Aproperty is expected to have NOI of $103,000 the first year. The NOI is expected to
increase by § percent per year thereafter. The appraised value of the property is currently $
1.30 million and th

Question

Aproperty is expected to have NOI of $103,000 the first year. The NOI is expected to
increase by § percent per year thereafter. The appraised value of the property is currently $
1.30 million and the lender is willing to make a $1,170,000 participation loan with a
contract interest rate of 5.5 percent. The loan will be amortized with monthly payments
over a 20 - year term. In addition to the regular mortgage payments, the lender will receive
50 percent of the NOI in excess of $103,000 each year until the loan is repaid. The lender
also will receive 50 percent of any increase in the value of the property. The loan includes a
substantial prepayment penalty for repayment before year 5, and the balance of the loan
is due in year 10. (If the property has not been sold, the participation wil be based on the
appraised value of the property.) Assume that the appraiser would estimate the value in
year 10 by dividing the NOI for year 11 by an 8 percent capitalization rate.

Assume that another alternative is a convertible mortgage (instead of a participation loan)
that gives the lender the option to convert the mortgage balance into a 60 percent equity
position at the end of ye

VIPSolutions · Accepted Answer

Let's break down the **convertible mortgage** scenario and address the questions step by step.

---

### Key Points

- **Loan Amount:** $1,170,000
- **Interest Rate:** 6.5%
- **Amortization:** 20 years (monthly)
- **At year 10:** Lender can convert to 60% equity **if** property value > loan balance.
- **If property value < loan balance in year 10:** Borrower defaults, lender gets property.

### 1. **Calculate Monthly Payment**

Formula for monthly payment (PMT) for amortizing loan:
$$
PMT = \frac{P \cdot r(1+r)^n}{(1+r)^n-1}
$$
Where:
- $ P = \$1,170,000 $
- $ r = \frac{6.5\%}{12} = 0.54167\% = 0.0054167 $
- $ n = 20 \times 12 = 240 $

$$
PMT = \frac{1,170,000 \times 0.0054167 \times (1+0.0054167)^{240}}{(1+0.0054167)^{240} - 1}
$$

$$
(1+0.0054167)^{240} \approx 3.6386
$$
So,

$$
PMT = \frac{1,170,000 \times 0.0054167 \times 3.6386}{3.6386 - 1}
= \frac{1,170,000 \times 0.019706}{2.6386}
= \frac{23,060.02}{2.6386}
= \$8,742.50\ (\text{monthly})
$$

### 2. **Find Loan Balance at Year 10**

Remaining balance after 10 years (120 payments):

$$
B = P \times \frac{(1+r)^n - (1+r)^p}{(1+r)^n-1}
$$
Where $ p = 120 $:

$$
B = 1,170,000 \times \frac{3.6386 - 1.8957}{2.6386}
= 1,170,000 \times \frac{1.7429}{2.6386}
= 1,170,000 \times 0.6607
= \$772,019
$$

### 3. **Calculate Property Value in Year 10 (for Q.a)**

Assume NOI grows 5% per year:

$$
NOI_{11} = 103,000 \times 1.05^{10} = 103,000 \times 1.6289 = \$167,776
$$
$$
Property\ Value_{10} = \frac{NOI_{11}}{0.08} = \frac{167,776}{0.08} = \$2,097,200
$$

---

## a. **Lender’s IRR if property sells for $2,097,200 in year 10**

- **Lender receives:** 60% of $2,097,200 = $1,258,320
- **Lender's cash flows:**
  - Outflow: $1,170,000 (Year 0)
  - Inflow: 120 monthly payments of $8,742.50 (Years 1-10)
  - Inflow: $1,258,320 (Year 10 equity conversion)

You'd use Excel/financial calculator for exact IRR (since monthly PMTs), but here's the outline:

1. Cash Flows (monthly for 120 months): +$8,742.50
2. Final Cash Flow (month 120): +$1,258,320
3. Initial Outflow (month 0): -$1,170,000

**In Excel:**
$$
\text{IRR} = \text{IRR}([-1,170,000, 8,742.50, 8,742.50, ..., 8,742.50 + 1,258,320])
$$
- The last payment month 120: $8,742.50 + $1,258,320 = $1,267,062.50

---

## b. **Lender’s IRR if property sells for only $1,175,000 after 10 years**

- **Lender receives:** 60% of $1,175,000 = $705,000
- **Cash flows as above, but final inflow is $705,000 instead of $1,258,320.**
- So, last payment is $8,742.50 + $705,000 = $713,742.50

---

## c. **Lender’s IRR if property sells for only $670,000 after 10 years**

- **Loan balance is $772,019**, which is more than the property value.
- **Borrower defaults, lender gets property:** Lender receives $670,000 (property value at year 10).
- Final cash flow is $8,742.50 + $670,000 = $678,742.50

---

### **Summary Table for IRR Calculation**
You'd set up the following cash flows:

| Month | Cash Flow (a) | Cash Flow (b) | Cash Flow (c) |
|-------|---------------|---------------|---------------|
| 0     | -1,170,000    | -1,170,000    | -1,170,000    |
| 1-119 | +8,742.50     | +8,742.50     | +8,742.50     |
| 120   | +1,267,062.50 | +713,742.50   | +678,742.50   |

**Use the IRR function in Excel/Calculator for actual percentages.**

---

### **If You Want a Quick Estimate (Annualized):**
- Use NPV/IRR functions, but since payments are monthly, adjust for 12 periods per year.

Would you like help with the Excel formula or a hand calculation for one scenario?

VIPSolutions · Answer

Certainly! Let's go through each part step-by-step with detailed calculations, focusing on the **convertible mortgage** scenario and addressing the three questions about the IRR under different sale prices after 10 years.

---

### **Given Data Recap**

- **Loan amount:** \$1,170,000
- **Interest rate:** 6.5% annually
- **Amortization period:** 20 years (240 months)
- **Monthly interest rate (r):** 6.5% / 12 = **0.54167% = 0.0054167**
- **Payments per year:** 12
- **Total payments:** 240
- **At year 10 (after 120 payments):**
  - **Remaining balance:** as calculated below
  - **Property value (based on NOI growth):** \$2,097,200
- **Sale prices after 10 years:**
  - **Q.a:** \$2,097,200
  - **Q.b:** \$1,175,000
  - **Q.c:** \$670,000

---

## Step 1: Calculate Monthly Mortgage Payment

Using the standard mortgage formula:

$$
PMT = P \times \frac{r (1 + r)^n}{(1 + r)^n - 1}
$$

Where:
- $ P = \$1,170,000 $
- $ r = 0.0054167 $
- $ n = 240 $

Calculate $(1 + r)^n$:

$$
(1 + 0.0054167)^{240} \approx e^{240 \times \ln(1.0054167)} \approx e^{240 \times 0.005404} \approx e^{1.297} \approx 3.658
$$

*(Close approximation)*

Now, compute $PMT$:

$$
PMT = 1,170,000 \times \frac{0.0054167 \times 3.658}{3.658 - 1} = 1,170,000 \times \frac{0.0198}{2.658} \approx 1,170,000 \times 0.007456
$$

$$
PMT \approx \$8,742.50
$$

**Thus, monthly payment is approximately \$8,742.50.**

---

## Step 2: Calculate Loan Balance After 10 Years (120 Payments)

Remaining balance after $p$ payments is:

$$
B_p = P \times \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1}
$$

Where:
- $ n = 240 $
- $ p = 120 $

Calculate $(1 + r)^{120}$:

$$
(1 + 0.0054167)^{120} \approx e^{120 \times 0.005404} \approx e^{0.648} \approx 1.912
$$

Now, substitute:

$$
B_{120} = 1,170,000 \times \frac{3.658 - 1.912}{3.658 - 1} = 1,170,000 \times \frac{1.746}{2.658} \approx 1,170,000 \times 0.657
$$

$$
B_{120} \approx \$769,200
$$

**Note:** Slight differences from earlier approximation (which was about \$772,019) are due to rounding; for precise calculations, use exact calculator or Excel.

---

## Step 3: Calculate Property Value at Year 10

- **NOI growth:** 5% annually
- **NOI in year 11:**

$$
NOI_{11} = 103,000 \times (1.05)^{10} \approx 103,000 \times 1.6289 \approx \$167,776
$$

- **Appraised property value in year 10:**

$$
Value_{10} = \frac{NOI_{11}}{0.08} = \frac{167,776}{0.08} \approx \$2,097,200
$$

---

## **Answer to Q.a: IRR if property sells for \$2,097,200**

### Step 4: Determine Cash Flows

- **Initial investment (Year 0):** $-\$1,170,000$
- **Monthly mortgage payments (Months 1-120):** \$8,742.50
- **Final cash inflow at Year 10:**

Since the property sells for \$2,097,200, and the lender owns 60%:

$$
\text{Proceeds} = 60\% \times 2,097,200 = \$1,258,320
$$

- **Total cash inflow at Year 10:**

The lender's "payoff" is the sum of the remaining mortgage balance plus the share of property value (via conversion):

**But note**: Since the lender converts to equity at Year 10, they effectively "own" 60% of the property, so they get 60% of sale proceeds.

### Step 5: Set Up Cash Flows for IRR Calculation

**Monthly cash flows:**

| Month | Cash Flow                        |
|---------|----------------------------------|
| 0       | $-\$1,170,000$ (initial loan) |
| 1-119   | $\$8,742.50$ (monthly payment) |
| 120     | $\$8,742.50 + \$1,258,320$ (final payment + sale proceeds) |

**Note:** The last month's total inflow is:

$$
\$8,742.50 + \$1,258,320 = \$1,267,062.50
$$

---

### Final step: **Calculate IRR**

Use Excel's IRR function with the following cash flows:

- **Month 0:** $-1,170,000$
- **Months 1-119:** \$8,742.50 each
- **Month 120:** \$1,267,062.50

Convert IRR to annualized IRR:

$$
IRR_{annual} = (1 + IRR_{monthly})^{12} - 1
$$

---

## **Q.b: Property sells for \$1,175,000**

- **Proceeds:** 60% of \$1,175,000 = \$705,000

- Final inflow:

$$
\$8,742.50 + \$705,000 = \$713,742.50
$$

Repeat the same process:

| Month | Cash Flow                        |
|---------|----------------------------------|
| 0       | $-\$1,170,000$                |
| 1-119   | \$8,742.50                      |
| 120     | \$713,742.50                   |

Calculate IRR as above.

---

## **Q.c: Property sells for \$670,000**

- **Loan balance at Year 10:** approx \$769,200 (from earlier)
- Since \$670,000 < \$769,200, the borrower defaults, and the lender gets the property value:

$$
\text{Inflow} = \$670,000
$$

- Final cash flow:

$$
\$8,742.50 + \$670,000 = \$678,742.50
$$

| Month | Cash Flow                        |
|---------|----------------------------------|
| 0       | $-\$1,170,000$                 |
| 1-119   | \$8,742.50                       |
| 120     | \$678,742.50                    |

Calculate IRR accordingly.

---

## **Final notes:**

- Use Excel or a financial calculator:
  - Input the monthly cash flows as described.
  - Compute the IRR for each scenario.
  - Convert the monthly IRR to annua

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