Time Value of MoneyFIL 240

Prepared by Keldon Bauer

Cash Flow Time Lines You win the American College Student

Publishers Contest. You have the option of taking $1.4 million

now or $250,000 per year for five years. Which should you take?

The answer comes through taking into consideration the time value of money.

Cash Flow Time Lines The first step is visualizing the cash flows by

drawing a cash flow time line. Time lines show when cash flows occur. Time 0 is now.

0 1 2 3 4 5

Cash Flow Time Lines Outflows are listed as negatives. Inflows are positive. State the appropriate “interest rate,” which

represents your opportunity costs

0 1 2 3 4 5

8%

$250K $250K $250K $250K $250K

Future Value Future value is higher than today, because if I

had the money I would put it to work, it would earn interest.

The interest could then earn interest. Compounding: allowing interest to earn interest

on itself.

Future Value - Example If you invest 1,000 today at 8% interest per

year, how much should you have in five years (in thousands).

0 1 2 3 4 5

8%

PrincipalInterest

TotalPrev. Interest

-10.080.001.08

0.08640.08001.1664

0.09330.16641.2597

0.10080.25971.3605

0.10880.36051.4693

Future Value For one year, the future value can be defined

as:

F u tu re V alu e P resen t V alu e + I n terestY ear 1

F V P V P V i

F V P V i1

1 1

Future Value The second year, the future value can be

stated as follows:

F V F V F V i

F V i

P V i i P V i

2 1 1

1

2

1

1 1 1

Future Value Therefore, the general solution to the future

value problem is:

F V P V i

P V F V IF

P V

n

n

i n

1

,

F u tu re V alu e I n terest F acto r

Future Value Interest can be seen as

the opportunity growth rate of money.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5 10 15

Periods

Fu

ture

Val

ue

of $

1

i=15%

i=10%

i=5%

i=0%

Present Value Present value is the value in today’s dollars of

a future cash flow. If we are interested in the present value of

$500 delivered in 5 years:

0 1 2 3 4 5

8%

$500PV=?

Present Value The general solution to this problem follows

from the solution to the future value problem:

F V P V i

P VF V

iF V

i

n

n

nn n n

1

1

1

1

Present Value Since the discount rate

is the opportunity cost, the present value represents what I would have to give up now to get the future value specified.

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15

Periods

Pre

sen

t V

alu

e of

$1

i=0%

i=5%

i=10%

i=15%

Interest Rates If we know the amount we need at time n and

the amount we can invest at time zero, then we must only solve for the interest rate.

0 1 2 3 4 5

?%

$500$100

Interest Rates Solving for interest rates algebraically:

F V P V i

F V

P Vi

F V

P Vi

F V

P Vi

n

n

n n

nn

nn

1

1

1

1

1

1

Interest Rates - Example0 1 2 3 4 5

?%

$500$100

5 0 0

1 0 01

15

i

5 1 1 3797 1 0 3797 37 97%1

5 . . .i

Time Periods If the present value, future value and interest

rate are known, but the number of time periods is not. Then n can be found algebraically:

F V P V i

F V P V n i

F V P V

in

n

n

n

n

1

1

1

ln ( ) ln ( ) ln [( )]

ln ( ) ln ( )

ln [( )]

Time Periods - Example If we use the last example of investing $100, we

want $500 in future, and the current market interest is 8%, n can be found:

9124.20)08.1ln(

)100ln()500ln(

n

Annuities Definition: A series of equal payments at a

fixed interval. Two types:

Ordinary annuity: Payments occur at the end of each period.

Annuity due: Payments occur at the beginning of each period.

Ordinary Annuity Example is a regular payment of $100 for five

years earning 8% interest.

0 1 2 3 4 5

$100 $100 $100 $100 $100

8%

What is the future value?

FV=?

Ordinary Annuity – Future Value The future value of an ordinary annuity can be

found as follows:

i

iPMTFVA

iPMTFVA

n

n

n

tn

11

derivationWithout

11

0

Ordinary Annuity - Example

0 1 2 3 4 5

$100 $100 $100 $100 $100

8%

66.586

08.0

108.01100

5

nFVA

Annuity Due Example is a regular payment of $100 for five

years earning 8% interest.

0 1 2 3 4 5

$100 $100 $100 $100$100

8%

What is the future value?

FV=?

Annuity Due – Future Value The future value of an annuity due can be found by

noticing that the annuity due is the same as an ordinary annuity, with one more compounding period:

i

i

iPMTFVA

n

n 111

Annuity Due - Example

0 1 2 3 4 5

$100 $100 $100 $100$100

8%

59.633

08.0108.0

108.01100

5

nFVA

Ordinary Annuity - Present Value Example is a regular payment of $100 for five

years earning 8% interest.

0 1 2 3 4 5

$100 $100 $100 $100 $100

8%

What is the present value?

PV=?

Ordinary Annuity - Present Value The present value of an ordinary annuity can be

found as follows:

P V P M Ti

P V P M Ti

i

t

t

n

n

1

1

11

1

1

W ith o u t D eriv atio n

Ordinary Annuity - Example

0 1 2 3 4 5

$100 $100 $100 $100 $100

8%

P V

1 0 0

11

1 0 0 8

0 0 83 9 9 2 7

5.

..

Annuity Due - Present Value Example is a regular payment of $100 for five

years earning 8% interest.

0 1 2 3 4 5

$100 $100 $100 $100$100

8%

What is the present value?

PV=?

Annuity Due - Present Value The future value of an annuity due can be found as

follows:

P V P M Ti

P V P M Ti

ii

t

t

n

n

1

1

11

11

0

1

W ith o u t D eriv atio n

Annuity Due - Example

0 1 2 3 4 5

$100 $100 $100 $100$100

8%

P V

1 0 0

11

1 0 0 8

0 0 81 0 0 8 4 3 1 2 1

5.

.. .

Annuities - Finding Interest Rate Interest rates cannot be solved directly. Calculators search for the correct answer

(there is only one correct answer). It guesses and then iteratively goes higher or

lower.

Perpetuities What would you have to pay to be paid

$2,000 per year forever (given a market rate of 8%)?

P V i P M T

P VP M T

i

$ 2 ,

.$ 2 5,

0 0 0

0 0 80 0 0

Uneven Cash Flow Streams If payments are irregular or come at irregular

intervals, we can still find the PV (or FV). Take the present value (or future value) of

individual payments and sum them together.

Uneven Cash Flows - Example

0 1 2 3 4 5

$100 $200 $300 $400 $500

8%

$ 92.59

$171.47

$238.15

$294.01

$340.29

$1,136.51 = Present Value

Uneven Cash Flow - Example

0 1 2 3 4 5

$100 $200 $300 $400 $500

8%

$432.00

$349.92

$251.94

$136.05

Future Value = $1,669.91

$1,136.51 = Present Value

Finding Interest Rate As with annuities, interest rates for uneven

cash flow streams cannot be solved directly. Calculators search for the correct answer,

called an IRR (there may be more than one correct answer). It guesses and then iteratively goes higher or

lower.

Compounding The more often one compounds interest, the

faster it grows.

0 1 2 3 4 5

8%

-$100 $146.93

0 1 2 3 4 105 6 7 8 94%

-$100 $148.02

Annual

Semi-Annual

Compounding Why is there a difference in future value?

Because interest is earned on itself faster! How would you adjust to compound

monthly? How would you make an adjustment in

annuities?

Effective Annual Rate To convert the other compounding periods to

an effective annual compounding rate (EAR) use the following formula:

E A Ri

m

m

m

1 1

C o m p o u n d in g p erio d s / year

Effective Annual Rate - Example 8% monthly compounding loan is equal to

what in effective annual rate?

E A R

10 0 8

1 21

1 0 8 3 1 0 0 0 8 3 8 3 %

1 2.

. . . .

Fractional Time Periods If you invest $100 for nine months at an EAR

of 8%, how does one calculate the future value? The same way one did before.

F V P V i

F V

n

1

1 0 0 1 0 0 8 1 0 0 1 0 5 9 4 1 9 9 40 7 5

w h ere n = # o f co m p o u n d in g p erio d s

. . $ 1 0 5..

Amortized Loans A loan with equal payments over the life of

the loan is called an amortized loan. Loan mathematics are the same as an annuity.

Loan amounts are the present value. Periodic loan payments are the payments.

Amortized Loans The present value of a monthly loan uses the

annuity formula adjusted for monthly payments:

L oan P M Ti

m

L oan P M Ti

mi

m

n m

t

n

n m

1

1

11

1

1

W ith o u t D eriv atio n

Amortized Loans Payments on a given loan can be found by solving

for PMT in the previous equation:

P M T

L oan

im

im

n m

11

1

Amortized Loans - Example What is the payment on a 30 year loan of

$150,000?

P M T

1 5 0 0 0 0

11

1 0 0 81 2

0 0 81 2

1 1 0 0 6 5

3 0 1 2

,

.

.

, .

Amortization Schedules Amortization schedules show how much of

each payment goes toward principal and how much toward interest.

The easiest way of calculating one by hand is by calculating the outstanding loan balance month-by-month, and then taking the difference in loan balance from month to month as the principal portion of the payment.

Amortization Schedules The portion in the amortized loan formula that

says n×m can be interpreted as months remaining.

So to find the part of the first $1,100.65 that is paid toward the principal one would realize that at the beginning one had all $150,000 outstanding.

Amortization Schedules After the first month, one has 359 payments left.

Therefore the loan principal outstanding is:

L oan

1 100 65

11

1 0 0812

0 0812

149 899 35

3 5 9

, ..

., .

Amortization Schedules The difference in principal outstanding is the

part of the payment that went toward principal. In this instance, 150,000-149,899.35=$100.65

The rest of the payment went toward interest. In this instance that would be 1,100.65-100.65=$1,000.

Amortization Schedules

$0

$200

$400

$600

$800

$1,000

$1,200

1 6 11 16 21 26 31

Years

InterestPrincipal

Different Types of Interest Simple Interest (i or isimple) - The rate we have

used thus far to calculate interest. Periodic Interest (i or iperiodic) - The interest

paid over a certain period.

ii

m

i

period ic

sim p le

period ic

F o r E x am p le: 8% co m p o u n d ed m o n th ly

0 0 8

1 20 6 6 6 6 %

..

Different Types of Interest Effective Annual Rate (EAR): Described

earlier as the rate that would be charged to get the same compounded annual rate.

Annual Percentage Rate (APR):

A P R i mi

mm iperiod ic

sim p le

sim p le