Reading 6 The Time Value of Money
LEARNING OUTCOMES
The candidate should be able to:
A. interpret interest rates as required rates of return, discount rates, or opportunity costs;从回报率、贴现率以及机会成本的角度去理解利息;
B. explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk;将利率解释为实际无风险利率与溢价之和，以补偿投资者承担不同类型的风险；
C. calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;给定年利率和复利频率，计算并解释有效年利率；
D. solve time value of money problems for different frequencies of compounding;解决不同复利频率的货币时间价值问题；
E. calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;计算并解释单笔款项，普通年金，到期年金，永久年金（仅PV）和一系列不平等现金流量的终值（FV）和现值（PV）；
F. demonstrate the use of a time line in modeling and solving time value of money problems.演示在建模和解决货币时间价值问题中使用时间线。
G. An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. 利率是反应不同时间的现金流之间的关系。
利率可以从以下3个方面理解：
1、回报率required rates of return：一个投资者可以接受的最低回报率。
2、贴现率 discount rates：现值9500，一年后10000，则500/9500=5.26%即贴现率。所谓贴现，是指将一年后的10000折回现值9500的比率。结合1、2，可以知道回报率与贴现率是一个可以互换的概念（interchangeably terms）
3、机会成本 opportunity cost：即投资者为了拿到现在的9500而放弃将来获得10000的成本是5.26%
r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
r = f+i+d+l+m
r = FIDLM
即
利率r=无风险回报率+4个溢价
（通胀溢价+违约溢价+流动性溢价+到期风险溢价）
通货膨胀溢价（Inflation Premium）
预期通货膨胀率与当前的通货膨胀率不一定相等。例如，国家统计局发布最近的通货膨胀率为2%，目前发行的一年期国债利率为5%，则当前的无风险真实利率为5%-2%=3%，称为“当前真实利率”。在财务领域里，除非特别指明“通货膨胀率”，均指“预期的通货膨胀率”。财务领域的“真实利率”是根据预期通货膨胀率计算的。例如，预期未来一年通货膨胀率为4%，则上述债券的“真实利率”为5%-4%=1%。
无风险真实利率与通货膨胀率之和，称为“无风险名义利率”，并简称“无风险利率”。
无风险(名义)利率=无风险真实利率+通货膨胀溢价
通常，人们在说“无风险利率”时，如果前面没有“真实”作定语，则是指“无风险名义利率”(nominal risk-free interest rate)。这种习惯的形成，是因为人们可以观察到的现实中的利率，都是含有通货膨胀溢价的。经常使用的“无风险利率”，通常就是指无风险名义利率，即包含了通货膨胀溢价。而对于排除了风险和通货膨胀的利率，应称“无风险真实利率”，简称“真实利率”或“纯粹利率”。
原文：Many countries have governmental short-term debt whose interest rate can be considered to represent the nominal risk-free interest rate in that country. The interest rate on a 90-day US Treasury bill (T-bill), for example, represents the nominal risk-free interest rate over that time horizon. US T-bills can be bought and sold in large quantities with minimal transaction costs and are backed by the full faith and credit of the US government.
到期风险溢价（Maturity Risk Premium）
什么是到期风险溢价
到期风险溢价（Maturity Risk Premium）是指一般债权人可能偏好短期的债务，因此对愈长期的债券所要求的补偿愈多，同一种类的债券的长期及短期利率之差，即为到期风险溢价。
到期风险溢价主要是当未来利率上升时，长期债券价格会相应下降。所以到期期限越长，风险越大，到期风险溢价越高。
证券的到期日越长，其本金收回的不确定性越大，在此期间市场利率等其他因素不确定性也增多。因此证券随着到期日的增强给持有者带来的风险增大。为了弥补这个风险，证券发行人必须要给予一定的补偿。在一般情况下，短期债券的到期日低于长期债券的到期日，长期债券到期日低于股票到期日（股票无到期日，即到期日无穷大），因此短期债券的利息最低，长期债券高于短期债券利息，股票投资回报率超过债券投资回报率。
根据上述论述，我们得到如下结论：
r=r*+IP+DRP+LP+MRP
其中，r为某证券利率（投资报酬率），r*为无通货膨胀环境，无其他风险条件下的纯利率即无风险真实利率，IP为投资者要求的通货膨胀风险溢价，DRP为投资者要求的违约风险溢价，LP为投资者要求的流动性溢价，MRP为投资者要求的到期风险溢价。
The default risk premium compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.
The liquidity premium compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly. US T-bills, for example, do not bear a liquidity premium because large amounts can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position.
PV FV的计算
单一现金流投资即整笔（单一）投资
Single cash flow = lump-sum investment
PV = initial investment = 初始投资
r = 回报率（利率、贴现率）=一段时间内的利率（rate of interest per period）
FVN= future value 未来N期的价值
公式1
FV1 = PV(1 + r)
此时教材引入复利(compounded interest)概念，相对于每一期的简单利率(simple interest, 即每一期的利息/本金)，举的例子是美国的interest-bearing bank account（一种对储户的deposit支付利息的账户），举例利率是5%。
The most important point to remember about using the future value equation is that the stated interest rate, r, and the number of compounding periods, N, must be compatible. Both variables must be defined in the same time units. For example, if N is stated in months, then r should be the one-month interest rate, unannualized.
意思是，公式1中的利率r与期数N必须是同一口径的（compatible），月利率就按月计算，年利率就按年计算。
现值PV与FV的关系图，注意index是从0开始的，0=即期
future value factor = (1+r)N
注意此时教材抛出3个概念
· 在上图中添加金额
· 利率不变，期限越长则FV越大
· 期限不变，利率越高则FV越大
其中第一个概念，在任意index（期限）中增加金额并计算是教材的考查重点（拿手好戏），这叫做interim cash reinvested.
The Frequency of Compounding 复利周期
这里引入3个概念：
· 真实年利率（effective annual rate）EAR
· 名义年利率（stated annual rate）rs
· 复利周期(frequency of compounding)
In this section, we examine investments paying interest more than once a year. For instance, many banks offer a monthly interest rate that compounds 12 times a year. In such an arrangement, they pay interest on interest every month. Rather than quote the periodic monthly interest rate, financial institutions often quote an annual interest rate that we refer to as the stated annual interest rate or quoted interest rate. We denote the stated annual interest rate by rs. For instance, your bank might state that a particular CD pays 8 percent compounded monthly. The stated annual interest rate equals the monthly interest rate multiplied by 12. In this example, the monthly interest rate is 0.08/12 = 0.0067 or 0.67 percent. This rate is strictly a quoting convention because
(1 + 0.0067)12 = 1.083, not 1.08; the term (1 + rs) is not meant to be a future value factor when compounding is more frequent than annual.
通常金融机构的利率报价都是名义年利率，考虑到周期，名义年利率不能作为投资收益的真实参考。
例：某银行报出的某金融产品利率为8%，季度付息，产品的实际年利率是多少？
复利周期＝4，每期利率为8%／4=2%。
所以实际年利率为： [公式见下方] =8.24%
也就是说，如果储户年初投资$10,000，年末他将获得$10,824，而不是$10,800。
With more than one compounding period per year, the future value formula can be expressed as
mN= number of compounding periods = 期数(m) 
年(N)
年(N)
期数(frequency)越多，实际利率(EAR, effective annual rate)越高，终值(FV)上升，现值(PV)下降
EAR =
注意得出来的是年利率
此利率（EAR）+1后即可与PV相乘，得到FV。注意如果多于一年，则指数为期数*年数
此公式的根本原因见教材的summary，即教材认为名义利率并不是真正意义上的复利，而是需要还原回单期利率，再套入基本终值公式。
The stated annual interest rate is a quoted interest rate that does not account for compounding within the year.
The periodic rate is the quoted interest rate per period; it equals the stated annual interest rate divided by the number of compounding periods per year.
教材举例：
Continuing with the CD example, suppose your bank offers you a CD with a two-year maturity, a stated annual interest rate of 8 percent compounded quarterly, and a feature allowing reinvestment of the interest at the same interest rate. You decide to invest $10,000. What will the CD be worth at maturity?
两年期定期存单，名义利率为8%，复利频率为按季，允许按照同样利率投资所得利息，初始投资本金为一万美金，则到期后存单价值多少？
