I. Return Calculations (20 pts, 4 points each)

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1 Universiy of Washingon Spring 015 Deparmen of Economics Eric Zivo Econ 44 Miderm Exam Soluions This is a closed book and closed noe exam. However, you are allowed one page of noes (8.5 by 11 or A4 double-sided) and he use of a calculaor. Answer all quesions and wrie all answers direcly on he exam in he space provided. If you need more space, you may use exra shees of paper. Time limi is 1 hour and 50 minues. Toal poins = 116. I. Reurn Calculaions (0 ps, 4 poins each) Consider a 60-monh (5 year) invesmen in wo asses: he Vanguard S&P 500 index (VFINX) and Apple sock (AAPL). Suppose you buy one share of he S&P 500 fund and one share of Apple sock a he end of January, 010 for P vfinx, , Paapl, , and hen sell hese shares a he end of January, 015 for P vfinx, 184., Paapl, (Noe: hese are acual adjused closing prices aken from Yahoo!). In his quesion, you will see how much money you could have made if you invesed in hese asses righ afer he financial crisis. a. Wha are he simple 60-monh (5-year) reurns for he wo invesmens? > r.vfinx = (p.vfinx. - p.vfinx.1)/p.vfinx.1 > r.aapl = (p.aapl. - p.aapl.1)/p.aapl.1 > r.vfinx [1] 1.05 > r.aapl [1] 3.51 b. Wha are he coninuously compounded (cc) 60-monh (5-year) reurns for he wo invesmens? > log(1 + r.vfinx) [1] > log(1 + r.aapl) [1] 1.51 c. Suppose you invesed $1,000 in each asse a he end of January, 010. How much would each invesmen be worh a he end of January, 015? > w0 = 1000 > w1.vfinx = w0*(1 + r.vfinx) > w1.aapl = w0*(1 + r.aapl) > w1.vfinx [1] 049 > w1.aapl [1] 4509

2 d. Wha is he compound annual reurn on he wo 5 year invesmens? > r.vfinx.a = (1 + r.vfinx)^(1/5) - 1 > r.aapl.a = (1 + r.aapl)^(1/5) - 1 > r.vfinx.a [1] > r.aapl.a [1] 0.35 e. A he end of January, 010, suppose you have $1,000 o inves in VFINX and AAPL over he nex 60 monhs (5 years). Suppose you purchase $400 worh of VFINX and he remainder in AAPL. Wha are he porfolio weighs in he wo asses? Using he resuls from pars a. and b. compue he 5-year simple and cc porfolio reurns. > w0 = 1000 > x.vfinx = 400/w0 > x.aapl = 1 - x.vfinx > x.vfinx [1] 0.4 > x.aapl [1] 0.6 > r.p = x.vfinx*r.vfinx + x.aapl*r.aapl > r.p [1].53 > log(1 + r.p) [1] 1.6

3 II. Probabiliy Theory (0 poins, 4 poins each) Le Rvfinx and RAAPL denoe he monhly simple reurns on VFINX and AAPL and suppose ha R ~ iid N (0.013,(0.037) ), R ~ iid N (0.08,(0.073) ). vfinx a. Skech he normal disribuions for he wo asses on he same graph. Show he mean values and he ranges mean ± sd. Which asse appears o be he mos risky? aapl AAPL has a higher SD which means is normal pdf is more spread ou han he normal pdf of VFINX. There is more uncerainiy, hence more risk, in he reurn for AAPL. b. Plo he risk-reurn radeoff for he wo asses. Tha is, plo he mean values of each asse on he y-axis and plo he sd values on he x-axis. Wha relaionship do you see? AAPL has boh a higher expeced reurn and sandard deviaion (risk) han VFINX. This is he sylized risk-reurn radeoff.

4 c. Le W0 = $1,000 be he iniial wealh invesed in each asse. Compue he 1% monhly Z Value-a-Risk values for each asse. (Hin: q ). > w0 = 1000 > q.vfinx.01 = mu.vfinx + sigma.vfinx*(-.36) > q.aapl.01 = mu.aapl + sigma.aapl*(-.36) > VaR.01.vfinx = w0*q.vfinx.01 > VaR.01.AAPL = w0*q.aapl.01 > VaR.01.vfinx [1] > VaR.01.AAPL [1] -14 d. Coninuing wih c., sae in words wha he 1% Value-a-Risk numbers represen (i.e., explain wha 1% Value-a-Risk for a one monh $1,000 invesmen means) Wih 1% probabiliy (or one monh in every 100 monhs), a one monh $1000 invesmen in VFINX will lose $73.1 or more. Wih 1% probabiliy (or one monh in every 100 monhs), a one monh $1000 invesmen in AAPL will lose $14 or more. e. The normal disribuion can be used o characerize he probabiliy disribuion of monhly simple reurns or monhly coninuously compounded reurns. Wha are wo problems wih using he normal disribuion for simple reurns? Given hese wo problems, why migh i be beer o use he normal disribuion for coninuously compounded reurns? Problem 1: Simple reurns are bounded from below by -1. The normal disribuion is defined over - o + and so i is possible reurns o be less han -1 wih posiive probabiliy if hey are normally disribued Problem : Muli-period simple reurns are muliplicaive, no addiive. So if simple reurns are normally disribued hen muli-period reurns are no normally disribued (because he produc of wo normal random variables is no normally disribued). The normal disribuion is more appropriae for coninuously compounded reurns because coninuously compounded reurns are defined over - o + and muli-period coninuously compounded reurns are addiive.

5 III. Time Series Conceps (16 poins, 4 poins each) a. Le { Y } represen a sochasic process. Under wha condiions is { Y } covariance saionary? EY [ ] var( Y ) cov( Y, Y ) (depends on j and no ) j j b. Realizaions from four sochasic processes are given in Figure 1 below. Figure 1: Realizaions from four sochasic processes. Which processes appear o be covariance saionary and which processes appear o be non-saionary? Briefly jusify your answers. Processes 1 and 3 appear o be covariance saionary. The means and volailiies appear consan over ime and he series exhibi mean reversion (when he series ges above or below he mean i revers back o he mean) Processes and 4 appear o be non-saionary. Process has a clear deerminisic rend, so he mean is no consan over ime. Process 4 appears o have wo disinc volailiies (low in he firs half and high in he second half).

6 c. The CER model for cc reurns r, ~ iid N(0, ), implies ha he log price follows a random walk wih drif ln P ln P r ln P. 1 0 s s1 Show ha E[ln P ] and var(ln P ) depend on so ha ln P is non-saionary. Eln P E[ln P] E[ ] E ln P which depends on. 0 s 0 s1 var(ln P) var ln P var var( ) 0 s s s which depends on. s1 s1 s1 d. Suppose he ime series { X } is independen whie noise. Tha is, X iid ~ (0, ) Define wo new ime series { Y } and { Z } where Y X and Z Y. Are { Y } and { Z } also independen whie noise processes? Why or why no? { Y } and { Z } are independen processes because any funcion of independen random variables are also independen. The process are covariance saionary because any funcion of covariance saionary processes are covariance saionary (provided he mean and variance and auocovariances are finie). Hence, he processes will have consan variance. The processes will no have mean zero. So hey will behave like a whie noise process wih a non-zero mean.

7 IV. Marix Algebra (16 poins, 4 poins each) Le Ri denoe he simple reurn on asse i (i = 1,, N) wih E[Ri] = i, var(ri) = i and cov(ri, Rj) = ij. Define he (N 1) vecors x, y, and x,, 1 x N y,, 1 y N R R1,, R N, ( 1,, N ), 1 1,,1 and he (N N) covariance marix 1 1 1N 1 N. 1N N N The vecors x and y conain porfolio weighs (invesmen shares) ha sum o one. Using simple marix algebra, answer he following quesions. a. For he porfolios defined by he vecors x and y give he marix algebra expression for he porfolio reurns, (Rp,x and Rp,y) and he porfolio expeced reurns (p,x and p,y). Rpx, xr, Rpy, yr ER [ ] xμ, ER [ ] yμ px, px, py, py, b. For he porfolios defined by he vecors x and y give he marix algebra expression for he consrain ha he porfolio weighs sum o one. x1 1 and y1 1 c. For he porfolios defined by he vecors x and y give he marix algebra expression for he porfolio variances ( and ), and he covariance beween Rp,x and Rp,y ( xy ). p, x p, y var( R ) xσx, var( R ) yσy px, px, py, py, cov( R, R ) xσy xy p, x p, y d. In he expression for he porfolio variance How many covariance erms are here? p, x, how many variance erms are here? There are N variance erms and N(N-1) oal covariance erms (N(N-1)/ unique covariance erms).

8 V. Descripive Saisics (3 poins, 4 poins each) Figure 3 shows monhly simple reurns on he Vanguard S&P 500 index (VFINX) and Apple sock (AAPL) over he 5-year period January 010, hrough January 015. For his period here are T=60 monhly reurns. Figure : Monhly simple reurns on wo asses. a. Do he monhly reurns from he wo asses look like realizaions from a covariance saionary sochasic process? Why or why no? Recall, covariance saionariy implies ha he mean, variance and auocovariances are consan over ime. Visually i looks like boh mean values are consan over ime and boh series exhibi mean reversion (boh series flucuae up and down abou he mean value). The volailiies of VFINX and AAPL look fairly consan over ime. Overall, boh series look like hey could be realizaions from a covariance saionary process.

9 b. Compare and conras he reurn characerisics of he wo asses. In addiion, commen on any common feaures, if any, of he wo reurn series. AAPL looks o have a slighly larger mean han VFINX and clearly a larger volailiy. AAPL also has many more large reurns (posiive and negaive) han VFINX. The wo reurns do no look like hey move very closely ogeher (e.g. when VFINX goes up (down) i is no always he case ha AAPL goes up (down)). Hence, i looks like he correlaion beween VFINX and AAPL is a small posiive number. c. Figure 4 below gives he cumulaive simple reurns (equiy curve) of each fund which represens he growh of $1 invesed in each fund over he sample period. Which fund performed beer over he sample? A $1 invesmen in AAPL clearly did beer han a $1 invesmen in VFINX. For AAPL $1 grew o abou $5, and for VFINX $1 grew o only abou $. Figure 3 Cumulaive simple reurns on wo asses.

10 The figures below give some graphical diagnosics of he reurn disribuions for VFINX and AAPL.

11 The following able summarizes some sample descripive saisics of he T=60 monhly reurns. Table 1 Descripive Saisics and CER model esimaes Saisic VFINX AAPL Mean ( ) Sandard Deviaion ( ˆ ) Skewness Excess Kurosis % empirical quanile ( ˆq 0.01 ) Lag 1 auocorrelaion ( 1 ˆ ) Lag auocorrelaion ( ˆ ) Lag 3 auocorrelaion ( ˆ 3) Correlaion beween VFINX and AAPL ( ˆVFINX, AMZN ) 0.435

12 d. Do he reurns on VFINX and AAPL look normally disribued? Briefly jusify your answer. Boh he reurns on VFINX and AAPL look prey normally disribued. The hisograms are roughly bell shaped (VFINX has a sligh negaive skewness bu no oo large). Also, he normal qq-plos for boh asses are mosly linear and he boxplos only show one moderae oulier for VFINX. Finally, he excess kurosis values for boh series are close o zero (abou 0.86 for VFINX and for AAPL) which is consisen wih he normal disribuion. e. Which asse appears o be riskier? Briefly jusify your answer. AAPL has a larger SD han VFINX (0.075 vs ) so here is more uncerainy in he reurn for AAPL. Also, he 1% quanile for for AAPL (-0.189) is more negaive han he 1% quanile for VFINX ( ) This means here is more ail risk wih AAPL. f. Does here appear o be any conemporaneous linear dependence beween he reurns on VFINX and AAPL? Briefly jusify your answer. The scaerplo of he AMZN and VFINX reurns looks prey much like a shogun blas wih a moderae righward il indicaing weak o moderae posiive linear dependence. This is confirmed by he sample correlaion value of g. Do he monhly reurns show any evidence of (linear) ime dependence? Briefly jusify your answer. For boh series he SACF plos show no significan auocorrelaions for lags less han 15 (no bars are above he doed lines). All of he sample auocorrelaions are small (close o zero) and show no disinc paern over ime. Hence, here does no appear o be any linear ime dependence in he reurns. h. Le W0 = $1,000 be he iniial wealh invesed in each asse. Compue he 1% monhly hisorical Value-a-Risk for each asse. (Recall, he daa are simple reurns here) > W0 = 1000 > q.01 = apply(re,, quanile, probs=0.01) > VaR.vfinx.01 = W0*q.01["VFINX"] > VaR.AAPL.01 = W0*q.01["AAPL"] > VaR.vfinx.01 VFINX > VaR.AAPL.01 AAPL -19

13 VI. Consan Expeced Reurn Model (16 poins) Consider he consan expeced reurn (CER) model r i i i, i ~ iid N(0, i ) cov( r, r ), cor( r, r ) i j ij i j ij for he monhly simple reurns on he Vanguard S&P500 index (VFINX) and Apple sock (AAPL) presened in par V above. Below are simulaed reurns and some graphical descripive saisics for VFINX and AAPL from he CER model calibraed using he sample esimaes of he CER model parameers for he wo asses (hese are he sample saisics from he able of he previous quesion).

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15 a. Briefly describe how o creae he simulaed reurns on VFINX and AAPL from he CER model. Tha is, ell me he algorihm for creaing he simulaed daa. You don have o show me R code o answer his quesion. Jus ell me he seps required (e.g. pseudocode) o simulae he daa. Here we wan o do a mulivariae simulaion from he CER model in marix form R μ ε, ε ~ ( 0, Σ ) N To do his we need values for he x 1 mean vecor μ and x covariance marix Σ. We can ge hese values from he sample saisics in Table 1 vfinx ˆ vfinx vfinx, aapl μˆ and Σ aapl vfinx, aapl aapl where ), ( ) and vfinx ( vfinx, aapl vfinx, aapl vfinx aapl aapl Then we can simulae T observaion on ε ~ N( 0, Σ )(using he R funcion mvnorm, for example) giving ε and creae he simulaed reurns using r μˆ ε.

16 b. Which feaures of he acual reurns shown in par V are capured by he simulaed CER model reurns and which feaures are no? Almos all feaures of he VFINX and AAPL reurns are capured by he simulaed CER model reurns. There are only a few noable differences Time plo of simulaed reurns looks very close o acual reurns (AAPL has higher mean and volailiy han VFINX; reurns do no move very closely ogeher) Hisograms are roughly bell shaped and qq-plos are mosly linear. The negaive skewness of acual VFINX reurns are no shown in he simulaed reurns. This hisogram of he simulaed VFINX reurns shows posiive skewness. Boxplos of simulaed reurns mach closely he boxplos of he acual reurns. Scaerplo of simulaed reurns looks like scaerplo of acual reurns SACF of simulaed reurns look like SACF of acual reurns. c. Does he CER model appear o be a good model for VFINX and AAPL reurns? Why or why no? Yes, he CER model appears o be a good model for VFINX and AMZN reurns. Mos of he imporan feaures of he acual daa are capured by simulaed daa from he CER model.