1) Diversification Problem
You are in charge of investing 10,000 dollars and you are considering three different investment strategies.
- Strategy 1: Invest the entire 10, 000 in a single asset. You believe that there is a 10% chance that you may lose your entire investment and a 90% chance that you will double your money.
- Strategy 2: Distribute the 10, 000 evenly across 10 different assets, that is, to invest 1,000 in each of 10 different assets. You believe that all 10 investments are statistically independent of each other. You also believe that for each individual asset there is a 10% chance that you may lose your entire investment and a 90% chance that you will double your money.
- Strategy 3: In each of five days, you may invest 2,000. You believe that the investments each day are statistically independent of each other. You also believe that for each individual asset there is a 10% chance that you may lose your entire investment and a 90% chance that you will double your money.
Questions:
1. What is the expected net payoff for Strategy 1, 2 and 3? (For strategy 3, assume you accept all bets with positive expected return)
2. What is the probability you lose all your money following strategy 2?
2) BAYES Probability
1. You believe that 1% of mutual fund managers can "beat the market" and deliver a positive mean daily return of .5%, also with standard deviation of 1%.
The remaining 99% of funds cannot beat the market; they all have identical daily mean return of 0% and standard deviation of 1%.
A broker calls you and claims his fund is one that can beat the market. You follow the fund for the day and see that today it has a positive (greater than 0%) return.
Use Bayes' Theorem to compute the probability that this fund actually can beat the market.
[Hint: this is a base rate question. You want to calculate "the probability the fund can beat the market conditional on having above average returns."
What you are given is "the probability of having above average returns conditional on the fund being able to beat the market." What is missing is the base rate.]
Answer: Probability of having above average returns conditional on the fund being able to beat the market is: %
Label the events and present them in the correct formula.
Using the means and standard deviations above, you can compute the conditional probability that a fund will have a positive return conditional on it being of the type that beats the market or of the normal type.
3)Bond Rating Geometric distribution.
A firm with a BBB bond rating has a 10.29 percent chance of default over a period of 40 years. The conditional probability of default (hazard rate) p is the same in all years. Find p.
4) Mean and Variance Flaw of Averages; data tables.
An investor puts 15, 000 dollars into each of four stocks, labeled A, B, C, and D. The table shown below contains the means and standard deviations of the annual returns of these four stocks.
|
Stock
|
Mean Annual Return μ
|
Standard Deviation σ
|
|
A
|
0.15
|
0.05
|
|
B
|
0.18
|
0.07
|
|
C
|
0.14
|
0.03
|
|
D
|
0.17
|
0.06
|
Questions:
1. Assume that the returns of these four stocks are independent of each other. Use a spreadsheet to calculate the mean and standard deviation of the total amount that this investor earns in one year from these four investments as a function of the information in the table.
2. Let v denote a market volatility index. The standard deviation of a stock n is now vσn, where σn is its base volatility level. The volatility index impacts all stocks in the same way. ?If v = 1 then the SD's of the four stocks A,B,C,D are as shown in the table. In general, v can be lower than 1 (low volatility) or higher.
For example, if v = 1.1, then stock A has volatility 1.1 × 0.05, stock B has volatility 1.1 × 0.07, and so on.
Modify the spreadsheet above to accommodate the possibility of v ?= 1. Use Excel's data table (one dimensional) to check the sensitivity of the portfolio's SD as a function of v. Vary v from 0.50 to 1.50 in 0.05 increments.