Time Value Of Money Part 2

ALAN ANDERSON, Ph.D.
ECI RISK TRAINING
www.ecirisktraining.com

The time value of money formulas can be
used to solve for the appropriate rate of
interest or time horizon given the present
and future value of a sum.

(c) ECI RISK TRAINING 2009
www.ecirisktraining.com 40

The present and future value
formulas can be used to solve
for the rate of interest.


Suppose that an investor deposits $10,000
in a bank account.

The investor plans to keep these funds in
the bank for ten years, with a goal of having
$20,000 at the end of that time. What rate
of interest would he have to earn to double
his money in ten years?


This can be determined
algebraically as follows:

FVN = PV(1 + I)N

FVN
= (1 + I ) N

PV


FVN
N = (1 + I )
PV

FVN
N −1= I
PV


In this example,

20, 000
10 − 1 = 0.07177 = 7.177%
10, 000


The present and future value formulas can
also be used to solve for the time horizon.


Suppose that an investor deposits
$5,000 in a bank account that pays 6%
interest per year. The investor wants
to know how long it will take for these
funds to be worth $10,000.


This can be determined
algebraically as follows:

FVN = PV(1 + I)N

FVN
= (1 + I ) N

PV


⎛ FVN ⎞
ln ⎜
⎝ PV ⎠⎟ = N ln(1 + I )

⎛ FVN ⎞
ln ⎜ ⎟
⎝ PV ⎠
N=
ln(1 + I )


In this example,

⎛ 10, 000 ⎞
ln ⎜ ⎟
⎝ 5, 000 ⎠
N= = 11.896
ln(1 + .06)


The Rule of 72 is a quick method for estimating
the time horizon or the interest rate needed to
double the value of an investment.


Dividing the interest rate into 72 gives the
approximate number of years that it would
take to double the value of an investment.

For the example of the investor who needs
to know how many years it would take to
double his money at an interest rate of 6%,
dividing 72 by 6 gives a result of 12, which
is very close to the actual value of 11.896
years.


Dividing the number of years into 72 gives the
approximate interest rate that would be required
to double the value of an investment.

For the example of the investor who needs to
know what rate of interest is required to double
his money in ten years, dividing 72 by 10 gives a
result of 7.2%, which is very close to the actual
value of 7.177%.


In the case of a stream of cash flows that
are not equal, computing the future and
present value of the cash flows is a more
complex process.


The two basic types of uneven cash
flows of interest in finance are:

1) an annuity with an additional
payment during the final period

2) a cash flow stream with no pattern,
known as an irregular stream of cash
flows


The cash flows from most bonds take
the form of an annuity with an additional
payment during the final period.

Investment projects often generate
irregular streams of cash flows to firms.


Suppose that a bond offers investors cash
flows of $100 each year for the next three
years, with an additional payment of $1,000
at the end of the third year. If the periodic
rate of interest is 5%, what is the present
value of this stream of cash flows?


In this case,

N=3
I=5
PMT = $100
FV3 = $1,000


⎡ 1 ⎤
1−
⎢ (1 + I )N ⎥
PVAN = PMT ⎢ ⎥
⎢ I ⎥
⎢
⎣ ⎥
⎦


⎡ 1 ⎤
1−
⎢ (1 + .05)3 ⎥
PVA3 = 100 ⎢ ⎥ = $272.32
⎢ .05 ⎥
⎢
⎣ ⎥
⎦


FVN 1, 000
PV = =
(1 + I ) N
(1.05) 3

1, 000
= = $863.84
1.1576


Combining these results gives the present
value of the cash flow stream:

PVA3 + PV = 272.32 + 863.84 = $1,136.16


Suppose that an investment project
produces cash flows of $200 at the end
of the next two years, and $300 at the
end of the following three years.


If the periodic rate of interest is 4%, what is
the present value of these cash flows?

In this case, the present value of each cash
flow is computed using the PV formula; these
results are combined to give the present value
of the stream of irregular cash flows.


In this case, the present value is:

200 200 300 300 300
1
+ 2
+ 3
+ 4
+ 5
(1.04) (1.04) (1.04) (1.04) (1.04)

= 192.31 + 184.91 + 266.67 + 256.44 +
246.58 = $1,146.91


Each of the examples considered so far has
been based on the assumption that interest
is paid annually.

When interest is paid more often than once
per year, the present value and future value
formulas must be adjusted.


Two adjustments must be made:

1) the periodic interest rate
2) the number of periods


 The periodic interest rate equals:

annual rate / number of periods per year

 The number of periods equals:

(number of years)(number of periods per year)


Suppose that a sum of $1,000 is invested for
two years at an annual rate of interest of 4%.
Compute the future value of this sum based
on the assumption of:

a) annual compounding
b) semi-annual compounding


With annual compounding,

N=2
I=4
PV = $1,000


Using the future value formula,

FVN = PV(1+I)N
FV2 = 1,000(1+.04)2
FV2 = 1,000(1.081600)
FV2 = $1081.60


With semi-annual compounding,

N=4
I=2
PV = $1,000


Using the future value formula,

FVN = PV(1+I)N
FV4 = 1,000(1+.02)4
FV4 = 1,000(1.082432)
FV4 = $1082.43


The more frequently interest is paid
each year, the greater will be the
future value of a sum or an annuity.


Compute the present value of $1,000 to
be received in four years using an annual
interest rate of 6% with:

a) annual compounding
b) semi-annual compounding


With annual compounding,

N=4
I=6
FV4 = $1,000


Using the present value formula,

FVN 1000
PV = = = $792.09
(1 + I ) N
(1 + .06) 4


With semi-annual compounding,

N=8
I=3
FV8 = $1,000


Using the present value formula,

FVN 1000
PV = = = $789.41
(1 + I ) N
(1 + .03) 8


The more frequently interest is paid
each year, the smaller will be the
present value of a sum or an annuity.


As the frequency of compounding
increases, the present value of a sum or
annuity decreases, while the future value
of a sum or annuity increases.


The limiting compounding frequency is known as
continuous compounding. In this case, interest is
compounded at every instant in time. As a result,
the number of compounding periods is infinite.

The present and future value formulas with
continuous compounding are:


FVN = eIN

FVN − IN
PV = IN = FVN e
e

e = 2.7182818......


The present value of $1,000 to be
received in four years with an annual
rate of interest of 5% compounded
continuously is computed as follows:


PV = 1,000e-(0.05)(4) =

1,000e-(0.20) = $818.73


The future value of $1,000 invested for
three years at an annual rate of interest of
4% compounded continuously is computed
as follows:


FV3 = 1,000e(0.04)(3) =

1,000e(0.12) = $1,127.50


In order to compare interest rates with different
compounding frequencies, they can be converted
into the effective annual rate (EAR).


This is done with the following formula:

M
⎛ APR ⎞
EAR = ⎜ 1 + ⎟ −1
⎝ M ⎠


where:

APR = the annual percentage rate


If a bank charges an APR of 6% per year,
compounded quarterly for a loan, what is
the effective annual rate?


This can be determined with the
formula, as follows:

M
⎛ APR ⎞
EAR = ⎜ 1 + ⎟ −1
⎝ M ⎠


4
⎛ .06 ⎞
EAR = ⎜ 1 + ⎟ − 1 = 0.06136
⎝ 4 ⎠


This indicates that the borrower is actually
paying 6.136% per year for this loan.


With continuous compounding,
the EAR formula becomes:

EAR = eAPR - 1


If a bank charges an APR of 5% per year,
continuously compounded, what is the
effective annual rate?

EAR = eAPR – 1
= e.05 – 1
= 0.051271 = 5.1271%


For free problem sets based on this material
along with worked-out solutions, write to
info@ecirisktraining.com. To learn about
training opportunities in finance and risk
management, visit www.ecirisktraining.com


Time Value Of Money Part 2

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