Given an interest rate of 7.15 percent per year, what is the value at year t = 8 of a perpetual stream of $3,468 payments that begin at year t = 20
The correct answer and explanation is:
To determine the value at year t = 8 of a perpetual stream of $3,468 payments that begins at year t = 20, with an interest rate of 7.15% per year, we follow these steps:
Step 1: Present value of a perpetuity at year 19
The perpetuity starts at t = 20, so its value at t = 19 is: PV at t = 19=Cr=3,4680.0715=48,487.41\text{PV at t = 19} = \frac{C}{r} = \frac{3,468}{0.0715} = 48,487.41
Step 2: Discount this amount back to year t = 8
We now discount the value at year 19 back to year 8: PV at t = 8=48,487.41(1+0.0715)11≈48,487.412.1052≈23,038.23\text{PV at t = 8} = \frac{48,487.41}{(1 + 0.0715)^{11}} \approx \frac{48,487.41}{2.1052} \approx 23,038.23
✅ Final Answer: $23,038.23
Explanation (300 words)
To find the value at year 8 of a perpetuity starting at year 20, we use the time value of money, a core principle in finance stating that money today is worth more than the same amount in the future due to its earning potential.
A perpetuity is a stream of equal payments that continue forever. The present value (PV) of a perpetuity is calculated using the formula: PV=CrPV = \frac{C}{r}
where:
- CC is the annual cash flow ($3,468)
- rr is the annual interest rate (7.15% or 0.0715)
Since the perpetuity begins in year 20, its present value is calculated as of year 19, right before the first payment. That gives us: PV19=3,4680.0715=48,487.41PV_{19} = \frac{3,468}{0.0715} = 48,487.41
To determine the value of this perpetuity from the perspective of year 8, we need to discount this amount back 11 years (from t = 19 to t = 8) using the compound interest formula: PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
Substituting the known values: PV8=48,487.41(1.0715)11≈48,487.412.1052=23,038.23PV_{8} = \frac{48,487.41}{(1.0715)^{11}} \approx \frac{48,487.41}{2.1052} = 23,038.23
So, the value at year 8 of the perpetual stream starting in year 20 is $23,038.23. This approach correctly accounts for both the delay in cash flow and the discounting over time.