Interest Rate

#Simple interest(单利)

Interest earned on only the principal amount during each interest period.

仅用本金计算利息，而不计算利息所产生的利息。

🌰如果有n个计息周期，周期利率为 i，本金为 P，则到投资期末，本利和 F 为：$F=P(1+i\times n)$

#Compound interest(复利)

The interest earned in each period is calculated based on the total amount at the end of the previous period.

除了本金的计算外，还要计算利息所产生的利息。

🌰如果有 n 个计息周期，周期利率为 i，本金为 P，则到投资期末，本利和 F 为：$F=P(1+i)^n$

#Cyclic Interest Rate(周期利率)

Cyclic interest rate refers to the interest rate which is calculated more than once in a year, and the average interest rate is the cyclic interest rate.

周期利率是指一年内计息超过一次以上，平均每次计息的利率就是周期利率。

$i=r/n$，$i$ 为周期利率，$r$ 为名义利率，$n$ 为一年内的计息周期(interest period)数

#Nominal Interest Rate(名义利率)

Interest rate quoted based on an annual period---annual percentage rate(APR) 年百分率

#Effective Interest Rate(实际利率)

Actual interest earned or paid in a year or some other time period.

如果计息周期和支付周期一致，$i_a=(1+r/n)^n-1=(1+i)^n-1$，$i_a$ 为实际利率，$r$ 为名义利率，$i$ 为周期利率

如果一年有 K 个支付周期 (payment period)，每个支付周期内有 C 个计息周期 (interest period)，那么一个支付周期内的实际利率为 $i_p = [ 1 + r / C K ] ^ { C } - 1$，当 C 特别大趋近于无穷时，$i_p = e^{r/K}-1$

#Equivalence Analysis using Effective Interest Rates

#Two concepts

payment period: 支付周期

interest period: 计息周期

#Steps

In all financial analysis, we need to convert the APR into an appropriate effective interest rate based on a payment period.(which is the interest amount accumulated over a given period).

• Identify the payment period (e.g., annual, quarter, month, week, etc)
• Identify the interest period (e.g., annually, quarterly, monthly, etc)
• Find the effective interest rate that covers the payment period.

#Example

1. Suppose your savings account pays 9% interest compounded quarterly. If you deposit ￥10,000 for one year, how much would you have?

   $i = \frac { 9 \% } { 4 } = 2.25 \%$

   $i _ { a } = ( 1 + 0.0225 ) ^ { 4 } - 1 = 9.31 \%$

   $F= 10,000 (F/P,2.25 \% ,4)=10,000(F/P,9.31 \% ,1) = 10,931$

2. A series of equal quarterly payments of ￥5,000 for 10 years is equivalent to what present amount at an interest rate of 9% compounded. (a) quarterly (b) monthly (c) continuously

   (a) Payment period: Quarterly; Interest Period: Quarterly $i = \frac { 9 \% } { 4 } = 2.25 \%$ per quarter

   $N = 40 quarters$

   $P = 5,000 ( P / A , 2.25 \% , 40 ) = 130,968$

   (b) Payment period: Quarterly; Interest Period: Monthly $i = \frac { 9 \% } { 12 } = 0.75 \%$ per month

   $i _ { p } = ( 1 + 0.0075 ) ^ { 3 } = 2.267 \%$ per quarter

   $N = 40 quarters$

   $P = 5,000 ( P / A , 2.267 \% , 40 ) = 130,586$

   (c) Payment period: Quarterly; Interest Period: Continuously

   $i = e ^ { 0.09 / 4 } - 1 = 2.276 \%$ per quarter

   $N = 40 quarters$

   $P = 5,000 ( P / A , 2.276 \% , 40 ) = 130,384$