(a) Calculate the perpetual equivalent annual worth in future dollars for years 1 through ∞ for income of $50,000 now and $5000 per year thereafter. Assume the market interest rate is 8% per year and inflation averages 4% per year. All amounts are quoted as future dollars. (b) If the amounts had been quoted in CV dollars, what is the annual worth in future dollars?
The Correct Answer and Explanation is :
To solve this problem, we’ll break it down into two parts:
Part (a) – Perpetual Equivalent Annual Worth in Future Dollars
We are given:
- An income of $50,000 now (this is a lump sum at time 0),
- A recurring income of $5,000 per year starting from year 1 and continuing perpetually,
- An interest rate of 8% per year,
- Inflation rate of 4% per year.
We are required to calculate the perpetual equivalent annual worth (in future dollars) for years 1 through ∞.
- Adjust for Inflation:
We need to account for the fact that future cash flows are quoted in future dollars, meaning that the amounts will increase due to inflation. To calculate the future worth of $50,000 and $5,000 at future times, we will first adjust them for inflation. - Present Worth of Perpetual Income:
To calculate the present worth of a perpetuity (the $5,000 per year), we use the formula for the present value of a perpetuity: [
PV_{\text{perpetuity}} = \frac{C}{r – i}
] Where:
- (C) is the annual cash flow ($5,000),
- (r) is the nominal interest rate (8% or 0.08),
- (i) is the inflation rate (4% or 0.04). Substituting the values: [
PV_{\text{perpetuity}} = \frac{5000}{0.08 – 0.04} = \frac{5000}{0.04} = 125,000
] This is the present value of the perpetuity in future dollars. Now we will adjust this value for inflation to get the future value.
- Future Worth of Perpetuity:
Since we are interested in future dollars, the perpetuity’s future worth (in year 1) can be calculated by adjusting the present value using the inflation rate: [
FV_{\text{perpetuity}} = PV_{\text{perpetuity}} \times (1 + i) = 125,000 \times (1 + 0.04) = 130,000
] So, the future value of the perpetual cash flow is $130,000 per year. - Future Worth of $50,000 Initial Lump Sum:
The lump sum of $50,000 is received now, and we need to calculate its future value in year 1 (as a future lump sum) using the market interest rate. The future value of the lump sum is: [
FV_{\text{lump sum}} = 50,000 \times (1 + 0.08) = 54,000
] So, the future worth of the lump sum in year 1 is $54,000. - Total Future Worth:
The total future worth, combining both the lump sum and the perpetuity, is: [
FV_{\text{total}} = 54,000 + 130,000 = 184,000
]
Thus, the perpetual equivalent annual worth in future dollars for years 1 through ∞ is $184,000.
Part (b) – Annual Worth in CV Dollars (Constant Value Dollars)
In CV dollars, we need to adjust the future worths to account for inflation.
- Present Worth in CV Dollars:
The lump sum of $50,000 is quoted in constant value dollars, so it doesn’t require any adjustments for inflation. However, the perpetuity requires us to use the inflation-adjusted interest rate to find its present value in CV dollars. - Adjusted Perpetuity in CV Dollars:
The formula for the present value of a perpetuity in CV dollars is: [
PV_{\text{perpetuity (CV)}} = \frac{C}{r} = \frac{5000}{0.08} = 62,500
] This is the present value of the perpetuity in CV dollars. - Total Present Worth in CV Dollars:
The total present value in CV dollars (sum of the lump sum and perpetuity in CV dollars) is: [
PV_{\text{total (CV)}} = 50,000 + 62,500 = 112,500
] - Convert to Future Dollars:
To convert this amount to future dollars, we multiply by the inflation factor: [
FV_{\text{total (CV)}} = 112,500 \times (1 + 0.04) = 117,000
]
Thus, the annual worth in future dollars, when quoted in constant value (CV) dollars, is $117,000.
Conclusion:
- The perpetual equivalent annual worth in future dollars (for years 1 through ∞) is $184,000.
- The annual worth in future dollars, when quoted in CV dollars, is $117,000.
In essence, the future value of the lump sum and perpetuity is greater in future dollars due to inflation adjustments, whereas the constant value dollars reflect the current value of money.