Investing and borrowing · Investissements et emprunts
| English | Français |
|---|---|
| present value/ˈprezənt ˈvæljuː/ | valeur actualisée |
| annuity/əˈnjuːɪti/ | annuité |
| loan repayment/ləʊn rɪˈpeɪmənt/ | remboursement de prêt |
| effective rate/ɪˈfektɪv reɪt/ | taux effectif |
| nominal rate/ˈnɒmɪnl reɪt/ | taux nominal |
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 · utilise $PV=FV/(1+r)^n$ 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. · Quelle est la valeur actuelle de 11000 reçue dans 2 ans, à un taux de 5 % par an ? Arrondissez au nombre entier le plus proche.
11000 ÷ 1.05² = 9977. It is the compound formula rearranged. · 11000 ÷ 1,05² = 9977. C'est la formule de capitalisation réarrangée.
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. · Pour tout prêt productif d'intérêts, la majeure partie de chaque remboursement anticipé va nécessairement aux intérêts plutôt qu'au capital.
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. · La répartition dépend du solde, du taux par période et du paiement. Avec un solde de 1000, un taux de 2 % et un paiement de 300 après intérêt, seuls 20 vont aux intérêts et 280 réduisent le capital.
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? · Selon un modèle sans paiement intermédiaire, comparez 10000 à 9 % capitalisés annuellement pendant 3 ans avec un taux nominal de 8,7 % annuel capitalisé mensuellement pendant 3 ans. Quel montant final est le plus élevé ?
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. · Le capitalisation annuelle donne 12950,29 ; le capitalisation mensuelle donne environ 12970,06. Le second est supérieur d'environ 19,77 selon le modèle stipulé sans paiement et sans frais.
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... · 12 % par an capitalisés mensuellement ont un taux effectif annuel d'environ...
1.01¹² = 1.1268. Compounding more often makes the effective rate exceed the nominal one. · 1,01¹² = 1,1268. Capitaliser plus souvent fait dépasser le taux effectif au taux nominal.
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? · Sur 5 ans avec un taux capitalisé mensuellement, quel doit être l'exposant ?
12 periods a year for 5 years. Dividing the rate by 12 without multiplying the exponent is the standard error. · 12 périodes par an sur 5 ans. Diviser le taux par 12 sans multiplier l'exposant est l'erreur classique.