is planning to retire in 18 years time. He currently has
GHC250,000, and he would like to have GHC1,000,000 when he
retires. You are required to compute the annual rate of interest he
would have to earn on his GHC250,000 in order to reach his goal,
assuming he saves no more money.
Finding Number of Periods
𝑭𝑽
𝑷𝑽
𝒏=
𝒍𝒐𝒈 𝟏 + 𝒊
𝒍𝒐𝒈
At 9% interest, how long does it take to double your money? To
quadruple it?
Multi-Period Compounding
• If compounding is done more than once a
year, the actual annualised rate of interest
would be higher than the nominal interest
rate and it is called the effective interest rate.