How to Calculate a BiWeekly Repayment Loan?

Question

I want to calculate the biweekly repayment loan. I have gotten the monthly using this formula

(D) = {[(1 + i) ^n] - 1} / [i(1 + i)^n]
Monthly Payment = Loan_Amount / D

How can I convert this to biweekly? Someone said this online, although I am having difficulty placing this into the equation:

You cannot simply divide the .050625 by 26; that will result in a rate higher than .050625,
since you are compounding 26 times.
You need a rate that when compounded 26 times results in .050625; this way:
(1 + i)^26 = 1.050625 ; 1 + i = 1.050625^(1/26) ; i = .0019012368...
Use that rate and you'll get 492.091735... as payment.

I've tried to implement this and here is what my answers are (I think biweekly is wrong)

Loan Amount: $20,000
Term: 3 Years
Rate 5.50%

My Answers:

MONTHLY: $603
Total: $21,741

BiWeekly: $278
Total: $21,716

Answer 1

In a spreadsheet, the rate per period is usually Rate/1200 here, 5.50/1200, giving you the decimal version per month. i.e. per time elapsed between payments.

You want Rate/2600 or 5.5/2600 and the term, usually say, 360, you want 78 as there are 78 payments for the loan you propose.

Using these numbers, I get $278.42 as a payment per 2 week period.

Keep in mind, when a bank offers a bi-weekly, the most common practice is to calculate a 30 year fixed monthly payment, and then have the borrower pay 1/2 that number every 2 weeks. This drops the amortization time by 6-8 years depending on the rate. The key thing to recognize is that the 'bi-weekly' i.e. frequent payments isn't the real benefit. The benefit comes from the fact that you are making 13 full payments per year. You would derive nearly all the benefit from simply paying 8.3% more on your payment each month on a 30 year loan.

Answer 2

You need to be clear what type of rate the 5.5% is. In the absence of a phrase such as "compounded monthly" or "compounded biweekly" the general assumption has to be that the rate is an effective rate, not a nominal rate, and that affects how the periodic rate (i) is calculated.

See the effective interest rate calculation.

Nominal rates are convenient for finding the periodic rate because they are simply the periodic rate multiplied by the compounding frequency, e.g. 0.458333 % monthly * 12 = nominal 5.5 % compounded monthly.

However, a nominal rate of 5.5% compounded monthly does not equal a nominal rate of 5.5% compounded biweekly. If you start a year with £100,000 an effective annual rate of 5.5% produces £105,500 by the end of the year whereas nominal rates of 5.5% compounded monthly and biweekly result in £105,366.04 and £105,359.59 respectively.

Calculating nominal and periodic rates equivalent to 5.5% effective annual interest

eff = 5.5/100
nom = 12 ((1 + eff)^(1/12) - 1) = 5.36604 % compounded monthly

For the monthly calculation the periodic rate based on an effective rate of 5.5% is

i = (1 + eff)^(1/12) - 1 = 0.0044717

For the biweekly calculation, taking the number of weeks in the year as 52 (an approximation), the periodic rate is

i = (1 + eff)^(1/26) - 1 = 0.00206138

and the nominal rate is

nom = 26 i = 5.35959 % compounded biweekly

Formula derivation

The loan formula can be derived from the summation of payments discounted to present value.

If i is the periodic rate and n is the number of periods

Rearranging the equation

periodicpayment = loanamount/(((1 + i)^n - 1)/(i (1 + i)^n))

If d = ((1 + i)^n - 1)/(i (1 + i)^n)

periodicpayment =  loanamount/d

Calculating periodic payments based on 5.5% effective interest

Monthly

loanamount = 20000
eff = 5.5/100
i = ((1 + eff)^(1/12) - 1)
n = 3*12
d = ((1 + i)^n - 1)/(i (1 + i)^n)
periodicpayment = loanamount/d = 602.71

Biweekly

loanamount = 20000
eff = 5.5/100
i = ((1 + eff)^(1/26) - 1)
n = 3*26
d = ((1 + i)^n - 1)/(i (1 + i)^n)
periodicpayment = loanamount/d = 277.84

You could use nominal rates

i = 5.5/100/12 for monthly
i = 5.5/100/26 for biweekly

but then you are using different rates, which doesn't help for comparing loans. If you want to use nominal rates in calculations for different compounding periods for meaningful comparisons you need to convert so that they are equivalent to a single effective rate.

To compare how the results from using different nominal rates differ from the results above, here is the calculation with 5.5% nominal compounded monthly

loanamount = 20000
nom = 5.5/100
i = nom/12
n = 3*12
d = ((1 + i)^n - 1)/(i (1 + i)^n)
periodicpayment = loanamount/d = 603.918

and 5.5% nominal compounded biweekly

loanamount = 20000
nom = 5.5/100
i = nom/26
n = 3*26
d = ((1 + i)^n - 1)/(i (1 + i)^n)
periodicpayment = loanamount/d = 278.416