Iii A Time Value Of Money

Time Value of Money

The chief value of money lies in the fact that one lives in a world in which it is overestimated.

HL Mencken

A firm is contemplating investing one lakh rupees in a project that is expected to pay rupees twenty thousands per annum over the next seven years. Should the firm accept the proposal?

• Time value of money

• Risk

Sale of land

• Two offers

10,000

11424 next year

• Bank interest rate =12%

10000 grows to 11200

Invest 10200 today at 12% to get 11424 next year.

Time Value of Money

• Money has a time value associated with it and therefore, a rupee received today is worth more than a rupee received in the future.

• The future value and present value of a rupee are based on the number of periods involved and the interest rate applicable.

The Manhattan Islands

• The Indians have always been ridiculed for selling the Manhattan Island for dollars 24 in 1624. Was it really ridiculous? If the Indians had invested dollars 24 at 6 per cent per annum, they would have had ……..

• If the Indians had been a little more astute and invested dollars 24 at 7.5%, they would now have had ……..

• 1000 crores

• 20,00,000 crores

Applications• Value maximisation • Value a stream of cashflows

Mortgage payments on a housing loan

Periodical savings required to accumulate a certain sum

Effective rate of borrowing for a car

Return on investment

Investment required to ensure pension after retirement• Capital budgeting and financing decisions• Valuation of bonds

Required Rate of Return

• Postponing consumption

• Inflation

• Risk

• Cost to the company

Simple Interest

• Interest paid or earned only on the original amount

• Simple interest = Principle* Interest rate* No of Periods

• Future value =Interest + Principle

Simple Interest

• Rs 100. 10%. 7th year

• Interest = 100 * 10% = 10

• Value at the end of the 7th year equals

100 + 10*7 = 100 +70 = 170

Compound Interest

• Interest earned during a period is added to the original principle to get the amount on which interest will be calculated in the next period.

• Interest earned during a period on the principle as well as the previous periods’ interest.

Compound Interest• Rs 100. 10%. 7th year• 100 (1+0.10)7-1 * 0.1 =17.72• Value = 100 (1+0.1)7 = 194.87• Future Value Interest Factor (FVIF) = (1+r)n

• FVIF 10% 7Years = 1.949• Future Value = 100 * 1.949 = 194.90• FVn = PV0 * FVIFr,n

Benjamin Franklin

Money makes money and the money that money makes, makes more money.

Ibbotson & Sinquefield

• Stock market return from 1926

• $1 invested in 1926 grows to $2279 in 2001

• 10.71% compounded annually

The power of compounding can explain why well to do families bequeath their wealth to their grandchildren rather than their children. Parents would rather make their grandchildren very rich than make their children moderately rich. In these families, the grandchildren have a more positive view of the power of compounding than do the children.

Four Values

• Future value of a single amount

• Present value of a single amount

• Future value of an annuity

• Present value of an annuity

Future Value of a Single Amount

• What is an amount equal to in future

FV = P0 (1+r)n

• (1+r)n is the Future Value Interest Factor

• FVn = PV0 * FVIFr,n

Present Value

• Present worth of a future cashflow• Amount today that will grow to the specified

amount at the end of the specified period• Calculate present value of a future cashflow to

make it comparable• Calculation of present value of a future amount is

called discounting. • The rate at which it is discounted is the discount

rate or the capitalisation rate.

Present Value (cont…)

• FV = P0 (1+r)n

• P0 = FV / (1+r)n

• P0 = FV* (1 / (1+r)n)• At 10%, Present value of 200 received 7

years hence is 102.62.• (1 / (1+r)n) is the present value interest factor• PV0 = FVn * (PVIFr,n)

Four Key Parameters

Present value

Future value

Discount rate

Periods

The Multiperiod Case:Future Value

• Suppose that Jay Ritter invested in the initial public offering of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year for the next five years.

• What will the dividend be in five years?

FV = P0×(1 + r)n

$5.92 = $1.10×(1.40)5

Future Value and Compounding

• Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30

This is due to compounding.

Future Value and Compounding

0 1 2 3 4 5

10.1$

3)40.1(10.1$

02.3$

)40.1(10.1$

54.1$

2)40.1(10.1$

16.2$

5)40.1(10.1$

92.5$

4)40.1(10.1$

23.4$

Present Value and Compounding

• How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

5)15.1(

000,20$53.943,9$

If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

TrCFV )1(0 T)10.1(000,5$000,10$

2000,5$

000,10$)10.1( T

2ln)10.1ln( T

years 27.70953.0

6931.0

)10.1ln(

2lnT

How Long is the Wait?

Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child’s education?

TrCFV )1(0 12)1(000,5$000,50$ r

10000,5$

000,50$)1( 12 r 12110)1( r

2115.12115.1110 121 r

About 21.15%.

What Rate Is Enough?

Special Considerations in TVM

• Annuity

• Perpetuity

• Growing perpetuity

• Intra year compounding

Annuities

• A series of equal cashflows over a specified number of periods

• Ordinary annuities : Payments start at the end of the current period.

• Future value of an annuity (FVIFAi,n)

• Present value of an annuity (PVIFAi,n)

Perpetuity

• Equal cashflows which continue forever

• Perpetual bonds. Preferred stock

• British railroad bonds

Growing Perpetuity

• A perpetuity that grows at a constant rate per annum

• Growth rate of companies

• Inflation

Perpetuity

A constant stream of cash flows that lasts forever.

0

…1

C

2

C

3

C

The formula for the present value of a perpetuity is:

32 )1()1()1( r

C

r

C

r

CPV

r

CPV

Perpetuity

What is the value of a British consol that promises to pay £15 each year, every year until the sun turns into a red giant and burns the planet to a crisp?

The interest rate is 10-percent.

0

…1

£15

2

£15

3

£15

£15010.

£15PV

Growing Perpetuity

A growing stream of cash flows that lasts forever.

0

…1

C

2

C×(1+g)

3

C ×(1+g)2

The formula for the present value of a growing perpetuity is:

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

Growing Perpetuity

The expected dividend next year is $1.30 and dividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value of this promised dividend stream?

0

…1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

00.26$05.10.

30.1$

PV

AnnuityA constant stream of cash flows with a fixed maturity.

The formula for the present value of an annuity is:

Tr

C

r

C

r

C

r

CPV

)1()1()1()1( 32

Trr

CPV

)1(

11

0 1

C

2

C

3

C

T

C

Annuity

If you can afford a $400 monthly car payment, what value of car can you afford if interest rates are 7% on 36-month loans?

59.954,12$)1207.1(

11

12/07.

400$36

PV

0 1

$400

2

$400

3

$400

36

$400

What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

22.297$09.1

97.327$0

PV0 1 2 3 4 5

$100 $100 $100 $100$327.97$297.22

97.327$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

11

tt

PV

PV of Annuity

• You wish to buy a photocopier and the supplier has quoted a price of 11,000 cash or 3000 per year for five years. If your cost of capital is 12%, which alternative would you prefer? What if the cost of capital is 8%

Lottery• New York State has started a lottery scheme

wherin it will give away 40 million (2million per year for the next 20 years) to the winner. It plans to donate 50% of the collections of 36 million to charities. If the discount rate is 10%, will it save anything from the scheme or will it have to fund the charity donations from other sources?

Retirement Benefit

• Savings in pension fund up to 2000 per annum are tax free. You are starting your career at 25 years. You plan to retire at 65 years. Your investments are likely to yield 8% per annum. How much money will you have at retirement? What amount will you have if your annual returns are taxed at 25%.

Intra Year Compounding

• Annually m = 1 100

• Semi annually m = 2 110.25

• Quarterly m = 4 110.38

• Monthly m = 12 110.47

• Weekly m = 52 110.51

• Daily m = 365 110.52

Compounding Periods

Compounding an investment m times a year for T years :

Tm

m

rCFV

10

For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to

93.70$)06.1(50$2

12.150$ 6

32

FV

Effective Annual Interest Rates

A reasonable question to ask in the above example is what is the effective annual rate of interest on that investment?

The Effective Annual Interest Rate (EAR) is the annual rate that would give us the same end-of-investment wealth after 3 years:

93.70$)06.1(50$)2

12.1(50$ 632 FV

93.70$)1(50$ 3 EAR

Effective Annual Interest Rates (continued)

• Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.

• What we have is a loan with a monthly interest rate rate of 1½ percent.

• This is equivalent to a loan with an annual interest rate of 19.56 percent

19561817.1)015.1(12

18.11 12

12

mn

m

r

Continuous Compounding

• A = P ( 1+ r ) n

In case of intra year compounding*

• A = P ( 1 + r/m ) nm

The limit as m tends to infinity is known as continuous compounding.

• A = P e r n

Amortising a Loan

• Repayment in Equated Monthly Installment

• Principal and interest payment

• Mortgage. Auto. Consumer loans

Key Concept

Managers should use the present value of cashflows to compare and evaluate cash payments made and received at different times.

Corporate Finance Lessons

• A rupee today is worth more than a rupee tomorrow and a rupee today cannot be worth more than a rupee yesterday.

• A safe rupee is worth more than a risky one.

Concepts

• Time value of money

• Present and Future value. Compounding and Discounting.

• Amortisation

• Annuity

Start Saving!

• 21 years. $1million on retirement at 65. 10% return

• Annual investments= $1532.24

• 8% return = $2801.52

• 12% return = $825.21

• Age 40 = $10168 per annum

• If Inflation = 5%, $1 million = $116861