# Calculate a weekly payment on a loan when payment is a month away

I'm trying to calculate a payment on a loan given the standard `P=r(PV)/1-(1+r)^-n` formula. How do I factor in interest accumulated if the first payment isn't due for 31 days? For reference, interest rate = 21, No pmts = 52, Amt financed = 8764.65. I'm anticipating a payment of 189.78 (based on a third party program's calculation)

• Is 21% the APR or the AER? Is interest compounded daily, weekly, or monthly? – CactusCake May 23 '17 at 19:31
• 21% is APR. I don't know about how often it's compounded. How can I tell? – Patrick Schomburg May 23 '17 at 19:33
• Your anticipated monthly payment sounds right. You'll pay something like 156.32 interest / 33.46 principal in the first payment. Then 35.26i/154.52p in the next, then gradually amortizing towards mostly principal with each subsequent payment. Your final payment should be 0.06 cheaper than all the others, woot! – CactusCake May 23 '17 at 19:47
• Fairly close to what their amortization schedule looks like. – Patrick Schomburg May 23 '17 at 20:03

Using the standard loan formula with 21% APR nominal, compounded weekly.

Calculate an adjusted loan start value by adding `31 - 7 = 24` extra days of daily interest (by converting the nominal compounded weekly rate to a daily rate).

For details see Converting between compounding frequencies

`````` dailyrate = ((1 + 0.21/52)^(52/365)) - 1

pv = 8764.65*(1 + dailyrate)^24 = 8886.27

n = 52
r = 0.21/52
``````

Applying the standard formula `r (pv)/(1 - (1 + r)^-n) = 189.80`

So every weekly payment will be 189.80

Alternatively

Directly arriving at the same result by using the loan formula described here, The extension `x` is `31 - 7 = 24` daily fractions of an average week (where 7 daily fractions of an average week equal one average week).

``````x = (31 - 7)/(365/52)

pv = 8764.65
n = 52
r = 0.21/52

pv = (c (1 + r)^(-n - x) (-1 + (1 + r)^n))/r

∴ c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)

∴ c = 189.80
``````

As before, the weekly payment will be 189.80

Both methods are effectively the same calculation.

• Thank you, this always gets within a penny or two of their payment. – Patrick Schomburg May 24 '17 at 12:39
• Do you know where that small discrepancy might be coming from? – Patrick Schomburg May 24 '17 at 13:02
• @PatrickSchomburg Nope, I tried a few things. – Chris Degnen May 24 '17 at 13:26
• If the payment was made weekly but the interest accrued daily, does that make any difference? – Patrick Schomburg May 24 '17 at 14:28
• Maybe this is better put in a different question, but they have a seperate spot where they list "U.S. Rule APR" at 20.9698. I noticed using that rather than 21 gets me closer to their payment. Do they somehow find the APR then the payment? – Patrick Schomburg May 24 '17 at 14:46

At time = 0, no interest has accrued. That's normal. And the first payment is due after a month, when there's a month's interest and a bit of principal due.

Note - I missed weekly payments. You'd have to account for this manually, add a month's interest, then calculate based on weekly payments.

• It affects the weekly payment though, somehow, right? – Patrick Schomburg May 23 '17 at 19:36
• I missed 'weekly'. Will edit. – JTP - Apologise to Monica May 23 '17 at 19:37
• Okay so here's what I did after reading this. I calculated daily interest at `principal * (interest/360)` I multiplied that by 24 days. This gives me the same answer as the program I'm comparing it to, but I still don't know why it charges 24 days instead of the full 31. – Patrick Schomburg May 23 '17 at 20:09
• @PatrickSchomburg You are extending the first payment period from 7 days to 31 days which is 24 extra days. – Chris Degnen May 23 '17 at 21:35

You'd have to look at the terms of the loan to be sure, but if the interest compounds weekly then you'd have to calculate the effect of 3 compounding periods, then compute for weekly payments. The balance after 3 weeks would be:

``````8764.65 * (1 + .21/52) ^ 3 = 8871.27
``````

Using Excel's `PMT` function for that principal balance, I get a weekly payment of \$189.48.

If the interest doesn't compound, the principal balance will be about \$8888.37 and the weekly payment would be \$189.85.

Note, however, that the terms of the loan could be completely customized, so you'd need to be sure that the payment and the amortization schedule make sense to you before you agree to the loan. Since the interest is very high, I suspect this is a "no credit needed" car loan which are notorious for unfavorable (to the borrower) terms.

• We call them "subprime". – Patrick Schomburg May 23 '17 at 20:24
• @PatrickSchomburg Are you the lender or the borrower? – D Stanley May 23 '17 at 20:28
• Neither really, I'm just verifying the numbers of the software the lender is using. – Patrick Schomburg May 23 '17 at 20:29