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Chapter Four

Chapter 4, Part 1

Interest Rates

Future Value - Present Value

Preview

• Develop understanding of interest rates.

• Develop the concept of yield to maturity (YTM) which is the most accurate measure of interest rate.

Learning Objectives

• Time and the value of payments

• Present value versus future value

• Nominal versus real interest rates

• Interest rates link the present to the future. – the future reward for lending today.

– the cost of borrowing now and repaying later.

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Why Do Lenders Charge Interest on Loans?

The interest rate on a loan should cover the opportunity cost of supplying credit.

It should include: •Compensation for the opportunity cost of waiting to spend the money.

•Compensation for inflation: if prices rise, the payments received will buy fewer goods and services.

•Compensation for default risk: the borrower might default on the loan.

Valuing Monetary Payments Now and in the Future

• Tools that link the present to the future:

– Future value

– Present value

Future Value and Compound Interest

• Future value - the value at some future time of an investment made today.

– $100 invested today at 5% interest gives $105 in a year.

– So the future value of $100 today at 5% interest is $105 one year from now.

– The $100 investment “yields” $5, which is why interest rates are called a yield.

– The above is an example of a simple loan of $100 for a year at 5% interest.

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Future Value and Compound Interest

• If the present value (PV) is $100 and the interest rate is 5%, then the future value (FV) one year from now is:

$105 = $100 + $100(0.05) = $100(1.05)

• This also shows that the higher the interest rate, the higher the future value.

• In general:

FV = PV + PV x i = PV(1 + i)

Future Value and Compound Interest

• Most financial instruments are not this simple.

• When using one-year interest rates to

compute the value repaid more than one year from now, we must consider compound interest.

• Compound interest is the interest on the interest.

Future Value and Compound Interest

• What if you leave your $100 in the bank for two years at 5% yearly interest rate?

• The future value is:

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

FV = $100(1.05)(1.05) = $100(1.05)2

• In general

FVn = PV(1 + i)n

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Future Value and Compound Interest • Computing the future value of $100 at

5% annual interest

Future Value and Compound Interest

• i and n must be in the same units.

• If the annual interest rate is 5%, what is the monthly rate?

– Assume im is the one-month interest rate and n is the number of months, then a deposit made for one year will have a future value of $100(1 + im)12. (NOTE: i and n are monthly)

Future Value and Compound Interest

• We know that in one year the future value is $100(1.05) so we can solve for im:

(1 + im)12 = (1.05)

(1 + im) = (1.05)1/12 = 1.00407, which is 0.407%

• These fractions of percentage points are called basis points. – A basis point is one one-hundredth of a percentage

point, 0.01 % – 0.407% is 40.7 basis points – Note: .05/12 = .00417 > .00407

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Present Value

• Present value: the value today (in the present) of a payment that is promised to be made in the future.

• Or, present value is the amount that must be invested today in order to realize a specific amount on a given future date.

Present Value

• Solve the future value formula for PV:

FV = PV x (1+i)

• This is just the future value calculation inverted.

)1( i

FVPV

Present Value • We can generalize the process as we did for

future value. Present Value of payment received n years in the future:

n

n

i

FVPV

)1(

• We can see that present value is higher: 1. The higher future value of the payment, FVn

2. The shorter time period until payment, n. 3. The lower the interest rate, i.

• Present value is the single most important relationship in our study of financial instruments.

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• We can turn a monthly growth rate into a compound-annual rate. – Investment grows 0.5% per month

– What is the compound annual rate? (1.005)12 = 1.0617

Compound annual rate = 6.17%

Note: 6.17 > (12 x 0.05) = 6.0%

Computing Compound Annual Returns

• We can also compute the percentage change per year when we know how much an investment has grown over a number of years.

• An investment has increased 20 percent over five years from $100 to $120.

FVn = PV(1 + i)n

120 = 100(1 + i)5

Solve for i i = 0.0371

Computing Compound Annual Rates

ni

FVPV

)1(

11)1(

)/1()/1(

nn

n

PV

FVi

PV

FVi

PV

FVi

1100

120)5/1(

i

i = 1.0371 -1=> i =3.71%

Computing Compound Annual Rates

Given FV and PV, solve for i:

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Internal Rate of Return • Business decision - you run a firm and you are

considering purchasing a new machine.

– Machine costs $1 million and can produce 4000 units of product per year.

– You sell the product for $30, generating $120,000 in revenue per year.

– Assume the machine is the only input (keeping this simple), you have certainty about the revenue (very simple), no maintenance (very, very simple) and a 10 year lifespan.

Internal Rate of Return

• If you borrow $1 million, is the 10 year revenue stream enough to make the payments?

• Balance the cost of the machine against the revenue.

– $1 million today versus $120,000 a year for ten years.

• IRR is the interest rate that equates the present value of an investment with its cost.

Internal Rate of Return Example

10321 )1(

000,120$......

)1(

000,120$

)1(

000,120$

)1(

000,120$000,000,1$

iiii

Solve for i: i = 0.0508 or 5.08%

• So long as the interest rate at which you borrow is less than 5.08%, then you should buy the machine

• Or, if IRR is greater than opportunity cost, you should buy the machine.

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Bond Basics

• A bond is a promise to make a series of payments on specific future dates.

• Bonds create obligations, and are therefore legal contracts that:

– Require the borrower to make payments to the lender, and

– Specify what happens if the borrower fails to do so.

Bond Basics • The most common type of bond is a coupon

bond. – Issuer (borrower) is required to make annual

payments, called coupon payments.

– The stated annual interest the borrower pays is called the coupon rate.

– The date on which the payments stop and the loan is repaid (n), is the maturity date or term to maturity.

– The final payment is the principal, face value, or par value of the bond.

Coupon Bond: the good-ole days

Coupons

Called a

coupon bond

as buyer

would receive

a certificate

with a number

of dated

coupons

attached.

Principal

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Valuing the Principal

• Assume a bond has a principle payment of $1000 and its maturity date is n years in the future.

• The present value of the bond principal is:

nnBPii

FVP

)1(

1000$

)1(

nCPi

C

i

C

i

C

i

CP

)1(......

)1()1()1( 321

Valuing the Coupon Payments • These resemble loan payments.

• The longer the payments go, the higher their total value.

• The higher the interest rate, the lower the present value.

• The present value expression gives us a general formula for the string of yearly coupon payments made over n years.

Valuing the Coupon Payments plus Principal

nnBPCPCBi

F

i

C

i

C

i

C

i

CPPP

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

)1()1()1( 321

• We can just combine the previous two equations to get:

Annuity + Face Value

• The value of the coupon bond, PCB, rises when

– The yearly coupon payments, C, rise and

– The interest rate, i, falls.

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Bond Pricing

• The relationship between the bond price and interest rates is very important.

– Bonds promise fixed payments on future dates, so the higher the interest rate, the lower their present value.

• The value of a bond varies inversely with the interest rate used to calculate the present value of the promised payment.

Real and Nominal Interest Rates

• Nominal Interest Rates (i) – The interest rate expressed in current-dollar terms.

• Real Interest Rates (r) – The inflation adjusted interest rate

• Borrowers care about the resources required to repay.

• Lenders care about the purchasing power of the payments they received.

• Neither cares solely about the number of dollars, they care about what the dollars buy.

Real and Nominal Interest Rates • The nominal interest rate you agree on (i) must be based on

expected inflation (e) over the term of the loan plus the real interest rate you agree on (r).

i = r + e • This is called the Fisher Equation.

• The higher expected inflation, the higher the nominal interest rate.

• This equation is an approximation that works well only when expected inflation and the real interest rate are low.

• Exact formula: (1 + i) = (1 + r)(1 + πe)

(1 + i) = 1 + r + πe +r πe

• Subtract 1 from each side and ignore the cross-term.

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Mankiw

Inflation and Nominal Interest Rates

Inflation and Nominal Interest rates

Real and Nominal Interest Rates

• Financial markets quote nominal interest rates.

• When people use the term interest rate, they are referring to the nominal rate.

• We cannot directly observe the real interest rate; we have to estimate it.

r = i - e

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Real and Nominal Interest Rates • Nominal interest rate (i) makes no allowance for

inflation

• Real interest rate (r) is adjusted for changes in price level

so it more accurately reflects the cost of borrowing

• Ex ante real interest rate is adjusted for expected changes

in the price level (πe)

• Ex post real interest rate is adjusted for actual changes in

the price level (π)

Real and Nominal Interest Rates

Fisher Equation:

i = r + πe

From this we get -

rex ante = i - πe

rex post = i - π

Real and Nominal Interest Rates

Real Interest Rate: Interest rate that is adjusted for expected changes in

the price level

r = i -πe

if i = 5% and πe = 3%:

r = 5% - 3% = 2%

if i = 8% and πe = 10%:

r = 8% - 10% = -2%

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Nominal Interest Rate, Inflation Rate and Real Interest Rate

A measure in inflationary expectations

i = r + πe

πe = i - r

http://www.bloomberg.com/markets/rates-bonds/government-

bonds/us/

http://research.stlouisfed.org/fred2/