Ch. 4 - The Time Value of Money

Topics Covered

Future ValuesPresent ValuesMultiple Cash FlowsPerpetuities and AnnuitiesEffective Annual Interest Rate Inflation & Time Value

The Time Value of Money

Compounding and Discounting Single

Sums

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.

Future Values

Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.

Interest Earned Per Year = 100 x .06 = $ 6

Future ValuesExample - Simple Interest

Interest earned at a rate of 6% for three years on a principal balance of $100.

Today Future Years1 2 3

Interest Earned 6 6 6Value 100 106 112 118

Value at the end of Year 3 = $118

Future Values

Example - Compound InterestInterest earned at a rate of 6% for three years on the previous year’s balance.

Interest Earned Per Year =Prior Year Balance x .06

Future ValuesExample - Compound Interest

Interest earned at a rate of 6% for three years on the previous year’s balance.

Today Future Years1 2 3

Interest Earned 6.00 6.36 6.74Value 100 106.00 112.36 119.10

Future Value of $100 compounded at 6% for three years = $119.10

Future Value of Single Cash Flow

trPVFV )1( +×=

Future Values

FV r t= × +$100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for three years?

10.119$)06.1(100$ 3 =+×=FV

0

1000

2000

3000

4000

5000

6000

7000

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

FV

of $

100

0%

5%10%

15%

Future Values with CompoundingInterest Rates

Example: Mutual Fund Fees and Retirement Savings

Prof. Finance moves to a new university and has $100,000 in retirement savings to invest (rollover) into a new retirement account.Prof. Finance wants to invest this money for 25 years into an indexed stock fund, which is expected to return 9% annually.Prof. has two choices: Vanguard Total Equity Fund with a 0.4% annual expense fee and Onguard Total Fencing Fund with an 1.2% annual expense fee.What is the difference in Prof. Finance’s expected future retirement savings between the two funds?

Present ValuesPresent Value

Value today of a future cash

flow.

Discount Rate

Interest rate used to compute

present values of future cash flows.

Discount Factor

Present value of a $1 future payment.

Present Values

Present Value = PV

PV = Future Value after t periods

(1+r) t

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 $85,000 for year 1 and $89,000 for year 2.

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 8% interest compounded annually.How large must this deposit be?

Value of Free CreditImplied Interest RatesInternal Rate of ReturnTime necessary to accumulate funds

Time Value of Money(applications)

Example : Finding Rate of Return or Interest Rate

A broker offers you an investment (a zero coupon bond) that pays you $5,000 five years from now for the cost of $3,700 today.What is your annual rate of return?

Important Time Value Relationships

Increasing interest rate and time increases future value. POSITIVE RELATIONSHIP.Increasing interest rate and time decreases present value. INVERSE RELATIONSHIP.

The Time Value of Money

Compounding and Discounting

Cash Flow Streams

0 1 2 3 4

Perpetuities

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

Perpetuities

PV of Perpetuity Formula

C = cash paymentr = interest rate

PV Cr=

Perpetuities & Annuities

Example - PerpetuityYou 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 in the rate of interest is 5%?

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

000,15 ==PV

Perpetuities & Annuities

Example - continuedIf the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

151,259$3)05.1(000,300 ==

+PV

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

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 pay a stream of equal payments.

Annuity-due

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

0 1 2 3 4PV FV

Examples of Annuities-due

Monthly Rent payments: due at the beginning of each month.Car lease payments.Cable TV and most internet service bills.

Perpetuities & Annuities

PV of Ordinary Annuity Formula

C = cash paymentr = interest rate t = Number of years cash payment is

received

[ ]PV C r r r t= −+

1 11( )

Perpetuities & Annuities

PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

[ ]PVAF r r r t= −+

1 11( )

Perpetuities & Annuities

ApplicationsValue of paymentsImplied interest rate for an annuityCalculation of periodic payments n Mortgage paymentn Annual income from an investment payoutn Future Value of annual payments

[ ]FV C PVAF r t= × × +( )1

Perpetuities & Annuities

PV (and FV) of Annuity-dues = PV (or FV) of ordinary annuity x (1 + r) or BGN mode on financial calculator.

C = cash paymentr = interest rate t = Number of years cash payment is

received

[ ] )1()1(

11 rCPV trrr +−=+

Example: Invest Early in an IRAHow much would you have at age 65 if you deposit $2,500 at the end of each year in an account paying 9% annually starting at:n (A) age 41?n (B) age 22?

Why an IRA?Imagine in the last example, you didn’t take advantage of the tax-sheltered environment of an IRA.Your annual investment return would be taxed!With a 28% tax rate, our annual after-tax return would fall from 9% to 6.48% (=9%(1-.28)).At age 65: I would have $135,519 vs. $191,975.You would have $535,392 vs. $1,102,114: 52% less!! The IRS killed Kenny,…!

Example: Enjoying your Retirement

You go ahead and make the contributions starting at age 22 in the last example, giving you $1,102,114 at age 65.You expect to live to age 85. So, you want to make 20 annual withdrawals from your IRA paying 9% at the beginning of each year starting at age 65.How large can this annual withdrawal be?

More annuity fun, enjoying your release from baseball

Bob B. is released from the last year of his guaranteed contract from a New York baseball team. He is due $5.9 million from the last year of this contract. Bob and the team agree to defer the $5.9 million at 8% interest for 15 years. At this time (15 years from today), the team will begin the first of 15 equal annual payments at 8% interest.The press is reporting the payments will total $30 million. Are they correct?

Non-Annual Interest Compounding

When interest is compounded more frequently than once a year.Important non-annual compounding terms and things to know:n Quoted Annual Rate, or Annual Percentage Rate (APR):

Stated nominal annual rate before compounding.n Effective Annual Rate (EAR): the actual (effective) annual

interest rate earned or paid.n Periodic Interest Rate: the interest rate paid or charged

each interest compounding period = quoted rate/m, where m = number of compounding periods per year.

Effective Interest Rates

exampleGiven a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

12.00%or .12=12 x .01=APR

12.68%or .1268=1-.01)+(1=EAR

r=1-.01)+(1=EAR12

12

FV and PV with non-annual interest compounding

n = number of yearsm = number of times interest is paid per yearAPR = nominal annual rate (APR)APR/m = periodic rate

Single CFFVnm = PV(1+ARR/m)nm

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

Non-annual annuitiesOrdinary:

PV = C(PVAFAPR/m,nm)FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm

Annuity-Due:PV = C(PVAFAPR/m,nm)(1+APR/m)FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm+1

Example: Low Rate or Rebate?

The Frontier family want to buy a sport ut (SUV). They decide on a 4 -wheel drive Jeep Grand Cherokee. The purchase price with tax of the vehicle is $32,500. The Frontiers have $4,000 as a down payment.Jeep offers the choice of two incentives on the 4-door Grand Cherokee.n 0% APR Financing for 60 months, orn $3,000 rebate which would be applied toward the purchase price.

If the Frontiers elect to take the rebate, they can get 4.49% APR financing for 60 months.

n Question: Which incentive would give the Frontiers the lowest monthly payment?

Example: The $200 national ISP signup credit: good deal for whom?

A national ISP all provide $200 for new customers to use at a particular electronics store chain if they sign-up for a 2-year internet service contract at $21.95/month.What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for free? (200 PV, -21.95 PMT, 24 N 0 FV, CPT I/Y = 9.8%/month x 12= 117.8% APR!!)What interest rate (APR) are you paying on this “free money” if you wanted internet service and could get it for $9.95/month?(-12 PMT CPT I/Y = 3.15%/mo x 12 = 37.8%! Thanks, but no thanks!

Inflation

Inflation - Rate at which prices as a whole are increasing.

Nominal Interest Rate - Rate at which money invested grows.

Real Interest Rate - Rate at which the purchasing power of an investment increases.

Inflation

1 + real interest rate = 1+ nominal interest rate1+inflation rate

approximation formula

Real int. rate nominal int. rate - inflation rate≈

InflationExample

If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?

2.8%or .028=.022-.050=ionApproximat

2.7%or .027 = rateinterest real

1.027 =rateinterest real1

=rateinterest real1 .022+1.050+1

+

+ Savings

Bond

Example: Real retirement income

Going back to your retirement in 43 years, you expect 3% inflation along with your 9% nominal investment rate annually and want to withdraw $32,000 in real terms at the beginning of each year for 20 years once you retire.How will this change your retirement saving plans?