An insurance company says you can pay for your life insurance by paying $100 at the beginning of each year or $51.50 at the beginning of each half-year. They say the rate of interest underlying this calculation is j2=3%. What is the true value of j2?
The answer in the textbook says j2=12.37%
This is what I have tried:
j=(1+0.03/2)^(2) - 1 = 0.0302% ....j2=6.05%
To find the true value of j2, we can set up an equation based on the given information. Let's start by calculating the present value of paying $100 at the beginning of each year.
Using the formula for present value of an ordinary annuity, we have:
PV1 = (PMT/r) * (1 - (1 + r)^(-n))
= (100/0.03) * (1 - (1 + 0.03)^(-1))
= 100/0.03 * (1 - 1.0303^(-1))
= 3333.33 * (1 - 0.9709)
= 3333.33 * 0.0291
= 96.63
Now, let's calculate the present value of paying $51.50 at the beginning of each half-year.
Using the same formula, we have:
PV2 = (PMT/r) * (1 - (1 + r)^(-n))
= (51.5/0.03) * (1 - (1 + 0.03)^(-2))
= 1716.67 * (1 - 1.0609^(-1))
= 1716.67 * 0.0112
= 19.23
Now, we can set up an equation to find the true value of j2:
PV1 = PV2
96.63 = 19.23 * (1 + j2/2)^(-2)
Rearranging the equation, we have:
(1 + j2/2)^(-2) = 96.63/19.23
(1 + j2/2)^(-2) = 5.023
Taking the square root of both sides to eliminate the exponent, we get:
1 + j2/2 = sqrt(5.023)
j2/2 = sqrt(5.023) - 1
j2 = 2 * (sqrt(5.023) - 1)
Evaluating this expression, we find:
j2 ≈ 12.37%
Therefore, the true value of j2 is approximately 12.37%, which matches the answer given in the textbook.