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5-Answer - The University of Hong Kong Department of...

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Unformatted text preview: The University of Hong Kong Department of Statistics and Actuarial Science STAT 2309 The Statistics of Investment Risk [First Semester 2009-20101 Problem Sheet 5 Suggested Solution (a) Consider the portfolio constructed by buying 1 share of stock A, 1 share of stock C and short selling 2 shares of stock B. Note that the initial investment is \$0. Under good economic condition, the portfolio will be worth \$10-2(\$8)+(\$9) = \$3 whereas under bad economic condition, the portfolio will be worth \$8—2(\$0)+\$12 = \$20. Hence, this is an arbitrage opportunity with \$0 initial investment and positive profit. ‘ (b) No arbitrage means that any portfolio with zero net cash investment and no risk should have zero return. Suppose we invest xA units of stock A, x3 units of stock B and 1 unit of stock C. Zero net cash investment: 3 xA+ 2 x3 + PC = 0. Zero risk: 10 xA+ 8 xB + 9 = 8 xA+ 12 Zero profit: 10 xA+ 8 x3 + 9 = 0 and 8 xA+ 12 = 0 This implies PC = \$3. 2. (a) £(R! )= 5 *8ﬂ11_ 65!} 14¢le l= ﬂéli’mif} 31+ ,BﬁlVarl‘F: l) CovlerR’, = A1 £4.1(Var(F-_ )lw- ﬁgﬂﬁlVarle )) C01'(RA.R5 )= 66 Cavity. RC _) = 190 C011R3.Rc ,)= 210 (b) Factor-mimicking portfolio for factors 1: 0x1+x3+xc=l x4=05 00.7.1} +1.31} +2.01}: =1 -) 1‘, =05 0—0.5x1+0.5x5+l.0xc =0 xc = F actor—mimicking portfolio to: factors 2: 0xJ+x,+xc=l xdsl 00.7.13 +1.3x, +2.0xc =0 -) x, =1 0-051"4 +0.51) +1.0xc -l xc - -l (C) we, )= 5+8(1)-6(0)= 13% HR: )= 5+8(0)—6{1)= —1% Var“?l )=100 Var(R2)=100 CoriR1.R2_)= o . :‘100 o 21‘» 13—5 8 T encyPortfohozl '= -= mg 1 o mil-j [-1—5] [-6] :1 =0.08 9 x1=4 .- E(R)-4{13_)—3(-l)-559'a Varma(4)3(100)+(—3)1(100)=2500 R, =(l-x)Rf +xR E(R,)=(l—xx5)+.\1'55)=50x-5 Var(R, ) a x3(3500) Expected Utility is: E(U(R1-))= Em, )— 0.025141%, 1 = 50x-5-(0.025)x=(2soop =—62.5x2 +50x-5 Thus, ElUlR, }) is maximized when -lle + 50 or x - 0.4. You should shon sell \$1.2 million of factor mimicking portfolio 2. invest \$0.6 million for risk-ﬂee lending and \$1.6 million in factot mimicking portfolio 1. 3. (a) Currentportfolio x}=(o.5263 0.1579 0.3158) (b) Denote x' Z 0 as the proportion of amount purchased: 1' 2. 0 as the propomon of amount sold. Min :x' E: subject to (1) x = x. +x" —x’ (No redemption nor further mvestment) 2) Ix’ =l=>Z x. =1 (3) p7: -p"x° -q’x' - .115 -10 (Return after adjustmg for the transaction costs) (4)x’203ndx'20 whereu’=(15 9 4).p’=q’={0.03 0.02 0.01:).I’=(l l l).and 122 0.2x12x10 0.1x12x3' '144 24 3.6' g: 0.2x12x10 101 0.3x10x3 = 24 100 9 .\0.1x12x3 0.3x10x3 31 3.6 9 9, . (18.38 11.38 36.00 49.70 10.81 (a) Annuahzed return= 13.24 9.87 14.59 -0.37 15.17) ‘ ' (29.30 26.55 57.67 76.64 16.62 Annuahzed rash 21.52 16.75 20.22 22.47 13.57) _ (19.56 11.50 39.61 28.92 11.31 (b) Annuahzed return: . 13.25 9.90 14.32 0.03 15.45) . . [28.75 26.48 55.32 59.98 16.55 Annuahzed nsk= _ 21.14 16.56 19.78 22.38 13.51) There is a slight diﬁ'erence compared to results in (a) (c) (i) Markowrtz method: x - 1‘ 0.1874 -0.1229 0.0219 0.0565 0.1168 0.0536 0.0717 0.0843 -0.0499 0.5807) (ii) Robust Markowitz method: x=(0.2141 —0.1416 0.0413 0.0330 0.0978 with up =18.38% mth 11,, =18.26% 0.0509 0.0914 0.1091 —-0.0698 0.5738) (d) Theportfolio risk is . .r 2930‘ 0.27x29.30x13.57‘ -' 0.7 " 07.40.? 0.3) , )) ) _)0.27x29.30x13.57 13.57- ‘03., = 0,. = 21.96 Hence. rebalance and hold the portfolio in C(i). 0'J 25 ' Fund A performs as well as the market, M 3 is the same as the expected return of the market portfolio. The M 3 of fund A says that if you had combined the fund with risk-free asset so that the standard deviation of your portfolio was equal to the market risk. then you would have obtained an expected return of 15%. (b) Appraisal Ratio 5; or Information Ratio 30-5-2(15-5)=5 0-3333 5 53,3“, -SR§‘,+.-1.Rj-==13+O.33332 =1.1 =>sxm -1054093 (d) , . 12112, )—R., = a], + ppm)?” )-R,J a? = £[Rpl—R, - approveM )—R,] a, =EiRn)‘R/ ‘ﬂaiﬂRJ/i—Rf] = 40-5 - (3.0115 — 5] - 5% Investors are Willing to pay at most 5% for Fund B‘s manager. As the current fee that is charged is 29"is"'year, the fee can be raised by 3% at most. (e) (a) (b) (C) (d) Selectivity =EiR, l“ {Rf 4‘ .3; [5 1R” )— R; ii [HRH Hz, 10, 0;: ;~ {Rx + ﬁplﬂﬁu 1- R; 1} .— Diversiﬁcation =[R_, + Net selectivrty = Selectivity - Diversification o In terms of selectivrry measures. Managers of Fund A and Fund B perform equally well and they perform better than Manager of Fund C. o In terms of diversiﬁcation measures. Manager of Fund B has the best diversiﬁcation skills than the Managers of Fund A and Fund C and the Managers of Fund A and Fund C perform equally. o In terms of net selectivity measures. Manager of Fund A performs the best and the Managers of Fund C performs better than that of Fund B. Fama overall performance = excess return = R, — Rf For ﬁmd 1, Fama overall performance = 26.4% - 6.2% = 20.2%. For fund 2, Fama overall performance = 13.22% - 6.2% = 7.02%. Under CAPM, p, = Rf+ Bp[uM — Rf] For fund 1, u1= 6.2% +1.351[15.71% - 6.2%] = 19.05%. For fund 2, pg: 6.2% + 0.905[15.71% - 6.2%] = 14.81%. Selectivity component = Ra — E[R(Ba)] E[R(Ba)] = Rf+ [Elle - Rf] Ba For fund 1, selectivity component = 26.4% - { 6.2% + (15.71% — 6.2%)(l.351) } = 7.35% For ﬁmd 2, selectivity component = 13.22% - { 6.2% + (15.71% - 6.2%)(0.905) } = -l.59% Diversiﬁcation = E[R(o,,)] - E[R(B,)] E[R(6a)] = Rr+ [ElRul - Rf] 63/ 6M For fund 1, diversiﬁcation = 6.2% + (15.71% - 6.2%)(20.67/ 13.25) — 19.05% = 1.99% For ﬁmd 2, diversiﬁcation = 6.2% + (15.71% - 6.2%)(14.2/13.25) — 14.81% = 1.58% Net selectivity = Selectivity — Diversiﬁcation For ﬁmd 1, net selectivity = 7.35% - 1.99% = 5.36% For ftmd 2, net selectivity = -1.59% - 1.58% = -3. 17% Fund 1 had better performance than fund 2. (a) Fund A has negative selecmity ability (-3). but positive timing ability (0007). Fund B has no timing ability but positive selectivity (0:5). (b) The Value of the timing ability = cVarlRu) The value of the selectivity ability = a The most that you should be willing to pay the Fund A manager is — 3 «— 0.007( 5)1 = —2.825. Thus. you should not invest when the manager pay you this much per month The most that you should be willing to pay the Fund B manager is 5% per month. (a) Manager‘s return = (0.7)(20) — (0.2)(1) — (0.1)(05) = 1.65% Benchmark return = (0.6)(25) "‘ (0.3)(1.2) + (0.1)(0.5) = 1.91% It was underpeﬁormed by 1.91 - 1.65 = 0.26% (1)) Security selection: ”mud return Manager’s Contribution to Sector mum market Portfolio Weight relative performance (Mana — benchmark) Equity 2% - 2.5%: 05% 70% (-0.5)(07) = 035% Bonds 1% - 1.2% = 02% 20% (-0.2)(0.2) = 0.04% Cash 05% - 0.5% = 092. 10% (0)(o.1) = 0.00% Contribution of security selection: 039% (c) Asset allocation: Excess we: t Contribution to Sector (Mana - benflli‘mark) BM Return relative performance Equity 70% - . = 109;. 2.59». (o.1)(2.5) = 0.25% Bonds 20% - 10% = 40% 1.2% (-o.1)(1.2) = 012% Cash 10% - 10% = 0% 0.5% 0 0.5 = 0.00% Contribution of security selection: 0.13% ...
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