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Ch. 2 - Time Value of Money

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Topics Covered

Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Non-annual interest compounding

Effective Annual Interest Rate

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The Time Value of Money

Compounding and Discounting Single Sums

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“The greatest mathematical discovery of all time is compound interest.”

Albert Einstein

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Future Values

Future Value - Amount to which an investment will grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the original investment.

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Example: Simple vs. Compound Interest Compare $100 invested at 10% interest

compound annually vs. 10% simple annual interest for 3 years.

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Future Value of Single Cash Flow

niPVFV )1(

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0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of Years

FV

of

$100

0%

5%

10%

15%

Future Values of $100 with Compounding

Interest Rates

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Example: Futurama Value?

Fry is frozen in the year 2000 with $0.93 in his checking account that pays 2.25% compounded annually. How much does Fry have in his account when he “awakes” a thousand years later in the year 3000?

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Present ValueToday's value of a lump sum received at a future point in

time:

nn iPVFV 1

n

n

i

FVPV

)1(

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Example: Paying for Baby’s MBA Just had a baby. You think the baby will take

after you and earn academic scholarships to attend college to earn a Bachelor’s degree. However, you want send your baby to a top-notch 2-year MBA program when baby is 25. You have estimated the future cost of the MBA at $102,000 for year 1 and $107,000 for year 2.

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Example: Paying for Baby’s MBA Today, you want to finance both years of

baby’s MBA program with one payment (deposit) into an account paying 6.5% interest compounded annually.

How large must this deposit be?

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The Power of High Discount Rates

Periods

Pre

sen

t V

a lu

e o

f O

ne

Do

llar

($)

0 2 4 6 8 10 12 14 16 18 20 22 24

0.5

0.75

1.00

0.25 10%

5%

15%20%

0%

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Implied Interest Rates Internal Rate of Return Time necessary to accumulate funds

Time Value of Money(applications)

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Example : Finding Rate of Return or Interest Rate A broker offers you an investment (a zero

coupon bond) that pays you $1,000 five years from now for the cost of $740 today.

What is your annual rate of return?

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The Time Value of Money

Compounding and Discounting

Cash Flow Streams

0 1 2 3 4

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Annuities

Annuity: a sequence of equal cash flows, occurring at the end of each period. This is known as an ordinary annuity.

0 1 2 3 4PV FV

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Examples of Ordinary Annuities:

If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond.

If you borrow money to buy a house or a car, you will re-pay the loan with a stream of equal payments.

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Annuity-due

A sequence of periodic cash flows occurring at the beginning of each period.

0 1 2 3 4PV FV

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Examples of Annuities-due

Monthly Rent payments: due at the beginning of each month.

Car lease payments. Cable & Satellite TV and most internet

service bills.

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What is the difference between an ordinary annuity and an annuity due?

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

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Solving for FV:3-year ordinary annuity of $100 at 10%

$100 payments occur at the end of each period, but there is no PV.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 -100

331

0

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Solving for PV:3-year ordinary annuity of $100 at 10%

$100 payments still occur at the end of each period, but now there is no FV.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 100 0

-248.69

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Solving for FV:3-year annuity due of $100 at 10%

Now, $100 payments occur at the beginning of each period.

FVAdue= FVAord(1+I) = $331(1.10) = $364.10. Alternatively, set calculator to “BEGIN” mode and

solve for the FV of the annuity:

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 -100

364.10

0BEGIN

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Solving for PV:3-year annuity due of $100 at 10%

Again, $100 payments occur at the beginning of each period.

PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. Alternatively, set calculator to “BEGIN” mode and solve for

the PV of the annuity:

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 100 0

-273.55

BEGIN

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Annuities

Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments

Mortgage payment Annual income from an investment payout Future Value of annual payments

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Example: Invest Early in an IRA How much would you have at age 65 if you

deposit $2,400 at the end of each year in an investment account with a 9% expected annual return starting at: (A) age 44? (B) age 22?

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A) Start at age 44

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B) Start at age 22

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Solving for PMT:How much must the 44-year old deposit annually to catch the 22-year old?

To find the required annual contribution, enter the number of years until retirement and the final goal of $1,058,030 and solve for PMT.

INPUTS

OUTPUT

N I/YR PMTPV FV

21 9

-18,639

1,058,0300

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Now about this?

Let’s assume that the 44-year old has already accumulated $120,000 in the IRA account. How much would he have to deposit on an annual basis at the 9% expected annual return to catch up with the 22-year old and be a millionaire at age 65?

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More Annuity Fun!

Springfield mogul Montgomery Burns, age 85, wants to retire at age 100 so he can steal candy from babies full time. Once Mr. Burns retires, he wants to withdraw $100 million at the beginning of each year for 10 years from a special off-shore account that will pay 20% annually. In order to fund his retirement, Mr. Burns will make 15 equal end-of-the-year deposits in this same special account that will pay 20% annually. How large of an annual deposit must be made to fund Mr. Burns’ retirement plans?

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Perpetuities

Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

PV of Perpetuity Formula

PMT = periodic cash payment

i = interest rate

iPMTPV

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Perpetuities & Annuities

Example - Perpetuity

You want to create an endowment to fund a football scholarship, which pays $15,000 per year, forever, how much money must be set aside today if the rate of interest is 5%?

000,300$05.000,15 PV 000,300$05.

000,15 PV

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What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = PV

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Solving for PV:Uneven cash flow stream Input cash flows in the calculator’s “CF” register:

CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50

Under NPV, enter I = 10, down arrow, and press CPT button to get NPV = $530.087. (Here NPV = PV.)

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The Time Value of Money

Non-annual Interest Compounding and

Discounting

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Classifications of interest rates

Nominal rate (INOM) – also called the quoted or state rate. An annual rate that ignores compounding effects. INOM is stated in contracts. Periods must also be given,

e.g. 8% Quarterly or 8% Daily interest.

Periodic rate (IPER) – amount of interest charged each period, e.g. monthly or quarterly. IPER = INOM / M, where M is the number of compounding

periods per year. M = 4 for quarterly and M = 12 for monthly compounding.

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Classifications of interest rates

Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, accounting for compounding.

EFF% for 10% semiannual investment

EFF% = ( 1 + INOM / M )M - 1

= ( 1 + 0.10 / 2 )2 – 1 = 10.25%

Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually.

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Why is it important to consider effective rates of return?

Investments with different compounding intervals provide different effective returns.

To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR).

See how the effective return varies between investments with the same nominal rate, but different compounding intervals.

EARANNUAL 10.00%EARQUARTERLY 10.38%EARMONTHLY 10.47%EARDAILY (365) 10.52%

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When is each rate used?

INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

IPER Used in calculations and shown on time lines. If M = 1, INOM = IPER = EAR.

EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match

compounding periods.

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FV and PV with non-annual interest compounding n = number of years m = number of times interest is paid per year inom = stated annual rate (APR) inom /m = periodic rateSingle CF

FVnm = PV(1 + inom/m)nm

PV = FVnm/(1 + inom/m)nm

Annuities: Use periodic rate and number of annuity payment and

compounding periods if interest compounding period and annuity payment period are the same.

Otherwise, need to find effective interest rate for each annuity payment period.

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What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

$134.49 (1.025) $100 FV

$134.01 (1.05) $100 FV

) 2

0.10 1 ( $100 FV

) M

I 1 (PV FV

123Q

63S

323S

NMNOMn

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Futurama Value Revisited

How much money would Fry have in his bank account in the year 3000 from the $0.93 deposited in the year 2000 if the 2.25% annual rate was compounded quarterly?

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Let’s buy a car!

Prof. Outback decides to purchase a brand-new 2007 Jeep Liberty Limited 4WD with heated premium leather seats, sunroof, and satellite radio for $28,800. After paying tax and license, Prof. Outback has $4,000 as a down payment. Jeep offers Prof the choice of 3.9% APR financing for 60 months or a $3,000 rebate. Prof. Outback can receive 6.25% APR financing for 60 months through E-Loan if the rebate option is selected. Which option would result in the lower monthly

payment? At what APR along without the rebate would the Prof.

be indifferent between the two options?

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Monthly Payments

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Indifference APR