1

The Time Value of Money

A core concept in financial managementA core concept in financial management

2

Lesson Objectives

To introduce the time value concept

Calculate present and future values of any set of expected future cash flows.

3

Time Time Value ???Value ???

Rs.1000 you received today or Rs.1000 will be received tomorrow.

What do you prefer? Simple reason is your time preference for money. Therefore, you may expect to get an extra cash amount as

compensation for delaying. It is called the interest.

This Interest for the Time Preference for Money is called the Time Value of Money

4

Why we need a premium for Why we need a premium for future cash flows?future cash flows?

Alternative uses of money. (Investment opportunities)

Individual’s preference for early consumption (time preference theory )

The risk associated with future cash flows

The interest rate for the time value of money can be regarded as the opportunity cost

5

• A rupee received today is more valuable than a rupee received tomorrow.

• Thus, cash flows in different time periods cannot be compared as they are.

• There are two ways

• Future value

• Present value

Comparison of CF at different Comparison of CF at different Time intervalsTime intervals

6

?

• Translate Rs.1 today into its equivalent in the future (COMPOUNDING).

• Translate Rs.1 in the future into its equivalent today (DISCOUNTING).

Today Future

?

?

Today Future

7

ASSUMPTIONS

A point in time is denoted by the letter “t”. Unless otherwise stated, t=0 represents today (the

decision point). Unless otherwise stated, cash flows occur at the end of a

time interval. Cash inflows are treated as positive amounts, while cash

outflows are treated as negative amounts. Compounding frequency is the same as the cash flow

frequency.

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The Time Line

t=0 t=1 t=2 t=3 t=4

Today Beginning of the fourth year

End of thethird year

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Cash Flows

Single cash flow

Annuity

Multiple/ uneven cash flows

10

Future Value and Compounding Process

Future value Is the total of the principle amount and the

interest accumulated on the principle for a given period.

Is the sum which an initial amount of principle (or present value (PV)) is expected to grow over a given (n) period at a given interest rate.

11

Example 1:

Suppose you place Rs.100 in a savings account that earns 6% interest compounded annually.

How much can you get at the end of each period?

12

Future Value Formula

Let PV = Present Value

FVn= Future Value at time n

r = interest rate (or discount rate) per period.

nn rPVFV 1

13

Future Value Interest Formula

Year Rs. FVIF0.1,n FVIF0.1,n

0 1 (1.1)0 1.00000

1 1 (1.1)1 1.10000

2 1 (1.1)2 1.21000

3 1 (1.1)3 1.33100

4 1 (1.1)4 1.46410

5 1 (1.1)5 1.61051

6 1 (1.1)6 1.77156

7 1 (1.1)7 1.94872

8 1 (1.1)8 2.14359

9 1 (1.1)9 2.35795

10 1 (1.1)10 2.59374

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r = 0%

r = 10%

Future Value Factor

0.00

2.00

4.00

6.00

8.00

10.00

0 5 10 15Time

FV

Fa

cto

r

r = 5%

r = 15%

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More Frequent Compounding

Interest may be compounded more than once a year.

The Nominal Rate (Annual Percentage Rate (APR)) is the periodic rate times the number of periods per year.

The Effective rate (Annual Percentage Yield (APY)) is the “true” annually compounded interest rate.

16

Effect of Compounding Frequency on Future Value

Find the future value at the end of one year if the present value is Rs.20,000 and the interest rate is 16%. Use the following compounding frequencies:

• Annual Compounding

• Semiannual Compounding

• Quarterly Compounding

• Monthly Compounding

• Daily Compounding

• Continuous Compounding

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Annual Compounding - Once a year

The periodic rate is 16%.

200,23.16.1000,20. 11 RsRsFV

APY = APR = 16%

Compounding m-times in a yearmn

n m

rPVFV

1

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Semi - Annual Compounding

Since m = 2, the periodic rate is 8%.

2328,232

16.01000,20

2

1

FV

%64.161664.012

16.01

2

orAPY

APY > APR

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Quarterly Compounding

Since m = 4, the periodic rate is 4%.

17.397,234

16.01000,20

4

1

FV

%986.1616986.014

16.01

4

orAPY

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Effect of Compounding Frequency on Future Value

Compounding m FV APY

Annual

Semi-Annual

Quarterly

Monthly

Daily

1

2

4

12

365

23,200.00

23,328.00

23,397.17

23,445.42

23,469.39

16.000%

16.640%

16.986%

17.227%

17.347%

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Continuous Compounding

With continuous compounding, m becomes very large.mn

n m

rPVFV

1

• As m approaches infinity, the value of (1+r/m)mn goes

to er n. Thus,

• Then the effective rate = er - 1, where e = 2.71828.• Thus, effective rate = (2.71828)0.16 - 1 = 0.17351 or

17.351%.• FV = Rs.20,000 (1.17351)1 = Rs.23,469.39

rnn ePVFV

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Present Value

Future value

Thus the Present Value

nn rPVFV 1

nnr

FVPV1

1

When we get the present value, the interest rate is referred as the Discount Rate and this process is called as Discounting

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Present Value Interest Formula

Year Rs. PVIF0.1,n PVIF0.1,n

0 1 1/(1.1)0 1.00000

1 1 1/(1.1)1 0.90909

2 1 1/(1.1)2 0.82645

3 1 1/(1.1)3 0.75131

4 1 1/(1.1)4 0.68301

5 1 1/(1.1)5 0.62092

6 1 1/(1.1)6 0.56447

7 1 1/(1.1)7 0.51316

8 1 1/(1.1)8 0.46651

9 1 1/(1.1)9 0.42410

10 1 1/(1.1)10 0.38554

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1. If you will receive Rs.100 one year from now, what is the PV of that Rs.100 if your opportunity cost is 6%

2. If you receive Rs.100 5 years from now, what is the PV of that Rs.100, if your opportunity cost is 6%?

3. What is the PV of Rs.1,000 to be received 15 years from now if your opportunity cost is 7%?

Present Value - single sums

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Present Value Factor and Time

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15Time

PV

Fac

tor

r = 5%

r = 10%

r = 0%

r = 15%

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1. If you sold land for Rs.11,933 that you bought 5 years ago for Rs.5,000, what is your annual rate of return?

2. Suppose you placed Rs.100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to Rs.500?

Present Value - single sumsEx.

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Multiple Cash Flows

PV of multiple cash flows = the sum of the present values of the individual cash flows.

FV of multiple cash flows at a common point in time = the sum of the future values of the individual cash flows at that point in time.

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How do we find the FV/PV of a cash flow stream when cash flows are different? (Use a 10% interest rate).

Uneven Cash FlowsUneven Cash FlowsFVFV

0 1 2 3 4

-10,000 2,000 4,000 6,000 7,000

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Annuities

An annuity is a series of identical cash flows that are expected to occur each period for a specified number of periods.

Thus, CF1 = CF2 = CF3 = Cf4 = ... = CFn

Examples of annuities: Installment loans (car loans, mortgages). Coupon payment on corporate bonds. Rent payment on your apartment.

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Types of Annuities

Ordinary Annuity: An annuity with End-of-Period cash flows, beginning

one period from today.

Annuity Due: An annuity with Beginning-of-Period cash flows.

Deferred Annuity: An annuity that begins more than one period from

today.

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Future Value of an Annuity

3210 11000110001100011000 rrrrFVAn

0 1 2 3 4

1,000 1,000 1,000 1,000

1210 1111 nn rCFrCFrCFrCFFVA

When it has n periods, the equation is

32

Simplification

)1(

)1(

),1(,1,

)1(........)1()1(1

111112

1210

R

RaS

nnrRaWhere

rrrCF

rCFrCFrCFrCFFVA

n

n

FVIFAGP

n

nn

When you substitute above variables and simplify the equation, You can arrive at

nr

n

n

FVIFACF

r

rCFFVA

,

)1)1((

33

If you invest Rs.1,000 at the end of each next 3 years, at 8%, how much would you have after 3 years?

40.3246

08.0

108.11000

)1)1((

3

r

rCFFVA

n

n

Future Value Interest Factor for Annuity (FVIFA) Tables can be used to get the answer

Future Value - annuity

34

FVIFA Table for 8%

Year Rs. FVIF0.08,n FVIF0.08,n FVIFA0.08,n

1 1 (1.08)0=1 1.00000 1.00000

2 1 (1.08)1 1.08000 2.08000

3 1 (1.08)2 1.16640 3.24640

4 1 (1.08)3 1.25971 4.50611

5 1 (1.08)4 1.36049 5.86660

6 1 (1.08)5 1.46933 7.33593

7 1 (1.08)6 1.58687 8.92280

8 1 (1.08)7 1.71382 10.63663

9 1 (1.08)8 1.85093 12.48756

10 1 (1.08)9 1.99900 14.48656

2464.310003,08.0

FVIFACFFVAn

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Present Value Of an Annuity

00 11 22 33 44

1,000 1,000 1,000 1,000

432 )1(

11000

)1(

11000

)1(

11000

1

11000

rrrrPVAn

When it has n periods, the equation is

nn rCF

rCF

rCFPVA

)1(

1....

)1(

1

1

12

36

Simplification

)1(

)1(

,)1(

1,1,

)1(

1........

1

11

1

1

)1(

1

)1(

1

1

1

1

2

R

RaS

nnr

RaWhere

rrrCF

rCF

rCF

rCFPVA

n

n

PVIFA

GP

n

nn

When you substitute above variables and simplify the equation, You can arrive at

nr

n

n

PVIFACF

rr

CFPVA

,

)1(1

1

37

What is the PV of Rs.1,000 CF at the end of each of the next 3 years, if the opportunity cost is 8%?

1.2577

5771.21000

08.0)08.1(

11

1000

)1(1

1

3

rr

CFPVAn

n

Present Value Interest Factor for Annuity (PVIFA) Tables can be used to get the answer

Present Value Of an Annuity

38

PVIFA Table for 8%

Year Rs. PVIF0.08,n PVIF0.08,n PVIFA0.08,n

1 1 1/(1.08)1 0.92593 0.92593

2 1 1/(1.08)2 0.85734 1.78326

3 1 1/(1.08)3 0.79383 2.57710

4 1 1/(1.08)4 0.73503 3.31213

5 1 1/(1.08)5 0.68058 3.99271

6 1 1/(1.08)6 0.63017 4.62288

7 1 1/(1.08)7 0.58349 5.20637

8 1 1/(1.08)8 0.54027 5.74664

9 1 1/(1.08)9 0.50025 6.24689

10 1 1/(1.08)10 0.46319 6.71008

5771.210003,08.0

PVIFACFPVAn

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Using an interest rate of 8%, we find that:

The Future Value (at 3) is Rs.3,246.40. The Present Value (at 0) is Rs.2,577.10.

0 1 2 3

1000 1000 1000

Earlier, we examined this Earlier, we examined this “ordinary” annuity:“ordinary” annuity:

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Same 3-year time line, Same 3 Rs.1000 cash flows, but The cash flows occur at the beginning of each year,

rather than at the end of each year. This is an “annuity due.”

0 1 2 3

1000 1000 1000

Annuity DueAnnuity Due

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If you invest Rs.1,000 at the beginning of each of the next 3 years at 8%,

how much would you have at the end of year 3?

)1(

)1()1)1((

, rFVIFACF

rr

rCFFVAD

nr

n

n

11.3506

)08.1(2464.31000

)08.1(08.0

)1)08.1((1000

n

nFVAD

Equation and the Solution is

Future Value - annuity due

42

What is the PV of Rs.1,000 at the beginning of each of the next 3

years, if your opportunity cost is 8%?

)1(

)1()1(

11

, rPVIFACF

rr

rCFPVA

nr

n

n

0 1 2 3

1000 1000 1000

Equation and the solution is

26.2783

)08.1(5771.21000

)08.1(08.0

)08.1(1

11000

3

nPVA

Present Value - Annuity due

43

Deferred Annuity

The first cash flow in a deferred annuity is expected to occur later than t=1.

The PV of the deferred annuity can be computed as the difference in the PVs of two annuities.

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Deferred Annuity

An annuity’s first cash flow is expected to occur 3 years from today. There are 4 cash flows in this annuity, with each cash flow being Rs.500. At an interest rate of 10% per year, find the annuity’s present value.

0 1 2 3 4 5 6

Rs.500 Rs.500 Rs.500 Rs.500

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Example

Cash flows from an investment are expected to be Rs.40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

After graduation, you plan to invest Rs. 400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years?

If you borrow Rs.100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

46

Upon retirement, your goal is to spend 5 years travelling around the world. To travel in style, it will require Rs.250,000 per year at the beginning of each year.

If you plan to retire in 30 years, what are the equal monthly (end of month) payments necessary to achieve this goal?

The funds in your retirement account will compound

at 10% per annum on monthly basis.

Team AssignmentTeam Assignment

47

Present Value of Your Bank Loan

Cynthia Smart agrees to repay her loan in 24 monthly installments of Rs.250 each. If the interest rate on the loan is 0.75% per month, what is the present value of her loan payments? You wish to retire 25 years from today with Rs.2,000,000 in

the bank. If the bank pays 10% interest per year, how much should you save each year to reach your goal?

Rob Steinberg borrows Rs.10,000 to be repaid in four equal annual installments, beginning one year from today. What

is Rob’s annual payment on this loan if the bank charges him 14% interest per year?

48

Loan Amortization Schedule

It shows how a loan is paid off over time.

It breaks down each payment into the interest component and the principal component.

We will illustrate this using Rob Steinberg’s 4-year Rs.10,000 loan which calls for annual payments of Rs.3,432.05. Recall that the interest rate on this loan is 14% per year.

49

Loan Amortization Schedule

Period: 1 2 3 4

Principal @ Start

of Period

Interest for Period

Balance

Payment

Principal Repaid

Principal @ End

of Period

10000.00

1,400.00

11,400.00

3,432.05

2,032.05

7,967.95

50

Loan Amortization Schedule

Period: 1 2 3 4

Principal @ Start

of Period

Interest for Period

Balance

Payment

Principal Repaid

Principal @ End

of Period

10000.00

1,400.00

11,400.00

3,432.05

2,032.05

7,967.95

7,967.95

1,115,51

9,083.47

3,432.05

2,316.53

5,651.42

51

Loan Amortization Schedule

Period: 1 2 3 4

Principal @ Start

of Period

Interest for Period

Balance

Payment

Principal Repaid

Principal @ End

of Period

10000.00

1,400.00

11,400.00

3,432.05

2,032.05

7,967.95

7,967.95

1,115,51

9,083.47

3,432.05

2,316.53

5,651.42

5,651.42

791.20

6,442.62

3,432.05

2,640.85

3,010.57

52

Loan Amortization Schedule

asPeriod: 1 2 3 4

Principal @ Start

of Period

Interest for Period

Balance

Payment

Principal Repaid

Principal @ End

of Period

10000.00

1,400.00

11,400.00

3,432.05

2,032.05

7,967.95

7,967.95

1,115,51

9,083.47

3,432.05

2,316.53

5,651.42

5,651.42

791.20

6,442.62

3,432.05

2,640.85

3,010.57

3,010.57

421.48

3,432.05

3,432.05

3,010.57

0.00

53

APR and APY for an Installment Loan

Suppose you borrow Rs.5,000 from the bank and promise to repay the loan in 12 equal monthly installments of Rs.437.25 each, with the first payment to be made one month from today.

What is the APR? What is the APY?

54

Perpetuity

A perpetuity is an annuity with an infinite number of cash flows.

The present value of cash flows occurring in the distant future is very close to zero. At 10% interest, the PV of Rs.100 cash flow occurring 50

years from today is Rs.0.85! The PV of Rs.100 cash flow occurring 100 years from

today is less than one penny Suppose you will receive a fixed payment every period

(month, year, etc.) forever. This is an example of a perpetuity.

55

Mathematically,

(PVIFA r, n ) = 1 - 1

(1 + r)n

r

We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?

56

1 - 1

(1 + r)n

r

1 1 r r

When n gets very large,When n gets very large,

this reaches to this reaches to zero.zero.

So we’re left with PVIFA =So we’re left with PVIFA =

57

PMT r

PV =

So, the PV of a perpetuity is very simple to find:

Present Value of a PerpetuityPresent Value of a Perpetuity

58

What should you be willing to pay in order to receive Rs.10,000 annually forever, if you require 8% per year on the investment?

Find the present value of a perpetuity of Rs.270 per year if the interest rate is 12% per year.

The First Commerce Bank offers a Certificate of Deposit (CD) that pays you Rs.5,000 in four years. The CD can be purchased today for Rs.3,477.87. Assuming you hold the CD to maturity, what annual interest rate is the bank paying on this CD?

Ex.Ex.

59

Solving for an Unknown Interest Rate on a Loan

Erin Clapton borrows Rs.10,000 from her bank, and agrees to repay the loan in six equal annual installments of Rs.2,100 each. The first payment will be made one year from today. What interest rate is the bank charging her?

60

Solving for an Unknown Interest Rate on a Loan

at r = 8%, PVA6 = Rs.9,708.05

at r = 6%, PVA6 = Rs.10,326.38

at r = 7%, PVA6 = Rs.10,009.83 The exact answer is 7.0315%

0 1 2 3 4 5 6

Rs.2,100Rs.2,100Rs.2,100 Rs.2,100 Rs.2,100 Rs.2,100

Rs.10,000