WEEK 3 and WEEK 4: WORKSHOPS
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This workshop is to be completed during Week 3 and Week 4 workshops and will depend on how quickly we get through the lecture material.

Part A: Week 3
Exercise 4.2, Exercise 4.4.
Exercise 4.5, Exercise 4.6, Exercise 4.7. Your tutor will discuss these 3 questions with you in the class. Exercise 4.8, Exercise 4.9.
Exercise 4.10, Exercise 4.12, Exercise 4.14.
Exercise 4.15, Exercise 4.16, Exercise 4.18.

Part B: Week 3 or 4
Exercise 5.1. Now what is the expected value and variance of Y=5X-3 for each distribution? (Hint: Use the ‘Summary of the Laws of Expected Value and Variance’ slide in the lecture notes.) Exercise 5.3. Now assume the manager receives a daily salary of $200 plus $85 per car sold. What is the expected value and standard deviation of her salary? (Hint: Use the ‘Summary of the Laws of Expected Value and Variance’ slide in the lecture notes.) Exercise 5.4. Now assume the company managing the network gets penalised $900 for each interruption. Calculate the expected daily penalty. Calculate the standard deviation of the daily penalty. (Hint: Use the ‘Summary of the Laws of Expected Value and Variance’ slide in the lecture notes.)

...Mean-Variance Analysis
Mean-variance portfolio theory is based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return. Markowitz called this approach to portfolio formation mean-variance analysis. Mean-variance analysis is based on the following assumptions:
1. All investors are risk averse; they prefer less risk to more for the same level of expected return.
2. Expected returns for all assets are known.
3. The variances and covariances of all asset returns are known.
4. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes.
5. There are no transaction costs or taxes.
The Mean-Variance Approach
The mean-variance theory postulated that in determining a strategic asset allocation an investor should choose from among the efficient portfolios consistent with that investor’s risk tolerance amongst other constraints and objectives. Efficient portfolios make efficient use of risk by offering the maximum expected return for specific level of variance or standard deviation of return. Therefore, the asset returns are considered to be normally distributed. Efficient portfolios plot graphically on the efficient frontier, which is...

...Sub: Finance Question:
Calculation of variance of portfolio.
Topic: Portfolio management
ClassOf1 provides expert guidance to College, Graduate, and High school students on homework and assignment problems in Math, Sciences, Finance, Marketing, Statistics, Economics, Engineering, and many other subjects.
Suppose there are three risky assets, A, B and C with the following expected returns, standard deviations of returns and correlation coefficients. E (rA)= 4% E (rB)=5% E (rC) =15% S.DEVA=5% S.DEVB=7% S.DEVC=10%
A, B=0.7 A, C=-0.2 B,C=0.3
QUESTION 1: Solving for the Global Minimum Variance Portfolio Consider a world where there are no risk free assets, and just these three risky assets. Suppose short sales are permitted. Solve for the weights and variance of the global minimum variance portfolio. If short sales are not permitted is the solution affected?
Solution:
Global Minimum Variance Portfolio is that set of portfolios that will provide the minimum level of risk for a given level of expected return. Given a world with just the three given risk assets we can use the Solver function in Excel to ascertain the weights and variance of the global minimum variance portfolio. We have to calculate the Variance Covariance matrix for the given set of assets. While variance is given by the square of Standard Deviation and is thus entered...

...Case on Mean-Variance Frontiers
1. Ignoring the risk-free asset, draw the frontier in mean-std space.
We solve the problem by Matlab:
clear; clc;
% input data
temp = xlsread('30_Industry_Portfolios');
ret = temp(:,2:end)/100; （this step is to get all the returns from the file）
rf = 0.01/100; (The risk free rate is rf =0.01%= 0.0001 per month.)
% compute moments
er = (mean(ret))'; (the (30.1)vector of returns on the 30 industries)
V = cov(ret); (the covariance matrix of the returns)
% draw the frontier
reqrets = 0.00:0.001:0.02;
for i = 1:length(reqrets)
[trash, rfront(i), varfront(i)] = frontierp(reqrets(i),V,er);
end
figure (1);
plot (varfront.^.5,rfront);
title ('Mean-Std Frontier');
ylabel('E[ret]'); xlabel('\sigma(ret)');
2. Now also consider the risk-free rate. Draw the eﬃcient frontier (do it on the same ﬁgure as 1).
% now add frontier with rf
for i = 1:length(reqrets)
[trash, rfront(i), varfront(i)] = rffrontierp(reqrets(i),V,er,+rf);
end
hold on;
plot(varfront.^.5,rfront);
hold off
3. Compute the tangency portfolio and plot it in the ﬁgure. Write the expected return and standard deviation of the portfolio.
clear;
% input data
temp = xlsread('30_Industry_Portfolios');
ret = temp(:,2:end)/100;
rf = 0.01/100;
% compute moments
er = (mean(ret))';
V = cov(ret);
>> one = ones(T,1); % unit vector
>> T = length(er)
T =
30
>> one = ones(T,1); % unit vector
>> A = one'*inv(V)*er;
>> C = one'*inv(V)*one
C =...

...13. Variance and Standard Deviation (expected). Using the data from problem 13, calculate the variance and standard deviation of the three investments, stock, corporate bond, and government bond. If the estimates for both the probabilities of the economy and the returns in each state of the economy are correct, which investment would you choose considering both risk and return? Why?
ANSWER
Variance of Stock = 0.10 x (0.25 – 0.033)2 + 0.15 x (0.12 – 0.033)2
+ 0.50 x (0.04 – 0.033)2 + 0.25 x (-0.12 – 0.033)2
= 0.10 x 0.0471 + 0.15 x 0.0076 + 0.50 x 0.0000 + 0.25 x 0.0234
= 0.0047 + 0.0011 + 0.0000 + 0.0059 = 0.0117 or 1.17%
Standard Deviation of Stock = (0.0117)1/2 = 0.1083 or 10.83%
Variance of Corp. Bond = 0.10 x (0.09 – 0.052)2 + 0.15 x (0.07 – 0.052)2
+ 0.50 x (0.05 – 0.052)2 + 0.25 x (0.03 – 0.052)2
= 0.10 x 0.0014 + 0.15 x 0.0003 + 0.50 x 0.0000 + 0.25 x 0.0005
= 0.0001 + 0.0000 + 0.0000 + 0.0001 = 0.000316 or 0.00316%
Standard Deviation of Corp. Bond = (0.000316)1/2 = 0.017776 or 1.78%
Variance of Gov. Bond = 0.10 x (0.08 – 0.042)2 + 0.15 x (0.06 – 0.042)2
+ 0.50 x (0.04 – 0.042)2 + 0.25 x (0.02 – 0.042)2
= 0.10 x 0.0014 + 0.15 x 0.0003 + 0.50 x 0.0000 + 0.25 x 0.0005
= 0.0001 + 0.0000 + 0.0000 + 0.0001 = 0.000316 or 0.0316%
Standard Deviation of Gov. Bond = (0.000316)1/2 = 0.017776 or 1.78%
The best...

...Prepared For:
Md. Siddique Hossain
(Sqh)
Answer to the question no 01
Inference Regarding the population variance, σ 2
An important area of statistic is concern with making inference about the population variance. Knowledge of population variability is an important element of statistical analysis. Two possibilities arise
For example.
A) For a car rental agency .
* Tires with low variability’s is preferred compared with durable lives with high variability.
B) A bank policy favor a single waiting line that feeds into several tellers.
* may remain same whether more than one lines formal.
* σ 2 may be lower in single line.
* σ 2 is higher in more than one line.
Like the population mean µ , σ 2 is ordinarily unknown, and its value must estimated using sample data.
Sample variance
A random sample of n observations drawn from a population with unknown mean and unknown varience σ 2 .Denote the sample x 1, x2 ,…….xn
The population variance is the expectation
σ 2 = E [ ( x - µ) ] , Which sagely that we consider the mean of ( xi – ) and n observation Since µ is unknown the sample mean is used to compute a sample variance.
The quantity
S2= 1/n-1i=1n (xi-) is called a sample variance ,and its square root s is called the sample standard deviation .
Given a specific random sample variance and the sample...

...Harry W. Markowitz, the father of “Modern Portfolio theory”, developed the mean-variance analysis, which focuses on creating portfolios of assets that minimizes the variance of returns i.e. risk, given a level of desired return, or maximizes the returns given a level of risk tolerance. This theory aids the process of portfolio construction by providing a quantitative take on it. It integrates the field of quantitative analysis with portfolio management. Meanvariance analysis has found wide applications both inside and outside financial economics. However it is based on certain assumptions which do not hold good in practice. Hence there have been certain revisions to it, so as to make it a more useful tool in portfolio management.
Mean Variance Analysis
Within the mean variance approach of Markowitz, the basic assumption is that risk is measured by variance, and the investment decision is based on the trade-off between higher mean and lower variance of the returns. The locus of optimal mean-variance combinations is called the efficient frontier, on which all rational investors would desire to be positioned. Asset returns are assumed to be (jointly) normally distributed random variables, and correlations between assets are also assumed to be fixed and constant forever. (Mz)
Other assumptions include: (Mz)
All investors aim to maximize economic utility (in...

...Terms: Forecast vs Forecast Error We clarify the terms used in the practice problems and the final exam problems. Some statisticians speak of the standard deviation or variance of the forecast. The forecast here is the distribution of future values. It is a random variable, which has a standard error (standard deviation and variance). Other statisticians use the term forecast for the mean of the distribution of future values. The forecast error (the error term in the forecast) is the distribution of future values minus its mean. Using these terms: ! The forecast is a scalar with a non-zero value (unless the forecasted value is zero). ! The forecast error is a random variable with a mean of zero and a non-zero standard error. The final exam problems may use either view. The meaning is clear from the context.
Objectives Actuaries forecast loss costs, mortality rates, reserves, and other values. We want a measure of the quality of the forecast. Illustration: An actuary uses a time series to estimate the average claim severity next year as $10,000. We use this forecast to set rates for auto insurance policies. The procedure used to estimate the future average claim severity may be unbiased, bu the actual claim severity next year will not be exactly $10,000. If the actuary’s estimate is a normal distribution with a mean of $10,000 and a standard deviation of $500, we are 95% confident that the true average claim severity will lie between $9,000...

...(additive model) or it may depend on the level of the other variable (interaction model). One common naming convention for a model incorporating a k-level categorical explanatory variable and an m-level categorical explanatory variable is “k by m ANOVA” or “k x m ANOVA”. ANOVA with more that two explanatory variables is often called multi-way ANOVA. If a quantitative explanatory variable is also included, that variable is usually called a covariate. In two-way ANOVA, the error model is the usual one of Normal distribution with equal variance for all subjects that share levels of both (all) of the explanatory variables. Again, we will call that common variance σ 2 . And we assume independent errors.
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CHAPTER 11. TWO-WAY ANOVA
Two-way (or multi-way) ANOVA is an appropriate analysis method for a study with a quantitative outcome and two (or more) categorical explanatory variables. The usual assumptions of Normality, equal variance, and independent errors apply.
The structural model for two-way ANOVA with interaction is that each combination of levels of the explanatory variables has its own population mean with no restrictions on the patterns. One common notation is to call the population mean of the outcome for subjects with level a of the ﬁrst explanatory variable and level b of the second explanatory variable as µab . The interaction model says that any pattern of µ’s is possible, and a plot of those µ’s...