You currently have two personal loans which you wish to consolidate into just one loan.

For the first loan you agreed to pay $
5
0
0
per month for seven years at an interest rate of
1
0
%
per

Question

You currently have two personal loans which you wish to consolidate into just one loan.
 
 ﻿For the first loan you agreed to pay $
5
0
0
 ﻿per month for seven years at an interest rate of 
1
0
%
 ﻿per annum compounded monthly. While for the second loan you agreed to pay $
4
0
0
 ﻿per month for 
1
0
 ﻿years at an interest rate of 
8
%
 ﻿per annum compounded monthly. You have been given the opportunity to consolidate the two loans into one, where the consolidated loan is for five years at an interest rate of 
1
2
%
 ﻿per annum compounded monthly. What is the monthly repayment on the consolidated loan? Answer with step by step working out.

VIPSolutions · Accepted Answer

Let's break down the problem step by step:

### **Step 1: Find the Present Value (PV) of Each Loan**

#### **1. Loan 1**
- **Monthly payment ($PMT_1$)**: \$500
- **Term ($n_1$)**: 7 years = 84 months
- **Monthly interest rate ($r_1$)**: $10\%/12 = 0.008333$

The present value of an ordinary annuity:

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

$$
PV_1 = 500 \times \frac{1 - (1 + 0.008333)^{-84}}{0.008333}
$$

Calculate $(1 + 0.008333)^{-84}$:

$$
(1 + 0.008333)^{-84} = (1.008333)^{-84}
$$

$$
\ln(1.008333) = 0.008298
$$
$$
-84 \times 0.008298 = -0.697
$$
$$
e^{-0.697} \approx 0.498
$$

So,

$$
PV_1 = 500 \times \frac{1 - 0.498}{0.008333}
$$
$$
PV_1 = 500 \times \frac{0.502}{0.008333}
$$
$$
PV_1 = 500 \times 60.24 = \$30,120
$$

---

#### **2. Loan 2**
- **Monthly payment ($PMT_2$)**: \$400
- **Term ($n_2$)**: 10 years = 120 months
- **Monthly interest rate ($r_2$)**: $8\%/12 = 0.0066667$

$$
PV_2 = 400 \times \frac{1 - (1 + 0.0066667)^{-120}}{0.0066667}
$$

$$
(1 + 0.0066667)^{-120} = (1.0066667)^{-120}
$$
$$
\ln(1.0066667) = 0.006645
$$
$$
-120 \times 0.006645 = -0.7974
$$
$$
e^{-0.7974} \approx 0.450
$$

$$
PV_2 = 400 \times \frac{1 - 0.450}{0.0066667}
$$
$$
PV_2 = 400 \times \frac{0.55}{0.0066667}
$$
$$
PV_2 = 400 \times 82.5 = \$33,000
$$

---

### **Step 2: Total Amount to Consolidate**

$$
PV_{total} = PV_1 + PV_2 = 30,120 + 33,000 = \$63,120
$$

---

### **Step 3: Find the Monthly Payment for the Consolidated Loan**

- **Principal ($PV_{total}$)**: \$63,120
- **Term ($n$)**: 5 years = 60 months
- **Monthly interest rate ($r$)**: $12\%/12 = 0.01$

The formula for the payment:

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

$$
PMT = 63,120 \times \frac{0.01}{1 - (1.01)^{-60}}
$$

First, calculate $(1.01)^{-60}$:

$$
\ln(1.01) = 0.00995
$$
$$
-60 \times 0.00995 = -0.597
$$
$$
e^{-0.597} \approx 0.550
$$

$$
1 - 0.550 = 0.450
$$

So,

$$
PMT = 63,120 \times \frac{0.01}{0.450}
$$
$$
PMT = 63,120 \times 0.02222
$$
$$
PMT \approx \$1,403
$$

---

## **Final Answer**

### **The monthly repayment on the consolidated loan is approximately $\boxed{\$1,403}$ (rounded to the nearest dollar).**

---

#### **Summary Table**

| Step       | Value     |
|------------|-----------|
| PV Loan 1  | \$30,120  |
| PV Loan 2  | \$33,000  |
| Total PV   | \$63,120  |
| Term       | 60 months |
| Interest   | 12%/year  |
| **Payment**| **\$1,403** |

**Let me know if you need to see the calculations in spreadsheet form or need them double-checked for more decimal places!**

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