An n-year bond with semiannual coupons has the following characteristics:

• The par value and redemption value are 2500;
• The annual coupon rate is 7% payable semi-annually;
• The annual nominal yield to maturity is 8% convertible semiannually; and
• The book value immediately after the fourth coupon is 8.44 greater than the book value immediately after the third coupon.

Calculate n.

• 6.5
• 7.0
• 9.5
• 12.0
• 14.0

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

Book values are linked by BV3(1 + i) – Fr = BV4. Thus BV3(1.04) – 2500(0.035) = BV3 + 8.44. Therefore, BV3 = [2500(0.035) + 8.44]/0.04 = 2398.5. The prospective formula for the book value at time 3 is, where m is the number of six-month periods.

[[math]] \begin{array}{l}{{2398.5=2500(0.035){\frac{1-1.04^{-(m-3)}}{0.04}}+2500(1.04)^{-(m-3)}}}\\ {{\mathrm{211=312.5(1.04)^{-(m-3)}}+2500(1.04)^{-(m-3)}}}\\ {{m-3={\frac{\ln(211/312.5)}{-\ln(1.04)}}=10.}}\end{array} [[/math]]

Thus, m = 13 and n = m/2 = 6.5. Note that the financial calculator can be used to solve for m – 3.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.