It is now January 1, 2009, and you are considering the purchase of an outstandin
ID: 2712456 • Letter: I
Question
It is now January 1, 2009, and you are considering the purchase of an outstanding bond that was issued on January 1, 2007. It has a 9.5% annual coupon and had a 30-year original maturity. (It matures on December 31, 2036.) There is 5 years of call protection (until December 31, 2011), after which time it can be called at 109—that is, at 109% of par, or $1,090. Interest rates have declined since it was issued; and it is now selling at 116.575% of par, or $1,165.75.
a.What is the yield to maturity? What is the yield to call?
b.If you bought this bond, which return would you actually earn? Explain your reasoning.
c.Suppose the bond had been selling at a discount rather than a premium. Would the yield to maturity have been the most likely return, or would the yield to call have been most likely?
Explanation / Answer
Answer (a)
Yield to maturity = 8%
Yield to call = 6.13%
Answer (b)
If the bond is held to maturity, present value of total returns at ytm 8% = $ 1030.64
If the bond is called after 3 years, present value of total returns at yield to call 6.13% = $ 178.04
As the ytm is higher than yield to call, there is a possibility of the company may call the bonds. Hence the expected total return is $178.04
Answer (c)
Suppose the bond is selling at a discount instead of premium, the yield to maturity will be lower than the yield to call. Hence the company will not utilize the call option of the bond and the bond may be allowed run its full maturity. Hence the total return earned by the investor will ytm.
Annual Coupon rate = 9.5%
Annual Coupon amount = 1000 * 9.5% = 95
Original Maturity = 30 Years (Date of maturity – 12/31/2036)
Current Price = $1165.75
Time to maturity at the time of purchase = 12/31/2036 – 1/1/2009 = 28 years
Calculation of Yield to Maturity
1165.75 = 95 * [(1-(1/(1+r)^28)/r] + 1000/(1+r)^28
1165.75 - 95 * [(1-(1/(1+r)^28)/r] - 1000/(1+r)^28 = 0
Let r = 8%, LHS will be
= 1165.75 - 95 * [(1-(1/(1.08)^28)/0.08] - 1000/(1.08)^28
= 1165.75 - 95 * [(1-(1/(8.627106)/0.08] - 1000/8.627106
= 1165.75 – 95 * (1-0.115914)/0.08 – 1000 * 0.115914
= 1165.75 – 95 * (0.88486/0.08) – 115.9137
=1165.75 – 95 * 11.05108 – 115.9137
= 1165.75 – 1049.852 – 115.9137
= -0.01618
Let r =8.05%, then LHS will be
= 1165.75 - 95 * [(1-(1/(1.0805)^28)/0.0805] - 1000/(1.0805)^28
= 1165.75 - 95 * [(1-(1/(8.739641)/0.0805] - 1000/8.739641
= 1165.75 – 95 * (1-0.114421)/0.0805 – 1000 * 0.114421
= 1165.75 – 95 * (0.885579/0.0805) – 114.4212
= 1165.75 – 95 * 11.0098 – 114.4212
= 6.235802
r = 0.08 + (-0.01618 * (0.08-0.0805))/(6.235802-(-0.01618)
r = 0.08 + (0.00000809/6.251982)
r = 0.08 + 0.000001293 = 0.080001293 or 8%
Calculation of yield to call
Date of expiry of call = 12/31/2011
Time to expiry of call = 3 years
Call Price = $1090 (109% of Par)
Let r be yield to call
1090 = 95 * [(1-(1/(1+r)^3)/r] + 1000/(1+r)^3
1090 - 95 * [(1-(1/(1+r)^3)/r] - 1000/(1+r)^3 = 0
Let r = 7%, LHS will be
= 1090 - 95 * [(1-(1/(1.07)^3)/0.07] - 1000/(1.07)^3
= 1090 – 95 * [(1-(1/(1.225043)/0.07] - 1000/1.225043
= 1090 – 95 * [(1-0.816298)/0.07] – 1000 * 0.816298
= 1090 – 95 * (0.183702 / 0.07) – 816.2979
= 1090 – 95 *2.624316 – 816.2979
= 24.3921
Let r = 6%, LHS will be
= 1090 - 95 * [(1-(1/(1.06)^3)/0.06] - 1000/(1.06)^3
= 1090 – 95 * [(1-(1/(1.191016)/0.06] - 1000/1.191016
= 1090 – 95 * [(1-0.839619)/0.06] – 1000 * 0.839619
= 1090 – 95 * 2.673012 – 839.6193
= 1090 – 253.9361 – 839.6193
= -3.55542
r = 0.06 + [(-3.55542 * (0.06-0.07)]/(24.3921-(-3.55542))
r = 0.06 +(0.0355542/ 27.94752)
r = 0.06 + 0.00127 = 0.06127 or 6.13% (rounded off)
Calculation of actual return – held to maturity
Annual coupon Amount = 95
Time to maturity = 28 years
Total coupon amount = 95 * 28 = $ 2660
Capital loss on the bond = par value – purchse price = 1000 – 1165.75 = -165.75
Total return on the bond = $2660 - $ 165.75 = $ 2494.25 (without taking time value of money into account)
Present value of coupon payments at ytm = 95 * 11.05108 = $ 1049.85
Where 11.05108 is the PVIFA factor derived from the formula (1-(1/1.08)^28/0.08)
Present value of capital loss = $165.75/1.08^28 = $ 165.75 * 0.11591372 = -$19.21
Present value of total return on the bond = $ 1049.85 - $19.21 = $ 1030.64
If the bond is called after 3 years
Present value of coupon payments = 95 * PVIFA (6.13%, 3 years)
= 95 * 2.666596 = 253.3266 or $253.33
Present value of capital loss = (1000-1090)/(1.0613)^3 = -90/1.195403 = -75.2884
Present Value of total returns = 253.3266 – 75.2884 = $178.0382 or $ 178.04 (rounded off)
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