B9339 Homework Assignment #4

1.（夏普比率与最优资本配置之间的关系）在动量崩溃中，丹尼尔和
Moskowitz的“动态”策略基于他们为Sharpe比率和

𝑔（𝑤）∗𝑇

（a）假设现金一文不值（其收益为0），则以（a）的期望来近似𝑔（𝑤）
（适当选择的）随机变量。
（b）在Daniel和Moskowitz的推导中使用近似值来推导

[提示：您可能想对𝑥= 0周围的ln（1 +𝑥）使用二阶泰勒展开式。]
（c）用（b）的𝑔（𝑤）近似值得出最佳分配an

𝑔（𝑤）。
[提示：您的最佳分配应与Daniel和Moskowitz获得的分配一致。]
（d）使用最优分配𝑤

𝑔
∗ =𝑔（𝑤

），您的资金。您可以将此表达式识别为熟悉的指标吗？
（e）重复步骤（a）-（d），并假设现金在每个投资期间的利率为𝑟。
（f）在推导中，您在哪里使用了收益的独立性（连续期间）？

capital将您的资金分配给所有（全部）风险投资，以及您的资金分配

（g）使用您从上述（d）（或（e））得出的结果来论证您的分配决策减少到

[注意：对于这一部分，您的论点不取决于现金收益率。

Momentum Crashes（动量崩溃）为您提供了一个投资多种股票的框架（已知

2.（波动性：不仅不好）在这个问题上，你要弄清楚波动性既有害又有害

（a）得出每个投资期间的预期投资收益（以𝑎为单位）。给
quantity的此数量的值= 10％，30％，50％。这是有利的（不利的，中立的）

（b）推导每个投资期的投资波动率（以terms为单位）。赋予价值
for = 10％，30％，50％。波动率如何取决于𝑎？
（c）在您将全部资本投资于有风险的人之后，得出您的财富的中值

（d）使用以上问题1中（b）的答案得出
(f) Derive the expected return of the investment for each investment period (in terms of 𝑎). How
does the expected return depend on 𝑎? Is this a favorable (unfavorable, neutral) investment for
any one period, when 𝑎 = 100%?
(g) Derive the volatility of the investment for each investment period (in terms of 𝑎). How does
volatility depend on 𝑎? What is the volatility of this risky investment when 𝑎 = 100%?
(h) Derive the median value of your wealth after you have invested all your capital in the risky
investment for two successive periods. Do you expect long-term growth if you repeatedly invest
in this risky investment when 𝑎 = 100%?
Consider now a change to your investment process. Before each investment period, you split your
capital into two halves and you invest one half in the risky investment and the other half in cash, which
earns nothing (its return is 0).
(i) Derive the median value of your wealth after you have followed this investment process for two
successive periods. Do you expect long-term growth if you repeatedly follow this process when
𝑎 = 100%?
Replace now the cash investment in your process with a second risky investment, which is
independent of the first and has an identical return distribution. Before each investment period, you
split your capital into two halves and you invest each half in one of the two risky investments.
(j) Derive the median value of your wealth after you have followed this investment process for four
successive periods. Explain why we are using four successive, instead of two, to evaluate this
process. Do you expect long-term growth if you repeatedly follow this investment process when
𝑎 = 100%?
(k) Compare your answers in (j), (i) and (h). Which investment process is more attractive long-term?
(l) Explain how your comparison in (k) suggests that volatility can be beneficial.
3. (The 72 Rule) In this problem, you are to derive a rule of thumb that practitioners have been using for