The YTM on a bond is the interest rate you earn on your investment if interest r
ID: 2748757 • Letter: T
Question
The YTM on a bond is the interest rate you earn on your investment if interest rates don’t change. If you actually sell the bond before it matures, your realized return is known as the holding period yield (HPY).
Suppose that today you buy a bond with an annual coupon of 7 percent for $1,090. The bond has 14 years to maturity. What rate of return do you expect to earn on your investment?(Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))
Two years from now, the YTM on your bond has declined by 1 percent, and you decide to sell. What price will your bond sell for? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))
The YTM on a bond is the interest rate you earn on your investment if interest rates don’t change. If you actually sell the bond before it matures, your realized return is known as the holding period yield (HPY).
Explanation / Answer
Answer (a)
Expected rate of return = 6.04%
Answer (b)
Bond Price = 1173.33
Answer (c)
Holding Period Yield = 9.77%
Bond Price = $ 1090
Coupon rate = 7%
Par Value = $ 1000
Annual Coupon amount = $ 1000* 7% = $ 70
Time to Maturity = 14 years
Let r be the yield to maturity, then price of the bond can be calculated as
Bond Price = Coupon Payment * [(1-(1/(1+r)^n))/r] + Par Value / (1+r)^n
1090 = 70 * [(1-(1/(1+r)^14))/r] + 1000 / (1+r)^14
1090 – 70 * [(1-(1/(1+r)^14))/r] - 1000 / (1+r)^14 = 0
Let r = 6%, LHS will be
= 1090 – 70 * [(1-(1/(1.06)^14))/0.06] - 1000 / (1.06)^14
= 1090 – 70 * [(1-(1/2.260904)/0.06] - 1000 / 2.260904
= 1090 – 70 * [(1-0.442301)/0.06] - 1000 * 0.442301
= 1000 – 70 * (0.557699/0.06) – 1000 * 0.442301
= 1090 – 70 * 9.294984 – 1000 * 0.442301
= 1090 – 650.6489 – 442.301
= -2.94984
Let r = 6.1%, then LHS will be
= 1090 – 70 * [(1-(1/(1.061)^14))/0.061] - 1000 / (1.061)^14
= 1090 – 70 * [(1-(1/2.290949)/0.061] - 1000 / 2.290949
= 1090 – 70 * [(1-0.4365)/0.061] - 1000 * 0.4365
= 1090 – 70 * (0.5635/0.061) – 1000 * 0.4365
= 1090 – 70 * 9.237699 – 1000 * 0.4365
= 1090 – 646.6389 – 436.5004
= 6.860712
r = 0.06 + [(-2.94984 * (0.06-0.061))/(6.860712-(-2.94984))
r = 0.06 + [0.00294984/6.810552]
r = 0.06 + 0.0004331 = 0.060433 or 6.04%
After two years, YTM declined by 1% to 5.04%
Time to maturity = 14 – 2 = 12 years
Annual coupon amount = $ 70
Price of the Bond = 70 * [(1-(1/(1+0.0504)^12))/0.0504] + 1000 / (1+0.0504)^12
= 70 * [(1-(1/(1.0504)^12))/0.0504] + 1000 / (1.0504)^12
= 70 * [(1-(1/1.804083)/0.0504] + 1000 / 1.804083
= 70 * [(1-0.554298)/0.0504] + 1000 * 0.554298
= 70 * (0.445702/0.0504) + 1000 * 0.554298
= 70 * 8.84329 + 1000 * 0.554298
= 619.0303 + 554.2982
= 1173.328 or 1173.33 (rounded off)
Calculation of HPY
Purchase price = $ 1090
Annual coupon amount = $ 70
Sale Price = $ 1173.33
Holding Period = 2 years
Total coupon income = 2* $ 70 = $ 140
Annualized HPY = [1+{(Income + (Sale price - purchase Price)})/purchase price]^(1/2 –1
= [1+{(140 + (1173.33 – 1090))/1090}]^(1/2) – 1
= [1+{(140 + 83.33)/1090]^0.5 - 1
= [1+(223.33/1090)]^0.5 – 1
= (1+0.2048899)^0.5 – 1
= 1.2048899^0.5 – 1
= 1.097675 – 1
= 0.097675 or 9.77% (rounded off)
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