Introduction to Fixed Income – part 1

Finance 30233 Fall 2004Advanced InvestmentsAssociate Professor Steven C. MannThe Neeley School of Business at TCU

Money market rate definitionsdiscount factorsspot zero-coupon yieldsusing the zero curve to price coupon bonds

Bill prices and interest rate definitions

Default free bonds (Treasuries)zero coupon bond price, stated as price per dollar:

B(t,T) = price, at time t, for dollar to be received at T

Interest ratesdiscount rate (T-bill market)simple interestdiscrete compoundingcontinuous compounding

Rate differences due to:compoundingday-count conventions

actual/actual; 30/360; actual/360; etc.

Discount rate: id (T)

B(0,T) = 1 - id (T)

T 360

T (days) B(0,T) id (T)30 0.9967 3.9660 0.9931 4.1490 0.9894 4.24

180 0.9784 4.32

T(days) id(T) B(0,T)30 3.96 0.996760 4.14 0.993190 4.24 0.9894

180 4.32 0.9784

Example:30-day discount rate id = 3.96%B(0,30) = 1 - (0.0396)(30/360)

= 0.9967

Current quotes: www.bloomberg.comid = 100 (1 - B(0,T)) 360

T

Example:90-day bill price B(0,90) = 0.9894 id (90) = 100 (1- 0.9894)(360/90)

= 4.24%

Simple interest rate: is (T)

B(0,T) =

T (days) B(0,T) is (T)

30 0.9967 4.0360 0.9931 4.2390 0.9894 4.34

180 0.9784 4.48

T(days) is(T) B(0,T)

30 4.03 0.996760 4.28 0.993090 4.34 0.9894

180 4.48 0.9784

Example:30-day simple rate is = 4.03%B(0,30) = 1/ [1+ (0.0403)(30/365)]

= 0.9967

Current quotes: www.bloomberg.comis= 100 [ - 1] 365

T

Example:90-day bill price B(0,90) = 0.9894 is (90) = 100 [(1/0.9894) -1](365/90)

= 4.34%

1

1 + is (T)(T/365)

1 B(0,T)

Discretely compounded rate: rt(h) compounding for h periods

B(t,t+h) =

rt(h) = h [ (1/B)(1/h) - 1 ]

Example:1 year zero-coupon bond price = 0.9560

semiannually compounded rate

r1(2) = 2 [ (1/0.9560) (1/2)

- 1 ]

= 4.551%

1

[1 + rt(h)/h] h 0.9560

periods rate1 4.603%2 4.551%3 4.534%4 4.525%5 4.520%6 4.517%

12 4.508%52 4.502%

365 4.500%8760 4.500%

1-year zero price =

Term structure (yield curve)

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 yield curve

“Term structure” may refer to various yields:

At first we will focus on the

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

Direct relationship between zero coupon spot yield, 0yt , and the price today of a riskless dollar delivered later, B(0,t):

Define 0yt such that B(0,t) = (1 + 0yt)–t

Remember: B(0,t) is the present value (time 0) of $1.00 at some later time (t)

Example A: use spot zero coupon yields to find B(0,t) vector

t 0y t (1 +0y t) t (1 +0y t)

-t B(0,t)1 8.00% 1.080 0.9259 0.92592 9.00% 1.188 0.8417 0.84173 10.00% 1.331 0.7513 0.75134 11.00% 1.518 0.6587 0.6587

6.00%

7.00%

8.00%

9.00%

10.00%

11.00%

12.00%

1 2 3 4

Use B(0,t) = (1 + 0yt)–t

Example B: use B(0,t) vector (discount factors) to find zero-coupon yields

t B(0,t) B(0,t)-1 B(0,t)-1/t B(0,t)-1/t -1 0y t

1 0.9259 1.080 1.080 0.0800 8.00%2 0.8417 1.188 1.090 0.0900 9.00%3 0.7513 1.331 1.100 0.1000 10.00%4 0.6587 1.518 1.110 0.1100 11.00%

Solve B(0,t) = (1 + 0yt)–t for 0yt:

0yt = [B(0,t)] –(1/t) -1

Example C: use B(0,t) vector (discount factors) to value a 10% annual coupon bond

10%

t B(0,t) 0y t B(0,t)Ct B(0,4)$1001 0.9259 8.00% 9.262 0.8418 9.00% 8.423 0.7514 10.00% 7.514 0.6588 11.00% 6.59 65.88 Bond Value

Bond yield = #NAME? 31.78 65.88 97.66

bond: 3-year $100 face; annual coupon =

Note that the bond yield (10.96%) is found FROM THE PRICEvia the excel function “yield(arguments)”

Make sure you know how the excel yield function worksYou also need to learn the “price” function. Use the excel help function for details.

0 1 2 3 4 time

$10 $10 $10 $110

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: Annual interest payment

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)

U.S. T-Bill (matures 9/23/99) prices & discount ratesfrom issue though 8/31/99 (source: Dow Jones )

96.50

97.00

97.50

98.00

98.50

99.00

99.50

100.00

4.0%

4.1%

4.2%

4.3%

4.4%

4.5%

4.6%

4.7%

discount Bid

U.S. T-Bills (maturing 9/23/99 & 4/27/00 ) prices & discount ratesfrom issue though 8/31/99 (source: Dow Jones )

95.0

95.5

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

100.0

4.0%

4.1%

4.2%

4.3%

4.4%

4.5%

4.6%

4.7%

4.8%

4.9%

5.0%

discount rate:9.23.99 bill

discount rate:4/27/00 bill

Bid price:9.23.99 bill

Bid Price:4.27.00 bill

U.S. T-Notes issued 8/15/97- prices & yields through 9/1/99

Issues: 6% of 8/15/00 & 6.125% of 8/15/07 (source: Dow Jones )

80

85

90

95

100

105

110

115

3.5%

4.5%

5.5%

6.5%

7.5%

8.5%

9.5%

07 Bid 00 Bid 07 yield 00 yield

prices

yields

Price versus yield: U.S. T-Note - 6.125% of 8/15/07daily observations 7/30/97 through 9/1/99 (source: Dow Jones )

95

100

105

110

115

4.5% 4.7% 4.9% 5.1% 5.3% 5.5% 5.7% 5.9% 6.1% 6.3% 6.5%

yield-to-maturity

Pri

ce

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.