Question

You purchased a bond with the following characteristics:
$1,000 par value 6.5% coupon, annual payments
25 years to maturity Callable in 7 years at $1,065.
You paid $1063.92 for the bond. Macaulay duration is 13.34 years
a. (5 pts) Calculate the yield to maturity.
b. (5 pts) Calculate the yield to call.
c. (5 pts) Assume market rates drop by one-half of one percent, what will be the new bond price?
d. (5 pts) Using modified duration, estimate the value of the bond following the drop in interest rates.
e. (5 pts) The estimate (from part d), is fairly close to the actual (in part c). What explains the difference in the two values? Be specific.
f. (5 pts) Of the three duration measures (Macaulay, modified, effective) which is the most appropriate measure for this bond? Why?

Answer 1

a. Given: FV=1000, PV=1063.92, PMT=Coupon=6.5%*1000=65, N=25, Compute I/Y

Using financial calculator we get, I/Y = 5.999975% = 6%

So Yield to Maturity = YTM = 6%

b. Yield to Call - Excercisable at 7 years at price of 1065, so we use FV=1065 & N=7

Using financial calculator,

FV=1065, PV=1063.92, PMT=Coupon=6.5%*1000=65, N=7, Compute I/Y

I/Y=6.1215%

So Yield to Call = YTC = 6.1215%

c. Originally YTM = 5.9999%.

{note we calculated YTM of 6% in part a}

If market rate drops by 0.5%, new YTM for the bond will be 5.9999% - 0.5% = 5.4999%

Using financial Calculator,

Given: FV=1000, PMT = Coupon = 6.5%*1000 = 65, N=25, I/Y=5.49999%, Compute PV

PV = -1134.1408

Hence, New price of the bond = $1134.1408

d.

Modified duration = (Maculay Duration)/[1+ { ytm / (number of copoun period in a year) } ]

=13.34/[1+(5.9999%/1)] = 12.58491753

% change in bond = -(Modified Duration) * (yield change)

So if interest yield lowers by 0.5%:

% change in bond = -12.58491753 * -0.5% = 0.06292458766

New bond price = Old bond price (1 + % change in bond)

=1063.92 * 1.06292458766 = 1130.866727

{Modified duration explains effect of parallel shift in the curve. Changes in Steepness and Curvature (convexity) is unexplained which leads to the difference. Effective duration would be a better measure for a bond with embedded option as it takes into account that the future interest changes will impact callable bond}