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 Universiy of Washingon Winer 00 Deparmen of Economics Eric Zivo Economics 483 Miderm Exam This is a closed book and closed noe exam. However, you are allowed one page of handwrien noes. Answer all quesions and wrie all answers in a blue book or on separae shees of paper. Time limi is hours and 10 minues. Toal poins = 100. I. Reurn Calculaions (0 ps) Use he end of monh price daa for he S&P 500 index in he able below o answer he following quesions. Close Dae price December January February March April May June July Augus Sepember Ocober November December Wha is he simple 1 monh reurn beween December, 000 and December, 001? Suppose you can ge his reurn every monh for he nex year. Wha is he simple 1 year reurn? Compare his reurn o he acual 1 year simple reurn beween December, 000 and December, Suppose he consumer price index (CPI) for December 000 is equal o 100 and ha he CPI for December 001 is 105. Compue he simple 1 year real reurn beween December 000 and December Wha is he coninuously compounded 1 monh reurn beween December, 000 and December, 001? Suppose you can ge his reurn every monh for he nex year. Wha is he coninuously compounded annual year reurn? Compare his reurn o he acual 1 year coninuously compounded reurn beween December, 000 and December, 001.

2 4. Suppose he consumer price index (CPI) for December 000 is equal o 100 and ha he CPI for December 001 is 105. Compue he coninuously compounded annual real reurn beween December 000 and December 001. II. Random Variables and Probabiliy (3 ps) A. Normal disribuion Le X be a normally disribued random variable wih mean µ = 0.05 and variance σ = (0.10). 1. Skech he pdf (probabiliy curve) of X. On your skech indicae he locaion of µ, µ + σ, and µ σ.. Wha are he skewness and kurosis values for X? 3. Wha is he approximae.5% quanile of he pdf for X? 4. Briefly explain how you would compue Pr( 1.5 X < 0.5) using Excel. 5. Consider he new random variable W = X. Compue E[ W], var( W) and SD( W ). B. Value-a-Risk Consider an invesmen of $100,000 in he S&P 500 index for a period of one day. Le R denoe he coninuously compounded daily reurn on he S&P 500 index and assume ha R ~ N (0.0004,(0.0015) ). Tha is, E[ R ] = 0.04% and SD( R ) = 0.15%. 6. Compue he daily 1% and 5% value-a-risk (VaR). FYI, he 1% and 5% quaniles of he N (0.0004,(0.0015) ) pdf are and , respecively. C. Aggregaing reurns Assume ha R ~ iid N (0.0004,(0.0015) ) for every day over he nex monh and assume here are 30 days in a monh. Le R m denoe he monhly coninuously compounded reurn. 7. Express he monhly coninuously compounded reurn R m in erms of he daily coninuously compounded reurns R. 8. Compue E[ Rm], var( Rm), and SD( R m) III. Descripive Saisics and he CER Model (8 ps) Consider he monhly coninuously compounded reurns on Washingon Muual sock and he S&P 500 compued using end of monh closing prices over he period December 1991 December 001. Descripive saisics for hese reurns are given in he able below and hisograms, boxplos and scaerplos are presened on he following pages. Based on he descripive saisics and graphs, answer he following quesions.

3 1. Compare he reurn risk properies of he wo asses. Which asse appears o be safes asse and which asse appears o be he mos risky asse? Jusify your answer.. Do he reurn disribuions of he wo asses look like hey could be normal disribuions? Use he univariae saisics in he able below and he boxplos o jusify your answers. 3. Describe he direcion and srengh of linear associaion beween he wo asses. 4. For he consan expeced reurn model Ri = µ i + εi, εi ~ iid N(0, σi ) cov( Ri, Rj ) = σij, corr( Ri, Rj ) = ρij give he mehod of momens esimaes for µ i, σi, σi, σij and ρ ij for he S&P 500 index and for Washingon Muual. 5. For each asse, use he approximae analyical formulas o compue esimaed sandard error values for µ i, σi,and ρ ij. Tha is compue SE( µ ˆ ), ( ˆ ),and ( ˆ i SE σi SE ρ ij). Commen on he size of hese esimaed sandard errors. 6. For each asse, compue an approximae 95% confidence inerval for µ i. Use he widhs of he inervals o evaluae he precision of esimae µ ˆi. 7. The monhly reurns on he S&P 500 index and 4-monh rolling esimaes of µ ˆ ˆ sp500 and σ sp500 are illusraed in figure 1. Based on his figure, wha can you say abou he appropriaeness of he CER model for he reurns on he S&P 500 index over he en year period December 1991 December 001? Univariae Saisics sp500 wamu Coun Average 0.84% 1.56% Minimum % -3.49% Maximum 9.3% 3.13% Range 4.99% 46.61% Sandard Deviaion 4.06% 8.80% Variance Skewness Kurosis Covariance Correlaion

4 sp wamu Boxplos for reurns on WAMU and SP monhly reurn sp500 asses wamu

5 Scaerplo of monhly reurns 30.00% 0.00% WAMU 10.00% 0.00% % -0.00% % -0.00% % % -5.00% 0.00% 5.00% 10.00% 15.00% S&P 500 Reurns on S&P 500 index wih 4-monh rolling means and SDs 15.00% 10.00% reurns, means, SDs 5.00% 0.00% -5.00% % % -0.00% Jan-9 Jul-9 Jan-93 Jul-93 Jan-94 Jul-94 sp500 reurn rolling means rolling SDs Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 monh Figure 1 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 Jan-01 Jul-01

6 IV. The CER Model and Mone Carlo Simulaion (0 ps) Consider he consan expeced reurn (CER) model R = µ + ε, = 1,...,T ε i ~ iid N(0, σ ) where R i denoes he coninuously compounded reurn on asse i and ε is a normally disribued random error erm. For specificiy, assume ha µ = 0.01 and σ = Wha are he assumpions of he CER model?. Wha is he inerpreaion of ε in he CER model? 3. The mehod of momens esimaor of σ in he CER model is he square roo of he sample variance 1 σ ˆ = µ T T ( R ˆ ), 1 = 1 where µ ˆ 1 T = T R = 1 is he sample mean. Using he concep of Mone Carlo simulaions from he CER model, briefly describe how you could deermine if σ ˆ is an unbiased esimae of σ. 4. The precision of σ ˆ is measured by he sandard error, SE( σ ˆ). Wha is he approximae formula for compuing SE( σ ˆ)? Using he concep of Mone Carlo simulaions from he CER model, briefly describe how you could compue SE( σ ˆ) exacly.