Fixed Income Basics - part 2

Finance 70520, Spring 2002The Neeley School of Business at TCU©Steven C. Mann, 2002

Forward interest rates

spot, forward, and par bond yield curves

Intro to Term Structure

Term structure

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7.0

6.5

6.0

5.5

5.0

yield

Maturity (years)

Typical interest rateterm structure

“Term structure” may refer to various yields:

“spot zero curve”: yield-to-maturity for zero-coupon bonds source: current market bond prices (spot prices)

“forward curve”: forward short-term interest rates: “short rates” source: zero curve, current market forward rates

“par bond curve”: yield to maturity for bonds selling at par source: current market bond prices

Forward rates

Introductory example (annual compounding) :

one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) = 0.944733

two-year zero yield: 0y2 =6.03% ; B(0,2) = 1/(1.0603)2 = 0.889493

$1 investment in two-year bond produces $1(1+0.0603)2 = $1.1242 at year 2.

$1 invested in one-year zero produces $1(1+0.0585) = $1.0585 at year 1.

What “breakeven” rate at year 1 equates two outcomes?

(1 + 0.0603)2 = (1 + 0.0585) [ 1 + f (1,2) ]

breakeven rate = forward interest rate from year 1 to year 2 = f (1,2) (one year forward, one-year rate)

1 + f (1,2) = (1.0603)2/(1.0585) = 1.062103 f (1,2) = 1.0621 - 1 = 6.21%

and $1.0585 (1.0621) = $1.1242.

Forward and spot rate relationships : annualized rates

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Example: Using forward rates to find spot rates

n spot rate

(year) 0yn+1

0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993%

f (n,n+1)

forward rate

B(0,n+1)

bill price

6%7%8%9%

10%11%12%

0 1 2 3

Forward rates Spot rates

Given forward rates, find zero-coupon bond prices, and zero curve

Bond paying $1,000:maturity Price yield-to-maturityyear 1 $1,000/(1.08) = $925.93 0y1=[1.08] (1/1) -1 =8%

year 2 $1,000/[(1.08)(1.10)] = $841.75 0y2 = [(1.08)(1.10)](1/2)- 1 =8.995%

year 3 $1,000/[(1.08)(1.10)(1.11)] = $758.33 0y3 =[(1.08)(1.10)(1.11)] (1/3) = 9.660%

year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $683.18 0y4 =[(1.08)(1.10)(1.11)(1.11)] (1/4) = 9.993%

Yield curves

maturity

maturity

rate

rate

Forward ratezero-coupon yieldcoupon bond yield

Coupon bond yieldzero-coupon yieldforward rate

Typical upward slopingyield curve

Typical downward slopingyield curve

Coupon bond yield is “average” of zero-coupon yields

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Coupon bond yield-to maturity, y, is solution to:

8.000%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 7.412 0.84175 9.00% 6.733 0.75833 9.66% 6.07 75.83 Bond Value

totals: 20.21 75.83 96.0419.581%

bond: $100 par, 3-year; annual coupon =

Bond yield =

Par bond yield is yield for bond priced at par: coupon = ytm

12.000%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 11.112 0.84175 9.00% 10.103 0.75833 9.66% 9.10 75.83 Bond Value

totals: 30.31 75.83 106.1459.546%

bond: $100 par, 3-year; annual coupon =

Bond yield (ytm) =

9.567%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 8.862 0.84175 9.00% 8.053 0.75833 9.66% 7.26 75.83 Bond Value

totals: 24.17 75.83 100.0009.567%

bond: $100 par, 3-year; annual coupon =

Bond yield (ytm) =

Example: Assume corporate yield is determined as: Treasury + 300 b.p.

12.000%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 11.112 0.84175 9.00% 10.103 0.75833 9.66% 9.10 75.83 Bond Value

totals: 30.31 75.83 106.1459.546%

bond: $100 par, 3-year; annual coupon =

Bond yield (ytm) =

9.567%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 8.862 0.84175 9.00% 8.053 0.75833 9.66% 7.26 75.83 Bond Value

totals: 24.17 75.83 100.0009.567%

bond: $100 par, 3-year; annual coupon =

Bond yield (ytm) =

Discount bond ( 8.000%) : Treasury ytm = 9.811% 12.811%Par bond ( 9.567% ): Treasury ytm = 9.567% 12.567%Premium bond (12.00 %): Treasury ytm = 9.546% 12.546%

$100 million 3-year bond issue:

Borrower: use of 8% instead of par:(12.581-12.567) x $100mm

= $14,000 annual cost

Lender:use of 12% instead of par:(12.567-12.546) x $100mm

= $23,000 annual cost

n spot rate

(year) 0yn+1

0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993%

f (n,n+1)

forward rate

B(0,n+1)

bill price

Holding period returns under certainty (forward rates are future short rates)

One year later:f (0,1) = 0y1 = 10%f (1,2) = 11%f (2,3) = 11%

One-year holding period returns of zero-coupons:invest $100:one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.

At end of 1 year, value = $108.00 ; return = (108/100)-1 = 8.0%

two-year zero: $100 investment buys $100/84.175 = $118.80 Face value.at end of 1 year, Value = $118.80/1.10 = $108.00 ;

return = (108/100) -1 = 8.0%three-year zero: $100 investement buys $100/75.833 = $131.87 face value

at end of 1 year, value = $131.87/[(1.10)(1.11)] = $108.00 ;return = (108/100) -1 = 8.0%

If future short rates are certain, all bonds have same holding period return

n spot rate

(year) 0yn+1 now0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995% 11.00%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660% 8.00%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993% 9.00%

one year later

possible short rate (0y1) evolution:

f (n,n+1)

forward rate

B(0,n+1)

bill price

Holding period returns when future short rates are uncertain

One year holding period returns of $100 investment in zero-coupons:one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.

1 year later, value = $108.00 ; return = (108/100)-1 = 8.0% (no risk)

two-year zero: $100 investment buys $118.80 face value. 1 year later: short rate = 11%, value = 118.80/1.11 = 107.03 7.03% return

short rate = 9%, value = 118.80/1.09 = 108.99 8.99% return

Risk-averse investor with one-year horizon holds two-year zero only if expected holding period return is greater than 8%:only if forward rate is higher than expected future short rate.

Liquidity preference: investor demands risk premium for longer maturity

Term Structure Theories

1) Expectations: forward rates = expected future short rates2) Market segmentation: supply and demand at different maturities3) Liquidity preference: short-term investors demand risk premium

maturity

rate

Expected short rate is constant

Forward rate = expected short rate + constant

Yield curve is upward sloping

Yield Curve: constant expected short ratesconstant risk premium

Possible yield curves with liquidity preference

rate

Expected short rate is declining

Forward rate

Yield curve

Liquidity premiumincreasing with maturity

maturity

maturity

rate

Expected short rate is declining

Forward rateHumped yield curve

Constant Liquidity premium