1

Time Value of MoneyTime Value of Money

2

CompoundingCompounding

• Assume that the interest rate is 10% p.a.Assume that the interest rate is 10% p.a.

• What this means is that if you invest $1 What this means is that if you invest $1 for one year, you have been promised for one year, you have been promised $1*(1+10/100) or $1.10 next year$1*(1+10/100) or $1.10 next year

• Investing $1 for yet another year Investing $1 for yet another year promises to produce 1.10 *(1+10/100) or promises to produce 1.10 *(1+10/100) or $1.21 in 2-years$1.21 in 2-years

3

Value of Investing $1Value of Investing $1

1 Year $1.1

2 Years $1.21

3 Years $1.331

4 Years $1.4641

– Continuing in this manner you will find Continuing in this manner you will find that the following amounts will be that the following amounts will be earned:earned:

4

Generalizing the methodGeneralizing the method

• Generalizing the method requires Generalizing the method requires some definitions. Letsome definitions. Let– i be the i be the iinterest rate nterest rate

– n be the life of the lump sum n be the life of the lump sum investmentinvestment

– PV be the PV be the ppresent resent vvalue alue

– FV be the FV be the ffuture uture vvaluealue

5

Future Value of a Lump Future Value of a Lump SumSum

niPVFV )1(* FV with growths from -6% to +6%

0

500

1,000

1,500

2,000

2,500

3,000

3,500

0 2 4 6 8 10 12 14 16 18 20

Years

Fu

ture

Va

lue

of

$1

00

0

6%

4%

2%

0%

-2%

-4%-6%

6

Example: Future Value of Example: Future Value of a Lump Suma Lump Sum• Your bank offers a

CD with an interest rate of 3% for a 5 year investment.

• You wish to invest $1,500 for 5 years, how much will your investment be worth?

1111145.1738$

)03.01(*1500$

)1(*5

niPVFV

n 5i 3%PV 1,500FV ?Result 1738.911111

7

Present Value of a Lump Present Value of a Lump SumSum

nn

n

n

iFVi

FVPV

i

iPVFV

)1(*)1(

:obtain to)1(by sidesboth Divide

)1(*

8

Example: Present Value of Example: Present Value of a Lump Suma Lump Sum

• You have been You have been offered $40,000 for offered $40,000 for your printing your printing business, payable business, payable in 2 years. Given in 2 years. Given the risk, you require the risk, you require a return of 8%. a return of 8%. What is the present What is the present value of the offer?value of the offer? today55.293,34$

55281.34293

)08.01(

000,40

)1(

2

ni

FVPV

9

Lump Sums FormulaeLump Sums Formulae

• You have solved a You have solved a present valuepresent value and a and a future valuefuture value of a of a lump sumlump sum. . There remains two other variables There remains two other variables that may be solved forthat may be solved for– interest, iinterest, i

– number of periods, nnumber of periods, n

10

Solving Lump Sum Cash Solving Lump Sum Cash Flow for Interest RateFlow for Interest Rate

1

)1(

)1(

)1(*

n

n

n

n

PVFV

i

PVFV

i

iPVFV

iPVFV

11

Example: Interest Rate on Example: Interest Rate on a Lump Sum Investmenta Lump Sum Investment

• If you invest If you invest $15,000 for ten $15,000 for ten years, you receive years, you receive $30,000. What is $30,000. What is your annual your annual return?return?

point) basisnearest the(to %18.7

071773463.0

121211500030000

1

101

1010

n

PVFV

i

12

Solving Lump Sum Cash Solving Lump Sum Cash Flow for Number of Flow for Number of PeriodsPeriods

iPVFV

iPVFV

n

iniPV

FV

iPV

FV

iPVFV

n

n

n

1ln

lnln

1ln

ln

1ln*)1(lnln

)1(

)1(*

13

The Frequency of The Frequency of CompoundingCompounding

• You have a credit card that carries a You have a credit card that carries a rate of interest of 18% per year rate of interest of 18% per year compounded monthly. What is the compounded monthly. What is the interest rate compounded annually? interest rate compounded annually?

• That is, if you borrowed $1 with the That is, if you borrowed $1 with the card, what would you owe at the end card, what would you owe at the end of a year?of a year?

14

The Frequency of The Frequency of Compounding-continuedCompounding-continued

• 18% per year compounded monthly 18% per year compounded monthly is just code for 18%/12 = 1.5% per is just code for 18%/12 = 1.5% per month month

• The year is the macroperiod, and the The year is the macroperiod, and the month is the microperiodmonth is the microperiod

• In this case there are 12 In this case there are 12 microperiods in one macroperiodmicroperiods in one macroperiod

15

The Frequency of The Frequency of Compounding-continuedCompounding-continued• When a rate is expressed in terms of a When a rate is expressed in terms of a

macroperiod compounded with a different macroperiod compounded with a different microperiod, then it is a nominal or annual microperiod, then it is a nominal or annual percentage rate (APR). In the credit card percentage rate (APR). In the credit card example, it is 18%.example, it is 18%.

• The (real) monthly rate is 18%/12 = 1.5% so The (real) monthly rate is 18%/12 = 1.5% so the real annual rate (Effective Annual Rate, the real annual rate (Effective Annual Rate, EFF) is (1+0.015)EFF) is (1+0.015)1212 - 1 = 19.56% - 1 = 19.56%

• The two equal APR with different frequency of The two equal APR with different frequency of compounding have different effective annual compounding have different effective annual rates:rates:

16

Effective Annual Rates of Effective Annual Rates of an APR of 1an APR of 1

AnnualPercentagerate

Frequency ofCompounding

AnnualEffective Rate

18 1 18.00

18 2 18.81

18 4 19.25

18 12 19.56

18 52 19.68

18 365 19.72

17

AnnuitiesAnnuities

• Financial analysts use several Financial analysts use several annuities with differing assumptions annuities with differing assumptions about the first payment. We will about the first payment. We will examine just two:examine just two:– regular annuityregular annuity with its first coupon with its first coupon

one period from nowone period from now

– annuity dueannuity due with its first coupon today with its first coupon today

18

Assumptions Regular Assumptions Regular Annuity Annuity

– the first cash flow will occur exactly the first cash flow will occur exactly one period form nowone period form now

– all subsequent cash flows are all subsequent cash flows are separated by exactly one periodseparated by exactly one period

– all periods are of equal lengthall periods are of equal length

– the term structure of interest is flatthe term structure of interest is flat

– all cash flows have the same (nominal) all cash flows have the same (nominal) valuevalue

– the present value of a sum of present the present value of a sum of present values is the sum of the present valuesvalues is the sum of the present values

19

Annuity Formula NotationAnnuity Formula Notation

• PV = the present value of the PV = the present value of the annuityannuity

• i = interest rate to be earned over i = interest rate to be earned over the life of the annuitythe life of the annuity

• n = the number of paymentsn = the number of payments

• pmt = the periodic paymentpmt = the periodic payment

20

PV of Annuity FormulaPV of Annuity Formula

n

n

iipmt

ii

pmt

PV

1

11*

}1

11{*

21

PV Annuity Formula: PV Annuity Formula: PaymentPayment

n

n

n

i

iPVpmt

iipmt

iipmt

PV

11

*

11*

1

11*

22

PV Annuity Formula: PV Annuity Formula: Number of PaymentsNumber of Payments

ipmt

iPV

npmt

iPVi

pmtiPV

inpmt

iPVi

ipmt

iPVi

ipmt

PV

n

n

nn

1ln

*1ln

;*

11

*1ln1ln*;

*11

11*

;11*

23

Perpetual Annuities / Perpetual Annuities / PerpetuitiesPerpetuities

• Recall the annuity formula:Recall the annuity formula:

niipmt

PV1

11*

• Let n -> infinity with i > 0:

ipmt

PV

24

Excel Exercise 1Excel Exercise 1

• Taking out a loan: borrow $100,000 Taking out a loan: borrow $100,000 from a bank, 30 year, 360 month from a bank, 30 year, 360 month payment, interest rate is 12% APR, payment, interest rate is 12% APR, what is PMT?what is PMT?

25

Excel Exercise 2Excel Exercise 2

• Another loan, 15 year loan, PMT is Another loan, 15 year loan, PMT is $1100 per month, what is the $1100 per month, what is the interest rate? (interest rate interest rate? (interest rate calculate of this kind will not be on calculate of this kind will not be on the exam)the exam)