Understanding interest rates

The Economics of Money, Banking, and Financial Markets Mishkin, 7th ed.

Chapter 4

Present value

The present value of a given future payment is smaller when: the interest rate is higher, and/or the payment is further in the

future.

rate.interest annual simple a is where

(1) )1(

years) in received ($

i

i

FVnFVPV

n

Simple loan of $1 at 10% interest

Year 1 2 3 n $1.10 $1.21 $1.33 $1x(1 + i)n

$1PV of future $1 =

(1 + i)n

Present value

Types of credit market instruments

Simple loan Fixed-payment loan Coupon bond Discount (zero coupon)

bond

Yield to maturity Yield to maturity for a debt

instrument = the interest rate at which the present value of the payment stream equals the current price of the instrument, i.e., interest rate that equates today’s value with present value of all future payments

Yield to maturity – simple loan (i =10%)

$100 = $110/(1 + i )

$110 – $100 $10i = = = 0.10 = 10%

$100 $100

ni)1(

payment FuturePrice

Yield to maturity – fixed-payment loan (i =12%)

2532 )(1

$126

)(1

$126

)(1

$126

1

$126000$1

:example

maturity n payment fixed FP loan value LV where

iiii

(2) )(1

FP

)(1

FP

)(1

FP

1

FPLV

32

niiii

Yield to maturity – coupon bond

101032

32

)(1

$1000

)(1

$100

)(1

$100

)(1

$100

1

$100000$1

C/F)10% rate(coupon :example

bondcoupon of price P

maturity maturity toyield

valueface F payment coupon C :where

(3) )(1

F

)(1

C

)(1

C

)(1

C

1

CP

iiiii

ni

iiiii nn

Yield to maturity – coupon bond

When bond is sold at face value, the yield to maturity = coupon rate

Price of bond and yield to maturity are inversely related Yield to maturity is greater than the coupon rate when

bond price is below the face value

Consol (perpetuity)

maturity toyield

paymentyearly C

consol theof price P :where

(5) P

Cor (4),

CP

i

ii

Yield to maturity – discount bond

%1.11111.0900

9001000then

year, one $1000, F and $900 Pfor if

(6) P

PF then ,1for

maturity maturity, toyield

bonddiscount theof pricecurrent P

bonddiscount theof valueface F :where

)(1

FP

i

in

ni

i n

Overview Current bond prices and interest

rates (yields to maturity) are inversely related for all types of bonds.

Bond Page of the Newspaper (a)

Both coupon bondsbond:Maturity>10yrsnote: Maturity<10yrs

Change on bid price from yesterday

For long-term bonds, the YTM is almost the

same as current yield, but this is not true for short-term

bonds. Differences represent dealer’s profits

Current yield=C/P

10 55%. coupon rate

asked price

=11.125

105 1732

Bond Page of the Newspaper (b), (c)

discount bondsMaturity<1yr

Change on asked yield from yesterday

Both discount yields=((F-P)/F)*(360/days to Mat.)

Coupon rate 5 5/8

Current yield=C/P

8 4%. coupon rate

asked price

=8 5

8102.75

138*$1,000

Mature at 2031

True discount yield=((F-P)/P)*(365/days to Mat.)

Other measures of interest rates

Current yield: ic = C/P ...(7) P=price of the coupon bond, C=yearly payment Only accurate for consols A reasonable approximation to the yield to

maturity for bonds with a maturity of 20+ years.

Relatively poor approximation to the yield to maturity for short-term bonds.

A better approximation when the bond price is close to the face value.

Changes in current yield are always in the same direction as changes in the yield to maturity.

Other measures of interest rate

idb: yeild on a discount basis, F: face value

P: purchase price of the discount bond Always understates the yield to maturity since:

this formula uses F instead of P(<F) in denominator a year (=365 days) has more than 360 days

Changes in the same direction as the yield to maturity. (larger days to maturity has smaller i-idb )

(8) maturity todays

360

F

PFidb

Interest rates and rate of return

Rate of return includes capital gains and losses as well as current interest receipts

gain capital of rate the: yield,current :

1 , at time bond theof price :,

paymentcoupon : return, of rate : where

(10) (9),

:year onefor bond theholding fromReturn

1

11

gi

ttPP

CRET

giP

PP

P

C

P

PPCRET

c

tt

ct

tt

tt

tt

One-year returns on 10%-coupon-rate bond

Maturity and the Volatility of Bond Returns

Key Findings from Table 21. Only bond whose return = yield is one with maturity = holding

period2. For bonds with maturity > holding period, i P implying capital loss3. Longer is maturity, greater is % price change associated with

interest rate change4. Longer is maturity, more return changes with change in interest

rate5. Bond with high initial interest rate can still have negative return if i

Conclusion from Table 2 Analysis1. Prices and returns more volatile for long-term bonds

because have higher interest-rate risk2. No interest-rate risk for any bond whose maturity equals

holding period

Bond prices and rate of return

Rate of return = initial yield to maturity if a bond is held to maturity

Bond prices decline as interest rates rise, leading to capital losses

Long-term bonds exhibit more variability in price in response to interest-rate changes

Longer-term bonds experience more capital gains and losses when interest rates change (interest-rate risk)

A negative return can occur for bonds when the interest rate rise.

Real and nominal interest rates

Fisher equation:

inflation of rate expected the:

rateinterest real the:

rateinterest nominal the: where

(12)

or (11),

e

r

er

er

i

i

ii

ii

U.S. Real and Nominal Interest Rates

Financial innovation In January 1997, the U.S. Treasury

introduced indexed bonds. Interest and principal payments are adjusted for changes in the price level.