Question

Gordon borrows $5000 at an effective rate of interest of 8% per annum. He agrees to pay interest payment at the end of each year for n years, and repays the capital after n years by making sinking fund deposits at the end of each year for n years. The first sinking fund payment is $800, the subsequent n-1 payments increase by 5% each, while the payment at the end of n years is an irregular payment (less than the regular payment) such that the capital can be repaid. The sinking fund earns interest at 7.5% per annum. Find the final irregular sinking fund deposit

Answer 1

(a) Find the final irregular sinking fund deposit:

`\[ P_{\text{irregular}} = (5000 - \left((0.075)/(1)\right))/(\left(1 + (0.075)/(1)\right)^(1 \cdot n) - 1) \]`

(b) Find the growth rate of the sinking fund deposit if the loan is to be paid at the end of
`\( n \)` years and the growth rate is always regular:

`\[ P_{\text{regular}} = 800 * \left(1 + 0.05\right)^(n-1) \]`

`\[ r_{\text{regular}} = 1 * \left(\left(\frac{5000}{P_{\text{regular}}}\right)^{(1)/(n)} - 1\right) \]`


Let's break down the problem and solve it step by step:

(a) Find the final irregular sinking fund deposit:

The sinking fund deposit
`(\(P\))` is given by the formula:

`\[ P = (A - \left((r)/(k)\right))/(\left(1 + (r)/(k)\right)^(nk) - 1) \]`

where:

-
`\( P \)` is the sinking fund deposit,

-
`\( A \)` is the loan amount (in this case, $5,000),

-
`\( r \)` is the interest rate per period (in this case, 7.5% for the sinking fund),

-
`\( k \)` is the number of compounding periods per year (assuming once per year),

-
`\( n \)` is the total number of compounding periods.

Since the interest rate for the sinking fund is 7.5%,
`\( r = 0.075 \)`.

The growth rate of the sinking fund deposit is not explicitly given for the irregular payment. However, it mentions that the subsequent payments increase by 5% each. Assuming this increase applies to the growth rate, we can set up an equation:

`\[ r_{\text{irregular}} = r_{\text{regular}} + 0.05 \]`

Now let's solve for the final irregular sinking fund deposit
`(\( P_{\text{irregular}} \))`:

`\[ P_{\text{irregular}} = (5000 - \left((0.075)/(1)\right))/(\left(1 + (0.075)/(1)\right)^(1 \cdot n) - 1) \]`

(b) Find the growth rate of the sinking fund deposit if the loan is to be paid at the end of
`\( n \)` years and the growth rate is always regular:

For regular payments, the growth rate
`(\( r_{\text{regular}} \))` can be calculated using the formula:

`\[ r_{\text{regular}} = k * \left(\left(\frac{A}{P_{\text{regular}}}\right)^{(1)/(nk)} - 1\right) \]`

where
`\( P_{\text{regular}} \)` is the regular sinking fund deposit. This involves using the first sinking fund payment of $800 and the subsequent payments increasing by 5%.

Let's proceed with the solution without specifying the value of
`\( n \)`. We'll express the final irregular sinking fund deposit
`(\( P_{\text{irregular}} \))` and the growth rate of the sinking fund deposit if regular
`(\( r_{\text{regular}} \))` in terms of
`\( n \)`.

(a) Find the final irregular sinking fund deposit:

`\[ P_{\text{irregular}} = (5000 - \left((0.075)/(1)\right))/(\left(1 + (0.075)/(1)\right)^(1 \cdot n) - 1) \]`

(b) Find the growth rate of the sinking fund deposit if the loan is to be paid at the end of
`\( n \)` years and the growth rate is always regular:

`\[ P_{\text{regular}} = 800 * \left(1 + 0.05\right)^(n-1) \]`

`\[ r_{\text{regular}} = 1 * \left(\left(\frac{5000}{P_{\text{regular}}}\right)^{(1)/(n)} - 1\right) \]`

Now, these expressions can be used to find the final irregular sinking fund deposit and the growth rate for regular payments once the value of
`\( n \)` is provided. If you have a specific value for
`\( n \)` or any other details, feel free to share, and I can assist further.

The probable question can be: Gordon borrows $5,000 at an effective rate of interest of 8% per annum. He agrees to pay interest payment at the end of each year for n years, and repays the capital after n years by

making sinking fund deposits at the end of each year for n years. The first sinking fund payment is $800, the subsequent n − 1 payments increase by 5% each, while the payment at the end of n years is an irregular payment (less than the regular payment) such that the capital can be repaid. The sinking fund earns interest at 7.5% per annum.

(a) Find the final irregular sinking fund deposit.

(b) Find the growth rate of the sinking fund deposit if the loan is to be paid at the end of n years and the growth rate is always regular (i.e., there is no final smaller irregular payment).