Investing and borrowing
| English | Chinese | Pinyin |
|---|---|---|
| present value | 现值 | xiàn zhí |
| annuity | 年金 | nián jīn |
| loan repayment | 贷款还款 | dài kuǎn hái kuǎn |
| nominal rate | 名义利率 | míng yì lì lǜ |
| effective rate | 实际利率 | shí jì lì lǜ |
A loan is a sequence you owe
- Borrowing 10,000 at 9% for three years does not cost 2,700. It costs what compounding makes it cost.
- Every loan and every investment in this unit is the same geometric machinery as unit 1, pointed at money you owe.
- The skill assessed is comparing two offers honestly, not computing one.
Present value
- Money later is worth less than money now, because money now could be earning.
- The present value 现值 of a future amount is $PV = \dfrac{FV}{(1+r)^n}$.
- It is the compound-interest formula rearranged, and it is how any two payments at different times are compared.
What is the present value of 11000 received in 2 years, at 5% a year? Give it to the nearest whole number.
11000 ÷ 1.05² = 9977. It is the compound formula rearranged.
Annuities and repayments
- An annuity 年金 is a fixed payment each period.
- A loan repayment 贷款还款 is an annuity paid to a lender: part interest, part principal, and the split shifts every month.
- Early payments are mostly interest, which is why paying a loan off early saves more than the remaining balance suggests.
In the early years of a loan, most of each repayment goes to interest rather than principal.
The balance is largest early, so the interest on it is too — which is why early repayment saves so much.
Which is cheaper: 10,000 at 9% compounded annually for 3 years, or at 8.7% compounded monthly?
The lower headline rate costs more, because it compounds 36 times instead of 3.
Comparing the headline numbers is the trap the question is built on. Comparing the totals is the method.
10000 at 9% compounded annually for 3 years, or at 8.7% compounded monthly. Which costs more?
12970 against 11950. The lower headline rate compounds 36 times instead of 3.
The APR exists so two loans can be compared. A nominal rate 名义利率 quoted per year but compounded monthly is not what you pay; the effective rate 实际利率 is. 12% compounded monthly is 12.68% effective, and any question that says "monthly" is testing this.
12% a year compounded monthly has an effective annual rate of about...
1.01¹² = 1.1268. Compounding more often makes the effective rate exceed the nominal one.
Read what the exponent counts. In $\left(1 + \frac{r}{12}\right)^{12n}$ the rate is divided by 12 and the exponent is multiplied by 12. Doing one without the other is the commonest arithmetic failure in the unit, and it produces an answer that looks plausible.
For 5 years at a monthly compounded rate, what number should the exponent be?
12 periods a year for 5 years. Dividing the rate by 12 without multiplying the exponent is the standard error.