Business Finance

Business FinanceBA303 ♦ Fall 2012Michael Dimond

Michael DimondSchool of Business Administration

The Time Value of Money (TVM)


Compounding

• Compounding is the growth of value resulting from some sort of return (such as interest payments) being added to the original amount.

• If you put $100 in the bank and receive 10% annual interest• After 1 year: $100 x (1+10%) = $110

• After 2 years: $110 x (1+10%) = $121

• After 3 years: $121 x (1+10%) = $133.10

• The three-year compounding could be rewritten like this:• After 3 years: $100 x (1+10%) x (1+10%) x (1+10%) = $133.10

or

• $100 x (1+10%)3 = $133.10

• The general formula for compounding:

PV x (1+i)n = FVwhere PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate


Discounting

• Discounting is the opposite of compounding. Instead of growing an amount by a specific rate, we are taking that expected growth out of a future total to find what the starting figure would be.

• Since compounding multiplies by (1+i)n, discounting will do the opposite: divide by (1+i)n.

• If you will need $133.10 at the end of three years, and you can receive 10% annual interest, how much would you need to deposit today?• $133.10 ÷ (1+10%)3 = $100

• The general formula for discounting:

FV ÷ (1+i)n = PVwhere PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate


Moving parts of compounding & discounting• There are four “moving parts” in a compounding or

discounting computation:• PV (Present Value)

• FV (Future Value)

• n (Number of Periods)

• i (Rate of Return per Period)

• The general formula for compounding:

PV x (1+i)n = FV• The more periods something is compounded, the greater the future value is.

• The general formula for discounting:

FV ÷ (1+i)n = PV• The more periods something is discounted, the smaller the present value is.


What if compounding happens more frequently?• APR means Annual Percentage Rate

• For example: 12% APR means 12% interest rate for the year.

• If interest compounds more frequently, divide that rate by the periods per year.• 12 % APR compounded…Annually 1 period/yr 12% ÷ 1 = 12.00% interest/period

Quarterly 4 periods/yr 12% ÷ 4 = 3.00% interest/period

Monthly 12 periods/yr 12% ÷ 12 = 1.00% interest/period

Daily 360 periods/yr 12% ÷ 360 = 0.03% interest/period

• Why do financiers use 360 days instead of 365?

• After 1 year, how much will $100 be at 12% APR…• compounded at the end of the year? $100 x (1.1200)1 =

$112.00

• compounded at the end of each quarter? $100 x (1.0300)4 = $112.55

• compounded at the end of each month ? $100 x (1.0100)12 = $112.68

• compounded at the end of each day ? $100 x (1.0003)360 = $112.75

Remember to watch out for

rounding errors:12/360 = 0.03333…


Effective Annual Rate

• The Effective Annual Rate (EAR) is the APR adjusted for the value of compounding.

• EAR = (1+APR ÷ n)n - 1• 12% APR compounded annually = (1.1200)1 -1 = 12.00% EAR

• 12% APR compounded quarterly = (1.0300)4 -1 = 12.55% EAR

• 12% APR compounded monthly = (1.0100)12 -1 = 12.68% EAR

• 12% APR compounded daily = (1.0003)360 -1 = 12.75% EAR

• Sometimes this is called the APY (Annual Percent Yield)


Time vs Return: Basic TVM

• A dollar is worth more now than it will be at any time in the future. The concept is called the Time Value of Money (TVM).

• What makes money lose value over time?• How long an investment takes to pay out will affect the price

you would pay.• If you require a 12% annual return, how much would you pay

for $100 to be given to you in…• 1 year?• 3 years?• 10 years?• The further in the future a cash flow is, the less it is worth.


Understanding TVM problems

• Time Value of Money scenarios are examined with a timeline.• Each tick mark on the timeline represents the end of one period.

• The first tick mark on the left is labeled 0 because zero periods have elapsed. It indicates the present, or the planned beginning of a project.

• The last tick mark indicates the end of the last period being analyzed.

• Payments and compounding happen at the end of each period.

• Consider our basic compounding example:

0 101 2 3 4 5 6 7 8 9

0 1 2 3

-100 133.10

i = 10%

PV = -100i = 10%n = 3FV = 133.10

You could use this diagram to analyze the future value or the

present value


• You could use this diagram to analyze the future value or the present value.

• Notice the cash outflow (money you invested) is shown with a minus sign. Financial calculators require this to give you the correct answer. This is called the sign convention.

Understanding TVM problems

100 x (1+0.10)3 = 133.10

:. FV = 133.10

0 1 2 3

-100 ?

i = 10%0 1 2 3

? 133.10

i = 10%

133.1 ÷ (1+0.10)3 = 100

:. PV = -100


• A TVM problem has one more “moving part” than a simple compounding or discounting problem.• PV (Present Value)

• FV (Future Value)

• n (Number of Periods)

• i (Rate of Return per Period)

• PMT (Payment)

• There may be payments which happen between the beginning and end of the timeline.• Each payment is discounted separately.

• The PV of the stream of cash flows is the sum of the individual PVs.

Moving parts of TVM


• If you require a 12% annual return, what would you pay for…• …$100 to be delivered in 1 year? ($89.2857)• …$100 to be delivered in 2 years? ($79.7194)• …$100 to be delivered in 3 years? ($71.1780)• …all of the above (i.e. $100 to be paid at the end of each of

the next three years)?

• By adding together the present values, you find the value of allthe cash flows in the stream.

Discounting payments

0 1 2 3

? 100

i = 12%

100 ÷ (1+0.12)3

100

100 ÷ (1+0.12)2

100

100 ÷ (1+0.12)1

89.285779.7194

+ 71.1780240.1831


• Remember the magic machine?• $100 per month for 5 years. What if you require a 12% annual return?

Discounting a stream of cash flows

0 601 2 3 4 56 57 58 59

? 100

i = 1% monthly (12% APR)

100100 100 100 100 100 100 100

100 ÷ (1+0.01)60

100 ÷ (1+0.01)1

Each payment has its own

present value. Adding up those

PVs gives the total value of the

stream of cash flows.

100 ÷ (1+0.01)2

100 ÷ (1+0.01)3

100 ÷ (1+0.01)4

100 ÷ (1+0.01)56

100 ÷ (1+0.01)57

100 ÷ (1+0.01)58

100 ÷ (1+0.01)59

99.01

57.28

98.03

97.06

96.10...

56.71

56.15

55.60

55.04


Timelines & PMTs

• i and n are always in the same increment. • Monthly periods → monthly rate.

• Annual periods → annual rate.

• What happens to PV as n increases?• As n increases, PV becomes smaller

100 ÷ 1.012 = 98.03 100 ÷ 1.0260 = 55.04

• Value = Sum of PVs

• So if you demand a 12% rate of return, the value of the machine’s monthly payments is:

• There is also an easier way to compute that value…

100 100 100 100 1.01 1.012 1.013 1.0160+ + + = 4,495.50$ + . . .

PMT PMT PMT PMT (1+i) (1+i)2 (1+i)3 (1+i)n Σ PV+ + + . . . + =


Ordinary Annuity: FV & PV

• A stream of cash flows where all payments are equal is called an Annuity.• In an Ordinary Annuity, each payment happens at the end of the period.

• Your financial calculator can solve these easily and quickly.• Find PV given n, i, and PMT

• Find FV given n, i, and PMT

• For the magic machine, the inputs would be:• PV = ? (This is what we’re solving for)• n = 60 (monthly payments)• i = 12/12 (12% ÷ 12 months)• PMT = 100 (per month)• FV = 0 (This has no value once the final payment is delivered)

Notice that these three items must always be in the same timeframe: monthly annually, daily… whatever is in the scenario


Comments about Annuity Due

• An Annuity Due has payments which happen at the beginning of each period instead of the end.

• Typically used in real estate…• Timelines…• Calculator setting…• Always reset your calculator as soon as you are done. Good

habits help avoid mistakes.


Ordinary Annuity with an additional payout• What happens if there is a stream of payments, and also a

lump sum being paid at the end of the timeline?• Timeline…• Find PV given n, i, PMT & FV


• If you require a 12% annual return, what would you pay for…• …$90 to be delivered in 1 year? ($80.3571)• …$95 to be delivered in 2 years? ($75.7334)• …$99 to be delivered in 3 years? ($70.4662)• …all of the above?

• By adding together the present values, you find the value of allthe cash flows in the stream.

Discounting unequal payments

0 1 2 3

? 99

i = 12%

99 ÷ (1+0.12)3

95

95 ÷ (1+0.12)2

90

90 ÷ (1+0.12)1

80.357175.7334

+ 70.4662226.5567


Using the calculator (NPV function)


Using the calculator (NPV function)


Uneven cash flows

• Covered later in the quarter• “Part 2” of How Do I Use This Financial Calculator explains

how to use the calculator for uneven cash flows


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