Chapter 4

Understanding Interest Rates

Present Value: Discounting the Future

• Present Value (PV) - the value today of a payment promised in the future

• A dollar paid to you one year from now is less valuable than a dollar paid to you today.

Discounting the Future (PV) First need to Understand Future Value (FV)

• Let i = 0.10 (10 percent interest rate)

• In one year: $100 +(.10 x $100) = $100 x (1+.10) = $110

• In two years: $110+(.10 x $110) = $110 x (1+.10) = $121

OR $100 x (1+.10)2 = $121

Future value

• In three years: $121+(.10 x $121)= $121x(1+.10) = $133

• OR $100 x (1+.10)3 = $133

• In n years: FV = $100 x (1+ .10)n

Future Value

• Future value in n years of an investment of PV today at interest rate i

(i is measured as a decimal, 10% = .10)

FVn = PV x (1+i)n

• Calculate one plus the interest rate (measured as a

decimal), raised to the nth power, multiply by the

amount invested (present value PV).

Future Value

Note:

When computing Future Value, both n and i must be measured in same time units—if i is annual, then n must be in years.

The future value of $100 in 18 months at 5% annual interest rate is:

FV = 100 *(1+.05)1.5

Future Value• The future value of $100 in one month at a 5%

annual interest rate is:

FV = $100 *(1+.05)1/12 = $100.4074

• (1+.05)1/12 converts the annual interest rates to a monthly rate.

• (1+.05)1/12 = 1.004074, which converted to percentage is 0.4074% or 0.41%(rounded)

Note: 0.05/12 = .004167

Basis Point

• Faction of a percentage point is called basis point.

• A basis point is one-one hundredth of a percentage point

• One basis point (bp) = 0.01 percent.

• On the previous slide: 0.41% is 41 basis points.

Discounting the Future PV Reverses the FV Calculation

n

PV = today's (present) value

CF = future cash flow (payment)

= the interest rate

CFPV =

(1 + )

i

i

FVn = PV x (1+i)n

PV = FVn /(1+i)nIn General:

Present Value

Present ValueExample 1:• Present Value of $100 received in 5 years discounted at

an interest rate of 8%. PV = $100 / (1.08)5 = $68.05

Example2:• PV of $20,000 received 20 years from now discounted at

8% is: PV = $20,000 / (1+0.08)20 = $20,000/ 4.6609 = $4,291 • Discounted at 9%: PV = $20,000 / (1+0.09)20 = $20,000/5.6044 = $3,568

Present Value

Example3:• PV of $20,000 received 19 years from now discounted at

8% is: PV = $20,000 / (1+0.08)19 = $20,000/ 4.3157 = $4,634

In general, present value is higher:

1. The higher the future value of the payment (CF).

2. The shorter the time period until payment (n).

3. The lower the interest rate. (i)

ni

CFPV

)1(

Mishkin Discusses Four Types of Credit Market Instruments

• Simple Loan

• Fixed Payment Loan (I will just mention this)

• Coupon Bond

- Special case: consol bond

• Discount Bond

We will focus on Bonds and look at three types of bonds• Coupon Bonds: which make periodic interest

payments and repay the principal at maturity. • U.S. Treasury Bonds and most corporate bonds are

coupon bonds.

• Discount or Zero-coupon bonds: which promise a single future payment, such as a U.S. Treasury Bill.

• Consols: which make periodic interest payments forever, never repaying the principal that was borrowed. (There aren’t many examples of these.)

Yield to Maturity -YTM

• The interest rate that equates the present value (PV) of cash flow payments (CF) received from a debt instrument with its value today

ni

CFPV

)1(

Given values for PV, CF and n, solve for i.

Yield to Maturity - One Year Simple Loan

1

PV = amount borrowed = $100

CF = cash flow in one year = $110

= number of years = 1

$110$100 =

(1 + )

(1 + ) $100 = $110

$110(1 + ) =

$100 = 0.10 = 10%

For simple loans, the simple interest rate equ

n

i

i

i

i

als the

yield to maturity

Fixed Payment Loan - YTM

2 3

The same cash flow payment every period throughout

the life of the loan

LV = loan value

FP = fixed yearly payment

= number of years until maturity

FP FP FP FPLV = . . . +

1 + (1 + ) (1 + ) (1 + )n

n

i i i i

Bonds: Our Objectives

• Bond price is a present value calculation.

• YTM is the interest rate.

• Supply and Demand determine the price of bonds.

- We will also discuss loanable funds and

money supply/money demand.

• Explain why bonds are risky.

Coupon Bond Price

A simple contract-

Price of a Coupon Bond

Present Value of Coupon Bond (PB) =

Present Value of yearly coupon payments (C)

+ Present Value of the Face Value (FV),

where: i = interest rate and n = time to maturity

nnB i

FV

i

C

i

C

i

C

i

CP

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

)1()1()1( 321

Present Value of Coupon Payments

Present Value of Principal Payment

Example: Price of a n-year Coupon Bond

nnB iiiiP

)1(

1000$

)1(

100$......

)1(

100$

)1(

100$21

Coupon Payment (C) =$100, Face value (FV) = $1,000, and n = time to maturity

Definition: Coupon Payment / Face Value = Coupon Rate

Given values for i and n, we can determine the bond price PB

Price of a 10-year Coupon Bond

101021 )1.1(

1000$

)1.1(

100$......

)1.1(

100$

)1.1(

100$1000$

BP

Excel Example

If n = 10, i = 0.10, C = $100 and FV = $1000.

What’s the Coupon Rate?

Price of a 10-year Coupon Bond

101021 )12.1(

1000$

)12.1(

100$......

)12.1(

100$

)12.1(

100$887$

BP

If n = 10, i = 0.12, C = $100 and FV = $1000.

What’s the Coupon Rate?

YTM - Coupon Bond

101021 )1(

1000$

)1(

100$......

)1(

100$

)1(

100$950$

iiiiPB

What’s the Coupon Rate?

What’s the YTM?

YTM = .1085 or 10.85%

Suppose n = 10, PB = $950, C = $100 and FV = $1000.

2

.PFV

nPFV

CYTMApprox

C = the coupon paymentFV = Face Value P = Price n = years to maturity

Previous example:

n = 10, PB = $950, C = $100 and FV = $1000.

Using the Approximation Formula

Paid Price

PaymentCoupon Yearly YieldCurrent

For our 10-year, $1000 FV bond with a $100 coupon payment selling at $950:

Coupon rate = 10% YTM = 10.85% App. YTM = 10.77%Current Yield = 10.52%

Current Yield

Two Characteristics of Current Yield

1. Is a better approximation of yield to maturity, nearer price is to par (face value) and longer is maturity of bond

2. Change in current yield always signals change in same direction as yield to maturity

• When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate

• The price of a coupon bond and the yield to maturity are negatively related

• The yield to maturity is greater than the coupon rate when the bond price is below its face value

Consol – Special Case Coupon Bond

• Infinite maturity

• No face value.

• Fixed coupon payment of C forever.

• Pconsol = C/(1+i) + C/(1+i)2 + C/(1+i)3 + … + C/(1+i)t

• As t goes to infinity this collapses to:

PConsol = C / i => i = C / P

$2,000 = $100/.05

Discount or Zero Coupon Bond

• Definition: A discount bond is sold at some price P, and pays a larger amount (FV) after t years. There is no periodic interest payment.

Let P = price of the bond, i= interest rate, n = years to maturity, and FV = Face Value (the value at maturity):

ni

nFVP

)1(

Zero Coupon Bonds - PriceExamples: Assume i=4%

Price of a One-Year Treasury Bill with FV = $1,000:

Price of a Six-Month Treasury Bill with FV = $1,000:

53.961$)04.01(

1000

BP

58.980$)04.01(

10002/1

BP

Price of a 20-Year zero coupon bond at 8% and FV = $20,000:

96.290,4$)08.01(

20000$20

BP

YTM - Zero Coupon Bonds

ni

FVP

)1(

11)1()/1()/1(

nnn

P

FVi

P

FVi

P

FVi

Zero Coupon Bonds - YTM

• For a discount bond with FV = $15,000 and P = $4,200, and n = 20, the interest rate (or yield to maturity) would be:

i = 1.0657 -1=> i =6.57%

1)/1(

n

P

FVi

1200,4

000,15)20/1(

i

Note: This is the formula for compound annual rate of growth

Zero Coupon Bonds - YTM

1491,6

000,10)7/1(

i

i = 1.06368 – 1 = .06368 or 6.368%

• For a discount bond with FV = $10,000 and P = $6,491, and n = 7, the interest rate (or yield to maturity) would be:

From a Coupon Bond to Zero Coupon Bonds (called Strips)

nnB i

FV

i

C

i

C

i

C

i

CP

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

)1()1()1( 321

Create n+1 discount bonds

Yield on a Discount Basis

SKIP

Distinction Between Interest Rates and Return

The payments to the owner plus the change in value

expressed as a fraction of the purchase price

RET = C

Pt

+ P

t1 - P

t

Pt

RET = return from holding the bond from time t to time t + 1

Pt = price of bond at time t

Pt1

= price of the bond at time t + 1

C = coupon payment

C

Pt

= current yield = ic

P

t1 - P

t

Pt

= rate of capital gain = g

Holding Period Return

• the return from holding a bond and selling it before maturity.

• the holding period return can differ from the yield to maturity.

Holding Period Return Example:• You purchase a 10-year coupon bond, with a 10%

coupon rate, at face value ($1000) and sell one year later.

1 1( 1)

( )t t

t

tt

tt

tt

C P PR

C

P

P PET

PP

Current Yield Capital Gain

Holding Period Return

If the interest rate one year later is the same at 10%:

One year holding period return =

or 10.0%

10.1000$

100$

1000$

1000$1000$

1000$

100$

Holding Period Return

If the interest rate one year later is lower, say at 8%:

One year holding period return =

or 22.5%

Where did the $1125 come from?

225.1000$

225$

1000$

1000$1125$

1000$

100$

Holding Period Returns

If the interest rate in one year is higher at 12%:

One year holding period return =

or -.70%

007.1000$

7$

1000$

1000$893$

1000$

100$

Where did the $893 come from. You need to know how to calculate the $1125 and $893

Key Conclusions From Table 2

• The return equals the yield to maturity (YTM) only if the holding period equals the time to maturity

• A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the holding period is less than the time to maturity

• The more distant the maturity, the greater the percentage price change associated with an interest-rate change

Key Conclusions From Table 2

• The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rate

• Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise

Interest-Rate Risk

• Change in bond price due to change in interest rate

• Prices and returns for long-term bonds are more volatile than those for shorter-term bonds

• There is no interest-rate risk for a bond whose time to maturity matches the holding period

Reinvestment (interest rate) Risk• If investor’s holding period exceeds the term to maturity

proceeds from sale of bond are reinvested at new interest rate

the investor is exposed to reinvestment risk

• The investor benefits from rising interest rates, and suffers from falling interest rates

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

Mankiw

Inflation and Nominal Interest Rates

Inflation and Nominal Interest Rates

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% then:

r = 5% - 3% = 2%

if i = 8% and πe = 10% then

r = 8% - 10% = -2%

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/