The Term Structure

of Interest Rates

The relationship between yield to maturity and maturity.

Yield curve - a graph of the yields on bonds relative to the number of years to maturity

Information on expected future short term rates can be implied from yield curve.

Three major theories are proposed to explain the observed yield curve.

Measure of rate of return that accounts for both current income and the price increase over the life

YTM is the discount rate that makes the present value of a bond’s payments equal to its price◦ Proxy for average return

Solve the bond formula for r

1 (1 )(1 )

T

TtTt

t

BParValueCP

rr

Yields on different maturity bonds are not all equal◦ Need to consider each bond cash flow as a

stand-alone zero-coupon bond when valuing coupon bonds

◦ Example: 1-year maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero and a 12-month zero. If each cash flow can be sold off as a separate security, the value of the whole bond should be the same as the value of its cash flows bought piece by piece.

Bond stripping and bond reconstitution offer opportunities for arbitrage

Law of one price, identical cash flow bundles must sell for identical prices

Value each stripped cash flow◦ discount by using the yield appropriate to its

particular maturity

Treat each of the bond’s payments as a stand-alone zero-coupon security (bond stripping : portfolio of three zeros)

Pure yield curve◦ Relationship between YTM and time to maturity for

zero-coupon bonds On-the-run yield curve

◦ Plot of yield as a function of maturity for recently issued coupon bonds selling at or near par value

1 2 3

100 100 11001082.17

1 5% 1 6% 1 7%P

If interest rates are certain Considering two strategies

◦ Buying the 2-year zero, YTM=6%, hold until maturity. Price=890

◦ Invest 890 in a 1-year zero, YTM=5%. Reinvest the proceeds in another 1-year bond

◦ The proceeds after 2 years to either strategy must be equal

22890(1 6%) 890(1 0.5) (1 )r

2 7.01%r

Spot rate◦ The rate that prevails today for a time period

corresponding to the zero’s maturity Short rate

◦ For a given time interval (e.g. 1 year) refers to the interest rate for that interval available at different points in time

2-year spot rate is an average (geometric) of today’s short rate and next year’s short rate

22 1 2

1

22 1 2

(1 ) (1 ) (1 )

1 (1 ) (1 )

y r r

y r r

An upward sloping yield curve is evidence that short-term rates are going to be higher next year

When next year’s short rate (r2=7.01%) is greater than this year’s short rate, the average of the two rates is higher than today’s rate

22 1 2

1

22 1 2

(1 ) (1 ) (1 )

1 (1 ) (1 )

y r r

y r r

When interest rate with certainty, all bonds must offer identical rates of return over any holding period

Calculate HPR for 1-year maturity zero-coupon bond (YTM=5%)

The first 1-year HPR for 2-year maturity zero-coupon bond (YTM=6%)

For 1-year maturity bond◦ Rate of return=(1000-952.38)/952.38=5%

For 2-year maturity bond◦ Price of today=890◦ One year later, when next year’s interest

rate=7.01%, sell it for 1000/1.0701=934.49◦ Rate of return=(934.49-890)/890=5%

No access to short-term interest rate quotations for coming years---infer future short rates from yield curve of zeros

Two alternatives get same final payoff◦ 3-year zero

◦ 2-year zero, reinvest in 1-year bond

3100*(1 1.0966) 131.87

23100*(1 1.08995) * 1 r

233 2 3100*(1 ) 100* 1 * 1y y f

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n

11

(1 )(1 )

(1 )

nn

n nn

yf

y

11(1 ) (1 ) (1 )n n

n n ny y f

future short rates are uncertain Forward interest rate

◦ defined as the break-even interest rate that equates the return on an n-period zero-coupon bond to that of an (n-1)-period zero-coupon bond rolled over into a 1-year bond in year n

◦ Calculated from today’s data, interest rate that actually will prevail in the future need not equal the forward rate.

4 yr = 8.00% 3yr = 7.00% fn = ?

(1.08)4 = (1.07)3 (1+fn)

(1.3605) / (1.2250) = (1+fn)

fn = .1106 or 11.06%

12.3 12.3 Interest Rate Uncertainty and

Forward Rates

What can we say when future interest rates are not known today

Suppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6% then:

The rate of return on the 2-year bond is risky for if next year’s interest rate turns out to be above expectations, the price will lower and vice versa

22 1 2(1 ) (1 ) [1 ( )] 1.05 1.06

1000898.47

1.05 1.06

y r E r

p

Short-term-horizon investors◦ If invest only for 1 yearcertain return=(1000-952.38)/952.38=5%◦ If invest for 2-year zero, if expect the 1-year rate

be 6% at the end of the first year the price will be 1000/1.06=943.4 the first-year’s expected rate of return also is

5%=(943.4-898.47)/898.47 but the 2nd year’s rate is risky, 943.4 is not certain

If >6%, bond price<943.4 If <6%, bond price>943.4

2-year bond must offer an expected rate of return greater than riskless 5% return , sell at price lower than 898.47

If the investors will hold the bond when it falls to 881.83 ◦ Expected holding period return for the first year =(943.4-881.83)/881.83=7%◦ risk premium=7%-5%=2%◦ Forward rate:

Liquidity premium compensates short-term investors for the uncertainty about the price at which they will be able to sell their long-term bonds

2

2 2

1000881.83

1 5% (1 )

8% 6%

f

f E r

n nliquidity premium f E r

Short-term Investors require a risk premium to hold a longer-term bond

This liquidity premium compensates short-term investors for the uncertainty about future prices

If most individuals are short-term investors, bonds must have prices that make f2 greater than E(r2)

Wish to invest a full 2-year period◦ Purchase 2-year zeros at 841.75,

guaranteed YTM=8.995%◦ If roll over two 1-year investments, an

investment of 841.75 grow in 2 years to be

◦ The investor will require

◦ Offered as a reward for bearing interest rate risk

2841.75*1.08*(1 )r

22 2

2 2

1.08* 1 (1.08995) 1.08*(1 )E r f

E r f

Wish to invest a full 2-year period◦ Purchase 2-year zeros at 890,

guaranteed YTM=6%◦ If roll over two 1-year investments, an

investment of 841.75 grow in 2 years to be

◦ The investor will require

◦ Offered as a reward for bearing interest rate risk

2890*1.05*(1 )r

22 2

2 2

1.05* 1 (1.06) 1.05*(1 )E r f

E r f

12.412.4Theories of the Term

Structure

Expectations theories Liquidity Preference theories

◦ Upward bias over expectations Market Segmentation

Observed long-term rate is a function of today’s short-term rate and expected future short-term rates.

Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.

◦ An upward-sloping yield curve if investors anticipate increases in interest rates

◦ Upward slope means that the market is expecting higher future short term rates

◦ Downward slope means that the market is expecting lower future short term rates

2 2E r f

Short-term investors dominate the market

Forward rates contain a liquidity premium and are not equal to expected future short-term rates.

Investors will demand a premium for the risk associated with long-term bonds.

The yield curve has an upward bias built into the long-term rates because of the risk premium.

2 2f E r

Short- and long-term bonds are traded in distinct markets.

Trading in the distinct segments determines the various rates.

Observed rates are not directly influenced by expectations.

Expected One-Year Rates in Coming Years

Year Interest Rate

0 (today) 8%

1 10%

2 11%

3 11%

8% 10% 11% 11%

PVn = Present Value of $1 in n periods

r1 = One-year rate for period 1

r2 = One-year rate for period 2

rn = One-year rate for period n

1 2

1

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

PVr r r

1

1000925.93

(1 8%)PV

2

1000841.75

(1 8%)(1 10%)PV

3

1000758.33

(1 8%)(1 10%)(1 11%)PV

Price of 1-year maturity bond

Price of 2-year maturity bond

Price of 3-year maturity bond

1

1000925.93

(1 )y

22

1000841.75

(1 )y

33

1000758.33

(1 )y

YTM is average rate that is applied to discount all of the bond’s payments

Time to Maturity Price of Zero* Yield to Maturity

1 $925.93 8.00%

2 841.75 8.995

3 758.33 9.660

4 683.18 9.993

* $1,000 Par value zero

8% 10% 11% 11%

y1=8%

y2=8.995%

y3=9.660%

y4=9.993%

expectedShort rate

in each year

YTM for various maturities

(Current spot rate)

YTM, average of the interest rates in each period (geometric)

22

1000 1000841.75

(1 8%)(1 10%) (1 )y

22(1 ) (1 8%)(1 10%)y

1

22 1 21 (1 )(1 )y r r

1

44 1 2 3 41 (1 )(1 )(1 )(1 )y r r r r

12.4 12.4 Interpreting the Term

Structure

Direct relationship between YTM and forward rate

Under certainty

Uncertain

The yield curve is upward sloping at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity

1

1 21 (1 )(1 ) (1 ) nn ny r r r

1

1 21 (1 )(1 ) (1 ) nn ny r f f

If yield curve is rising, must exceed Example

◦ YTM on 3-year zero is 9%, YTM on 4-year zero

◦ If then◦ If , then

1nf ny

13 4

4 41 (1 9%) (1 )y f

4 9%,f 4 39%y y

4 9%f 4 39%y y

Given an upward-sloping yield curve, What account for the higher forward rate?

Expectations of increases in can result in rising yield curve; converse is not true .

n nf E r liquidity premium

nE r