1、CHAPTER 5,Introduction to Risk, Return, and the Historical Record 风险与收益入门及历史回顾,5-2,Interest Rate Determinants 利率水平的决定因素,Supply供给 Households家庭 Demand需求 Businesses企业 Governments Net Supply and/or Demand政府净供给或需求 Federal Reserve Actions美联储行为,5-3,Real and Nominal Rates of Interest 真实与名义利率,Nominal interes
2、t rate: Growth rate of your money 金钱增长 Real interest rate: Growth rate of your purchasing power 购买力增长,Let R = nominal rate, r = real rate and I = inflation rate. Then:,5-4,Equilibrium Real Rate of Interest 真实利率的均衡如何达成,Determined by: Supply供给 Demand需求 Government actions政府行为 Expected rate of inflation
3、通胀预期,5-5,Figure 5.1 Determination of the Equilibrium Real Rate of Interest,5-6,Equilibrium Nominal Rate of Interest 名义利率的均衡如何达成,As the inflation rate increases, investors will demand higher nominal rates of return通胀上升时投资者要求更高的名义利率 If E(i) denotes current expectations of inflation, then we get the Fi
4、sher Equation:费雪方程式 Nominal rate = real rate + inflation forecast,5-7,Taxes and the Real Rate of Interest 税收与真实利率,Tax liabilities are based on nominal income税赋是基于名义收入的 Given a tax rate (t) and nominal interest rate (R), the Real after-tax rate is: i是通胀率The after-tax real rate of return falls as the
5、inflation rate rises税后真实收益被通胀侵蚀,5-8,Rates of Return for Different Holding Periods比较不同持有期利率,Zero Coupon Bond, Par = $100, T=maturity, P=price, rf(T)=total risk free return,5-9,Example 5.2 Annualized Rates of Return 收益的年化率,5-10,Equation 5.7 EAR,EAR definition: percentage increase in funds invested ove
6、r a 1-year horizon投资1年所获得的收益,5-11,Equation 5.8 APR,APR: annualizing using simple interest 简单年化率,5-12,Table 5.1 APR vs. EAR两者相隔不大 前者计算简便，后者更为精确,5-13,Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2009短期国库券、通胀以及真实利率的统计数据,5-14,Bills and Inflation, 1926-2009通货膨胀与短期国库券,Modera
7、te inflation can offset most of the nominal gains on low-risk investments.低风险投资的大多数收益都会被通胀抵消 A dollar invested in T-bills from19262009 grew to $20.52, but with a real value of only $1.69.投资短期国库券名义收益1:20.52，真实收益只有1:1.69 Negative correlation between real rate and inflation rate means the nominal rate
8、responds less than 1:1 to changes in expected inflation.名义利率在实际上也没有随着预期通胀的上升完全上涨，这说明真实利率被侵蚀的更厉害,5-15,Figure 5.3 Interest Rates and Inflation, 1926-2009利率与通胀,5-16,Risk and Risk Premiums风险与溢价,HPR = Holding Period Return持有期 P0 = Beginning price初始价格 P1 = Ending price结束价格 D1 = Dividend during period one期
9、间红利,Rates of Return: Single Period单期收益,5-17,Ending Price = 110 Beginning Price = 100 Dividend = 4HPR = (110 - 100 + 4 )/ (100) = 14%,Rates of Return: Single Period Example 单期收益的例子,5-18,Expected returns,p(s) = probability of a state状态概率 r(s) = return if a state occurs状态下收益 s = state状态,Expected Return
10、 and Standard Deviation期望收益,5-19,State Prob. of State r in State Excellent .25 0.3100 Good .45 0.1400 Poor .25 -0.0675 Crash .05 -0.5200,E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675) + (0.05)(-0.52) E(r) = .0976 or 9.76%,Scenario Returns: Example案例,5-20,Variance (VAR):,Variance and Standard Deviati
11、on方差与标准差,Standard Deviation (STD):,5-21,Scenario VAR and STD如何计算,Example VAR calculation: 2 = .25(.31 - 0.0976)2+.45(.14 - .0976)2 + .25(-0.0675 - 0.0976)2 + .05(-.52 - .0976)2= .038 Example STD calculation:,5-22,Time Series Analysis of Past Rates of Return历史收益的时间序列分析,The Arithmetic Average of rate
12、of return: 收益率的算术平均值,5-23,Geometric Average Return几何平均值,TV = Terminal Value of the Investment最终投资价值,g= geometric average rate of return收益的几何平均值,5-24,Geometric Variance and Standard Deviation Formulas几何方差与标准差,Estimated Variance = expected value of squared deviations误差平方的期望,5-25,Geometric Variance and
13、 Standard Deviation Formulas几何方差与标准差公式,When eliminating the bias, Variance and Standard Deviation become:标准差公式,5-26,The Reward-to-Volatility (Sharpe) Ratio风险收益比：夏普比,Sharpe Ratio for Portfolios:组合的夏普比,5-27,The Normal Distribution 正态分布,Investment management is easier when returns are normal.如果收益正态分布投资
14、管理就比较轻松 Standard deviation is a good measure of risk when returns are symmetric.如果收益对称的话标准差是风险的一个好的度量 If security returns are symmetric, portfolio returns will be, too.如果证券收益对称那么组合的收益也是对称的 Future scenarios can be estimated using only the mean and the standard deviation.未来的收益就可以只用均值和方差估计出来,5-28,Figur
15、e 5.4 The Normal Distribution 正态分布,5-29,Normality and Risk Measures 正态性与风险测度,What if excess returns are not normally distributed?如果超额收益不是正态分布 Standard deviation is no longer a complete measure of risk标准差不再是风险的完全测度 Sharpe ratio is not a complete measure of portfolio performance夏普比也就不再是组合表现的完全测度 Need
16、to consider skew and kurtosis需要考虑的偏度和峰度,5-30,Skew and Kurtosis 偏度与峰度,Skew,Equation 5.19,Kurtosis,Equation 5.20,5-31,Figure 5.5A Normal and Skewed Distributions 正态与有偏分布,5-32,Figure 5.5B Normal and Fat-Tailed Distributions (mean = .1, SD =.2)正态与厚尾分布,5-33,Value at Risk (VaR)在险价值,A measure of loss most
17、frequently associated with extreme negative returns度量一定概率下发生极端负值所造成的损失 VaR is the quantile of a distribution below which lies q % of the possible values of that distribution在险价值的另一个名字是分布的分位数 The 5% VaR , commonly estimated in practice, is the return at the 5th percentile when returns are sorted from
18、 high to low.通常估计的5%的VaR是指收益从高到底排列后的，有95%的收益都会好过的那一个值。,5-34,Expected Shortfall (ES)预期损失,Also called conditional tail expectation (CTE)也叫做条件尾部期望 More conservative measure of downside risk than VaR比起在险价值是下侧风险更为保守的风险测度 VaR takes the highest return from the worst cases在险价值取所有最坏情况的最高值来表达风险 ES takes an av
19、erage return of the worst cases预期损失取所有最坏情况的平均值,5-35,Lower Partial Standard Deviation (LPSD) and the Sortino Ratio下标准偏差与Sortino比,Issues:分布不对称带来的问题 Need to consider negative deviations separately需要单独考察负值偏差 Need to consider deviations of returns from the risk-free rate.需要考察收益偏离无风险收益的水平 LPSD: similar to
20、 usual standard deviation, but uses only negative deviations from rf下标准和标准差的计算一样，不同的是它只用负收益为样本 Sortino Ratio replaces Sharpe Ratio在这种情况下我们用Sortino比率代替夏普比衡量风险收益比,5-36,Historic Returns on Risky Portfolios 风险组合的历史收益,Returns appear normally distributed收益通常是正态分布的Returns are lower over the most recent hal
21、f of the period (1986-2009)近年来收益降低了SD for small stocks became smaller; SD for long-term bonds got bigger小型股的标准差变小，但是长期债券的标准差变大,5-37,Historic Returns on Risky Portfolios 风险组合的历史收益,Better diversified portfolios have higher Sharpe Ratios分散较好的组合有较高的夏普比Negative skew右偏，或者说向负值偏斜,5-38,Figure 5.7 Nominal and
22、 Real Equity Returns Around the World, 1900-2000名义与真实收益,5-39,Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000股票与债券真实收益的标准差,5-40,Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal Distribution 投资经过25年后的概率结果：对数正态分布,5-41,Terminal Valu
23、e with Continuous Compounding,When the continuously compounded rate of return on an asset is normally distributed, the effective rate of return will be lognormally distributed.当连续复利正态分布时，有效有益是对数正态分布的，（最终资产价格） The Terminal Value will then be lognormally distributed.最终价值将是对数正态分布的,5-42,Figure 5.10 Annually Compounded, 25-Year HPRs投资复合收益分布（25年）,5-43,Figure 5.11 Annually Compounded, 25-Year HPRs投资复合收益分布（25年）,5-44,Figure 5.12 Wealth Indexes of Selected Outcomes of Large Stock Portfolios and the Average T-bill Portfolio 部分大盘股组合的财富指数和短期国库券组合,
