Fixed Income Basics

Finance 30233, Fall 2010The Neeley School of Business at TCU©Steven C. Mann, 2010

Spot Interest ratesThe zero-coupon yield curveBond yield-to-maturityDefault-free bond pricingForward RatesTerm Structure Theory

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 (0yt ) source: current market bond prices (spot prices)

“forward curve”: forward short-term interest rates: “forward rates: f(t,T)” source: zero curve, current market forward rates

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

Determination of the zero curve

B(0,t) is discount factor: price of $1 received at t; B(0,t) (1+ 0yt)-t .

Example: find 2-year zero yielduse 1-year zero-coupon bond price and 2-year coupon bond price:bond price per $100: yield1-year zero-coupon bond 94.7867 5.500%2-year 6% annual coupon bond 100.0000 6.000%

B(0,1) = 0.9479. Solve for B(0,2): 6% coupon bond value = B(0,1)($6) + B(0,2)($106)

$100 = 0.9479($6) + B(0,2)($106) 100 = 5.6872 + B(0,2)($106) 94.3128 = B(0,2)(106)

B(0,2) = 94.3128/106 = 0.8897

so that 0y2 = (1/B(0,2))(1/2) -1 = (1/0.8897)(1/2) -1 = 6.0151%

“Bootstrapping” the zero curve from Treasury prices

Example:six-month T-bill price B(0,6) = 0.974812-month T-bill price B(0,12) = 0.9493

18-month T-note with 8% coupon paid semi-annually price = 103.77

find “implied” B(0,18):

103.77 = 4 B(0,6) + 4 B(0,12) + (104)B(0,18)= 4 (0.9748+0.9493) + 104 B(0,18)= 7.6964 + 104 B(0,18)

96.0736 = 104 B(0,18)B(0,18) = 96.0736/104 = 0.9238

24-month T-note with 7% semi-annual coupon: Price = 101.25

101.25 = 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) + 103.5B(0,24)= 3.5(0.9748+0.9493+0.9238) + 103.5B(0,24)

B(0,24) = (101.25 - 9.9677)/103.5 = 0.9016

Coupon Bonds

Price = Ct B(0,t) + (Face) B(0,T)

where B(0,t) is price of 1 dollar to be received at time t

or

Price = Ct + (Face)

where rt is discretely compounded rate associated witha default-free cash flow (zero-coupon bond) at time t.

Define par bond as bond where Price=Face Value = (par value)

t=1

t=1

T

T 1 1 (1+rt)

t (1+rt)T

Yield to Maturity

Define yield-to-maturity, y, as:

Price Ct + (Face) t=1

T 1 1 (1+y)t (1+y)T

Solution by trial and error [calculator/computer algorithm]

Example: 2-year 7% annual coupon bond, price =104.52 per 100.by definition, yield-to-maturity y is solution to:

104.52 = 7/(1+y) + 7/(1+y)2 + 100/(1+y)2

initial guess : y = 0.05 price = 103.72 (guess too high)second guess: y = 0.045 price = 104.68 (guess too low)

eventually: when y = 0.04584 price = 104.52 y = 4.584%

If annual yield = annual coupon, then price=face (par bond)

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

Facey

Cy

FaceTBCtBValueBondT

Tt

T

tt

t

T

tt )1(

1

)1(

1),0(),0(

01 01

Facey

Cy

Facey

Cy

ValueBondT

Tt

T

tt

t

T

tTtt )1(

1

)1(

1

)1(

1

)1(

1

01 01

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

10%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 9.262 0.84175 9.00% 8.423 0.75833 9.66% 7.58 75.83 Bond Value

total: 25.26 75.83 101.099.56%

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

Bond yield =

Bonds with same maturity but different coupons will have different yields.

15%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 13.892 0.84175 9.00% 12.633 0.75833 9.66% 11.37 75.83 Bond Value

total: 37.89 75.83 113.729.52%

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

Bond yield =

5%

T B(0,T) 0y T B(0,t)Ct B(0,3)$1001 0.92593 8.00% 4.632 0.84175 9.00% 4.213 0.75833 9.66% 3.79 75.83 Bond Value

total: 12.63 75.83 88.469.61%

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

Bond yield =

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

1)1(

)1()2,1(

10

220

y

yf

)2,1(1

1)1,0()1,0(;1

)2,0(

)1,0()2,1(

fBB

B

Bf

1)1(

)1()1,(

0

110

n

n

nn

y

ynnf

)1,(1

1),0()1,0(

;1)1,0(

),0(1

)1(

)1()1,(

0

110

nnfnBnB

nB

nB

y

ynnf

nn

nn

1),0()1,0(

11

)1(

1

1

)1()1,(

0

110

nB

nBy

ynnf

nn

nn

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

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