Ch. 4 - The Time Value of Ch. 4 - The Time Value of MoneyMoney

Topics CoveredTopics Covered

Future ValuesFuture Values

Present ValuesPresent Values

Multiple Cash FlowsMultiple Cash Flows

Perpetuities and AnnuitiesPerpetuities and Annuities

Effective Annual Interest Rate Effective Annual Interest Rate

Inflation & Time ValueInflation & Time Value

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and Discounting Single SumsDiscounting Single Sums

Future ValuesFuture Values

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

Compound InterestCompound Interest - Interest earned on - Interest earned on interest.interest.

Simple InterestSimple Interest - Interest earned only on the - Interest earned only on the original investment.original investment.

Future ValuesFuture Values

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

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

Future ValuesFuture ValuesExample - Simple InterestExample - Simple Interest

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

TodayToday Future YearsFuture Years 11 22 33

Interest EarnedInterest Earned 66 6 6 66ValueValue 100100 106106 112 112 118118

Value at the end of Year 3 = $118Value at the end of Year 3 = $118

Future ValuesFuture Values

Example - Compound InterestExample - Compound Interest

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

Interest Earned Per Year =Prior Year Balance Interest Earned Per Year =Prior Year Balance x .06x .06

Future ValuesFuture ValuesExample - Compound InterestExample - Compound Interest

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

TodayToday Future YearsFuture Years

11 22 33

Interest EarnedInterest Earned 6.006.00 6.36 6.36 6.746.74

ValueValue 100100 106.00 106.00 112.36 112.36 119.10119.10

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

Future Value of Single Cash FlowFuture Value of Single Cash Flow

trPVFV )1(

Future ValuesFuture 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 30

Number of Years

FV

of

$100

0%

5%

10%

15%

Future Values with CompoundingFuture Values with Compounding

Interest Rates

Example: Mutual Fund Fees and Example: Mutual Fund Fees and Retirement SavingsRetirement Savings

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

Present ValuesPresent 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 ValuesPresent Values

Present Value = PV

PV = Future Value after t periods

(1+r) t

Example: Paying for Baby’s Example: Paying for Baby’s MBAMBA

Just had a baby. You think the baby will Just had a baby. You think the baby will take after you and earn academic take after you and earn academic scholarships to attend college to earn a scholarships to attend college to earn a Bachelor’s degree. However, you want Bachelor’s degree. However, you want send your baby to a top-notch 2-year MBA send your baby to a top-notch 2-year MBA program when baby is 25. You have program when baby is 25. You have estimated the future cost of the MBA at estimated the future cost of the MBA at $85,000 for year 1 and $89,000 for year 2. $85,000 for year 1 and $89,000 for year 2.

Example: Paying for Baby’s Example: Paying for Baby’s MBAMBA

Today, you want to finance both years of Today, you want to finance both years of baby’s MBA program with one payment baby’s MBA program with one payment (deposit) into an account paying 8% (deposit) into an account paying 8% interest compounded annually.interest compounded annually.

How large must this deposit be?How large must this deposit be?

Value of Free CreditValue of Free Credit

Implied Interest RatesImplied Interest Rates

Internal Rate of ReturnInternal Rate of Return

Time necessary to accumulate fundsTime necessary to accumulate funds

Time Value of MoneyTime Value of Money(applications)(applications)

Example : Finding Rate of Example : Finding Rate of Return or Interest RateReturn or Interest Rate

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

What is your annual rate of return?What is your annual rate of return?

Important Time Value Important Time Value RelationshipsRelationships

Increasing interest rate and time Increasing interest rate and time increasesincreases future value. POSITIVE future value. POSITIVE RELATIONSHIP.RELATIONSHIP.

Increasing interest rate and time Increasing interest rate and time decreasesdecreases present valuepresent value. INVERSE . INVERSE RELATIONSHIP.RELATIONSHIP.

The Time Value of MoneyThe Time Value of Money

Compounding and Compounding and DiscountingDiscounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4

PerpetuitiesPerpetuities

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

PerpetuitiesPerpetuities

PV of Perpetuity FormulaPV of Perpetuity Formula

C = cash paymentC = cash payment

r = interest rate r = interest rate

PV Cr

Perpetuities & AnnuitiesPerpetuities & Annuities

Example - PerpetuityExample - Perpetuity

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

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

000,15 PV

Perpetuities & AnnuitiesPerpetuities & Annuities

Example - continuedExample - continued

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

151,259$3)05.1(

000,300

PV

AnnuitiesAnnuities

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

0 1 2 3 4PV FV

Examples of Ordinary Annuities:Examples of Ordinary Annuities:

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

If you borrow money to buy a house or a If you borrow money to buy a house or a car, you will pay a stream of equal car, you will pay a stream of equal payments.payments.

Annuity-dueAnnuity-due

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

0 1 2 3 4PV FV

Examples of Annuities-dueExamples of Annuities-due

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

Car lease payments.Car lease payments.

Cable TV and most internet service bills.Cable TV and most internet service bills.

Perpetuities & AnnuitiesPerpetuities & Annuities

PV of Ordinary Annuity FormulaPV of Ordinary Annuity Formula

C = cash paymentC = cash payment r = interest rate r = interest rate t = Number of years cash payment is t = Number of years cash payment is

receivedreceived

PV C r r r t

1 11( )

Perpetuities & AnnuitiesPerpetuities & Annuities

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

PVAF r r r t

1 11( )

Perpetuities & AnnuitiesPerpetuities & Annuities

ApplicationsApplications

Value of paymentsValue of payments

Implied interest rate for an annuityImplied interest rate for an annuity

Calculation of periodic payments Calculation of periodic payments Mortgage paymentMortgage payment Annual income from an investment payoutAnnual income from an investment payout Future Value of annual paymentsFuture Value of annual payments

FV C PVAF r t ( )1

Perpetuities & AnnuitiesPerpetuities & Annuities

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

C = cash paymentC = cash payment r = interest rate r = interest rate t = Number of years cash payment is receivedt = Number of years cash payment is received

)1()1(

11 rCPV trrr

Example: Invest Early in an IRAExample: Invest Early in an IRA

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

Why an IRA?Why an IRA?

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

Example: Enjoying your Example: Enjoying your RetirementRetirement

You go ahead and make the contributions You go ahead and make the contributions starting at age 22 in the last example, giving you starting at age 22 in the last example, giving you $1,102,114 at age 65.$1,102,114 at age 65.

You expect to live to age 85. So, you want to You expect to live to age 85. So, you want to make 20 annual withdrawals from your IRA make 20 annual withdrawals from your IRA paying 9% at the paying 9% at the beginning of each yearbeginning of each year starting starting at age 65.at age 65.

How large can this annual withdrawal be?How large can this annual withdrawal be?

More annuity fun, enjoying your More annuity fun, enjoying your release from baseballrelease from baseball

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

Non-Annual Interest Non-Annual Interest CompoundingCompounding

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

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

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

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

Effective Interest RatesEffective Interest Rates

exampleexample

Given a monthly rate of 1%, what is the Effective Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?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 FV and PV with non-annual interest compoundinginterest compounding

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

Single CFSingle CF

FVFVnmnm = PV(1+ARR/m) = PV(1+ARR/m)nmnm

PV = FVPV = FVnmnm/(1+APR/m)/(1+APR/m)nmnm

Non-annual annuitiesNon-annual annuitiesOrdinary:Ordinary:

PVPV = C(PVAF= C(PVAFAPR/m,nmAPR/m,nm))

FVFVnm nm = C(PVAF= C(PVAFAPR/m,nmAPR/m,nm)(1+APR/m))(1+APR/m)nmnm

Annuity-Due:Annuity-Due:

PVPV = C(PVAF= C(PVAFAPR/m,nmAPR/m,nm)(1+APR/m))(1+APR/m)

FVFVnm nm = C(PVAF= C(PVAFAPR/m,nmAPR/m,nm)(1+APR/m))(1+APR/m)nm+1nm+1

Example: Low Rate or Rebate?Example: Low Rate or Rebate?

The Frontier family want to buy a sport ut (SUV). They decide The Frontier family want to buy a sport ut (SUV). They decide on a 4-wheel drive Jeep Grand Cherokee. The purchase price on a 4-wheel drive Jeep Grand Cherokee. The purchase price with tax of the vehicle is $32,500. The Frontiers have $4,000 with tax of the vehicle is $32,500. The Frontiers have $4,000 as a down payment.as a down payment.Jeep offers the choice of two incentives on the 4-door Grand Jeep offers the choice of two incentives on the 4-door Grand Cherokee.Cherokee. 0% APR Financing for 60 months, or0% APR Financing for 60 months, or $3,000 rebate which would be applied toward the purchase price. $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 If the Frontiers elect to take the rebate, they can get 4.49% APR financing for 60 months.financing for 60 months.

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

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

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

InflationInflation

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

Nominal Interest RateNominal Interest Rate - Rate at which - Rate at which money invested grows.money invested grows.

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

InflationInflation

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

approximation formula

Real int. rate nominal int. rate - inflation rate

InflationInflationExampleExample

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

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

How will this change your retirement How will this change your retirement saving plans?saving plans?