投资学第5章利率史与风险溢价1student

▪ 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.
them.
5.5.2 期望收益与算术平均
收益率的算术平均数 arithmetic average of
rates of return：
E(r) n p(s)r(s) 1 n r(s)
s 1
n s1
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5.5.2 几何收益率
Geometric Average Return
TVn (1 r1)(1 r2 )(1 rn )
▪ 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.
▪ If E(i) denotes current expectations of inflation, then we get the Fisher Equation:
▪ Nominal rate = real rate + inflation forecast
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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
1 EAR lim 1 T APR 1/T ercc T 0
即：1 EAR ercc rcc ln(1 EAR)
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Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009
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5.6 正态分布 - The Normal Distribution
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5.6 正态分布 - The Normal Distribution
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Normality and Risk Measures
▪ What if excess returns are not normally distributed?
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5.2 持有期收益率
Zero Coupon Bond, Par = $100, T=maturity, P=price, rf(T)=total risk free return
若T为持有期，P(T )为买入价格， 则贴现债券的持有期收益率为：
rf
(T )
100 P(T )
1
折为有效年利率(EAR, effectiveannual rate)，则为：
5.4.1 持有期收益 holding period return
▪ 股票收益包括两部分：红利收益(dividends) 与资本利得(capital gains)
▪ 持有期收益率(holding-period return)
HPR
股票期末价格 - 期初价格 期初价格
现金红利
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Risk and Risk Premiums
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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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
即：APR
(1 EAR )T
T
1
1 T APR 1 /T
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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
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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
利率
均衡的 真实利率
E’ ● E ●
供给
需求 均衡资金借出
资金
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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
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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
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%
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Variance and Standard Deviation
Variance (VAR):
2 p(s)r(s) E(r)2
s
Standard Deviation (STD):
STD 2
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Scenario VAR and STD
▪ Example VAR calculation:
投资学 第5章
历史数据中的收益与风险
Introduction to Risk, Return, and the Historical Record
本章主要内容
▪ 利率水平的确定 - Interest Rate Determinants
▪ 期望收益与波动性 – Expected Return and its Variance
▪ We measure the return of an investment with its E(r), we measure the risk of an investment with its risk premium’s standard deviation.
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5.4.3 超额收益与风险溢价 – Risk and Risk premiums
Figure Biblioteka Baidu.3 Interest Rates and Inflation, 1926-2009
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Figure 5.4 Nominal and Real Wealth Indexes for Investment in Treasury Bills,
1966-2005
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5.4 风险和风险溢价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
▪ 风险价值 – Value at Risk
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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
r(1) (140 100 4) /100 44%
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σ=450^0.5=21.2132
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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?
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)
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5.5.4 方差与标准差
▪
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5.5.5 报酬-风险比率(夏普比率) The Reward-to-Volatility (Sharpe) Ratio
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
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
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s
Scenario Returns: Example
State Excellent Good Poor Crash
1 EAR 1 rf (T) 1/T
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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
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5.5 历史收益率时间序列分析
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
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。
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
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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