Fixed Income Basics - part 1

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

Spot Interest rates

The zero-coupon yield curve

Bond yield-to-maturity

Default-free bond pricing

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

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 =

Semi-annual Yield-to-Maturity

Define semi-annual yield-to-maturity, ys, as:

Price Ct + (Face) t=1

T 1 1 (1+ys/2)t (1+ys/2)T

Example: 2-year 7% semi-annual coupon bond, price =103.79 per 100.by definition, semi-annual yield-to-maturity ys is solution to:

103.79 = 3.50/(1+ys/2)t + 100/(1+ys/2)4

eventually: when ys/2 = 0.0249 = 2.49%

effective annual yield-to-maturity is yA = (1 + 0.0249)2 - 1 = 5.04%

Note effective annual yield-to-maturity is yA (1+ys/2)2 - 1

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

Reinvestment assumptions and yield-to-maturity

Yield-to-maturity (ytm) is holding period rate of return only if coupons can be reinvested at the same rate as yield-to-maturity

Example: 6% semi-annual coupon Par bond (price=100.00)yield-to-maturity, ys, is defined as:

22 )2/1(

100

)2/1(

3

)2/1(

3100

sss yyy

So that ys = 0.06

6-month coupon re-invested at ytm becomes 3(1+ys/2) = 3(1.03) in one year. End-of-year value: 103 + 3(1.03) = 106.09.Holding period return: (106.09-100)/100 = 6.09% Effective annual yield: 6% semi-annual yield = (1+0.06/2)2-1 = 6.09%

When re-investment is compounded semi-annually:re-investment holding-period rate proceeds at one year return 5.0% 103 + 3.075 = 106.075 6.075% 7.0% 103 + 3.105 = 106.105 6.105%

Treasury bond quotes and prices

Coupon 11.625% Bid 129.875Maturity 11/15/04 Ask 130.000

Par value (Face) $100,000Settlement date 1/22/98days in coupon period 181days since last coupon 68accrued interest $2,183.70Total purchase price if bought at bid $132,058.70Total purchase price if bought at ask $132,183.70

Accrued interest = Coupon x [(days since last coupon)/(days in coupon period)]

Quotes are “clean prices” (no accrued interest)Actual price is “dirty price”

Coupon period

coupon coupon

Floating rate notes

Debt contract: face value, maturity, coupon payment dates

Interest payments (coupons) reset at each coupon date. Example:one-year floater, semi-annual payments, Face=$100.00payment based on six-month simple rate at beginning of coupon period

spot six-month rate coupon paid: end of perioddate zero (today) 5.25% c = 5.25/2 = 2.625 six months later 5.60% c = 5.60/2 = 2.80

Six months from now, value of note is:

102.80/[1+ 0.056 x (1/2)] = 102.80/1.028 = $100In six months bond will be valued at par.So value of note at time zero is:

(100 + 2.625)/[1 + 0.0525 x (1/2)] = 102.625/1.02625 = $100

Note value is at par each reset date.