第四章最优投资组合(实际)
第四章 最优投资组合（实际）
Estimation of the inputs a computer exercise using real data Implementation problem and some solutions Estimation results in uncertainty Shrinkage and Bayesian inference to deal with uncertainty Imposing restrictions
◼ Estimation of the inputs ◼ A computer exercise using real data ◼ Implementation problem and some solutions ◼ Estimation results in uncertainty ◼ Shrinkage and Bayesian inference to deal with uncertainty ◼ Imposing restrictions
1投资者的最优投资组合过程 ■投资者的最优投资组合过程可以分为三 步 ■数据的估计 ■寻找最优投资集 选择最优证券组合
1.投资者的最优投资组合过程 ◼ 投资者的最优投资组合过程可以分为三 步： ◼ 数据的估计 ◼ 寻找最优投资集 ◼ 选择最优证券组合
第一步:数据的估计 ■数据的估计:如何得到证券回报率的期 望值、方差、协方差 ■标准差相对稳定,利用历史数据估计 ■但期望值不一定 仅仅关注过去可能导致错误的结果 将来和过去可能完全不同
第一步：数据的估计 ◼ 数据的估计：如何得到证券回报率的期 望值、方差、协方差 ◼ 标准差相对稳定，利用历史数据估计 ◼ 但期望值不一定 ◼ 仅仅关注过去可能导致错误的结果 ◼ 将来和过去可能完全不同
期望值的估计方法 样本均值 在如下条件下合理 股票指标保持不变 将来的回报率和过去的回报率由同一个模型产生 否则,需要根据现在的环境调整 ■寻找特殊变量来预测回报率 ■寻找现在可以观测,可以预测将来回报率的变量 一般不容易找到 已有的几个例子:红利收益、红利价格比、市盈率、一些利 率变量 回归 例如 a +8 P
期望值的估计方法 ◼ 样本均值 ◼ 在如下条件下合理 ◼ 股票指标保持不变 ◼ 将来的回报率和过去的回报率由同一个模型产生 ◼ 否则，需要根据现在的环境调整 ◼ 寻找特殊变量来预测回报率 ◼ 寻找现在可以观测，可以预测将来回报率的变量 ◼ 一般不容易找到 ◼ 已有的几个例子：红利收益、红利价格比、市盈率、一些利 率变量 ◼ 回归 ◼ 例如 t t t p d r + = + −1
期望值的估计方法 ■利用CAPM、APT等模型
期望值的估计方法 ◼ 利用CAPM、APT等模型
A model weighting game in estimating expected returns(By Lubos Pastor) It is hard to overstate the importance of expected returns in investment. (money manager, corporate manager, ordinary consumer) Unfortunately, expected returns are as elusive as they are important. There is no absolute agreement among finance professionals on how expected returns should be estimated The best estimates are produced by combining finance theory with historical returns data and our own judgment
A model weighting game in estimating expected returns (By Lubos Pastor) ◼ It is hard to overstate the importance of expected returns in investment. (money manager, corporate manager, ordinary consumer). ◼ Unfortunately, expected returns are as elusive as they are important. There is no absolute agreement among finance professionals on how expected returns should be estimated. ◼ The best estimates are produced by combining finance theory with historical returns data and our own judgment
Does history repeat itself? Unless we suspect that expected return changes nontrivially over time, the sample average return is an unbiased estimator of expected return. The unbiasedness of the sample average return is its main advantage The main disadvantage of the sample average Is Its imprecision GM 1991-2000 the sample average return is 14% per year. The standard error is 10% per year. With 95% confidence, th le true expected return is within two standard errors of the sample average [-6%,34%]
Does history repeat itself? ◼ Unless we suspect that expected return changes nontrivially over time, the sample average return is an unbiased estimator of expected return. The unbiasedness of the sample average return is its main advantage. ◼ The main disadvantage of the sample average is its imprecision. ◼ GM: 1991-2000, the sample average return is 14% per year. The standard error is 10% per year. With 95% confidence, the true expected return is within two standard errors of the sample average: [-6%, 34%]
Does history repeat itself? Would the precision increase if we used Weekly instead of monthly data? although higher-frequency data helps in estimating variances and covariance of returns it does not help in estimating expected returns Intuitively, what matters for expected return is the beginning and ending levels of prices over a given period but not what happens in between
Does history repeat itself? ◼ Would the precision increase if we used weekly instead of monthly data? Although higher-frequency data helps in estimating variances and covariance of returns, it does not help in estimating expected returns. ◼ Intuitively, what matters for expected return is the beginning and ending levels of prices over a given period, but not what happens in between
Does history repeat itself? The only way to get a more precise sample average is to collect more data further back in time. But as we add older data, we gain recision at the expense of introducing potential bias GM: 1925-2000, historical average is 15.5% standard error is 3. 4%:8.7%,22.3%.The interval is still too wide for comfort Moreover, GM is very different from seventy years ago so the current estimate could be contaminated by old data
Does history repeat itself? ◼ The only way to get a more precise sample average is to collect more data further back in time. But as we add older data, we gain precision at the expense of introducing potential bias. ◼ GM: 1925-2000, historical average is 15.5%, standard error is 3.4%: [8.7%, 22.3%]. The interval is still too wide for comfort. ◼ Moreover, GM is very different from seventy years ago, so the current estimate could be contaminated by old data