Understanding Efficient Markets and Drug Patent Valuation

School

University of California, Berkeley**We aren't endorsed by this school

Course

ECON 136

Subject

Economics

Date

Dec 10, 2024

Pages

Uploaded by ChefKnowledge37692

1Problem Set 5 SolutionsEcon 136, Fall 20241.Efficient markets(a)True. Although earnings increased, the increase may have been below expecta-tions from available information. This would be bad news and would cause thestock price to fall.(b)False.If the market is efficient in the semi-strong sense, but not efficient inthe strong sense, then people with inside information will start purchasing thecompany’s shares before the good news is announced, driving up the stock price.Thus efficiency in the semi-strong sense is consistent with a price increase justbefore the announcement. This is what we observed for takeover announcementsin the data.(c)True.The efficient markets hypothesis states that expected abnormal returnsshould be zero.Compensation for risk and the time-value of money should bereflected in expected normal returns. So, risky stocks having higher returns onaverage than safe stocks is consistent with EMH.(d)False.As long as there are smart investors in the market, they can undo theactions of crazy investors and set prices so that there are no positive expectedabnormal returns.

22.A biotech company(a) Event tree:FailureFailureSuccessFailureSuccessSuccess$55M$25M$0$80M0.250.250.250.750.750.75(b) LetXdenote the drug patent value.E1[X|success]=0.25·80 + 0.75·55 = $61.25ME1[X|failure]=0.25·25 + 0.75·0 = $6.25M(c)E0[X] = 0.25·0.25·80 + 0.25·0.75·55 + 0.75·0.25·25 + 0.75·0.75·0 = $20M(d) We just showed thatE0[X] = $20MNow observe thatE0[E1[X]] = 0.25·E1[X|success] + 0.75·E1[X|failure] = $20MsoE0[X] =E0[E1[X]]This means that the Law of Iterated Expectations (LIE) holds. Your best guesstoday of your best guess in the future of the drug patent value is your best guesstoday of the drug patent value.

3Now we show thatE0[X-E1[X]] = 0:E0[X-E1[X]]=0.25·0.25·(80-61.25) + 0.25·0.75·(55-61.25)+ 0.75·0.25·(25-6.25) + 0.75·0.75·(0-6.25)=1.17-1.17 + 3.52-3.52 = 0This result shows that the expected forecast error next period is equal to zero.If you are rational, you cannot predict how you will change your forecast in thefuture.(e)Var0(X)=E0[X2]-E0[X]2=0.25·0.25·802+ 0.25·0.75·552+ 0.75·0.25·252+ 0.75·0.75·02-202=684.375Var0(E1[X])=E0[E1[X]2]-E0[E1[X]]2=0.25·61.252+ 0.75·6.252-202=567.1875Var0(E0[X])=E0[E0[X]2]-E0[E0[X]]2=202-202=0These numbers verify thatVar0(X)>Var0(E1[X])>Var0(E0[X]). The drugpatent market value,X, is the actual outcome att= 2,E1[X] is your best guessofXatt= 1, andE0[X] is your best guess ofXatt= 0.Xhas four possiblevalues:{0; 25; 55; 80}. The range of values forE1[X],{6.25,61.25}, is smaller thanthe range of values forX. As a result, our measure of spread, variance, should besmaller, i.e.Var0(E1[X])<Var0(X).E0[X] has only one value,E0[X] = 20, soVar(20) = 0. The spread ofXis greater than the spread of your best guess ofXfrom the prior period; the spread of your best guess ofXis greater than thespread of your best guess ofXfrom the prior period. The general point here isthat we haveVar0(X)>Var0(E1[X]) because there is more uncertainty aboutXthen there is aboutE1[X].

43.Covariance and correlationState (si)Prob (si)R1R20.5R1+ 0.5R2s11/30.08-0.060.01s21/3-0.040.140.05s31/30.110.070.09E0.050.050.05Var0.00420.00690.0011(a)E[R] =3Xi=1R(si)·Prob(si)E[R1] =13·(0.08) +13·(-0.04) +13·(0.11) = 0.05E[R2] =13·(-0.06) +13·(0.14) +13·(0.07) = 0.05Var(R) =3Xi=1Prob(si)·(R(si)-E[R])2Var(R1) =13·(0.08-0.05)2+13·(-0.04-0.05)2+13·(0.11-0.05)2= 0.0042Var(R2) =13·(-0.06-0.05)2+13·(0.14-0.05)2+13·(0.07-0.05)2= 0.0069Std(R1) =Var(R1)1/2= 0.065Std(R2) =Var(R2)1/2= 0.083(b)Cov(R1, R2) =3Xi=1Prob(si)·(R1(si)-E[R1])·(R2(si)-E[R2])Cov(R1, R2)=13·(0.08-0.05)·(-0.06-0.05) +13·(-0.04-0.05)·(0.14-0.05)+13·(0.11-0.05)·(0.07-0.05)=-0.0034

5ρR1,R2=Cov(R1, R2)Std(R1)·Std(R2)ρR1,R2=-0.00340.065·0.083=-0.63(c)Rp=12·R1+12·R2E[Rp] =13·(0.01) +13·(0.05) +13·(0.09) = 0.05Var(Rp) =13·(0.01-0.05)2+13·(0.05-0.05)2+13·(0.09-0.05)2= 0.001067Std(Rp) = (0.001067)1/2= 0.03266(d) A mean-variance optimizing investor maximizesE[R]-0.5A·Var(R), and prefersportfolios that generate greater values of this “utility” than portfolios that gen-erate lower values. The expected returns on the portfolio consisting only of asset1, the portfolio consisting only of asset 2, and the equally-weighted portfolio areall 0.05, so we can focus just on the risk term,-0.5A·Var(R), when rankingportfolios.–ForA= 0, the investor is risk-neutral and since they all give the sameexpected return, the investor gets the same utility from all the portfolios. Asa result, she is indifferent between them all.–ForA >0, the investor is risk-averse and prefers portfolios with lower risk tothose with higher risk, so Portfolio>Asset 1>Asset 2.–ForA <0, the investor is “risk-loving” and prefers portfolios with higher riskto those with lower risk, so Portfolio<Asset 1<Asset 2.

64.Historical returnsSee the spreadsheet “returnssolution.xlsx”.(a) As discussed in lecture, in the period 1927-2007, small-cap stocks have outper-formed large-cap stocks and value stocks have outperformed growth stocks. Theseare two of the so-called anomalies with respect to the efficient markets hypothesis:the value effect and the size effect.(e) See the figure below.(f) For the most part, we find that portfolios with higher average returns have higherrisk, as measured by the standard deviation on those returns. This makes sensebecause if investors are risk-averse, they require higher expected returns in ex-change for shouldering higher risk. Also, since the market contains a combinationof big growth, big value, small growth, and small value stocks, as well as someother stocks, the logic of diversification suggests that it should have a reducedstandard deviation.(g) The mean return of the equally weighted portfolio is the average of the meanreturns of the big growth, big value, small growth, and small value portfolios, asexpected.The risk (standard deviation) also lies between the risks of the fourtypes of portfolios, suggesting that big growth, big value, small growth, and smallvalue are positively correlated.