Chap 004.ppt

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Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter 4

Future Value,

Present Value andInterest Rates

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Future Value, Present Valueand Interest Rates:The Big Questions

1. How can we compare payments at differentdates?

2. What is an interest rate?

3. What is a bond?

4. What is the relationship between interestrates and inflation?

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Future Value, Present Valueand Interest Rates:

Roadmap

Future Value

Present Value

Internal Rate of Return Bond Basics

Real vs. Nominal Interest Rates

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Lending and Interest

Why do lenders charge interest?

The existence of alternatives means

that lenders face an opportunity cost. Borrowers rent resources from lenders.

Interest is the rent.

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Future Value:Definition

The value on a future date of an

investment made today.

If you invest $100 today at 5 percent interestper year, in one year you will have $105.

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Future Value:One Year

Future Value =Present Value + Interest

FV = PV + PVxi

$105 = $100 + $100x(0.05)

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Future Value:One Year

FV = PV + PVxi

= PVx(1+i)

Future Value in one year =Present Value x (one plus interest rate)

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Future Value:Two Years

$100+$100(0.05)+$100(0.05) + $5(0.05) =$110.25

Present Value of the Initial Investment

+ Interest on the initial investment in the 1

st

Yr+ Interest on the initial investment in the 2nd Yr+ Interest on the Interest from the 1st Yr in the 2nd Yr

= Future Value in Two Years

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Future Value:General Formula

Future value of an investment of PVin n years at interest rate i

FVn = PVx(1+i)n

(Remember: The interest rate ismeasured is a decimal so if 5%, i = .05)

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Future Value:$100 Investment at 5% Annual Interest

After 10 years, $100 as grown to $162.89thats the initial investment of$100 plus interest of $62.89. Ignoring compounding, you would have justmultiplied 5 percent times 10 years and gotten $50.The difference of $12.69 comes from compounding.

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Future Value:Caution

Time (n) & interest rate (i)must be in same time units

If i is at annual rate, then n must be in years.

Future Value of $100 in 18 months at

5% annual interest rate is

FV = 100 x (1+.05)1.5

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Invest $100 at 5% annual interest

How until you have $200?

The Rule of 72: Divide the annual interest rate into 72

So 72/5=14.4 years.

1.0514.4 = 2.02

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Present Value:Definition

Present Value (PV) is the value today

(in the present) of a payment that is

promised to be made in the future.

At a 5 percent interest rate, the present value of

$105 one year from now is $100. Reverses the future value calculation

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Present Value:One Year

Solve the Future Value Formula for PV:

FV = PV x (1+i)so

Present Value = Future Value divided byone plus interest rate

)1( i

FVPV

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Present Value:One Year Example

$100 received in one year, i=5%

Note: FV = PVx(1+i) = $95.24x(1.05) = $100

24.95$05.1100$

)1(

iFVPV

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Present Value:General Formula

Present Value of payment receivedn years in the future:

ni

FVPV

)1(

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Present Value:Example

Present Value of $100 received in2 yrs at interest rate of 8%.

Note: FV = PVx(1+i)n

=$82.50x (1.08)2.5

= $100

20.85$)08.1(

100$)1( 5.2

n

iFVPV

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Present Value:Important Properties

Present Value is higher:

1. The higher future value of the payment.

(FV bigger)2. The shorter time period until payment.

(n smaller)

3. The lower the interest rate.(i smaller)

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Present Value:$100 at 5% interest rate

Note rate of decline of Present Value. At a 5% interest rate, a $100payment made in 14.4 years has a PV=$50.

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Present Valueof $100 Payment

As the interest raterises from 1% to 5%,a payment due

1 year falls by $3.77

10 years falls by $29.14

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Divine law of Islamic religion (Sharia)forbids paying interest

Banks developed alternatives.

Liabilities Deposit accounts: No interest

Investment accounts: Share in banks profits orlosses

Assets Profit share in exchange for loan

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Investment grows 0.5% per month

What is the compound annual rate?

FVn=PV(1+i)n = 100x(1.005)12=106.17

Compound annual rate = 6.17%(Note: 6.17 > 12x0.05=6.0)

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To decide you need to compare

1. The value of the extra savings you willaccumulate from waiting that allows youto purchase a more expensive care

2. The value of having the new caresooner.

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Internal Rate of Return:Definition

The interest rate that equates the

present value of an investmentwith its cost.

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Internal Rate of Return:Example

You run a sports equipment factory.

Should you purchase new tennis racquet machine?

Cost: $1 million

Produces 3000 racquets per year

Sell racquets for $50 apiece

The machine lasts 10 years and collapses

with no resale value. Should you buy the machine?

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Balance the cost of the machine against therevenue

$1 million today vs.$150,000 a year for ten years.

Is the $150,000 revenue enough to makepayments on a $1 million loan?

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Solve for i:

$1,000,000

10321 )1(

000,150$......

)1(

000,150$

)1(

000,150$

)1(

000,150$

iiii

Solving for i, i=.0814 or 8.14%

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Can you retire when youre 40?

Assume

Live to 85 Interest rate = 4%

Want to have $100,000 per year

You will need004,072,2$

)04.1(

000,100$

)04.1(

000,100$

)04.1(

000,100$

)04.1(

000,100$

)04.1(

000,100$4544321

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Coupon Bond

Coupons

$1000 Face Value50-yr, 3% couponbond issued on

May 1, 1945.

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Coupon Bond

A type of loan: Interest paid during the life of the loan

Loan repaid at maturity

Coupon Rate: the annual interest theborrower pays (ic)

Maturity Date: when the payments stop

and the loan is repaid (n) Principal: the final payment (F)

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Coupon Bond:Valuing the Principal

nnBP

ii

FP

)1(

100$

)1(

Present value of Bond Principal =Payment divided by one plus the interest rate raised to n

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Coupon Bond:Valuing the Coupon Payments

nCP

i

C

i

C

i

C

i

CP

)1(......

)1()1()1( 321

Value of Coupon Payments = Present value of the sequence

Note that C= icx F

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Price of Coupon Bond:Principal + Coupons

Price of Coupon Bond (PCB

) =

Present value of Coupon Payments (PCP)

+ Present Value of the Principal (PBP)

nnBPCPCB

i

F

i

C

i

C

i

C

i

CPPP

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

)1()1()1( 321

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Credit cards are useful.

But lenders charge highinterest rates.

Pay off your balance as

fast as you can.

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Real and Nominal Interest Rates

Borrowers care about the resourcesrequired to repay.

Lenders care about the purchasingpower of the payments they received.

Neither cares solely about the number

of dollars, they care about what thedollars buy.

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Nominal Interest Rates (i)

Interest Rates expressed in current

dollar terms.

Real Interest Rates (r)

Nominal Interest Rate adjusted forinflation.

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Nominal interest rate =Real Interest Rate + Expected Inflation

i = r + e

(This is called the Fisher Equation)

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Nominal Interest Rate,Inflation Rate and Real Interest Rate

Nominal Interest Rate = Real Interest Rate + Expected Inflation

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Chapter 4

End of Chapter

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Chap 004.ppt