Investing and borrowing
Introduced| English | Meaning |
|---|---|
| present value/ˈprezənt ˈvæljuː/ | The value today of a future amount or series of payments, calculated using a specified discount rate. |
| annuity/əˈnjuːɪti/ | In financial mathematics, a series of equal payments made at regular time intervals. |
| loan repayment/ləʊn rɪˈpeɪmənt/ | A payment towards money borrowed, generally including principal and any interest due under the loan terms. |
| effective rate/ɪˈfektɪv reɪt/ | effective rate |
| nominal rate/ˈnɒmɪnl reɪt/ | nominal rate |
State what happens to the balance
- Compound growth without payments uses $A=P(1+r)^n$. A loan with repayments generally needs a changing-balance rule instead.
- For this practice convention, add period interest before subtracting the payment: $B_n=B_{n-1}(1+r)-M$. A different payment timing changes the calculation.
Discount a future amount under a stated model
- Present value 现值 uses $PV=FV/(1+r)^n$ for the specified discount rate and periods. It is a modelled equivalent, not an unconditional claim that all future money has one value today.
- With $FV=11000$, annual rate 5% and two years, $PV=11000/(1.05)^2\approx9977.32$. Round only as requested after calculating.
What is the present value of 11000 received in 2 years, at 5% a year? Give it to the nearest whole number.
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11000 ÷ 1.05² = 9977. It is the compound formula rearranged.
Payment schedules need explicit conditions
- An annuity 年金 has specified payments at regular intervals. A loan repayment 贷款还款 may include interest and principal, with an adjusted final payment to clear the balance.
- Interest need not be most of an early payment. At opening balance 1000, period rate 2% and payment 300, interest is 20 and principal reduction is 280.
For every interest-bearing loan, most of each early repayment necessarily goes to interest rather than principal.
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The split depends on balance, period rate and payment. With balance 1000, rate 2% and payment 300 after interest, only 20 goes to interest and 280 reduces principal.
Correct a rate comparison. Under a no-interim-payment model, 10000 at 9% annually for three years gives $A_A=10000(1.09)^3=12950.29$. A nominal 8.7% annual rate with monthly rate $0.087/12$ gives $A_B=10000(1+0.087/12)^{36}\approx12970.06$. B is higher by about 19.77 units under these terms. A regular-repayment loan needs its own schedule instead of this model.
Under a no-interim-payment model, compare 10000 at 9% compounded annually for 3 years with a nominal 8.7% annual rate compounded monthly for 3 years. Which final amount is higher?
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Annual compounding gives 12950.29; monthly compounding gives about 12970.06. The second is higher by about 19.77 under the stated no-payment and no-fee model.
A nominal and effective rate describe different calculations. A nominal rate 名义利率 of 12% with monthly rate 1% gives effective rate 实际利率 $(1.01)^{12}-1\approx12.68\%$ annually. Sheet 2.3 specifies every period rate and payment rather than relying on a headline label or a real lending convention.
12% a year compounded monthly has an effective annual rate of about...
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1.01¹² = 1.1268. Compounding more often makes the effective rate exceed the nominal one.
With a stated nominal annual rate split into 12 monthly periods, divide that rate by 12 and multiply years by 12. Do not use this conversion for a rate already specified as effective annually without deriving its monthly equivalent.
For 5 years at a monthly compounded rate, what number should the exponent be?
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12 periods a year for 5 years. Dividing the rate by 12 without multiplying the exponent is the standard error.