Midterm Exam. Use the end of month price data for the S&P 500 index in the table below to answer the following questions.

Transcription

1 Unversty of Washngton Summer 2001 Department of Economcs Erc Zvot Economcs 483 Mdterm Exam Ths s a closed book and closed note exam. However, you are allowed one page of handwrtten notes. Answer all questons and wrte all answers n a blue book or on separate sheets of paper. Tme lmt s 2 hours and 10 mnutes. Total ponts = 100. I. Return Calculatons (15 pts) Use the end of month prce data for the S&P 500 ndex n the table below to answer the followng questons. Month Prce Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec What s the smple return between December, 1998 and December, 1999? Suppose you can get ths return every year for the next fve years. What s the smple fve year return? 2. What s the contnuously compounded return between December, 1998 and December, 1999? Suppose you can get ths return every year for the next fve years. What s the contnuously compounded fve year return? 3. Why do we use contnuously compounded returns nstead of smple returns n our probablty models of returns? II. Random Varables and Probablty (10 pts) Let X be a contnuous random varable wth pdf p(x). Gven the followng shape characterstcs of p(x), draw a rough sketch of the pdfs for the followng cases.

2 1. E[X] = 0, var(x) = 1, skew(x) = 0, excess kurt(x) = E[X] = 0, var(x) = 1, skew(x) = -0.5, excess kurt(x) = E[X] = 0, var(x) = 1, skew(x) = 0.5, excess kurt(x) = E[X] = 0, var(x) = 1, skew(x) = 0, excess kurt(x) = E[X] = 0, var(x) = 1, skew(x) = 0, excess kurt(x) = -3. III. Descrptve Statstcs (20 pts) Consder the monthly contnuously compounded returns on Boeng, Mcrosoft and the S&P 500 computed over the perod June 1992 October Descrptve statstcs for these returns are gven n the table below and hstograms, boxplots and scatterplots are presented on the followng pages. Based on the descrptve statstcs and graphs, answer the followng questons. 1. Compare the return rsk propertes of the three assets. Whch asset appears to be safest asset and whch asset appears to be the most rsky asset? 2. Do the return dstrbutons of the three assets look lke they could be normal dstrbutons? Justfy your answers. 3. Descrbe the drecton and strength of lnear assocaton between the three assets. Whch assets appear to have the hghest and lowest correlatons? 4. Why s the monthly standard devaton for the S&P 500 lower than the standard devaton values for Boeng and Mcrosoft? Unvarate Statstcs rboeng rmsft rsp500 Count Average Medan Standard Devaton Varance Skewness Excess Kurtoss

3 Box plots for Boeng, Mcrosoft and S&P Returns rboeng rmsft rsp Asset

4

5 IV. The CER Model and Monte Carlo Smulaton (20 pts) Consder the constant expected return (CER) model R t = µ + ε t, = 1,, N; t = 1,...,T ε t ~ d N(0, σ ) 2 where R t denotes the return on asset and ε t s a normally dstrbuted random error term. For specfcty, assume that µ = and σ = What s the nterpretaton of ε t n the CER model? 2. Brefly explan how you could generate one Monte Carlo smulaton of T = 50 observatons from the CER model for an asset usng Excel. 3. Recall, the least squares estmator of µ n the CER model s the sample mean T 1 µ ˆ = Rt. T t= 1 The sample mean s an unbased estmator of µ ; that s, E [ µ ˆ] = µ. Usng the concept of Monte Carlo smulatons from the CER model, brefly descrbe what t means for µ ˆ to be an unbased estmate of µ. 4. The precson of µ ˆ s measured by the standard error, SE ( µ ˆ ). Usng the concept of Monte Carlo smulatons from the CER model, brefly descrbe what SE ( µ ˆ ) represents. V. Portfolo Theory (15 pts) Consder the problem of an nvestor tryng to determne the best portfolo of two rsky assets (stocks) and a rsk-free asset (T-bll). Let the two rsky assets be Boeng and Mcrosoft, the rskfree asset be a one-year T-bll and suppose the nvestment horzon s one year. It s assumed that nvestors lke assets wth hgh expected returns but dslke those wth hgh rsk (as measured by return standard devaton) and choose to hold effcent portfolos. Transfer the dagram below to your bluebook and answer the followng questons.

6 60.0% 50.0% 40.0% 30.0% Mcrosof t 20.0% Boeng 10.0% T-bll 0.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% Portfolo SD 1. Identfy the nvestment opportunty set consstng of only the two rsky assets. On ths dagram, dentfy the sets of neffcent and effcent portfolos. 2. Identfy the tangency portfolo and the set of effcent combnatons of T-blls and the two rsky assets. On ths effcent set, ndcate and descrbe the asset allocaton of the portfolos that a very rsk averse nvestor would choose to hold and the portfolos that a very rsk tolerant nvestor would choose to hold. 3. State the maxmzaton problem to be solved to determne the tangency portfolo. VI. Effcent Portfolos and Value-at-Rsk (15 pts) Consder an nvestor who has $100,000 to nvest over the next year. Assume that the nvestor ntally holds a portfolo consstng of 80% Boeng and 20% Mcrosoft. Ths asset s denoted asset A n the dagram below. Use the nformaton n the table below to answer the followng questons.

7 60.0% 50.0% 40.0% 30.0% Mcrosof t 20.0% Tangency 10.0% Asset A Boeng T-bll 0.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% Portfolo SD Asset E[R] SD(R) Asset A Tangency T-Bll What s the 5% Value-at-rsk over the next year on the $100,000 portfolo consstng of just asset A? Assume that returns are contnuously compounded. (FYI NORMINV(0.05, 0.186, 0.223) = ) 2. Fnd the effcent portfolo (combnaton of T-blls, Boeng and Mcrosoft) that has the same expected return as asset A. In ths effcent portfolo, how much s nvested n the T-bll, Boeng and Mcrosoft? What s the rsk (standard devaton) of ths portfolo? 3. Fnd the effcent portfolo (combnaton of T-blls, Boeng and Mcrosoft) that has the same rsk (standard devaton) as asset A. In ths effcent portfolo, how much s nvested n the T- bll, Boeng and Mcrosoft? What s the expected return of ths portfolo?