Dr. Sudhakar Raju FN 3000 ANSWERS TO CHAPTER 6: INTEREST RATES & BOND VALUATION
1.) YTM & Require Required d Return Return refer refer to the same same concept. concept. The fundament fundamental al differenc differencee between between the YTM and coupon rate is that while the YTM on a bond changes as economy wide interest rates rates change, the coupon rate is contractually contractually fixed and does not change over the life of the bond. A bond that is selling at par implies that the coupon rate & YTM are at the same level. Suppose two years from now the required return (or YTM) changes to 8%. This has no impact on the coupon rate. The coupon rate on the bond is fixed and continues to be 10%.
2.) Assume that the face value of the bond is $1000 & it is an annual bond. Thus: 1 Shift P/yr 1000 FV 70 PMT 20 N 7 I/yr (When bonds are issued they are issued at par. Thus, coupon rate equals YTM) PV = 1000 15 I/yr PV → $499.25 Thus, as interest rates rise, bond prices fall.
3.) 1 Shift P/yr 1000 FV 90 PMT 7N 8 I/Yr PV → $1052.06
4.) 1 Shift P/yr 1000 FV 80 PMT 9N 910.85 +/- PV I/Yr → 9.52%
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5.) 1 Shift p/yr 1000 FV +
1086 PV 14 N 6.80 I/yr PMT →$77.72 −
Coupon Rate =
$77.72 $1000
= 7.77 %
6.) 2 Shift P/yr 1000 FV 10 N (11 year bonds issued one year ago. Thus, remaining maturity is 10 yrs.) Shift N 42.50 PMT ($85/2 = $42.50) 7.90 I/yr PV → $1040.95
7.) 2 Shift P/Yr 1000 FV 39 PMT ($78/2 = $39) 13 N (Remaining maturity is 15 – 2 = 13 years) Shift N +
920 PV (92% of par value implies 92% of $1000) I/Yr→ 8.85% −
8.) 2 Shift P/Yr 1000 FV +
1080 PV 10.50 N Shift N 7.60 I/yr PMT → $43.60 (This is a semi-annual payment) Annual Payment = $43.60 x 2 = $87.20 Coupon Rate = ($87.20 / $1000) = 8.72% −
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9.)
Bond X 1 Shift P/yr 1000 FV 80 PMT 13 N 6 I/yr PV → $1177.05
Bond Y 1 Shift P/yr 1000 FV 60 PMT 13 N 8 I/Yr PV → $841.92
The price of Bond X today is $1177.05 whereas the price of Bond Y is $841.92. One year from now if interest rates remain unchanged, the remaining maturity of the bond is 12 years. Replace 13 N by 12 N. The prices of the bonds will then be: PV → $1167.68
PV → $849.28
In 3 years, N = 13-3 = 10 years the prices will be: PV → $1147.20
PV → $865.80
In 8 years, N = 13-8 = 5 years the prices will be: PV → $1084.25
PV → $920.15
In 12 years, N = 13-12 = 1 year the prices will be: PV → $1018.87
PV → $981.48
In 13 year, N = 13-13 = 0 years the prices will be: PV → $1000
PV → $1000
Conclusion: If interest rates remain unchanged over the life of the bond, as the bond gets closer to maturity, the bond price converges to its face value of $1000.
10.)
Bond J 2 Shift P/yr 1000 FV 25 PMT 8N Shift N
7 I/yr PV → $879.06
Bond S 2 Shift P/yr 1000 FV 55 PMT 8N Shift N
7 I/Yr PV → $1241.88
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9 I/yr PV → $775.32 5 I/Yr PV → $1000
9 I/Yr PV → $1112.34 5 I/yr PV →$1391.65
Percentage change for Bond J if rates rise from 7% to 9%: % ∆ = [$775.32- $879.06] / [$879.06] = -11.80% Percentage change for Bond J if rates rise fall from 7% to 5%: % ∆ = [$1000- $879.06] / [$879.06] = +13.76% Percentage change for Bond S if rates rise from 7% to 9%: % ∆ = [$1112.34 - $1241.88] / [$1241.88] = -10.43% Percentage change for Bond S if rates rise fall from 7% to 5%: % ∆ = [$1391.65- $1241.88] / [$1241.88] = +12.06% Conclusion: All else the same, the lower the coupon rate on a bond the greater its price volatility. Notice that Bond J (the lower coupon bond) exhibits higher price swings or volatility.
11.) 2 Shift P/yr 1000 FV 40 PMT 12 N Shift N +
1080 PV (108% of par means 1.08 x $1000 = $1080) I/Yr → 7% Shift Eff% →7.13% −
The YTM is 7%. The effective annual yield is similar to the EAR in that it takes into account the effect of compounding. Given the fact that this bond makes interest payments semiannually, the effective yield on the bond is a little higher at 7.13% as compared to the YTM of 7%. AnnualCoupon Current Yield (CY) = Pr iceofBond CY = [$80 / $1080] = 7.41%.
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12.) Identify the YTM on the company’s existing bonds. Thus: 2 Shift P/Yr 1000 FV 45 PMT +
1073 PV 20 N Shift N I/Yr → 8.25% −
Thus, the company’s existing bonds have a YTM of 8.25%. To get the company’s new bonds to sell at par, set the coupon rate equal to the YTM. Thus, the new coupon rate should be set equal to 8.25%, implying a semi-annual payment of [$82.50 / 2] = $41.25
13.) a.) Coupon Bond FV= $1000; CIR = 7%; Market Rate /YTM /Discount Rate = 7%; N = 20 years 1 Shift P/Yr 1000 FV 70 PMT 7 I/YR 20 N PV →$1000 Thus, each coupon bond should be priced at $1000 (notice that the bond is trading at par). To raise $25 million the company would need to issue: [$25,000,000 / $1000] = 25,000 bonds. Zero Coupon Bond FV= $1000; CIR = 7%; Market Rate /YTM /Discount Rate = 7%; N = 20 years
1 Shift P/Yr 1000 FV 0 PMT 7 I/YR 20 N PV →$258.42 Thus, each zero coupon bond should be priced at $258.42. To raise $25 million, the company would need to issue: [$25,000,000 / $258.42] = 96,742 bonds.
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b.) Coupon Bond In 20 years, the company would need to repay $70 (coupon) + $1000 (principal) or $1070 x 25,000 bonds = $26,750,000 or $26.75 million. Zero Coupon Bond In 20 years, the company would need to repay the face value of $1000 per zero coupon bond. Thus, the company would need to pay: $1000 (face value) x 96,742 bonds = $96,742,000 or $96.742 million.
14.) Note that corporate bonds are semi-annual coupon paying instruments. To solve for the YTM do the following: 2 Shift P/Yr 1000 FV 41.8750 PMT (or $83.75/2 = $41.8750) +
780.98 PV (78.098 implies 78.098% of the face value of $1000) 27 N (Remaining maturity is 2035- 2008 = 27 years) Shift N I/Yr→ 10.91% −
Thus, the YTM (or Internal Rate of Return) on this bond is 10.91%. This 10.91% will be correct it three conditions are satisfied: a.) The issuer pays the coupons and principal as promised and does not default on payments. b.) If the bond is held to maturity by the investor. That is, the bond is not sold off before 2035. c.) Interest rates remain unchanged over the life of the bond. The first two conditions can be satisfied. The final condition that interest rates remain unchanged over the life of the bond is never satisfied in real life financial markets. Thus, it is important to recognize that the YTM is only an approximation of the true yield that will eventually be received on the bond by the investor.
15.) The quotes refer to U.S. Treasury bonds. Recognize that the Ask or selling price of the dealer is the investor’s buying price. The investor’s buying price is thus: 80:12 (i.e. 80 points and 12 ticks) or 80 + (12/32) = 80.375 of the face value of the bond. For a $1000 face value T-bond, the purchase price equals $803.75.
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To solve for the yield to maturity (YTM) do the following:
2 shift P/Yr 1000 FV 803.75 +/- PV 32.50 PMT (Semi-annual coupon payment of $65/2 = $32.50) 30 N [Nov. 2038 – Nov. 2008 = 30 years] Shift N I/YR = > 8.28% p.a. [YTM or ASK YLD] Thus, an investor who purchases this bond and holds it to maturity can expect to receive 8.28% if yields remain unchanged over the life of the bond.
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