A bank issues three annual coupon bonds redeemable at par, all with the same term, price, and annual effective yield rate. The first bond has face value 1000 and annual coupon rate 5.28%. The second bond has face value 1100 and annual coupon rate 4.40%. The third bond has face value 1320 and annual coupon rate r.

Calculate r

• 2.46%
• 2.93%
• 3.52%
• 3.67%
• 4.00%

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

Solution: B

The present value equation for a par-valued annual coupon bond is [math]P = Fv^n + Fra_{\overline{n}|i}[/math]; solving for the coupon rate r yields

[[math]] r={\frac{P-F\nu_{i}^{\ n}}{F a_{\overline{{{n}}}|i}}}={\frac{P}{a_{\overline{{{n}}}|i}}}{\left(\frac{1}{F}\right)-{\frac{\nu_{i}^{\ n}}{a_{\overline{{{n}}}|i}}}}. [[/math]]

From the first two bonds: 0.0528 = x/1000 + y and 0.0440 = x/1100 + y. Then, 0.0528 – 0.044 = x(1/1000 – 1/1100) for x = 96.8 and y = 0.0528 – 96.8/1000 = –0.044. For the third bond, r = 96.8/1320 – 0.044 = 0.2933 = 2.93%.

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