Give step-by-step solution with explanation and final answer:
You have just purchased a home with a $
3
5
0
,
0
0
0
mortgage. The mortgage will be paid back over
1
0
years. The interest rate on

Question

Give step-by-step solution with explanation and final answer: 
You have just purchased a home with a $
3
5
0
,
0
0
0
 ﻿mortgage. The mortgage will be paid back over 
1
0
 ﻿years. The interest rate on the mortgage is initially set at 
6
.
0
%
 ﻿and will rise by 
0
.
5
%
 ﻿every 
2
 ﻿years. The payment will remain constant for the 
2
-
year time period once it is reset. Construct an amortization table that shows the cash flows for the loan. a
.
 ﻿What is the total interest paid on the loan? b
.
 ﻿How much more interest do you pay with this structure versus if the loan was fixed for the whole 
1
0
 ﻿years at 
6
.
0
%
?
 ﻿Assume your payments are made annually at the end of the year.

VIPSolutions · Accepted Answer

Let's break down your mortgage problem step by step.

---

**Given:**
- Principal (P): $350,000
- Term (n): 10 years
- Initial interest rate (r): 6.0% (compounded annually)
- Rate increases by 0.5% every 2 years
- Payments are **annual** and constant **for each 2-year period** (then reset)
- Payments are made at the **end of each year**

We will:
1. Build an amortization schedule (summary level)
2. Calculate total interest paid
3. Compare with a **fixed 6.0%** loan

---

## 1. **Step-by-step Amortization Table Construction**

### **Step 1: Determine the Interest Rates per Period**

| Years | Interest Rate |
|-------|--------------|
| 1-2   | 6.0%         |
| 3-4   | 6.5%         |
| 5-6   | 7.0%         |
| 7-8   | 7.5%         |
| 9-10  | 8.0%         |

---

### **Step 2: Calculate Annual Payment for Each 2-Year Block**

At the **start of each 2-year block**:
- Find the remaining principal
- Calculate the payment to amortize the balance **over the remaining term** at the new rate

#### **Amortization Formula:**

$$
PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}}
$$
Where:
- $ PMT $ = annual payment
- $ P $ = remaining principal
- $ r $ = annual interest rate (as decimal)
- $ n $ = number of years remaining

---

#### **Block 1: Years 1-2 (6.0%)**

- $ P = 350,000 $
- $ r = 6.0\% = 0.06 $
- $ n = 10 $

$$
PMT_1 = \frac{350,000 \times 0.06}{1 - (1 + 0.06)^{-10}}
$$

Calculate denominator:

$$
(1 + 0.06)^{-10} = (1.06)^{-10} \approx 0.5584
$$
$$
1 - 0.5584 = 0.4416
$$

So,

$$
PMT_1 = \frac{21,000}{0.4416} \approx \$47,569.49
$$

---

#### **Block 2: Years 3-4 (6.5%)**

First, compute the principal remaining after 2 years.

##### **Amortization after 2 years:**

For each year:

$$
\text{Interest}_t = \text{Principal}_{t-1} \times r
$$
$$
\text{Principal Paid}_t = PMT - \text{Interest}_t
$$
$$
\text{New Principal}_{t} = \text{Principal}_{t-1} - \text{Principal Paid}_t
$$

##### **Year 1:**
- Interest: $ 350,000 \times 0.06 = \$21,000 $
- Principal Paid: \$47,569.49 - \$21,000 = \$26,569.49
- New Principal: $ 350,000 - 26,569.49 = 323,430.51 $

##### **Year 2:**
- Interest: $ 323,430.51 \times 0.06 = \$19,405.83 $
- Principal Paid: \$47,569.49 - \$19,405.83 = \$28,163.66
- New Principal: $ 323,430.51 - 28,163.66 = \$295,266.85 $

So, **Principal after 2 years: $295,266.85$**

Now, calculate new payment for 8 years at 6.5%:

$$
PMT_2 = \frac{295,266.85 \times 0.065}{1 - (1 + 0.065)^{-8}}
$$

$$
(1 + 0.065)^{-8} = (1.065)^{-8} \approx 0.6046
$$
$$
1 - 0.6046 = 0.3954
$$

$$
PMT_2 = \frac{19,192.35}{0.3954} \approx \$48,558.42
$$

---

#### **Block 3: Years 5-6 (7.0%)**

**Find remaining principal after 4 years.**

##### **Years 3 & 4:**

**Year 3:**
- Interest: $ 295,266.85 \times 0.065 = \$19,192.35 $
- Principal Paid: \$48,558.42 - \$19,192.35 = \$29,366.07
- New Principal: $ 295,266.85 - 29,366.07 = 265,900.78 $

**Year 4:**
- Interest: $ 265,900.78 \times 0.065 = \$17,283.55 $
- Principal Paid: \$48,558.42 - \$17,283.55 = \$31,274.87
- New Principal: $ 265,900.78 - 31,274.87 = 234,625.91 $

**Principal after 4 years: $234,625.91$**

Now, recalculate payment for 6 years at 7.0%:

$$
PMT_3 = \frac{234,625.91 \times 0.07}{1 - (1 + 0.07)^{-6}}
$$

$$
(1.07)^{-6} \approx 0.6663
$$
$$
1 - 0.6663 = 0.3337
$$
$$
PMT_3 = \frac{16,423.81}{0.3337} \approx \$49,223.35
$$

---

#### **Block 4: Years 7-8 (7.5%)**

**Find principal after 6 years.**

##### **Years 5 & 6:**

**Year 5:**
- Interest: $ 234,625.91 \times 0.07 = \$16,423.81 $
- Principal Paid: \$49,223.35 - \$16,423.81 = \$32,799.54
- New Principal: $ 234,625.91 - 32,799.54 = 201,826.37 $

**Year 6:**
- Interest: $ 201,826.37 \times 0.07 = \$14,127.85 $
- Principal Paid: \$49,223.35 - \$14,127.85 = \$35,095.50
- New Principal: $ 201,826.37 - 35,095.50 = 166,730.87 $

**Principal after 6 years: $166,730.87$**

Now, recalculate payment for 4 years at 7.5%:

$$
PMT_4 = \frac{166,730.87 \times 0.075}{1 - (1 + 0.075)^{-4}}
$$

$$
(1.075)^{-4} \approx 0.7385
$$
$$
1 - 0.7385 = 0.2615
$$
$$
PMT_4 = \frac{12,505.82}{0.2615} \approx \$47,849.86
$$

---

#### **Block 5: Years 9-10 (8.0%)**

**Find principal after 8 years.**

##### **Years 7 & 8:**

**Year 7:**
- Interest: $ 166,730.87 \times 0.075 = \$12,505.82 $
- Principal Paid: \$47,849.86 - \$12,505.82 = \$35,344.04
- New Principal: $ 166,730.87 - 35,344.04 = 131,386.83 $

**Year 8:**
- Interest: $ 131,386.83 \times 0.075 = \$9,854.01 $
- Principal Paid: \$47,849.86 - \$9,854.01 = \$37,995.85
- New Principal: $ 131,386.83 - 37,995.85 = 93,390.98 $

**Principal after 8 years: $93,390.98$**

Now, recalculate payment for 2 years at 8.0%:

$$
PMT_5 = \frac{93,390.98 \times 0.08}{1 - (1 + 0.08)^{-2}}
$$

$$
(1.08)^{-2} = 0.8573
$$
$$
1 - 0.8573 = 0.1427
$$
$$
PMT_5 = \frac{7,471.28}{0.1427} \approx \$52,354.69
$$

---

##### **Years 9 & 10:**

**Year 9:**
- Interest: $ 93,390.98 \times 0.08 = \$7,471.28 $
- Principal Paid: \$52,354.69 - \$7,471.28 = \$44,883.41
- New Principal: \( 9

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