Bond Valuation ExampleWe wish to value a 30-year 10% coupon bond with a face value equal to $1000. Assume a discount rate equal to 6%. This worksheet can also be used to value other bonds with n between 1 and 53 by changing values in the yellow table.Bond Details t PV[CF(t)]F 1000c 0.1 1 94.33962 94.33962k 0.06 2 88.99964 183.3393n 30 3 83.96193 267.3012

4 79.20937 346.5106PV Bond 1550.593 5 74.72582 421.2364PV Bond 1550.593 6 70.49605 491.7324

7 66.50571 558.2381B9 is the sum of 8 62.74124 620.9794 discounted cash 9 59.18985 680.1692 flows on from the 10 55.83948 736.0087 bond. 11 52.67875 788.6875B10 is based on 12 49.69694 838.3844 the PV Annuity 13 46.8839 885.2683 as applied to bond 14 44.2301 929.4984 valuation. 15 41.72651 971.2249

16 39.36463 1010.5917 37.13644 1047.72618 35.03438 1082.7619 33.0513 1115.81220 31.18047 1146.99221 29.41554 1176.40822 27.75051 1204.15823 26.17973 1230.33824 24.69785 1255.03625 23.29986 1278.33626 21.981 1300.31727 20.7368 1321.05328 19.56301 1340.61629 18.45567 1359.07230 191.5211 1550.59331 0 1550.59332 0 1550.59333 0 1550.59334 0 1550.59335 0 1550.59336 0 1550.59337 0 1550.59338 0 1550.59339 0 1550.59340 0 1550.59341 0 1550.59342 0 1550.59343 0 1550.59344 0 1550.59345 0 1550.593

46 0 1550.59347 0 1550.59348 0 1550.59349 0 1550.59350 0 1550.59351 0 1550.59352 0 1550.59353 0 1550.593

We wish to value a 30-year 10% coupon bond with a face value equal to $1000. Assume a discount rate equal to 6%. This worksheet can also be used to value other bonds with n between 1 and 53 by changing values in the yellow table.

Yield to Maturity ExampleEnter Bond data into Column B of the Yellow Region. Revise your y estimate until you are sufficiently close to the bond's correct yield.Bond DetailsF 1000c 0.1Guess for y 0.15n 30Initial Bond Price 1000

PV Bond 671.701DECREASE YOUR y ESTIMATE

Note: This calculator assumes that coupon payments are made annually beginning in one year.

Enter Bond data into Column B of the Yellow Region. Revise your y estimate until you are sufficiently close to the bond's correct yield.

Fixed Income Arbitrage

Bond A Bond B Bond C Use Bonds A, B Bond DP0=1000 P0=1055.5 P0=889 and C to replicate P0 = 1360F=1000 F=1000 F=1000 Bond D: c = .20c=.04 c=.06 c=0 F = 1000 P0(D) should be:n=2 n=3 n=3 n = 3 1444

Bond A Bond B Bond C From Column Ft=1 40 60 0 0t=2 1040 60 0 3.333333333333t=3 0 1060 1000 -2.33333333333

^Original Cash Flow Matrix^ ^Weights^-0.001 0.001 0 200 0

0.0173333 -0.0006667 0 200 3.33333333333333-0.0183733 0.00070667 0.001 1200 -2.33333333333333^Inverse of Cash Flow Matrix^ ^Bond D CFs^ ^Weights^

-1000 -1055.5 -889 -1360 040 60 0 200 -3.33333333333333

1040 60 0 200 2.333333333333330 1060 1000 1200 1

^Cash Flow of each bond^ ^Arb.Port.Weights^

200200

1200Solve for Bond D CFs

84 CF Time 00 CF Time 10 CF Time 20 CF Time 3

Fixed Income Portfolio Dedication

Bond A Bond B Bond CP0=1000 P0=1055.5 P0=889F=1000 F=1000 F=1000c=.04 c=.06 c=0n=2 n=3 n=3

Fixed Income Arbitraget=1 40 60 0t=2 1040 60 0t=3 0 1060 1000

^Fixed Income Portfolio Cash Flows^40 60 0 2000 12000000 <Liability Cash Flows

1040 60 0 198666.666666667 = 140000000 1060 1000 -195586.666666667 15000000

^Original Cash Flow Matrix^ ^#'s of Each Bond^-0.001 0.001 0 12000000 2000

0.01733333 -0.0006667 0 14000000 198666.666667-0.0183733 0.0007067 0.001 15000000 -195586.66667

^Inverse of Cash Flow Matrix^ Liability Outflows # of each Bond-1000 -1055.5 -889 -2000 37816120 CF Time 0

40 60 0 -198666.66667 -12000000 CF Time 11040 60 0 195586.666667 -14000000 CF Time 2

0 1060 1000 -15000000 CF Time 3^Cash Flow of each bond^ Dedicated Port.Weights

210586670-195586670-195586.67

<Liability Cash Flows

Thesearerounded

Calculating Bond Duration and Immunized Portfolio WeightsCalculating DurationBond Details t t*CF(t) t*PV(CF(t))

1000 P(0) 1 100 90.909091000 F 2 200 165.2893

0.1 c 3 300 225.39445 n 4 400 273.2054

0.1 y 5 5500 3415.0674.169865 Bond Duration

-2.010686 Duration of the Liability Stream from Fixed Income Dedication; Cells F14:F16

-1.909091 Duration of some second Bond (B)

Calculating Portfolio Weights for Immunization

-4.169865 -1.909091 Durations of the 2 bonds1 1 1*w(A) + 1*w(B) must sum to 1

-0.442326 -0.844441 -2.010686 0.044938 Weight of Bond A0.442326 1.844441 1 0.955062 Weight of Bond B

Inverse Matrix

Calculating Bond Duration, Convexity and Immunized Portfolio WeightsCalculating Duration The Green sectionsBond Details t t*CF(t) t*PV(CF(t)) (t+1)*t*PV(CF(t))/(1+y)^2 deal with bond

1000 P(0) 1 100 90.9090909 150.262960180316 convexity.1000 F 2 200 165.289256 409.808073219042

0.1 c 3 300 225.39444 745.1055876709865 n 4 400 273.205382 1128.94786010755

0.1 y 5 5500 3415.06728 16934.21790161334.16986545 19.3683423827912

Bond Duration Bond Convexity-1.801667 Duration of the Liability Stream from Fixed Income Arbitrage Worksheet; Cells F19:F214.969424 Convexity of the Liability Stream from Fixed Income Arbitrage Worksheet; Cells F19:F21

-1.909091 Duration of a second Bond (B) Duration of Bond C -1.34.658152 Convexity of a second Bond (B) Convexity of Bond C 3.2

Calculating Portfolio Weights for Immunization

-4.169865 -1.909091 -1.3 Durations of the 3 bonds With three bonds, an infinite number1 1 1 1*w(A) + 1*w(B) + 1*w( C) must sum to 1 of weights solutions are possible

19.36834 4.658152 3.2 Convexities of the 3 bonds based on the duration and weightsconstraints. With a convexity

0.257474 -0.009446 0.107551 -2.010686 0.00731955 Weight of Bond A constraint, only one of these-2.854938 -2.089823 -0.506749 1 1.13230978 Weight of Bond B solutions are feasible in a three-2.597463 3.099269 0.399198 4.969424 -0.13962933 Weight of Bond C bond portfolio.

Inverse Matrix

Calculating Bond Duration, Convexity and Immunized Portfolio Weights

With three bonds, an infinite numberof weights solutions are possiblebased on the duration and weightsconstraints. With a convexityconstraint, only one of thesesolutions are feasible in a three-

Duration and ImmunizationBond A Bond B Bond C Liabilities

P(0) 900 ? ? CF(1)= 100,000F 1000 1000 1000 CF(2)= 400,000c 0.03 0 0 CF(3) 50,000n 3 4 1We will first use Bonds A and B to immunize interest rate risk of liabilities based on Duration only.First, we will calculate the ytm of Bond A. Then, assuming a flat yield curve, we calculate the prices of Bond B and of the liability stream.P(0) -900 P(0)(B)= 768.731 Here we solve for the price of Bond B.CF(1) 30 P(0)(o)= 485394.2 Here we solve for the sum value of the firm's liabilities..CF(2) 30CF(3) 1030ytm(A) 0.067963Next, we calculate the Durations of Bonds A and B and then the Duration of the liability stream.DURa -2.90835DURb -4DURo -1.891661Finally, we begin the process of computing the investment amounts in a portfolio comprised of Bonds A and B such that the duration of the portfolio matches the duration of the liability stream. Note that the total amount invested must sum to the value of the liability stream.

-2.90835 -41 1

Now, invert the matrix and solve for the amount of money to invest in each bond.0.916045 3.664178 -1.891661 1.93133299 = the fractional amount to invest in Bond A.

-0.916045 -2.664178 1 -0.931333 = the fractional amount to invest in Bond B.

The Green sections below deal with bond convexity.

Next, we introduce Bond C to the immunization process.Continuing to assume a flat yield curve, we calculate the price of Bond C:Value of Bond C = 936.3616ytm(A) = 0.067963 = all spot and forward ratesDURa -2.90835 Convexitya 10.09387 These convexityDURb -4 Convexityb 17.53546 figures are secondDURc -1 Convexityc 1.753546 derivativesdivided byDURo -1.891661 Convexityo 5.028976 the bond prices.

We will now determine how much to invest in each of the three bonds to immunize portfolio risk based on both Duration and Convexity. Note also that the total invested equals the PV of the liability stream.

-2.90835 -4 -11 1 1

10.09387 17.53546 1.753546Finally, invert the matrix and solve.-3.096653 -2.064435 -0.588646 -1.8916606 0.833095 = the fractional amount to invest in Bond A.1.636499 0.979888 0.374447 1 -0.232726 = the fractional amount to invest in Bond B.1.460154 2.084547 0.214198 5.02897607 0.39963 = the fractional amount to invest in Bond C.

We will first use Bonds A and B to immunize interest rate risk of liabilities based on Duration only.First, we will calculate the ytm of Bond A. Then, assuming a flat yield curve, we calculate the prices

Here we solve for the sum value of the firm's liabilities..

Finally, we begin the process of computing the investment amounts in a portfolio comprised of Bonds A and B such that the duration of the portfolio matches the duration of the liability stream. Note that

We will now determine how much to invest in each of the three bonds to immunize portfolio risk based on both

= the fractional amount to invest in Bond A.= the fractional amount to invest in Bond B.= the fractional amount to invest in Bond C.

Duration and ImmunizationBond A Bond B Bond C Liabilities

P(0) 1000 1055.9 889 CF(1)= 12,000,000F 1000 1000 1000 CF(2)= 14,000,000 y = .04 for all bonds; the yield curve is flat.c 0.04 0.06 0 CF(3) 15,000,000 The value of the liability stream isn 2 3 3We will use Bonds E, F and G to immunize interest rate risk of liabilities based on Duration only.Next, we calculate the Durations of Bonds E, F and G and then the Duration of the liability stream.DUR(A) -1.961538DUR(B) -2.837056DUR(C) -2.999988DUR(L) -1.974863Next, we begin the process of computing the investment amounts in a portfolio comprised of Bonds A, B and C such that the duration of the portfolio matches the duration of the liability stream. Note that the total amount invested must sum to the value of the liability stream. Also, notice that there are an infinity of solutions to our two-equation, three variable system.-1.961538 -2.837056 -2.999988 wA -1.974863

1 1 1 wB = 1wC

Suppose that the manager already has invested $3781719.4 (wA = .10) into Bond A and wants to hold this investment in Bond A constant. We solve for investment weights as follows:

-1 -2.837056 -2.999988 wA -1.9748631 1 1 wB = 11 0 0 wC 0.1

0 0 1 -1.974863 0.16.137542 18.41255 -12.27501 1 = 5.064243

-6.137542 -17.41255 11.27501 0.1 -4.164243

The Green sections below deal with bond convexity.Now, we will ignore the investment constraint (10%) on Bond E, but will seek to match portfolio convexity as well as duration.W will work with durations and compute convexities for our bonds and liability outflow stream.DUR(A) -1.961538 ConvexityA 5.405098 These convexityDUR(B) -2.837056 ConvexityB 10.29779 figures are secondDUR(C) -2.999988 ConvexityC 11.09463 derivatives divided byDUR(L) -1.974863 ConvexityL 6.375227 the bond prices.

We will now determine how much to invest in each of the three bonds to immunize portfolio risk based on both Duration and Convexity. Note also that the total invested equals the PV of the liability stream.-1.961538 -2.837056 -2.999988

1 1 15.405098 10.29779 11.09463

Finally, invert the matrix and solve.-8.005791 -5.855669 -1.636973 -1.9748634 -0.481398 = the fractional amount to invest in Bond A.57.16266 55.73381 10.43329 1 9.359945 = the fractional amount to invest in Bond B.

-49.15687 -48.87814 -8.796316 6.37522686 -7.878547 = the fractional amount to invest in Bond C.

y = .04 for all bonds; the yield curve is flat.The value of the liability stream is 37817194

Next, we calculate the Durations of Bonds E, F and G and then the Duration of the liability stream.

Next, we begin the process of computing the investment amounts in a portfolio comprised of Bonds A, B and C such that the duration of the portfolio matches the duration of the liability stream. Note that

Now, we will ignore the investment constraint (10%) on Bond E, but will seek to match portfolio convexity as well as duration.

We will now determine how much to invest in each of the three bonds to immunize portfolio risk based on both

= the fractional amount to invest in Bond A.= the fractional amount to invest in Bond B.= the fractional amount to invest in Bond C.

Bootstrapping The Yield Curve

Bond %Coupon Ask Price Spot Rate1 5.00 102 0.9714286 2.94%2 5.00 101 3/4 0.9227891 4.10%3 5.00 101 1/2 0.8764658 4.49%4 5.00 101 1/4 0.8323484 4.69%5 5.00 101 1/4 0.7927128 4.76%6 5.00 101 1/4 0.7549645 4.80%7 5.00 101 1/4 0.7190138 4.83%8 5.00 101 1/4 0.6847751 4.85%9 5.25 102 1/4 0.64455 5.00%

10 5.25 102 1/4 0.612399 5.03%11 5.25 102 1/4 0.5818518 5.05%12 5.25 102 1/4 0.5528283 5.06%13 5.50 104 0.5193962 5.17%14 5.50 104 0.4923187 5.19%15 5.50 104 0.4666528 5.21%16 5.75 105 3/4 0.4331835 5.37%

D t

0 2 4 6 8 10 120.00%

200.00%

400.00%

600.00%

800.00%

1000.00%

1200.00%

Years

Spot

Rat

e

Obtaining the Yield CurveSuppose that market prices of two two-year bonds are given to be 826.4 and 1005.The first bond is a zero coupon and the second is a 10% coupon issue. Assume that investors value bonds withpresent value frameworks. We set up two valuation equations with discount functions D1 and D2.

We will first solve the following simple linear system for D1 and D2:

0D1 + 1000D2 = 826.4100D1 + 1100D2 = 1005

0 1000 This represents the coefficients matrix C. We will invert this matrix below.

100 1100C

826.4 The Price D1 The discount

1005 D2 s d

Spot and Forward Rates-0.011 0.01 826.4 0.9596 0.04210.001 0 1005 = 0.8264 0.1

s d 0.1612

Vector s Vector d

y0,1 =y0,2 =

C-1 y1,2 =

The first bond is a zero coupon and the second is a 10% coupon issue. Assume that investors value bonds with

Spot and Forward Rates

Yield Curve Estimation with Coupon Bonds: Example 1Bond A Bond B*P0=1000 P0=1055.5 40 1040 Original Cash Flow MatrixF=1000 F=1000 60 1060c=.04 c=.06n=2 n=2

-0.0530 0.0520 1000 1.886 -0.469777 y010.0030 -0.0020 1055.5 0.889 0.060594 y02

Inverse Matrix Price Discount Spot y12Vector Functions Rates

Yield Curve Estimation with Coupon Bonds: Example 2Bond A Bond B Bond CP0=1000 P0=1055.5 P0=889 40 1040 0F=1000 F=1000 F=1000 60 60 1060c=.04 c=.06 c=0 0 0 1000n=2 n=3 n=3

-0.001 0.017333 -0.018373 1000 0.96144 0.0401070.001 -0.000667 0.000707 1055.5 0.92456 0.039998

0 0 0.001 889 0.889 0.039999 Inverse Matrix Price Discount Spot

Vector Functions Rates

Yield Curve Estimation with Coupon Bonds: Example 3Bond E Bond F Bond Gc=.05 c=.06 c=.09P0=947.376 P0=904.438 P0=980.999n=2 n=3 n=3F=1000 F=1000 F=1000The following illustrate computations for the example:Cash Flow Matrix

50 1050 060 60 106090 90 1090

-0.001 -0.03815 0.0371 947.376 0.943377 0.060021 y010.001 0.001816667 -0.001767 904.438 0.85734 0.079999 y02

0 0.003 -0.002 980.999 0.751316 0.099999 y03 Inverse Matrix Price Discount Spot y12 0.100353

Vector Functions Rates y23 0.141118y13 0.12055ForwardRates

Bond Hc=.02P0=? 802.3566667 is the price of Bond Hn=3F=1000

1.121485 Bad ExampleForwardRate

ytm(A) = 0.04ytm(B) = 0.046147ytm(C) = 0.046713

y01 1000y02 1055.5y03 889y12 0.039889 37817194y23 0.039999y13 0.039945ForwardRates