# Help with Loan Amortization Question

Hello I'm stuck on a question if anyone could walk me through on the steps that would be great. This is a loan Amortization question
`The principal amount is \$100,000`
`The Interest Rate is 3%`
`The Interest Is compounded Semi-Annually`
`The Payment Frequency Is Monthly`
`The Duration of the loan is 36 Months`

Find the information for Payment No. 1, The Payment Amount, Principal, Interest and remaining balance.

• Has anyone ever had a loan where the interest is compounded less frequently than the payments are made? – Nathan L Feb 14 '14 at 22:24
• Yes. This Question came from a real world Problem. – user3276954 Feb 14 '14 at 22:29
• It protects the borrower, a little. Lenders cannot hide a higher interest rate by quoting a low nominal rate with a high frequency of compounding. So a lender can quote 3% per annum, compounded semi-annually, and then find the equivalent rate for payments made weekly, bi-weekly, monthly, whatever.... And the borrower can make basic comparisons to offers of other potential lenders. – DJohnM Feb 15 '14 at 2:18
• @User58220 Effective annual interest rate disclosures suit me just fine. – user11865 Feb 15 '14 at 3:31
• @quantycuenta: Effective annual rate would be simper to understand...especially in these days of \$10 scientific calculators and Excel in a lot of homes... – DJohnM Feb 15 '14 at 17:49

With the correct interest rate:

The interest rate is 3% per year, compounded semi-annually. so the effective annual rate is (1.015)^2 - 1, or 3.0225%

To have this effective annual rate for a monthly compounded investment, the monthly rate, r, should be such that (1 + r)^12 = 1.030225; this yields a value of 0.2484517% per month.

You now have all the information for using the standard ordinary annuity formulas; principal, interest rate per payment period, number of payments.

• Sorry I'm sort of new at that term, not sure what the Standard Ordinary Annuity formulas are. Is it this? `P = PMT [((1 + r)^n - 1) / r]` – user3276954 Feb 18 '14 at 15:00
• How did you get `0.2484517% per month` – user3276954 Feb 18 '14 at 17:06
• 1) the formula you just gave is the value, on the date of the last payment, of a series of n payments of PMT each, which accumulate interest at a rate r per payment period...Another formula give the value of the same payments one period before the first payment. – DJohnM Feb 19 '14 at 2:55
• 2) Divide 3% by 2 (semi-annual compounding). Add 1, and square (for one year). Take the 12th root (or raise to the 1/12 power) ans subtract 1, to get the monthly compounding rate... – DJohnM Feb 19 '14 at 3:01

Nominal interest rate `i` is 3% compounded semi-annually

Using the formula for the effective annual rate here,

Effective rate `r = (1 + i/2)^2 - 1 = 0.030225 = 3.0225%`

Monthly rate `m = (1 + 0.030225)^(1/12) - 1 = 0.0024845`

Principal `p` is \$100,000

Using the example for an ordinary annuity from here: Calculating The Present And Future Value Of Annuities

The example demonstrates how a present value principal of \$4329.48 is paid down by five repayments of \$1000 each discounted to present value by the interest rate and period. The example shows

`p = Σ d (1 + m)^-k` for `k = 1 to n`

where `d` is the periodic deposit and `n` is the number of periods.

By induction this can be converted to a formula

``````p = (d - d (1 + m)^-n)/m

∴ d = (m (1 + m)^n p)/((1 + m)^n - 1)

∴ d = (0.0024845 (1 + 0.0024845)^36 * 100000)/((1 + 0.0024845)^36 - 1) =  2907.30
``````

The monthly deposit is \$2907.30

Check

``````(d - d (1 + m)^-n)/m = p

(2907.30 - 2907.30 (1 + 0.0024845)^-36)/0.0024845 = 100000
``````

Upon Payment No. 1 the interest accumulated on the principal is

``````100000 * 0.0024845 = 248.45
``````

therefore the balance remaining after the first repayment is

``````100000 + 248.45 - 2907.30 = \$97,341.15
``````