Q2 W23 sol
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School
Toronto Metropolitan University**We aren't endorsed by this school
Course
QMS 310
Subject
Accounting
Date
Dec 18, 2024
Pages
2
Uploaded by ConstableRose27721
STAD70 W23Quiz 2Name (f,l):Student #:1. (8 points) Below are 3 random samples of sizen= 1000 from six possible elliptical bi-variate copulas: Gaussian with correlation parameterρ=-0.75,0,+0.75, ort(df= 1)with correlation parameterρ=-0.75,0,+0.75. Identify the distribution andρparam-eter that generated of each sample (e.g.,twithρ=-.75).Copula:Copula:Copula:Solution:Copula:twithρ= 0Copula: Gaussian withρ= +0.75Copula:twithρ=-0.752. Consider a market consisting of 2 assets with bivariate Normal net returns:R=R1R2∼Nμ=0.050.1,Σ= 0.041111Note that the returns have the same variance and are perfectly positively correlated.(a) (6 points) Show thatanyportfolio consisting of the two assets will have varianceV[Rp] =V[wR1+ (1-w)R2] = 0.04,∀w∈R.(b) (6 points) Now restrict the set of feasible portfolios to thosewithout short-selling(i.e.,w∈[0,1]). Assuming the risk-free interest rate isμf= 0.02, find the Sharperatio of the tangency/market portfolio for this restricted model.(Hint: the answer can be found geometrically from the (σp, μp) risk-return diagram.)Solution:
STAD70 W23Quiz 2Name (f,l):Student #:(a)V[Rp] =w>Σw= 0.04w(1-w)1111w(1-w)= 0.04w+ (1-w)w+ (1-w)w(1-w)= 0.04[w2+ 2w(1-w) + (1-w)2]= 0.04[w2+ (HH2w-2w2) + (1-HH2w+w2)] = 0.04(b) Since every portfolio has the same variance, the feasible setwithout short-sellingis the vertical line atσp=√0.04 = 0.2, from returnsμp∈[0.05,0.1]. Obviously,the only optimal portfolio, which is also the tangency portfolio, consists of onlythe higher-return asset (R2). The slope of the line with the risk-free return, i.e.,the Sharpe ratio, is thusμ2-μfσ2=0.1-0.020.2=0.080.2= 0.4The following plot illustrates the situation:0σpμpμf(σ1, μ1)(σ2, μ2) = (σM, μM)Page 2