Practice Question
The Yotsuba Group (ヨツバグループ) pays a $320 perpetuity at the end of every 3 years starting the end of this year. In other words, the first payment is at the end of year 1, the second payment at the end of year 4, and so on. Note that the discount rate is 7.8860% compounded 12 times per year. What is the present value of the Yotsuba perpetuity?
Answer +
Final Answer: $1,408.17
Explanation +
Step 1: Highlight Key Information
- PMT: $320 every 3 years
- First payment at end of year 1
- Discount rate: 7.8860% compounded monthly
Step 2: Convert Quoted Rate to Effective Annual Rate (EAR)
\[ EAR = \left(1 + \frac{0.078860}{12} \right)^{12} - 1 \approx 0.081774 \text{ or } 8.1774\% \]Step 3: Convert EAR to Effective 3-Year Rate (EPR)
\[ EPR = (1 + 0.081774)^3 - 1 \approx 0.265930 \text{ or } 26.5930\% \]Step 4: Apply Perpetuity Formula (Value at t = -2)
\[ PV_{-2} = \frac{PMT}{EPR} = \frac{320}{0.26593} \approx 1,203.33 \]Step 5: Compound Forward 2 Years to Get PV at t = 0
\[ PV_0 = PV_{-2} \cdot (1 + EAR)^2 = 1,203.33 \cdot (1.081774)^2 \approx 1,408.17 \]Conclusion: The present value of this grouped perpetuity starting at year 1 is $1,408.17.