#math/applied #economics #flashcards/math #review
A fixed-rate, amortizing loan is a loan where the payments each period is a constant amount with varying towards-interest and towards-principal ratios. The payment loan formula is as follows:
$M = P \frac{r(1+r)^n}{(1+r)^n - 1}$
- $M=\text{Monthly Payment}$
- $P=\text{Principal}$
- $r=\text{Rate}$
- $n=\text{Number of Payments (usually in months i.e. 60, 120, 360)}$
Let $I_{k}$ represent the interest paid on the $k\text{-th}$ month. The interest paid per month decreases geometrically
$
I_{k}=M-P_{1}(1+r)^{(k-1)}.
$
And principal paid per month increases geometrically
$
P_{1}(1+r)^{(k-1)}.
$
Thus the start of a loan always has the least favorable interest-to-principal ratio from the borrower's perspective. Notice that as you increase $n$, the principal to interest ratio gets worse but you get lower monthlies. The ratio does not change as you change the starting $P$. The [[Amortizing Loan Derivation|formula is derived]] using the [[Geometric Series]] sum.
**Visual Difference of Interest vs Principal**
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\fill[red!30] (0.167,5.652) -- (0.167,4.190) -- (0.667,4.254) -- (1.333,4.339) -- (2.000,4.427) -- (2.833,4.538) -- (3.500,4.630) -- (4.167,4.723) -- (5.000,4.842) -- (5.667,4.940) -- (6.333,5.040) -- (7.000,5.141) -- (7.833,5.271) -- (8.500,5.377) -- (9.167,5.486) -- (10.000,5.624);
\draw[black, thick, dashed] (0,5.652) -- (10,5.652);
\node[right] at (10,5.652) {M=19333};
\draw[->] (-0.3,0) -- (10.8,0) node[right] {months};
\draw[->] (0,-0.3) -- (0,7.5) node[above] {USD/mo};
\draw (2.000,0.05) -- (2.000,-0.05) node[below] {12};
\draw (4.000,0.05) -- (4.000,-0.05) node[below] {24};
\draw (6.000,0.05) -- (6.000,-0.05) node[below] {36};
\draw (8.000,0.05) -- (8.000,-0.05) node[below] {48};
\draw (10.000,0.05) -- (10.000,-0.05) node[below] {60};
\draw (0.05,1.462) -- (-0.05,1.462) node[left] {5k};
\draw (0.05,2.924) -- (-0.05,2.924) node[left] {10k};
\draw (0.05,4.385) -- (-0.05,4.385) node[left] {15k};
\draw (0.05,5.847) -- (-0.05,5.847) node[left] {20k};
\node[red!70!black] at (2.000,5.039) {Interest};
\node[green!40!black] at (7.833,2.635) {Principal};
\node[above, font=\bfseries] at (5,7.9) {5-Year Loan, 1M at 6 pct};
\end{scope}
\begin{scope}[shift={(15.6,0.0)}, scale=0.55]
\fill[green!30] (0.167,0) -- (0.167,3.111) -- (0.667,3.206) -- (1.333,3.336) -- (2.000,3.471) -- (2.833,3.648) -- (3.500,3.796) -- (4.167,3.950) -- (5.000,4.152) -- (5.667,4.321) -- (6.333,4.496) -- (7.000,4.679) -- (7.833,4.917) -- (8.500,5.117) -- (9.167,5.325) -- (10.000,5.596) -- (10.000,0) -- cycle;
\fill[red!30] (0.167,5.652) -- (0.167,3.111) -- (0.667,3.206) -- (1.333,3.336) -- (2.000,3.471) -- (2.833,3.648) -- (3.500,3.796) -- (4.167,3.950) -- (5.000,4.152) -- (5.667,4.321) -- (6.333,4.496) -- (7.000,4.679) -- (7.833,4.917) -- (8.500,5.117) -- (9.167,5.325) -- (10.000,5.596) -- (10.000,5.652) -- cycle;
\draw[black, thick] (0.167,3.111) -- (0.667,3.206) -- (1.333,3.336) -- (2.000,3.471) -- (2.833,3.648) -- (3.500,3.796) -- (4.167,3.950) -- (5.000,4.152) -- (5.667,4.321) -- (6.333,4.496) -- (7.000,4.679) -- (7.833,4.917) -- (8.500,5.117) -- (9.167,5.325) -- (10.000,5.596);
\draw[black, thick, dashed] (0,5.652) -- (10,5.652);
\node[right] at (10,5.652) {M=22244};
\draw[->] (-0.3,0) -- (10.8,0) node[right] {months};
\draw[->] (0,-0.3) -- (0,7.5) node[above] {USD/mo};
\draw (2.000,0.05) -- (2.000,-0.05) node[below] {12};
\draw (4.000,0.05) -- (4.000,-0.05) node[below] {24};
\draw (6.000,0.05) -- (6.000,-0.05) node[below] {36};
\draw (8.000,0.05) -- (8.000,-0.05) node[below] {48};
\draw (10.000,0.05) -- (10.000,-0.05) node[below] {60};
\draw (0.05,1.270) -- (-0.05,1.270) node[left] {5k};
\draw (0.05,2.541) -- (-0.05,2.541) node[left] {10k};
\draw (0.05,3.811) -- (-0.05,3.811) node[left] {15k};
\draw (0.05,5.082) -- (-0.05,5.082) node[left] {20k};
\node[red!70!black] at (2.000,4.562) {Interest};
\node[green!40!black] at (7.833,2.459) {Principal};
\node[above, font=\bfseries] at (5,7.9) {5-Year Loan, 1M at 12 pct};
\end{scope}
\end{tikzpicture}
\end{document}
```