An investment offers to pay $100 a year forever starting at the end of year 6. If the interest rate is 8%, what is the investment's value today?

1. Break Down the Problem

To find the present value of an investment that pays $100 annually forever starting at the end of year 6, we will:

1. Calculate the present value at year 5 (the time just before the first payment).
2. Discount that present value back to today (year 0).

2. Relevant Concepts

The present value of a perpetuity can be calculated using the formula: $PV = \frac{C}{r}$ where:

• $PV$ = Present Value
• $C$ = Cash flow per period
• $r$ = Interest rate

3. Analysis and Detail

1. Calculate the Present Value at Year 5:

   • The cash flow $C = 100$
   • The interest rate $r = 0.08$

   Substituting into the perpetuity formula: $PV_{5} = \frac{100}{0.08} = 1250$

2. Discount the Present Value Back to Today:

   • The present value we calculated is at year 5, so we need to discount it to year 0 using the formula: $PV_{0} = \frac{PV_{5}}{(1+r)^n}$ where $n = 5$ (the number of years between year 0 and year 5).

   Substituting into the formula: $PV_{0} = \frac{1250}{(1 + 0.08)^5} = \frac{1250}{1.4693} \approx 852.31$

4. Verify and Summarize

• We calculated the present value at year 5 as $1250 and discounted it back to today (year 0) to get approximately$852.31.
• All calculations have been verified and are consistent with the principles of present value and discounting.

Final Answer

The investment's value today is approximately $852.31.