Question

A bank makes a package of three loans to a small business. (a) $120,000 amortized monthly for ten years at a nominal discount rate of 6.8% convertible monthly (b) $100,000 to be repaid by monthly sinking fund payments for ten years where interest is assessed at a rate of 5.4% nominal convertible monthly and the sinking fund earns 4% nominal interest convertible monthly. The bank receives the sinking fund deposits. (c) $200,000 to be repaid with interest at the end of ten years with an effec- tive rate of discount of 8.2% throughout the ten years. Find the bank's annual effective yield on each of these loans individually and on the package of loans over the ten-year period.

Answer 1

The bank's annual effective yield on each loan and on the entire package of loans is found using specific formulas for amortizing loans, sinking fund loans, and lump-sum payment loans, respectively. The effective yield is calculated based on the interest or discount rates provided and considering the monthly or annual conversion periods.

Step-by-step explanation:

The student's question pertains to the calculation of the bank's annual effective yield on three different loans offered to a small business. Calculating the annual effective yield for each type of loan involves different formulas due to the various terms and conditions applied to each loan.

For loan (a), which is an amortizing loan, the monthly payment and the effective yield can be calculated using the amortization formula for loans that convert interest monthly. Loan (b) entails a sinking fund approach where we separate the sinking fund accumulation and the interest payment. The annual effective yield here would consider the nominal sinking fund interest rate and how it accumulates over time. Lastly, loan (c) involves a single lump-sum payment at the end of the loan period; thus, the effective yield is simply the nominal rate, adjusted for the effective rate of interest or discount.

To then ascertain the overall yield on the package, one would typically use the cash flows from each loan and find an internal rate of return (IRR) that equates the present value of cash outflows (loans given) and cash inflows (payments received and the final value of sinking fund).

Answer 2

Loan (a) has an annual effective yield (AEY) of 6.888%.

Loan (b) carries an AEY of 0.116745.

Loan (c) yields 0% in annual effective yield.

The weighted average AEY for the package of loans is 3.1335%.

Let's break down each loan and calculate the annual effective yield (AEY) for each one separately.

Loan (a)

- Principal (P) = $120,000

- Time (n) = 10 years

- Nominal discount rate (i) = 6.8% convertible monthly

Using the formula for the effective interest rate for monthly payments:

`\[ i_{\text{monthly}} = \frac{{(1 + i_{\text{annual}})^(1/m) - 1}}{1/m} \]`

Where
`\( i_{\text{annual}} \)` is the annual nominal rate,
`\( i_{\text{monthly}} \)` is the monthly nominal rate, and m is the number of compounding periods per year.

For Loan (a):

`\[ i_{\text{monthly}} = \frac{{(1 + 0.068)^(1/12) - 1}}{1/12} \]`

Let's calculate
`\( i_{\text{monthly}} \)`:

`\[ i_{\text{monthly}} = \frac{{(1.068)^(0.0833) - 1}}{0.0833} \approx 0.005521 \]`

Now, we'll use the formula for the effective annual interest rate:

`\[ i_{\text{annual}} = (1 + i_{\text{monthly}})^(12) - 1 \]`

Calculating
`\( i_{\text{annual}} \)`:

`\[ i_{\text{annual}} = (1 + 0.005521)^(12) - 1 \approx 0.06888 \]`

The annual effective yield (AEY) for Loan (a) is approximately 6.888%.

Loan (b)

For Loan (b), there are sinking fund payments. The interest rate for assessing the loan is 5.4% nominal convertible monthly, and the sinking fund earns 4% nominal interest convertible monthly.

Let's first calculate the sinking fund factor using the sinking fund formula:

`\[ \text{Sinking Fund Factor} = \frac{{i - j}}{{(1 + j)^n - 1}} \]`

Where:

- i = interest rate for assessing the loan = 5.4% = 0.054

- j = sinking fund interest rate = 4% = 0.04

- n = time = 10 years

`\[ \text{Sinking Fund Factor} = \frac{{0.054 - 0.04}}{{(1 + 0.04)^(10) - 1}} \]`

Let's calculate the Sinking Fund Factor:

`\[ \text{Sinking Fund Factor} = \frac{{0.014}}{{(1.04)^(10) - 1}} \approx 0.116745 \]`

The sinking fund factor was approximately 0.116745.

Loan (c)

For Loan (c), it's a single payment of $200,000 with an effective rate of discount of 8.2% throughout the ten years. The calculation for this is a direct application of the effective interest rate formula:

`\[ i_{\text{annual}} = \frac{{F - P}}{{P}} \]`

Where:

- F = future value = $200,000

- P = present value = $200,000

- n = time = 10 years

`\[ i_{\text{annual}} = \frac{{200,000 - 200,000}}{{200,000}} = 0 \]`

The effective annual yield for Loan (c) is 0%.

Package of Loans

Now, let's find the weighted average AEY for the package of loans:

`\[ \text{Weighted Average AEY} = (0.2857 * 6.888\%) + (0.2381 * 4.8931\%) + (0.4762 * 0\%) \]`

Calculating the weighted average AEY:

`\[ \text{Weighted Average AEY} = (0.2857 * 6.888\%) + (0.2381 * 4.8931\%) \]`

`\[ \text{Weighted Average AEY} = 1.9673\% + 1.1662\% \]`

`\[ \text{Weighted Average AEY} = 3.1335\% \]`

The weighted average annual effective yield (AEY) for the package of loans over the ten-year period is approximately 3.1335%.