[Math] APR Calculation

averagefinance

I'm hoping someone can clarify this for me.

The model/example is this:

We lend an amount of 1498.50 (loan amount). Other fees total 39.95.
The term of the loan is for 12 months. There is no interest per month, per se, but a 10.00 fee is charged for each of the 12 months.

Using federally-approved software that calculates APR (to comply with Regulation Z, Appendix J), the calculation returns the APR to be 21.488%.

The monthly payment is calculated as follows:

(Loan Amount + (10.00 x 12 months)) / 12 = 134.87.

According to the amortization schedule, none of the interest for any particular month exceeds 14%. How can the APR for the entire term be ~ 21% when none of the effective monthly interest rates even come close to 21%?

My intuition tells me that there is some averaging going on, but to achieve an average of 21%, some months would have to exceed 21% by quite a bit, while being offset by smaller percentages on the fringes of the curve.

If you are familiar with Reg Z, Appendix J, and the given example above, would you say that the formula being used is not appropriate? (There are a few others that the feds provide; maybe we/the software applied the wrong one)

Additionally, playing around with the software, keeping all numbers the same, but changing the term (in months), I find that the APR also changes. I find this confusing and counter-intuitive.

For example:
3 months, APR is 39.126%
6 months, APR is 27.393%
12 months, APR is 21.488%.

Can someone please explain this?

*All numbers above are the input and output values of approved calculation software.

Huge thanks in advance!

PS: Apologies for the seemingly uninformative tag(s). I couldn't find one that was more meaningful.

Best Answer

First, if the other fees are paid as part of the loan, I find a payment of 138.1625. If they are paid in advance, I believe they still count as interest. Then the effective interest is 159.95. The average balance is about half of the amount borrowed, as you start off owing 1498.5 and end at 0. So the effective annual interest rate is about 159.95/(1498.5/2)=23.15%. I think they get a lower value because they do an amortization, which keeps the balance higher for longer, but it is not far off. For the shorter terms, the interest rate goes higher because the fixed 39.95 fee gets spread over fewer months.

Related Solutions

[Math] Formula for APR with Odd Days

Let $r$ denote the APR we seek to establish, and the payments are made each $\frac{1}{n}$ of the year (i.e. $n=12$ for monthly payments, and $n=26$ for biweekly). Let $a_k$ be the principal amount of the loan at the beginning of the $k$-th period. Then $a_0 = a$, and $$ a_{k+1} = a_k (1+r)^{1/n}-p $$ Such a recurrence equation is not hard to solve using generating function technique, i.e. multiplying both sides of the equation with $x^k$ (for some indeterminate $x$) and forming $g(x) = \sum_{k=0}^\infty a_k x^k$: $$\begin{eqnarray} \sum_{k=0}^\infty a_{k+1} x^k &=& (1+r)^{1/n} \sum_{k=0}^\infty a_k x^k - p \sum_{k=0}^\infty x^k \\ \frac{f(x)-a_0}{x} &=& (1+r)^{1/n} f(x) - p \frac{1}{1-x} \\ f(x) &=& \frac{1}{(1+r)^{1/n}-1} \left( \frac{p}{1-x} - \frac{a + p - a(1+r)^{1/n}}{1 - (1+r)^{1/n} x} \right) \\ a_{k} &=& \frac{1}{(1+r)^{1/n}-1} \left( p - \left( a + p - a (1+r)^{1/n} \right) (1+r)^{k/n} \right) \end{eqnarray} $$ Requirement that after $m$ equal payments the loan is paid off gives the equation for effective annual percentage rate $r$: $$ p = \left( a + p - a (1+r)^{1/n} \right) (1+r)^{m/n} $$ As this is a non-linear equation, solving for $r$ will need to be done numerically. Here is a code in Mathematica:

In[63]:= FindAPR[a0_, p_, m_, n_] := 
 Block[{sol}, 
  sol = FindRoot[
    p == (a0 + p - a0 (1 + r)^(1/n)) ((1 + r)^(m/n)), {r, 1/2}];
  If[sol === {}, Indeterminate, r /. First[sol]]]

In[64]:= FindAPR[1000, 256, 4 (* payments *), 12 (* paid monthly *)]*100

Out[64]= 12.0876

If the time to the first payment is different, $a_0$ needs to be replaced with different effective principal amount, $a_0^\ast = a_0 \left(1 + r \right)^{d/n}$, where $d$ is the length of the no-payment period given as a fraction of payment intervals. For example, if payments are made monthly $n=12$, there are 4 payments to be made $m=4$ in the amount of $p=\$260.00$ on the loan of $\$1000.00$ with 1 month no payment period, the effective APR is

In[3]:= FindAPR2[a0_, p_, m_, n_, d_] := 
 Block[{sol}, 
  sol = FindRoot[
    p == (p - a0 (1 + r)^(d/n) ( (1 + r)^(1/n) - 1)) ((1 + r)^(m/
       n)), {r, 1/2}];
  If[sol === {}, Indeterminate, r /. First[sol]]]

In[5]:= FindAPR2[1000, 260, 4, 12, 1]*100

Out[5]= 14.4241

[Math] How is APR calculated

Let P be the monthly amount that you need to pay for the total amount of Borrowing which is the loan amount C and the added cost, say, E ( fees especially). As a borrower you need to consider C+E as the total amount borrowed and find the monthly payment by treating the Borrowed funds C+E. Let us say you pay monthly an interest rate of "r". Then you will have to find the monthly payments discounted over the entire period that will equal the present value which is the borrowed funds C+E.

And that is :

$$C+E = \frac{P}{1+r} + \frac{P}{(1+r)^2} + \frac{P}{(1+r)^3} + \cdots + \frac{P}{(1+r)^N}$$

$$C+E = \frac{P}{1+r}\left(1+\frac{1}{1+r}+ \frac{P}{(1+r)^2}+\cdots+\frac{P}{(1+r)^{N-1}}\right)$$

$$C+E = \frac{P}{1+r}\left(\dfrac{\left(1-\frac{1}{(1+r)^N}\right)}{\left(1-\frac{1}{(1+r)}\right)}\right)$$

If you simplify this, you get

$$C+E = \dfrac{P((1+r)^N-1)}{r(1+r)^N}$$

Now rearraging, we get

$$P = (C+E)\dfrac{r(1+r)^N}{(1+r)^N-1}$$.

Using this monthly instalment and now the the loan amount "alone" you find the interest rate "a" that equals the monthly payment. i.e

$$P =C\dfrac{a(1+a)^N}{(1+a)^N-1}$$

$$\frac{P}{C} = \dfrac{a(1+a)^N}{(1+a)^N-1}$$

$$ \dfrac{a(1+a)^N}{(1+a)^N-1} - \frac{P}{C} = 0$$.

This can only be found iteratively by numerical methods ( you could use Newton Raphson method, Secant method or Binary search method) and find "a"

This "a" is called the APR according to the loan Advertisement.

Best Answer

Related Solutions

[Math] Formula for APR with Odd Days

[Math] How is APR calculated

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