Time Value Of Money Part 1

ALAN ANDERSON, Ph.D. ECI RISK TRAINING

www.ecirisktraining.com

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

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The time value of money is one of the most fundamental concepts in finance; it is based on the notion that receiving a sum of money in the future is less valuable than receiving that sum today.

This is because a sum received today can be invested and earn interest.

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The four basic time value of money concepts are:

 future value of a sum  present value of a sum  future value of an annuity  present value of an annuity

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If a sum is invested today, it will earn interest and increase in value over time. The value that the sum grows to is known as its future value.

Computing the future value of a sum is known as compounding.



The future value of a sum depends on the interest rate earned and the time horizon over which the sum is invested.

This is shown with the following formula:

FVN = PV(1+I)N



where:

FVN = future value of a sum invested for N periods

I = periodic rate of interest PV = the present or current

value of the sum invested



Suppose that a sum of $1,000 is invested for four years at an annual rate of interest of 3%. What is the future value of this sum?



In this case,

N = 4 I = 3 PV = $1,000



Using the future value formula,

FVN = PV(1+I)N

FV4 = 1,000(1+.03)4 FV4 = 1,000(1.125509) FV4 = $1,125.51



The present value of a sum is the amount that would need to be invested today in order to be worth that sum in the future.

Computing the present value of a sum is known as discounting.



The formula for computing the present value of a sum is:

PV =FVN(1+ I )N



How much must be deposited in a bank account that pays 5% interest per year in order to be worth $1,000 in three years?



In this case,

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



PV =

FVN(1+ I )N

=1,000(1.05)3

=1,0001.1576

= $863.84



An annuity is a periodic stream of equally-sized payments.

The two basic types of annuities are:

 ordinary annuity  annuity due



With an ordinary annuity, the first payment takes place one period in the future.

With an annuity due, the first payment takes place immediately.



The formulas used to compute the future value and present value of a sum can be easily extended to the case of an annuity.



The formula for computing the future value of an ordinary annuity is:

FVAN = PMT(1+ I )N −1

I⎡

⎣⎢

⎤

⎦⎥



where:

FVAN = future value of an N-period ordinary annuity

PMT = the value of the periodic payment



Suppose that a sum of $1,000 is invested at the end of each of the next four years at an annual rate of interest of 3%. What is the future value of this ordinary annuity?



In this case,

N = 4 I = 3 PMT = $1,000



Using the formula,

FVAN = PMT

(1+ I )N −1I

⎡

⎣⎢

⎤

⎦⎥



FVA4 = 1,000(1+ .03)4 −1

.03⎡

⎣⎢

⎤

⎦⎥ = $4,183.63



The future value of the annuity can also be obtained by computing the future value of each term and then combining the results:



1,000(1.03)3 + 1,000(1.03)2 + 1,000(1.03)1 + 1,000(1.03)0

= 1,092.73 + 1,060.90 + 1,030.00 + 1,000.00

= $4,183.63



The future value of an annuity due is computed as follows:

FVAdue = FVAordinary(1+I)



Referring to the previous example, the future value of an annuity due would be:

4,183.63(1+.03) = $4,309.14



The formula for computing the present value of an ordinary annuity is:

PVAN = PMT1− 1

(1+ I )N

I

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



where:

PVAN = future value of an N-period ordinary annuity

PMT = the value of the periodic payment



How much must be invested today in a bank account that pays 5% interest per year in order to generate a stream of payments of $1,000 at the end of each of the next three years?



In this case,

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



Using the formula,

PVAN = PMT1− 1

(1+ I )N

I

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



PVA3 = 1,0001− 1

(1+ .05)3

.05

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= $2,723.25



The present value of the annuity can also be obtained by computing the present value of each term and then combining the results:



1,000(1.05)-3 + 1,000(1.05)-2 + 1,000(1.05)-1 = 863.84 + 907.03 + 952.38 = $2723.25



The present value of an annuity due is computed as follows:

PVAdue = PVAordinary(1+I)



Referring to the previous example, the present value of an annuity due would be:

2,723.25(1+.05) = $2,859.41



Economy & Finance

Time Value Of Money Part 1