Today you purchase a $1,000 par value convertible bond of Bunky’s Burgers. The b
ID: 2651633 • Letter: T
Question
Today you purchase a $1,000 par value convertible bond of Bunky’s Burgers. The bond matures in 30 years and has an annual coupon of 12%, payable semiannually. The yield to the maturity on the bond is 10% a year, compounded semiannually. The bond is convertible into Bunky’s common stock at a conversion price of $100 a share. You forecast that the earnings and dividends of Bunky’s will grow at annual rates of 30% for the next 5 years and then 20% for another 5 years before settling at a 6% growth rate for the indefinite future. Yesterday the firm paid a dividend (D0) of $2.72. Stockholders require a return of 18% on stocks in Bunky’s risk class.
{a} You hold the bond for 7 years and then convert it. Assume that all your forecasts hold. What IRR did you earn over the 7 year period?
{b} If you hold the bond for 30 years and convert it the day it matures, what rate of return did you earn if all the forecasts come true? (Assume that you do receive the final coupon payment).
Explanation / Answer
Part A)
To calculate the annual rate of return, we will have to calculate the conversion value after 7 years and price of the bond today. The price of the bond today can be calculated with the use of PV function/formula of EXCEL/Financial Calculator. The function/formula for EXCEL/Financial Calculator is PV(Rate,Nper,PMT,FV) where Rate = Yield to Maturity, Nper = Period, PMT = Interest Payment and FV = Face Value
To calculate, the conversion value after 7 years, we need to calculate the stock price at the end of Year 7. The conversion value will be calculated by multiplying this stock price with the conversion ratio.
The internal rate of return can be calculated with the use of Rate function/formula of EXCEL/Financial Calculator. The function/formula for Rate is Rate(Nper,PMT,PV,FV) where Nper = Period, PMT = Interest Payment, PV = Present Value and FV = Face/Conversion Value
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Price of the Bond Today:
Here, Rate = 10%/2 = 5%, Nper = 30*2 = 60, PMT = 1,000*12%*1/2 = $60 and FV = $1,000 [we use 2, since the bond is semi-annual]
Using these values in the above function/formula for PV, we get,
Price of Bond Today = PV(5%,60,60,1000) = $1,189.29
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Price of the Stock at the End of 7 Years can be calculated with the use of following formula:
Price at the end of 7 Years = D8/(1+Required Return)^1 + D9/(1+Required Return)^2 + D10/(1+Required Return)^3 + P10/(1+Required Return)^3*(Required Return - Growth Rate) where D8 = Year 8 Dividend, D9 = Year 9 Dividend, D10 = Year 10 Dividend and P10 = Price at the End of Year 10
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Using the information provided in the question, we get,
Price at the End of 7 Years = 2.72*(1+30%)^5*(1+20%)^3/(1+18%)^1 + 2.72*(1+30%)^5*(1+20%)^4/(1+18%)^2 + 2.72*(1+30%)^5*(1+20%)^5/(1+18%)^3 + 2.72*(1+30%)^5*(1+20%)^5*(1+6%)/(1+18%)^3*(18% - 6%) = $180.23
Conversion Ratio = Face Value of the Bond/Conversion Rate = 1,000/100 = 10 shares for 1 bond
Total Conversion Value after 7 Years = Price at the End of 7 Years*Number of Shares = 180.23*10 = $1802.30
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Here, Nper = 7*2 = 14, PMT = 1,000*12%*1/2 = $60, PV = $1,189.29 and FV = $1,802.30 (same as conversion value)
Using these values in the function/formula for Rate, we get,
Internal Rate of Return = Rate(14,60,-1189.29,1802.30)*2 = 14.57% (answer for Part A)
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Part B)
The internal rate of return can be calculated with the use of Rate function/formula of EXCEL/Financial Calculator. The function/formula for Rate is Rate(Nper,PMT,PV,FV) where Nper = Period, PMT = Interest Payment, PV = Present Value and FV = Face/Conversion Value
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We will have to calculate the conversion value at the end of 30 years. For that, we need to determine the price at the end of 30 years with the use of following equation:
Price at the end of 30 Years = 2.72*(1+30%)^5*(1+20%)^5*(1+6%)^21/(18% - 6%) = $711.92
Conversion Value at the End of 30 Years = 711.92*10 = $7,119.20
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Here, Nper = 30*2 = 60, PMT = 1,000*12%*1/2 = $60, PV = $1,189.29 and FV = $7,119.20 (same as conversion value)
Using these values in the function/formula for Rate, we get,
Internal Rate of Return = Rate(60,60,-1189.29,7119.20)*2 = 11.97% (there can be a slight variation in the answer as a result of rounding off)