本文深入探讨了货币的时间价值概念及其对财务管理的重要性，并详细解释了利率的不同解释方式及组成部分，包括实际无风险利率、通货膨胀溢价等。此外，还介绍了单利与复利的区别，以及如何计算现值与未来值。
The Time Value of Money
6.1 利率
6.1.1 货币的时间价值
The cash flow additivity principle refers to the fact that present value of any stream of cash flows equals the sum of the present values of the cash flows.
additivity: 添加；相加性
refer to: 参考；涉及；指的是；适用于
stream: 流
6.1.2 利率的三种解释方式
一、要求回报率（Required Rate of Return）
二、折现率（Discounted Rate）
三、机会成本（Opportunity Cost）
6.1.3 利率的组成
利 率 ( r e q u i r e d i n t e r e s t r a t e o n a s e c u r i t y ) = 实 际 无 风 险 利 率 ( r e a l r i s k − f r e e r a t e ) + 预 期 通 货 膨 胀 率 ( e x p e c t e d i n f l a t i o n r a t e ) + 违 约 风 险 溢 价 ( d e f a u l t r i s k p r e m i u m ) + 流 动 性 风 险 溢 价 ( l i q u i d i t y r i s k p r e m i u m ) + 期 限 风 险 溢 价 ( m a t u r i t y r i s k p r e m i u m ) 利率(required \ interest \ rate \ on \ a \ security) = 实际无风险利率(real \ risk-free \ rate) + 预期通货膨胀率(expected \ inflation \ rate) + 违约风险溢价(default \ risk \ premium) + 流动性风险溢价(liquidity \ risk \ premium) + 期限风险溢价(maturity \ risk \ premium) 利率(required interest rate on a security)=实际无风险利率(real risk−free rate)+预期通货膨胀率(expected inflation rate)+违约风险溢价(default risk premium)+流动性风险溢价(liquidity risk premium)+期限风险溢价(maturity risk premium)
security: 证券；抵押品
inflation: 膨胀；通货膨胀；
一、实际无风险利率（Real Risk-Free Interest Rate）
二、通货膨胀溢价（Inflation Premium）
在银行查到的利率都是名义利率。
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名义利率(nominal \ risk-free \ rate) \approx 实际利率(real \ risk-free \ rate) + 预期通货膨胀率(expected \ inflation \ rate)
名义利率(nominal risk−free rate)≈实际利率(real risk−free rate)+预期通货膨胀率(expected inflation rate)
三、违约风险溢价（Default Risk Premium）
四、流动性风险溢价（Liquidity Premium）
五、期限风险溢价（Maturity Premium）
六、注意
- Differences between real and nominal interest rate: whether inflation rate is added
- Differences between risk-free and risky rate: whether risk premium is added
6.1.4 不同计息方式的利率
一、单利与复利（Single Interest vs Compound interest or Interest on interest）
二、报价利率与有效年利率
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报价利率
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有效年利率（Effective Annual Rate or Effective Annual Yield）
E A R = ( 1 + p e r i o d i c r a t e ) m – 1 p e r i o d i c ( e f f e c t i v e ) r a t e = s t a t e d a n n u a l r a t e / m m = t h e n u m b e r o f c o m p o u n d i n g p e r i o d s p e r y e a r EAR = (1 + periodic \ rate)^m – 1 \\ periodic \ (effective) \ rate = stated \ annual \ rate/m \\ m = the \ number \ of \ compounding \ periods \ per \ year EAR=(1+periodic rate)m–1periodic (effective) rate=stated annual rate/mm=the number of compounding periods per year
periodic rate: 周期性利率
stated annual rate: 名义年利率
6.2 现值（Present value）与未来值（Future value）
6.2.1 现值与未来值的关系
F V = P V ∗ ( 1 + I / Y ) N P V = a m o u n t o f m o n e y i n v e s t e d t o d a y ( t h e p r e s e n t v a l u e ) I / Y = r a t e o f r e t u r n p e r c o m p o u n d i n g p e r i o d N = t o t a l n u m b e r o f c o m p o u n d i n g p e r i o d s FV = PV * (1 + I/Y)^N \\ PV = amount \ of \ money \ invested \ today \ (the \ present \ value) \\ I/Y = rate \ of \ return \ per \ compounding \ period \\ N = total \ number \ of \ compounding \ periods \\ FV=PV∗(1+I/Y)NPV=amount of money invested today (the present value)I/Y=rate of return per compounding periodN=total number of compounding periods
P V = F V ∗ [ 1 ( 1 + I / Y ) N ] = F V ( 1 + I / Y ) N F V = a m o u n t o f m o n e y i n t h e f u t u r e ( t h e f u t u r e v a l u e ) I / Y = r a t e o f r e t u r n p e r c o m p o u n d i n g p e r i o d N = t o t a l n u m b e r o f c o m p o u n d i n g p e r i o d s PV = FV * [\frac{1}{(1 + I/Y)^N}] = \frac{FV}{(1 + I/Y)^N} \\ FV = amount \ of \ money \ in \ the \ future \ (the \ future \ value) \\ I/Y = rate \ of \ return \ per \ compounding \ period \\ N = total \ number \ of \ compounding \ periods PV=FV∗[(1+I/Y)N1]=(1+I/Y)NFVFV=amount of money in the future (the future value)I/Y=rate of return per compounding periodN=total number of compounding periods
- For a given discount rate, the farther in the future the amount to be received, the smaller that amount’s present value.
- Holding time constant, the larger the discount rate, the smaller the present value of a future amount.
- The stated rate and the actual (effective) rate of interest are equal only when interest is compounded annually.
- The greater the compounding frequency, the greater the EAR will be in comparison to the stated rate.
When continuous compounding, for a single cash flow：
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FV = PV * e^{r_sN} \\ r_s: the \ (nominal) \ continuous \ compounding \ interest \ rate
FV=PV∗ersNrs:the (nominal) continuous compounding interest rate
E A R = e r s − 1 EAR = e^{r_s} - 1 EAR=ers−1
frequency: 频率；频繁
comparison: 比较；对照；
continuous: 连续的，持续的；继续的；连绵不断的
6.2.2 年金
- An annuity is a stream of equal cash flows that occurs at equal intervals over a given period.
annuity: 年金
interval: 间隔；间距；
一、普通年金（Ordinary Annuity）
- ordinary annuities: cash flows occur at the end of each compounding period.
二、先付年金（Annuity Due）
- annuities due: payments or receipts occur at the beginning of each period.
- With an annuity due, there is one less discounting period since the first cash flow occurs at t = 0 and thus is already its PV.
- This implies that, all else equal, the PV of an annuity due will be greater than the PV of an ordinary annuity.
- Two ways to find annuities due:
- put the calculator in the BGN mode and then input all the relevant variables;
- multiply the resulting PV by [1 + periodic compounding rate (I/Y)] - This is the same for FV of Annuities Due.
receipts: （企业、银行等）收到的款，进款；收到( receipt的名词复数 )；收入；收据；收条
imply: 意味；暗示；隐含
relevant: 相关的；切题的；中肯的；有重大关系的；有意义的，目的明确的
variables: [数] 变量
multiply: 乘
三、延期年金（Deferred Annuity）
- deferred annuity: payments or receipts start in the future period.
四、永续年金（Perpetuity Annuity）
- perpetuity: a perpetual annuity, or a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now (year end).
P V p e r p e t u i t y = P M T I / Y PV_{perpetuity} = \frac{PMT}{I/Y} PVperpetuity=I/YPMT
sequential: 连续的；相继的；有顺序的
五、不规则现金流的现值与未来值
- unequal cash flows: To evaluate a cash flow stream that is not equal from period to period - FV
- unequal cash flows: To evaluate a cash flow stream that is not equal from period to period - PV
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