What is the present value of a four-year annuity of $100 per year that makes its first payment 2 years from today if the discount rate is 9%?

1. Break Down the Problem

We want to find the present value (PV) of an annuity that pays $100 per year for 4 years, starting 2 years from today, with a discount rate of 9%.

2. Relevant Concepts

The present value of an annuity can be calculated using the formula:

$PV = P \times \left(1 - (1 + r)^{-n}\right) / r$

Where:

• $P$ = payment per period ($100)
• $r$ = discount rate (0.09)
• $n$ = number of payments (4)

Since the first payment starts in 2 years, we need to discount the present value of the annuity back to today.

3. Analysis and Detail

Step 1: Calculate the present value of the annuity.

Using the formula:

$PV_{\text{annuity}} = 100 \times \left(1 - (1 + 0.09)^{-4}\right) / 0.09$

Calculating this step-by-step:

$1 + 0.09 = 1.09$

$(1.09)^{-4} \approx 0.42241$

$PV_{\text{annuity}} = 100 \times \left(1 - 0.42241\right) / 0.09$

$= 100 \times \left(0.57759\right) / 0.09$

$= 100 \times 6.41767 \approx 641.77$

Step 2: Discount this present value back to today, which is 2 years prior.

Using the formula:

$PV_{\text{today}} = PV_{\text{annuity}} / (1 + r)^{t}$

Where $t = 2$:

$PV_{\text{today}} = 641.77 / (1.09)^{2}$

Calculating $(1.09)^{2}$:

$(1.09)^{2} \approx 1.1881$

Now find $PV_{\text{today}}$:

$PV_{\text{today}} = 641.77 / 1.1881 \approx 540.55$

4. Verify and Summarize

After performing the calculations, we find that the present value of the four-year annuity starting 2 years from now, discounted at a rate of 9%, is approximately $540.55.

Final Answer

The present value of the annuity is approximately $540.55.