When running a financial loan payoff simulation, I realized that my calculated monthly payment differed from the calculator offered by popular loan service Sallie Mae.
I found two formulas for loan amortization when constraining number of payments. The one that matches Sallie Mae's output:
$$MonthlyPayment = \frac{Pr}{n(1-(1+\frac{r}{n})^{-nt})}$$
But when running this in a simulation (where interest is compounded daily, as Sallie Mae does) it will pay the loan off too early in cases of high principals and interest rates (ie $10k with 10%APR). In contrast, this formula works perfectly in my simulation:
$$MonthlyPayment = \frac{Pr(1+r)^{nt}}{(1+r)^{nt}-1}$$
Where P: Principal, r: interest rate, n: number of payments per time t, t: unit of time.
Can anyone explain the difference between these two formulas, and if I'm using mine wrong?
Your formula is almost right. The other is similar without the factor $n$ at the denominator and with $\frac{r}n$ instead of $r$ only. We can transform the first formula into the second formula in just one step:
$$\textrm{Monthly Payment} = \frac{Pr}{1-(1+\frac{r}{n})^{-nt}}$$
Expand the RHS by $(1+\frac{r}{n})^{nt}$
$$\textrm{Monthly Payment} = \frac{Pr\cdot (1+\frac{r}{n})^{nt}}{1\cdot (1+\frac{r}{n})^{nt}-\underbrace{(1+\frac{r}{n})^{-nt}\cdot (1+\frac{r}{n})^{nt}}_{=1}}$$
$$\textrm{Monthly Payment} = \frac{Pr\cdot (1+\frac{r}{n})^{nt}}{ (1+\frac{r}{n})^{nt}-1}$$
You see the interest rate has to be divided by $n$. This is missing at your formula. Numerical example:
Loan amount: $10,000$, yearly interest rate: $5\%$, years to repay:$5$ ($=60$ months, $t=5, n=12$)
$$\textrm{Monthly Payment} = \frac{10000\cdot \frac{0.05}{12}\cdot (1+\frac{0.05}{12})^{60}}{ (1+\frac{0.05}{12})^{60}-1}=188.71\approx189$$
This result matches the result Sallie Mae's output.