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Bond Valuation Solutions
Derivatives and Risk Management (Options, Futures and Derivatives)
SVKM's NMIMS
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Problems on Bond Valuation (Soln)
- A Rs 1000 par value bond, bearing a coupon rate of 11% will mature after 5 years. What is the value of the bond, if the discount rate is 12%?
Soln.
Value of the bond PV (bond) = PV of Coupon Annuity (PVA of C) + PV of Maturity Value (PV of M) PV (bond) = PVA of C + PV of M
Part 1 C = 11% of 1000 = Rs 110, M = Rs 1000, k = 12%, n = 5, PV =?
PV (bond) = 110 x PVA Factor (12%, 5) + 1000 x PV Discount Factor (12%,5) = 110 x [ଵ−ଵ. ଵଶ
− . ଵଶ ] +
ଵ ଵ. ଵଶ 5 = 396+567 = Rs 963.
Part 2 C = 5% of 1000 = Rs 55, M = Rs 1000, k = 6% using bond *, n = 10, PV =?
PV (bond) = 55 x PVA Factor (6%, 10) + 1000 x PV Discount Factor (6%,10) = 55 x [ଵ−ଵ.
−భబ . 6 ] +
ଵ ଵ. 6భబ = 404.+558 = Rs 963.
[* Note that yield of semi-annual bonds are quoted as 2 x semi-annual compounded yield. Thus 12% yield for semi-annual bond means effective discount rate of 6% compounded semi- annually.]
- A Rs 1000 par value bond bearing a coupon rate of 12% will mature after 7 years. What is the value of the bond if the discount rate is: 14%? 12%? 10%?
C = 12% of 1000 = Rs 120, M = Rs 1000, n = 7, PV =? PV (bond) = PVA of C + PV of M
0 1 2 3 4 5 k = 12%
C = 110 C = 110 C = 110 C = 110 C = 110 M = 1000
0 1 2 3 4 5 6 7
C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 M = 1000
At k = 14% PV (bond) = 120 x PVA Factor (14%, 7) + 1000 x PV Discount Factor (14%,7) = 514 + 399 = Rs 914.
At k = 12% PV (bond) = 547 + 452 = Rs 1,
At k = 10% PV (bond) = Rs 1,097.
[Observations A. As k declines, PV increase and vice versa. The fair value and hence price of a bond has an inverse with yield. B. When k > coupon rate, PV < M; when k = coupon rate, PV = M; when k < coupon rate, PV>M]
Homework: Take two other bonds with same coupon rate (12%) but with time to maturity of 5 years, 3 years and 1 year. Calculate first with discount rate of 14% and then with discount rate of 10%. Observe how the prices change when you reduce the maturity.
[Observations C. As time to maturity decreases, bond values converge towards the face value M. If the bond is trading at a discount to M, the value should increase towards M; if the bond is trading at a premium to M, the value should decrease towards M. D. The range of bond values for the same change in discount rate is higher for the bond with longer time to maturity, and lower for the bond with lower time to maturity. Bonds with longer time to maturity are more sensitive to interest rate changes than bonds with shorter time to maturity.]
- The market value of a Rs 1000 par value bond, carrying a coupon rate of 14% and maturing in 10 years is Rs 800. What is the yield to maturity of this bond? Soln
Bond price P = Rs 800, M = Rs 1000, C = Rs 140, n = 10 years YTM =?
Using approximation formula, YTM =
[𝠀+𠀀−𠰀𦰀 ] ሺ𠀀+𠰀ሻ/ଶ
0 1 2 3 4 5 6 7 8 9 10
C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 C = 140 M = 1000
Solution to Text Book Problems (Chapter 3)
5.
a. Fall. Example: Assume a one-year, 10% bond. If the interest rate is 10%, the bond is worth $110/1 = $100. If the interest rate rises to 15%, the bond is worth $110/1 = $95.
b. Less (e., see 5a—if the bond yield is 15% but the coupon rate is lower at 10%, the price of the bond is less than $100).
c. Less (e., with r = 5%, one-year 10% bond is worth $110/1 = $104).
d. Higher (e., if r = 10%, one-year 10% bond is worth $110/1 = $100, while one-year 8% bond is worth $108/1 = $98).
e. No. Low-coupon bonds have greater interest rate sensitivity and are therefore more volatile (e., if r falls from 10% to 5%, the value of a two-year 10% bond rises from $100 to $109 (a rise of 9%). The value of a two-year 5% bond rises from $91 to $100 (a rise of 9%).
10.
a. Price today is $108; price after one year is $106.
b. Return = (8 + 106)/108 - 1 = .06, or 6%. If a bond’s yield to maturity is unchanged, the return to the bondholder is equal to the yield.
7%; the extra return that you earn for investing for two years rather than one year is 1 2 /1. - 1 = .0701.
a. Real rate = 1.10/1 – 1 = .0476, or 4%.
b. The real rate does not change. The nominal rate increases to 1 × 1 – 1 = .1209, or 12%. 16.
a.
= − + =
2020
1 1 100
PV 3 102.
0 0 (1) (1)
`
b.
The bond price at a 5% yield is:
1,201. (1)
1,
0 (1)
1
0.
1
PV 1005 + 5 =$
= −
If the yield decreases to 5%, the price would rise to:
1,173.
(1)
1,
0 (1)
1
0.
1
PV 1005 + 5 =$
= −
Bond Valuation Solutions
Course: Derivatives and Risk Management (Options, Futures and Derivatives)
University: SVKM's NMIMS
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