Q:

# 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:

$$\text{Total PV} = PV_1 + PV_2$$

Calculating the present values:

$$PV_1=\frac{2000(1-(1+0.02)^{-10})}{0.04}$$
$$PV_2=\frac{1000(1-(1+0.02)^{-16})}{0.04}$$

Simplifying the equations and solving:

$$PV_1=17965.17$$
$$PV_2=13577.71$$

Now, we can find the total discounted value:

$$\text{Total PV} = PV_1 + PV_2$$
$$\text{Total PV}=17965.17+\frac{13577.71}{\left(1.02\right)^{16}}$$
$$\text{Total PV}=27855.80$$

Answer: The discounted value of these payments is 27855.80 dollars.