Given interest, time and principal - how do I figure out % interest rate

hussey

I borrow $100,000  if it takes me 200 days paying off $550 a day, so the total interest (calculated daily) was $10,000
What formula do I use to figure out what the interest rate was?
I had a look on the web, but most of the formula I found was to calculate the fee given the %.
Cheers

L'prof

Formula you found was I=PRT?

What you're looking for is R=I/PT

hussey

Thank you, does that formula take into account interest calculated daily? Or just simple interest?

hussey

I found this https://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm
I am wondering if my equation is right

given A = P[1+(r/n)]^nt
A/P = [1+r/n)]^nt
(A/P)^(1/nt) = 1 + r/n
(A/P)^(1/nt) - 1 = r/n

r = n[(A/P)^(1/nt) - 1]

so using the example above
n = 365 (interest daily compounded)
A = 110,000
P = 100,000
t = 200/365 (200 days to pay off)

so r = 365[1.1^(1/200) -1)
r = 365 *(1.00047666 - 1)
r = .17398

so 17.398%

MathsManiac

If you borrowed 100,000 and paid back 110,000 in a single lump sum after 200 days, then what you have calculated is the APR for that loan.

That's not the same as the APR for the loan / repayment schedule you described. If you're looking for the APR for what you described, then you need the "amortisation formula":

Imagine a loan of principal P paid off over t years using equal annual repayments A, with the first repayment occurring a year after the loan is drawn down. If the APR of the loan is i, then:
A=P[i(1+i)^t]/[(1+i)^t-1]
(i is expressed as a decimal, so that, for example, 8% means i=0.08)

Fundamentally, the same formula can be used for repayment intervals other than a year. Suppose that each day in the scenario you described was a year. Then find the value of i that would make the above formula work. This then gives you the true daily rate being paid and compounded. If that number is, say, r, then the APR for the loan is (1+r)^365.25.

MathsManiac

By my calculations, you're paying a compounding daily rate of about 1.163%, which is an APR of over 6700% !!!

Yakuza

Lump Payment
Paying off $110,000 off in one lump sum after 200 days implies that : 1.1 = (1+x)^200. (x is the daily interest rate)

Solving for x [ = exp (ln (1.1)/200) -1) ] yields 0.000477.

The annual rate is then (1+x)^365 (or 365.25 if you prefer) 18.9985% or 19.0127%

Thinking about it, 10% after 200 days (approximately half a year) is 21% after 400 days (1.1^2), so the APR should be somewhere below that.

Daily Payment
Paying off $550 per day for 200 days. The interest rate is found by equating the present values of the income stream and the loan:
$100000 = 550 (1 - (1+x)^-200) / x) (this is a standard annuity formula, where x is the interest rate in days).
This can only be solved by trial and error (or Goal Seek in Excel ) yielding x = 0.000964.
The annual rate in this case is 1.000964^365, or 42.1584%.

The interest rate has more or less doubled as you're effectively giving the lender more money. The same nominal amount is being transferred, but a stream of $550 a day starting from the end of today for 200 days is worth more than $110,000 to be paid in 200 days. It has a higher "present value".
You could bank each of those slices of $550 (for 199, 198, 197 etc. days) and by the time 200 days was up, you'd have more in the bank than $110,000 (assuming a positive interest rate!).

Edit : Added an amortisation schedule in Excel.

MathsManiac

My bad on two counts:

I didn't look carefully at what the OP had done, and just assumed from the first couple of lines that he had correctly worked out the APR for a single repayment after 200 days. Sorry.

My second error was on my second (shorter) post. I had done the calculations using a monthly repayment calculator and read from a wrong cell when picking out the daily rate. I should have spotted this when I got such an outlandish answer.

I agree with Yakuza's calculations.

Also, by the way, the formula that I gave as the "amortisation formula" is the same formula as Yakuza gave as the "annuity formula". (It's in the Formulae and Tables booklet on p31.)

Yakuza

Yep, they're different ways of expressing the same formula. I think mine is a little more concise as it only involves exponentiation once, and it's the one beaten into my 18-year-old actuarial trainee ass many, many years ago