Question

(2) On January 1, 1988, Wanda received a deferred perpetuity paying $3,000 on July I of even numbered years beginning on July l, 1996 and $1,200 on July 1 of odd years beginning on July l, 1997. The interest rate is 4% effective for even numbered years and 5% effective for odd numbered years. (That is, the interest rate is 4% effective for 1988, 1990, 1992, ... and 5% effective for 1989, 1991, 1993, . . . . ) Find the value of this perpetuity on January I, 1988.

Answer 1

Answer:

$33,954.85

Explanation:

First let's divide the issue into the number of two perpetuities per two years with payments.  

I consider the two year interest rate to do this,

Let July be j.

1 + j = (1 + 0.04)(1 + 0.05) j

= (1.04)(1.05) - 1 j

= 0.092

The value of the $3000 payments one period (2 years) before the first payment is 3000 / j = 3000/0.092 = 32,608.6956521739. This is the value on July 1, 1994.

To obtain the value January 1, 1988 discount the following way 32,608.6956521739(1 + j)^-3(1.04)^-.5

= 32,608.6956521739(1.092)^-3(1.04)^-.5

= 24555.5034456

Similarly, the value of the $1200 payments is 1200/j

= 1200/0.092

= 13043.4782608696 (Value on July 1, 1995)

Again discounting back to January 1, 1987 13043.4782608696(1.092)^-3(1.05)^-.5(1.04)^-1

= 9399.34336004 24555.5034456 + 9399.34336004

= 33954.8468056 = $33,954.85