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VIPSolutions · Accepted Answer

## Problem's Objective

**Find the annualized return (IRR) for the following investment:**
- Initial investment: $1,000
- Annual distribution: $60 (for 7 years)
- Sale price after 7 years: $775

---

## Step-by-Step Solution

### Step 1: Restate Cash Flows

- **Year :** -$1,000 (initial outlay)
- **Year 1 to 7:** +$60 per year (annual distributions)
- **Year 7:** +$775 (sale of shares at the end of Year 7)

So, Year 7's cash flow = $60 (distribution) + $775 (sale) = **$835**

#### List of Cash Flows:
- Year : -1,000
- Year 1: +60
- Year 2: +60
- Year 3: +60
- Year 4: +60
- Year 5: +60
 Year 6: +60
- Year 7: +835

---

### Step 2: Write the IRR Equation

The IRR ($r$) satisfies:

$$
 = -100 + \frac{60}{(1+r)^1} + \frac{60}{(1+r)^2} + \frac{60}{(1+r)^3} + \frac{60}{(1+r)^4} + \frac{60}{(1+r)^5} + \frac{60}{(1+r)^6} + \frac{835}{(1+r)^7}
$$

---

### Step 3: Guess and Check (Trial and Error with Present Value Table)

Let's try different rates until the Present Value (PV) of inflows equals the outflow ($1,000):

#### Try r = 6%

PVIF (Present Value Interest Factor) for 6%:

$$
\text{PV of $60$ annuity for 7 years at 6%:}
$$
$$
\text{PV} = 60 \times \left[\frac{1 - (1 + .06)^{-7}}{.06}\right]
$$
$$
= 60 \times \left[\frac{1 - (1.5036)^{-1}}{.06}\right]
$$
$$
= 60 \times \left[\frac{1 - .665}{.06}\right]
$$
$$
= 60 \times \left[\frac{.335}{.06}\right] = 60 \times 5.583 = 335
$$

But let's use the PVIF from tables for accuracy:
- PVIFA (6%, 7) ≈ 5.582
- PVIF (6%, 7) ≈ .665

$$
\text{Total PV} = 60 \times 5.582 + 775 \times .665
$$
$$
= 334.92 + 515.88 = 850.80
$$

This is too low. Try a lower rate.

---

#### Try r = 4%

- PVIFA (4%, 7) ≈ 6.003
- PVIF (4%, 7) ≈ .760

$$
\text{Total PV} = 60 \times 6.003 + 775 \times .760
$$
$$
= 360.18 + 589.00 = 949.18
$$

Still below $1,000. Try a lower rate.

---

#### Try r = 2%

- PVIFA (2%, 7) ≈ 6.573
- PVIF (2%, 7) ≈ .873

$$
\text{Total PV} = 60 \times 6.573 + 775 \times .873
$$
$$
= 394.38 + 677.58 = 1,071.96
$$

Now it's over $1,000. So IRR is between 2% and 4%.

---

### Step 4: Linear Interpolation

$$
\text{IRR} \approx 2\% + \frac{1,071.96 - 1,000}{1,071.96 - 949.18} \times (4\% - 2\%)
$$
$$
= 2\% + \frac{71.96}{122.78} \times 2\%
$$
$$
= 2\% + 1.17\% = 3.17\%
$$

---

## Final Answer

**The annualized return (IRR) on your investment is approximately:**

$$
\boxed{3\%}
$$

(Rounded to the nearest whole number, as instructed.)

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