Question

7. Company Triple A semi-annual par value bonds currently sell for $1,055. They have a 5.50% coupon rate and a 25-year maturity and are callable in 6 years at 8% premium. Assume that no costs other than the call premium would be incurred to call and refund the bonds, and also assume that the yield curve is horizontal, with rates expected to remain at current levels on into the future. Under these conditions, what rate of returns should an investor expect to earn if he or she purchases these bonds, the YTC or the YTM and why? Is this a discount or premium bond and why?

Answer 1

Let’s first calculate bond’s yield to maturity (YTM) and yield-to-call (YTC)

We have following formula for calculation of bond’s yield to maturity (YTM) for the case when it is not called (normal bond price calculation)

Bond price P0 = C* [1- 1/ (1+YTM) ^n] /YTM + M / (1+YTM) ^n

Where,

P0 = the current market price of bond = $1,055

M = value at maturity, or par value = $ 1000

C = coupon payment = 5.5% of $1000 = $55 but semiannual coupon, therefore C = $55/2 = $27.5

n = number of payments (time remaining to maturity) = 25 years; therefore number of payments n = 25 *2 = 50

YTM = interest rate, or yield to maturity =?

Now we have,

$1,055 = $27.5 * [1 – 1 / (1+YTM) ^50] /YTM+ 1000 / (1+YTM) ^50

By trial and error method we can calculate the value of YTM = 2.55% semiannual

Or annual YMT = 2 *2.55% = 5.11% per year

[Or you can use excel function for YTM calculation in following manner

“= Rate(N,PMT,PV,FV)”

“Rate(50,-27.5,1055,-1000)” = 2.55%]

The formula to calculate the bond's yield-to-call (YTC) is as follows

P = the current market price of bond = $1,055

M = value at maturity, or par value = $ 1000

C = coupon payment = 5.5% of $1000 = $55 but semiannual coupon, therefore C = $55/2 = $27.5

CP = the call price =8% premium of par value = $1000 *(1+8%) = $1,080 (assumed it as the maturity value if the bond is callable)

t = the number of years remaining until the call date = 6 years or 6 *2 = 12 semi-annual payments

YTC = the yield to call =?

The complete formula to calculate yield to call is:

P = C * {(1 – 1/ (1 + YTC) ^ t) / (YTC)} + (CP / (1 + YTC) ^t)

$1,055 = $27.5 *{(1- 1/ (1+ YTC) ^12)/ (YTC)} + ($1,080/ (1+YTC) ^12)

With the help of above equation and by trial and error method we can calculate the value of YTC = 3.18% per semiannual or 2 * 2.78% = 5.55% per year

[Or you can use excel function for YTC calculation in following manner

“= Rate(N,PMT,PV,FV)”

“Rate(12,-27.5,1055,-1080)” = 2.78%]

Now we know that

YTM = 5.11% per year

YTC = 5.55% per year

If interest rates are expected to remain constant, the best estimate for the remaining life is 25 years because the company would not call the bonds as YTM is less than the YTC.

Therefore investors expect to earn YTM if he or she purchases these bonds. This bond will be premium bond because coupon rates are more than YTM.