Interest Payments on Short Term Loans

For short-term loans (e.g. car loans), your average lifetime interest is roughly half of what you naively might expect. The average lifetime interest rate on a short-term low-interest rate loan (e.g. a car loan) is $\rho \approx 0.5-0.55$ of the quoted interest rate. “Short term” is under 8 years and “low interest” is under 7%. For a loan principal $P$, if you reduce your interest rate by $\mathrm{\Delta }$, your annual (yearly) payment will decrease by roughly $P\times \mathrm{\Delta }\rho$, for $\rho \approx 0.53$. Understand your savings when deciding to pursue a lower interest rate loan. In many cases, do not stress over saving 0.5% to 1% on your auto loan.

In a short term loan, your payments will be significant. If you conceptually invest the loan proceeds, you have to liquidate your investments uniformly over time. E.g. for a 5 year loan, after 2 years, you have paid back $\approx 40\%(\frac{2}{5})$ of the loan and might have had to liquidate 40% of your investment. Your investment return is also reduced by $\rho$.

Example: On a $60K 5-year loan with an 8% interest rate, how much total interest $\mathit{To}{t}_{\mathit{Interest}}$ do you pay over the loan lifetime?

1.

The naive answer is $\mathit{NX}=\mathit{Principal}\times \mathit{Interest}\text{\_}\mathit{rate}\times \mathit{Duration}$. In this example, NX= $60K * 5 years * (8% / year) = $24K. This answer NX is wrong by a factor of almost 2X.

2.

The actual interest paid is roughly $0.53\times \mathit{NX}\approx \$12.8K$, where 0.53 is the adjustment denoted $\rho$.

The value $\rho$ is the average fraction of the remaining principal balance over the entirety of the loan. For short low-interest loans, most of your monthly payment is toward principal reduction and the amount of principal remaining is reduced roughly linearly over time. I use $\rho \approx 0.53$, which is fairly accurate for most car loans. (If the interest was 0%, the payments are all principal and $\rho =0.50$.) Define: the average lifetime interest rate = $\rho \times$interest rate.

Cutting to the chase, if you can reduce the interest by $\mathrm{\Delta }$, your savings are:

• Per year: $P\times \mathrm{\Delta }\rho =P\times \mathrm{\Delta }\times 0.53$. (E.g for P=$50K and $\mathrm{\Delta }$ = 1%, the annual savings is $50K * 1% * 0.53 $\approx$ $265.)
• Loan Lifetime: $P\times \mathit{duration}\times \mathrm{\Delta }\times \rho =P\times \mathit{duration}\times \mathrm{\Delta }\times 0.53$.