Fixed Income Cheatsheet

LECTURE 1

• If the annualized simple interest rate for T years is x%, the period interest rate for the T years is T*x/100
• Let rAPR be the annual percentage rate and k be the number of compounding intervals per year. One dollar invested today yields (1+ rAPR/k)^k dollars in a year.
• Effective annual rate, rEAR is given by: rEAR=(1+ rAPR /k)^k-1
• continuously compounded rate of r will give e^rt after t years.
• In general, saving for t years at r% apr with n compoundings per year corresponds to a period rate of (1 + r /n )^tn-1; provided that tn is a whole number.
• Let d(t) denote the current price (at date 0) of a discount bond maturing at t.
• d(t)=1/(1+rt)^t, rt=1/(dt^(1/t))-1
• The price of a coupon bond must be
• B =sum (Ct *d(t)) + (P *d(T))=C1/1+r1+… +CT-1/(1+rT-1)^T-1 +(CT +P)/(1+rT )^T
• If there are n coupons per year, rt =1/n*(the APR with n compounding per year), and discount the t-th coupon by (1 + rt )^t
• Yield-to-maturity of a bond, denoted by y, B =sum Ct/ (1+y)^t +P/(1+y)^T :

forward interest rate between year t-1 and year t is(1 + rt )^t = (1 + rt-1)^t-1(1 + ft ),

ft =d(t-1)/d(t) - 1 =(1 + rt )^t/ (1 + rt-1) ^t-1 -1

LECTURE 2

• A bond is a promised to fixed cash flows. If there is less discounting of the fixed promise, the value of the promise goes up
• As yields goes up, more discounting of the fixed promises, the price goes down
• Realized return (or gross return in BTAS): Rt;t+1 =(Pt+1 + c-Pt)/ Pt
• P(R) = CF1/ (1+f1) + CF2/ (1+f1)(1+f2) + : : : + CFT/ (1+f1)_:::_(1+fT )
• The spread is defined by the s solving:

  actual price = CF1/ (1+f1+s) + : : : + CFT/ (1+f1+s)_:::_(1+fT+s)

• In general, if I hold a long maturity bond and rates go down, it is good news
• Duration = 􀀀-1dB/Bdy
• for a zero coupon bond: B = e^-rt, D = -1dB/Bdy = t, duration is maturity

A bond's duration is the weighted average of the maturity of individual cash flows, with the weights being proportional to their present values:

 D =sum PV(CFt )/B  *t = 1/B*sum CFte^-rt  *t:

• V = -D * V * y. V is the market value not the face value![pic 1][pic 2]
• If we use instead the EAR, we have: B =1/(1 + y)^t = (1 + y)^-t,                 dB/dy=-t(1+y)^-t-1=-t*B/(1+y)
• Modified duration is MD =D/(1 + y), y is the EAR and assume flat term structure.
• V = -MD *V* y.[pic 3][pic 4]
• For fixed cash-flows, define Macaulay durationMacD=sum t*PVt/PV
• Bond price is not a linear function of the yield.
• DV01: DV01=P/(10000*y),the dollar value of 01. 01 is a basis point (1/100th of 1%) change in the yield.[pic 5][pic 6]
• Convexity is the curvature of the bond price (per $ of market value) as a function of the yield: CX=(1*1*^2*B) / (2*B*y^2)[pic 7][pic 8]
•  B/B= -D* (y) +CX* (y)^2, if yield is in EAR, better to use MD[pic 9][pic 10][pic 11]
•  B = - B *MD * y[pic 12][pic 13]

Examples 1

1. Price of a 1-year bond: 997: 5 = B1 ＊(50 + 1000)，B1 =997.5/1050= 0.95, r1 = 5:26%
2. Price of a 2-year bond: 1048 = (80)(0.95) + B2 *1080, B2 =972/1080= 0.90, r2 = 5:41%:

• Current 1- and 2-year spot interest rates are 5% and 6% (EAR),respectively. The price of a 2-year Treasury coupon bond with face value of $100 and a coupon rate of 6%/year is  B =6/(1 + 0.05)+106/(1 + 0.06)^2 = 100.0539: Its YTM is 5.9706%.

• f4 =(1 + r4) ^4/ (1 + r3) ^3 – 1=(1.07)^4 / (1.065)^3 – 1 =8.51%

Examples 2

• Suppose that you buy $100M (market value) worth of 10-year zero coupon bonds. Immediately after you buy the bond, interest rates go up 20 basis point (0.20%) from 3% (EAR). What is the return on your bond? Bad news.            V = -MD *V* y=-100*10/1.03*0.002=-1.94.[pic 14][pic 15]

• Consider a 5-year coupon bond with 4% annual coupons and face value $100. Suppose that the term structure is at at 3% EAR.

[pic 16]

• Duration is D = 0: 037_ 1+0: 036_ 2+0: 035_ 3+0: 034_ 4+0: 858_ 5 = 4: 64 years.  Modified duration is MD = D= (1 + y ) = 4: 64= 1: 03 = 4: 50 years.   If yields move up by 0.1%, the bond price decreases roughly by 0.45%.

• Suppose that GM's pension liability is approximately: Year5/10/15: 30$B. and the term structure is at at 3% (EAR).
• What is the PV of the liabilities? V =30/(1 + 0.03)^5 +30/(1 + 0.03)^10 +30/(1 + 0.03)^15
• How does the value of the liability change if rates fall to 2.9%?   The fraction of value from the t = 5; 10; 15 cash flows are 38.4%, 33.1%,28.5%. So the duration is

D = 0.384 * 5 + 0.331* 10 + 0.285 *15= 9.51 years.

V =-MD * V * y=-9.51/(1 + 0.03) * 67.46 * (-0.0010)= $0.623B[pic 17][pic 18]

• If you wanted to fund the liability by purchasing a single maturity

of zero coupon bond, what maturity would you choose and how

much would you buy?

You want to match the value and the duration. The value is

$67.46B and the duration is 9.51 years.

So we should try to buy $67.46B (market value) in 9.5 year zero

coupon bonds.

• Suppose that the liability is currently only funded with $50B in

assets. What do you recommend to hold so that you do not risk