P&sw07 - ffggh PDF

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Question 1

Suppose yo u ha ve invest ed only in two sto cks, A a nd B. Yo u exp ect tha t retur ns o n the s to cks depend o n the fo llowing t hree states o f eco no my, which are equall y li kely to ha pp en. Sta te of Economy Bear Normal Bull

Return on Sto ck A ( %) 7. 3 % 11.5 16.6

Return on Stock B (%) -4 . 7% 5.4 24 . 3

1. Cal cul ate th e exp ected return o f ea ch stock. 2. Cal cul a te the sta nd ard devia ti o n o f returns o f ea ch sto ck. 3. Cal cul a te the cova ri a nce a nd co rrelation between the two stocks.

Question 2

There a re two stocks in the market, stock A a nd sto ck B. The price of stock A today is $50. The price of stock A next year will be $40 i f the eco no my is in a recession, $55 if the eco no my is no rma l, and $60 if the economy is expa nd ing. The a ttenda nt pro ba bilities of recessio n, no rma l times, and expa nsio n a re 0. 2 , 0 . 6, and 0. 2, resp ecti vely. Stock A pays no di vi dend . Assume the C APM is true and that ¯ i = C o v (i; M )=¾ 2M where i i s a ri sky asset and M is the market portfolio. Other info rma tion a b ou t th e market i ncl udes: SD(R M ) = standard deviatio n o f the market portfoli o = 0 . 10; SD(RB ) = standard devi ati on of stock’s B =0.12; R B = expected retu rn on stock B=0.09; Corr (R A; R M ) = the co rrel a tion of stock A and the market = 0.8; Corr (R B ; RM ) = the correlatio n o f stock B and the market = 0.2; Corr (RA ; R B ) = the correlation of stock A and stock B = 0.6; a . If yo u are a typical risk-averse investor, which stock would you prefer? Why? b. W ha t a re th e exp ect ed ret urn and sta ndard deviatio n o f a portfolio consi sting of 60% o f sto ck A and 40% of stock B? c. What is th e b eta of the portfolio in part (b)?

Question 3

Miss Ma pl e i s co nsidering two securi ti es, A and B, and the relevant i nfo rma tio n is g iven b elow: State of Economy Bea r Bull

Pro ba bility 0 .6 0.4

Return on Security A (%) 3.0% 15.0 %

Return on Security B (%) 6 . 5% 6 .5

1. Cal cul ate th e exp ected ret urns and st andard deviations of the two securiti es. 2 . Suppo se Miss Ma pl e i nvested $2,500 i n Securi ty A a nd $3,500 in security B. Calculate the exp ected retu rn and standard deviation of her portfolio. 3 . Supp o se Miss Ma ple borrowed from h er fri end 40 shares of security B, which is currently sold at $50, and sold all shares of the security. (Sh e promised her fri end she would pay her back in a year with the same number of s ha res o f securi ty B. ). Then s he bo ug ht security A with the proceeds obtai ned i n the sa l es of security B sha res a nd the ca sh o f $6 ,000 she ow ned. C al cul ate th e exp ected return and standard devia ti o n of th e p ortfolio.

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Question 4

Suppose th e cu rrent risk-free is 7.6 percent. Po tpo urri Inc. stock has a beta of 1.7 a nd a n exp ected retu rn of 16.7 percent. (Assume the CAPM is true) a . Wha t i s the ri sk premium on the market? b. Magnoli a Industri es sto ck has a beta of 1.8 . Wha t is the expected return on the Mag no lia stock? c. Suppose you have invested $100,000 in a po rtfo lio of Potpourri a nd Mag no lia , a nd the beta of the portfoli o i s 1 . 77 . Ho w much did you invest in each stock? what is th e exp ected r eturn o n the portfolio?

Question 5

Consider the follo wing two sto cks: Stock A B

i

Beta 1.4 0.7

Exp ected R eturn 25% 14%

Assume the CAPM hol ds. Based upon the C APM, w hat i s the return o n the market? What is the risk-free rate?

Question 6

1 . If a po rtfo lio has a positive weig ht fo r each asset, can th e exp ected r eturn o n the portfolio be g rea ter than the retur n o n the asset i n the portfolio that has the high est retu rn? Can t he expected return on the portfoli o be l ess t han t he ret urn on t he asset in t he po rtfo li o w it h t he lowest return? Explai n. 2. Comment on t he following quotation from a leading investment a nalyst. Stocks that move perfectly with the market have a beta o f 1 . Betas get hig her as volatility goes up and lower a s it go es down. Thus, So uthern Co, a utility whose share have traded close to $12 for most of the past t hree years, has a lo w beta . At the other extreme, th ere is True North Networks, which ha s been a s $150 and as low a s its curr ent $15 . 3. Given the followi ng si tua ti ons, determine in each case whether o r not the hypothesis o f a n e¢ci ent capital market (semistro ng fo rm) is contradicted. a ) Thro ug h the i ntro ductio n o f a co mplex co mputer program into the anal ysi s of pa st sto ck price changes, a brokerage …rm is able to predi ct pri ce movements w ell eno ugh to earn a consi stent 3% pro …t, adjusted for risk, above no rma l ma rket returns. b) On the averag e, investors in the stock market this yea r a re expected to ea rn a p o si ti ve r eturn (pr o…t) on thei r i nvestment. Some i nvestors will earn con si dera bl y mo re tha n o th ers. c) You have discovered that the squa re roo t o f a ny given sto ck price multiplied by the da y o f the mont h provi des an i ndi catio n o f the direction i n pri ce movement of that parti cul ar stock wi th a pro ba bility of 0. 7 . 4 . Wha t so rt of investor ratio na lly views the variance (or standard devia ti o n) o f a n indi vi dual security’s retu rn as the securi ty’ s pro per mea sure of risk? What sort of i nvesto r ra ti onall y vi ews th e b eta of a security as the security’ s pro per mea sure of ri sk?

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Answer 1:

1. E(R A ) = (7: 3 + 11: 5 + 16: 6)=3 = 11 : 8% E(R B ) = (¡4 : 7 + 5: 4 + 24 : 3) =3 = 8: 3% 2 = f(0 : 073 ¡ 0 :118)2 + (0: 115 ¡ 0 :118) 2 + (0 :166 ¡ 0 : 118)2 g= 3 = 0 : 001446 2. ¾A 1= 2 = 0 : 0380 = 3 :80% ¾ A = (0: 001446 ) 2 2 2 2 ¾ B = f(¡0 :047 ¡ 0 : 083 ) + (0 : 054 ¡ 0: 083 ) + ( 0: 243 ¡ 0 :083) g = 3 = 0 :014447 1 =2 = 0: 1202 = 12 :02% ¾ B = (0 :014447) 3 . Co v (R A ;R B ) = [(0 :073 ¡ 0 :118)(¡0 :047 ¡ 0 : 083) + ( 0: 115 ¡ 0: 118 )(0 : 054 ¡ 0: 083 )+(0 :166 ¡ 0 : 118(0: 243 ¡ 0 :083)] = 3 = =[0.00585+0.000087 +0 . 00768 ] /3 =0 .004539 Corr (R A ;R B ) = 0 :004539= (0 :0389 ¤ 0 : 1202) = 0 : 9937: Answer 2:

The typical risk-averse investor seeks high returns and low ris ks. To a ssess the two s tocks, …nd the risk a nd return pro …les fo r each stock. Sta te of Economy Pr. R A R A ¡ E (R A) (R A ¡ E (R A)) 2 P £(R A ¡ E (R A)) 2 . Recession 0.2 -0. 2 -0. 26 0 . 0676 0.01352 Normal 0.6 0. 10 0 . 04 0.0016 0.00096 Expa nsio n 0 .2 0. 20 0 . 14 0.0196 0.00392 * since security A pays no dividend, the return on A i s si mpl y ( P1 = P0 ) ¡ 1: E(R A ) = 0 : 2(¡0 :20 ) + 0 :6 (0 :10 ) + 0 :2 (0 : 20 ) = 0: 06 E(R B ) = 0 :09 Thi s i s gi ven i n the pro bl em. Risk i s ca l cul ated i n table : Var(R A ) = 0: 0184 Sta nd ard devia ti o n is (0 :0184 )1= 2 = 0 : 1356 ¯ A = fCorr(R A; R M )¾ ( R A)g =¾ (R M ) = 0: 8 (0 :1356 )= 0: 10 = 1 :085 ¯ B = fCorr(R B ; R M )¾ ( R B )g =¾ (R M ) = 0 : 2(0 : 12 )= 0: 10 = 0 :24 The return on stock B i s hi gher than t he return on stock A. The ri sk of stock B, as measured by its beta, i s l ower than the risk of A. Thu s, a typi cal ri sk-averse investor will prefer sto ck B. b. E(R P ) = 0: 6 E (RA ) + 0 : 4E (RB ) = 0 : 6(0 : 06 ) + 0 : 40 (0 : 09 ) = 0: 0576 2 2 2 2 2 ¾ P = (0: 6) ¾ A + (0 :4) ¾ B + 2 (0 : 6)(0: 4)Corr(RA ; RB )¾ A ¾ B = 0 :01361595 ¾ P = 0: 116687 c. The beta of a portfolio is the weighted averag e o f th e b etas o f the co mpo nents o f th e p ortfoli o : ¯ P = (0 :6 )¯ A + (0 : 4)¯ B = (0: 6 )(1 :085) + (0: 4 )(0 :240) = 0 : 747 Answer 3.

1. E(R A ) = 0: 6 (0: 03) + 0: 4 (0 :15 ) = 0 : 078 = 7 : 80% E(R B ) = 0 :6 (0 :065) + 0: 4 (0 :065 ) = 6 :5% 2 = 0 : 6(0 : 03 ¡ 0 : 078 )2 + 0 :4 (0 :15 ¡ 0 :078)2 = 0 :003456 ¾ A 1= 2 = 0 : 05878 ¾ A = (0: 003456 ) 2 ¾B = ¾B = 0 2. W A = $2; 500= $6; 000 = 0: 417 W B = 1 ¡ 0 : 417 = 0 : 583 E(R P ) = 0: 417 (0 :078) + 0: 583 (0 :065) = 0 : 0704 = 7 :04% 2 = W 2 ¾ 2 = 0 :0006 ¾ P A A 1= 2 = 0 : 0245 = 2: 45% ¾ P = (0: 0006 ) 3 . Amo unt borrowed = -40*$50 =$ -2000 W A = $8; 000 = $6 ; 000 = 4= 3 W B = 1¡W A = ¡1= 3 E(R P ) = (4= 3 )(0 :078) + (¡1 =3 )(0 : 065 ) = 0: 0823 = 8 : 23% 2 2 2 2 ¾ P =W A¾ A = (4= 3 ) (0 :003456) = 0 :006144 1 = 2 ¾ P = (0: 006144 ) = 0 : 07838 Questio n 4 :

a. The risk premium =R m ¡ Rf Potpourri stock return: 16.7=7.6 +1 .7[ Rm ¡ Rf ], then [ Rm ¡ Rf ] = [16 :7 ¡ 7 : 6] = 1: 7 = 5 : 3529% b. E(R M ag ) = 7: 6 + 1 :8 (5 :353) = 17 :2353% c. W P ot ¯ P ot + WM ag ¯ M ag = 1 : 77 1.7 WP ot + 1: 8(1 ¡ WP ot) = 1 : 77

3

= 0 03 =03 ) = 7 6 + 1 77(5 353 ) = 17 07 %

=07

0.1 WP ot : , then WPot : and W Mag : Thus invest $30,000 in Potpo urri sto ck a nd $70,000 in Mag no lia. E(R P : : : : No te: the o ther way to cal cul ate E(R P : : : :

Question 5:

+ 1 4[ + 0 7[

) = 0 30(0 167 ) + 0 7(0 17235 ) = 17 07%

] ]

:

0.25=R f : R m ¡ Rf 0.14=R f : R m ¡ Rf Substract the second equatio n fro m the …rst, we get: 0.11 =0 .4 [ Rm ¡ Rf o r [ Rm ¡ R f =0.1571 Put the above equation in a ny o f the previous two ones for either asset A or Asset B, so we g et: 0.25 = Rf : : a nd therefore R f Use [ Rm ¡ Rf =0.1571 to …nd Rm : Rm : : : :

+ 1 4[0 1571] ]

Answer 6:

]

]

= 3% = 0 1571 + 0 03 = 18 71 %

1 . The expected r eturn o n a ny po rtfo lio must be less than or equal to the return o n the s tock with th e h ighest return. It cannot be g rea ter than this stock’s return beca use all stocks with lower returns will pull down the value of the weighted average retu rn. Simil arly, t he exp ected ret urn on a ny p ortfolio must be greater than or equal to the return of the asset with the lowest retur n. T he portfolio retu rn cannot be less than the lowest r eturn in the portfolio beca use a ll hig her ea rni ng s sto cks will pull up the vl a ue o f the weighted averag e. 2. If we a ssume tha t the ma rket ha s no t sta yed constand during the past t hree years, then the low volatility o f So uthern Co.’ s stock pri ce onl y indicates that the sto ck has a b eta that is very near to zero. The hig h vo latility of Texas In stru ments’ stock p rice do es n ot i mpl y that the …rm’ s beta is hig h. Total volatility (the price ‡uctuation) is a function of both systematic and unsystema ti c ri sk. The beta only re‡ects the systematic risk. Observi ng pri ce volatility does not i nd icate whether it was due to systema ti c factors, or …rm speci…c factors. Thu s, if yo u o bserve a high price volatili ty like tha t o f TNN, yo u ca nno t claim that the beta of TNN’s stock i s hi gh. All you know is that the to tal risk of TNN i s hi gh. 3 a) The information which enables the brokerage …rm to ea rn a consistent 3 % a bno rma l pro …t is not costl ess. If the compu ter costs exceed the excess 3% pro…ts from s tocks, th e … rm is a ctua lly ea rni ng wo rse than no rma l returns. If the co mputer co sts a re less tha n the 3 % pro…t, semi-strong ca pita l market e¢ci ency may be refuted. Also, brokerage fees may wipe o ut a ny tra di ng pro …ts. b) T he hypothesis o f a n e¢cient ca pita l market is not co ntra di cted. Except for very bad years, the average (and exp ected) return on t he ma rket is p o sitive. Thi s i s consid ered a n ormal retu rn. It is also a fair game. The fa ct tha t some investors enjoy high er return s than others i s the result of the uncertainty i n stock return s. Given any probabili ty d istributio n, so me o bservations will lie above th e mea n and some will lie below. c) Semi-strong (as well as weak) ca pita l ma rket e¢cien cy is contradicted. You have discovered a tradi ng rul e based on past, nearly costl ess, price i nfo rma ti on that ena bles you to forecast future prices with better-tha n-random accuracy. Thus, all rel evant publicly available informatio n ha s no t been insta nta neo usly incorpora ted into sto ck p rices. 4. A goo d a nswer mi ght be something li ke th e fo ll owing: A rational, ri sk-averse investor views the va ri a nce (o r sta ndard deviation) of h er p ortfolio’s return as the proper mea sure o f the risk of her po rtfo lio. If for so me rea son or another the i nvestor can hold o nly o ne security, the varia nce o f tha t security’s return becomes t he varia nce o f the p o rtfolio’s retu rn. Hence, the varian ce of th e security’s return is the security’s pro per mea su re of ri sk. If an invi di vi du al holds a diversi…ed portfolio, she still views the variance (or standard deviation) of her portfolio’s retu rn as th e p roper measure of the risk of her portfolio. However, she is no longer interested in the variance of each indivi dual securi ty’s return. Rather, she i s i nterested i n the contri bution o f an individual security to the variance of the portfolio. Und er t he assumptio n o f ho mog eneo us exp ectatio ns, a ll i ndi vi dual s ho l d the market po rtfo li o. Thus, we measure risk a s the co ntributio n o f an individual security to the va ri a nce of th e market p ortfolio. This contributio n, w hen sta ndardized properly, i s the beta of the securi ty. Whil e very few i nvesto rs ho ld the market po rtfo lio exa ctly, ma ny ho ld reaonsably diversi…ed portfolio s. These p ortfolios are clo se eno ug h to the market portfolio so that the beta of a securi ty i s li kely to be a rea so na ble mea sure o f i ts risk.

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