an annuity pays $2,000 at the end of each half-year for 5 years and then $1,000 at the end of each half-year for the next 8 years. determine the discounted value of these payments if interest is 4% with semi-annual compounding
Accepted Solution
A:
To determine the discounted value of these payments, we need to find the present value of each payment and then sum them up.
For the first 5 years, the annuity pays 2,000 at the end of each half-year. We can use the formula for the present value of an annuity:
$$PV = \frac{P(1 - (1 + r)^{-n})}{r}$$
Where: - PV is the present value - P is the payment amount - r is the interest rate per period - n is the number of periods
In this case, the payment amount is 2,000, the interest rate per period is 4% / 2 = 0.02, and the number of periods is 5 * 2 = 10.
Calculating the present value for the first 5 years:
$$PV_1=\frac{2000(1-(1+0.02)^{-10})}{0.02}$$
For the next 8 years, the annuity pays 1,000 at the end of each half-year. Using the same formula with the adjusted payment amount and number of periods:
$$PV_2=\frac{1000(1-(1+0.02)^{-16})}{0.02}$$
To find the total discounted value, we simply sum up the present values: