Give step-by-step solution with explanation and final answer:
Jupiter Inc. has decided to acquire a new weather satellite. After considering several options it has narrowed its search to two satellit

Question

Give step-by-step solution with explanation and final answer: 
Jupiter Inc. has decided to acquire a new weather satellite. After considering several options it has narrowed its search to two satellites.

Satellite XPTO: purchase cost of $334,632 and operating costs of $20,342 per year (paid at the end of each year).

Satellite XYZ: purchase cost of $127,451 and operating costs of $57,272 per year (paid at the end of each year).

Both satellites have a service life of 11 years. Based on the defender-challenger approach and given that the MARR is 3%, reinvestment rate is 9%, and minimum external rate of return is 9%, compute the incremental external rate of return of choosing the most expensive satellite. Note: round your answer to two decimal places, and do not include spaces, percentage signs, plus or minus signs, nor commas. If your answer is 15%, write 15, not 0.15.

VIPSolutions · Accepted Answer

Let's solve the problem step by step:

### **Step 1: List Data**

- **Satellite XPTO** (Challenger):
  - Purchase cost ($P_X$): \$334,632
  - Annual operating costs ($A_X$): \$20,342
- **Satellite XYZ** (Defender):
  - Purchase cost ($P_Y$): \$127,451
  - Annual operating costs ($A_Y$): \$57,272
- **Service Life ($n$)**: 11 years
- **MARR**: 3% (not directly used in IRR calculation)
- **Reinvestment/External Rate of Return ($i_{ext}$)**: 9%
- **Goal**: Compute incremental external rate of return (ERR) for XPTO over XYZ.

---

### **Step 2: Calculate Incremental Cash Flows**

**Incremental = XPTO - XYZ**

- Initial cost: $ \Delta P = P_X - P_Y = 334,632 - 127,451 = \$207,181 $
- Annual cost: $ \Delta A = A_X - A_Y = 20,342 - 57,272 = -\$36,930 $

So the incremental project is:
- Year 0: $-207,181$
- Years 1-11: $+36,930$ (since XPTO saves you \$36,930/year compared to XYZ)

---

### **Step 3: External Rate of Return Calculation**

#### **ERR Steps:**

1. **All negative cash flows (here, just Year 0) are compounded forward to the end ($n=11$) at the external rate (9%).**
2. **All positive cash flows (here, Years 1–11) are compounded forward to the end ($n=11$) at the external rate (9%).**
3. **Set sum of compounded outflows (FV out) equal to sum of compounded inflows (FV in) discounted at ERR:**
   $$
   FV_{in}(ERR) = FV_{out}(ERR)
   $$
   Solving for ERR.

But in the **external rate of return** method, the positive cash flows are compounded forward at the external rate (9%), and then the ERR is the rate that equates the FV of outflows (compounded at ERR) to the FV of inflows (compounded at external rate).

But since the only outflow is at Year 0 (initial investment), its future value at year 11 at ERR is:
$$
FV_{out} = 207,181 \times (1 + ERR)^{11}
$$

The inflows are the annual savings (\$36,930) each year for 11 years, each compounded forward at the external rate (9%) to year 11:
$$
FV_{in} = 36,930 \times \frac{(1 + 0.09)^{11} - 1}{0.09}
$$

Set $FV_{in} = FV_{out}$:
$$
36,930 \times \frac{(1.09)^{11} - 1}{0.09} = 207,181 \times (1 + ERR)^{11}
$$

---

### **Step 4: Plug in Numbers**

- $ (1.09)^{11} = $
  - Let's compute:
    - $ (1.09)^{11} = e^{11 \cdot \ln(1.09)} $
    - $ \ln(1.09) \approx 0.08618 $
    - $ 11 \cdot 0.08618 = 0.948 $
    - $ e^{0.948} \approx 2.581 $
  - Or, quickly, $ 1.09^{11} \approx 2.5804 $

So,

$$
FV_{in} = 36,930 \times \frac{2.5804 - 1}{0.09}
= 36,930 \times \frac{1.5804}{0.09}
= 36,930 \times 17.560
= \$648,262.8
$$

$$
FV_{out} = 207,181 \times (1 + ERR)^{11}
$$

Set equal:

$$
648,262.8 = 207,181 \times (1 + ERR)^{11}
$$

$$
(1 + ERR)^{11} = \frac{648,262.8}{207,181} = 3.129
$$

$$
1 + ERR = (3.129)^{1/11}
$$

Now, take natural log:

$$
\ln(1 + ERR) = \frac{1}{11} \cdot \ln(3.129)
$$
$$
\ln(3.129) \approx 1.140
$$
$$
\frac{1.140}{11} = 0.1036
$$
$$
1 + ERR = e^{0.1036} \approx 1.1091
$$
$$
ERR = 1.1091 - 1 = 0.1091
$$

Express as a %: $0.1091 \times 100 = 10.91$

---

## **Final Answer**

$$
\boxed{10.91}
$$

**So, the incremental external rate of return (ERR) is 10.91.**

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