Practice Question
Your friends, Lee and Kuan, are struggling to solve for the missing cash flows of the following stream. They believe "x" must be $55.87.
The present value is $6,253.75, and the discount rate is 7.2441% compounded 6 times per year.
The stream includes:
- A normal annuity of $320 at the end of years 1 and 2
- A growing annuity starting at end of year 3 to year 6 with growth rate 5%
- A perpetuity beginning at end of year 7
Step 1: Convert Quoted Rate to Effective Annual Rate
\[ EAR = \left(1 + \frac{0.072441}{6}\right)^6 - 1 \approx 0.074663 \]
Step 2: Present Value of Normal Annuity
\[ PV_{normal} = \frac{320}{(1.074663)^1} + \frac{320}{(1.074663)^2} \approx 574.85 \]
Step 3: Present Value of Growing Annuity (Years 3–6)
\[ PV_2 = \frac{1108}{0.074663 - 0.05} \left[1 - \left(\frac{1.05}{1.074663}\right)^4\right] \approx 3,984.27 \] \[ PV_0 = \frac{3984.27}{(1.074663)^2} \approx 3449.88 \]
Step 4: Present Value of Perpetuity (Begins Year 7)
\[ PV_6 = \frac{2.5x}{0.074663}, \quad PV_0 = \frac{2.5x}{0.074663 \cdot (1.074663)^6} \approx \frac{2.5x}{11.5012} \approx 21.7369x \]
Step 5: Solve for x using total PV
\[ 6253.75 = 574.85 + 3449.88 + 21.7369x \] \[ 2229.02 = 21.7369x \Rightarrow x \approx \frac{2229.02}{21.7369} \approx 55.87 \]
Conclusion: Lee and Kuan are correct. The missing value of "x" is $55.87.