SOA Life Insurance Life Insurance Company has a portfolio of two bonds:

• Bond 1 is a bond with a Macaulay duration of 7.28 and a price of 35,000; and
• Bond 2 is a bond with a Macaulay duration of 12.74 and a price of 65,000

The price and Macaulay duration for both bonds were calculated using an annual effective interest rate of 4.32%. Bailey estimates the value of this portfolio at an interest rate of i using the first-order Macaulay approximation to be 105,000.

Determine i.

• 3.49%
• 3.62%
• 3.85%
• 3.92%
• 4.03%

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

Solution: C

The Macaulay duration of the portfolio is

[[math]]\frac{35, 000(7.28) + 65, 000(12.74)}{35, 000 + 65, 000} = 10.829.[[/math]]

Then

[[math]] 105,000=100,000{\left({\frac{1.0432}{1+i}}\right)}^{1.0432}\Rightarrow{\frac{1.0432}{1+i}}=\left({\frac{105,000}{100,000}}\right)^{1.029}=1.004516\Rightarrow i=0.0385. [[/math]]

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