32
5.6 正态分布 - The Normal Distribution
5-33
5.6 正态分布 - The Normal Distribution
34
Normality and Risk Measures
▪ What if excess returns are not normally distributed?
即：APR
(1 EAR )T
T
1
1 T APR 1 /T
11
Equation 5.8 年百分比率 - APR
▪
APR 1 EAR T 1
T
1 EAR 1 rf T 1T
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Table 5.1 APR vs. EAR
5-13
5.2.2 连续复利收益率
▪ 当T趋于无限小时，可得连续复利 (continuous compounding)概念
▪ 风险价值 – Value at Risk
2
5.1 利率水平的确定
利率水平的决定因素：
▪ 资金供给(居民) - Households ▪ 资金需求(企业) - Businesses
▪ 资金供求的外生影响(政府) - Government’s
Net Supply and/or Demand ➢Federal Reserve Actions
▪ If E(i) denotes current expectations of inflation, then we get the Fisher Equation:
▪ Nominal rate = real rate + inflation forecast
6
5.1.4 税收与实际利率
记税率为t，名义利率为R， 则税后名义利率为R(1 t) 税后真实利率为： R(1 t) i (r i)(1 t) i r(1 t) it 可见：税后实际利率随着通胀率的上升而下降
▪ rstockA-rf=excess return, or excess return=actual return – riskfree rate.
▪ The risk premium is the expected value of the excess return, then E(r)-rf=risk premium.
1
EAR
1
rf
T
1 T
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5.2.1 年百分比利率
短期投资利率常用年百分比利率
(APR，annual percentage rate)来表示，即若
一年为n
1 T
期，每期利率为rf(T
)，则有：
APR n rf(T )或rf(T ) T APR
更一般地，有：
1 EAR 1 rf(T )n 1 rf(T )1 /T
Rates of Return: Single Period Example
Ending Price =
110
Beginning Price = 100
Dividend =
4
HPR = (110 - 100 + 4 )/ (100) = 14%
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5.4.2 期望收益expected return与标 准差 standard deviation：E-V方法
利率
均衡的 真实利率
E’ ● E ●
供给
需求 均衡资金借出
资金
5
5.1.3 名义利率均衡 - Equilibrium Nominal Rate of Interest
费雪方程(Fisher equation)
R r E(i)
含义：名义利率应该随预期通胀率的增加而增加
▪ As the inflation rate increases, investors will demand higher nominal rates of return
We would like to know the trade-off between reward(the risk premium) and risk(as measured by standard deviation or SD)
Sharpe
Ratio
for
Portfolios
=
Risk Premium SD of Excess Return
σ2 = .25(.31 - 0.0976)2+.45(.14 - .0976)2 + .25(-0.0675 - 0.0976)2 + .05(-.52 .0976)2 = .038
▪ Example STD calculation: .038
.1949
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▪ 例：假定投资于某股票，初始价格1 0 0美元，持 有期1年，现金红利为4美元，预期股票价格由如 下三种可能，求其期望收益和方差。
1 EAR 1 rf (T) 1/T
8
Example 5.2 Annualized Rates of Return
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Equation 5.7 实际年利率 - EAR
▪ Effective annual rate definition: percentage increase in funds invested over a 1-year horizon
▪ 例：上例中我们得到股票的预期回报率E(r)为14 ％，若无风险收益率为rf8％。初始投资100元于 股票，其风险溢价(E(r)-rf)为6元，作为其承担风 险（标准差为21.2元）的补偿。
▪ 投资者对风险资产投资的满意度取决于其风险厌 恶(risk aversion)程度
28
5.5 历史收益率时间序列分析
Rates of Return: Single Period
HPR P1 P0 D1 P0
HPR = Holding Period Return P0 = Beginning price P1 = Ending price D1 = Dividend during period one
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▪ We are not sure about the eventual HPR, so we have to know the Probability Distribution of the future outcome.
▪ We will characterize PD in terms of their expected return E(r) and their standard deviation σ.
Prob. of State .25 .45 .25 .05
r in State 0.3100 0.1400 -0.0675 -0.5200
E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675) + (0.05)(-0.52)
E(r) = .0976 or 9.76%
1 EAR lim 1 T APR 1/T ercc T 0
即：1 EAR ercc rcc ln(1 EAR)
14
Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009
5-15
r(1) (140 100 4) /100 44%
25
σ=450^0.5=21.2132
26
5.4.3 超额收益与风险溢价 – Risk and Risk premiums
▪ Example: rf=6%, rstockA=14%, so what is 8% which equals to rstockA-rf?
5.5.1 时间序列与情景分析
We do not know the PD of future outcomes, as
well as their E(r) and σ. We must infer from
its history or time series in order to estimate
3
5.1.1 实际利率(real interest rate)与 名义利率(nominal interest rate)
消费者物价指数(CPI，consumer price index) ▪ Nominal interest rate(R): Growth rate of your
money ▪ Real interest rate(r): Growth rate of your
TV = 投资终值(Terminal Value of the Investment)
g TV 1/ n 1
g= 几何平均收益率(geometric average rate of return)
30
5.5.4 方差与标准差
▪
31
5.5.5 报酬-风险比率(夏普比率) The Reward-to-Volatility (Sharpe) Ratio
purchasing power
近似地看，有：r R i,
其中r为实际利率，R为名义利率，i为通胀率
严格上讲，有：1 r 1 R r R i
1 i
1 i
4
5.1.2 实际利率均衡 - Equilibrium Real Rate of Interest
四因素：供给、需求、政府行为和通胀率
记不确定情形的集合为s，p(s)为各情形的概率，
r(s)为各情形的HPR，E(r)为期望收益，为标准差
则有：E(r) p(s)r(s)
s
2 p(s)[r(s) E(r)]2
21
s
Scenario Returns: Example