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Chicago Booth BUSF 35150 - Problem Set 1

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Business 35150 John H. Coc hraneProblem Set 1Due in class, week 1Do the readings, as specified in the syllabus. Answer the following problems.Note: in this and following problem sets, make sure to answer the direct questions, as well as makethe indicated plots and tables. That’s how we grade them.Part IThese problems should help you to get up to speed with time series. Read the time series section ofthe notes first if this isn’t easy.1. A stock price follo w s an AR(1) process, i.e.= −1+ where are a sequence of i.i.d. normally distributed random variables with mean 0 and variance2.(a) Find the sequence of conditional means. I.e., if you have observed values {,−1 } find(+1)(+2)(+) in terms of those observed values. (Hints: You may hav e seen theconditional expectation (+) written as (+|−1 ) or (+|) I.i.d. meansthat (+)=0(+)=2. Start by writing +1= + +1; +2= 2++1+ +2, ... and then take means and variances given information available at time t,{−1}.)(b) Find the sequence of conditional variances.2(+1)2(+2) etc. (Again, you may have seenconditional variance written as 2(+|−1 )or 2(+|).)(c) Suppose  =095 ()=025, =0for all  ≤ 0, and at time 1 you see 1=1.i. Plot the conditional mean, where we expect to go in the future given we have seen1=1, i.e. plot 1() vs.  for  =1 2 20.ii. Add plus and minus one conditional standard deviation bands, i.e. plot 1() ± 1()for  =1 220. This plot gives you a sense of how uncertain the forecast is, the rangeof things that might happen. Risk management is as (if not more) important thanforecasting!iii. Add a simulation of to your plot — pick {=2 3 4} from a random numbergenerator and plot the resulting path of .(1=1of course.) (In matlab, the randncommand makes random normals. Then you can use for i=1:20, ... end to loop throughand make  from ). You should do this several times for yourself and watch how differentrandom numbers lead to different paths.The point of this is to get a feeling for how “mean-reve rts” and how uncertainty aboutits future builds up through time. By looking at the simulated values you should get somesense of ho w the conditional mean (forecast) and variance relate to the actual series.(d) Find the unconditional mean and variance () and 2(). There are two ways to do this.First, the unconditional mean is the limit of the conditional mean as you move informationbackwards through time, () = lim→∞−(). Second (better), you can just take theunconditional expectation of both sides of = −1+ yielding ()= (−1) .1Then, realize that the mean of and −1are the same — unconditional moments don’tdepend on calendar time.(e) How would you generalize the AR(1) model = −1+ to produce a mean ()=?2. In this problem, I want you to think about how returns look at different time horizons, and how pre-dictabilit y of returns might change that. Suppose first that one-year log returns =log() arenot correlated over time (+)=0and have mean ()= and variance 2()=2thatare constant over time. The compound log long-horizon return is log(+1+2+)=+1++2+++and the annualize d compound log long-horizon return is logh(+1+2+)1i=1[+1+ +2+  + +]. Now, let’s think about how these returns scale with horizon. The bigquestion underlying this analysis is, are returns in some sense “safer” for long-horizon investors?(a) Find the mean log long-horizon return  [+1+ +2+  + +] and the mean annualizedlog long horizon ¡1[+1+ +2+  + +]¢return as a function of the horizon  and theone-period mean . If the mean annual return is 6%, what is the mean monthly return? Whatis the mean annualized monthly return?(b) Find the variance of the log long-horizon return 2[+1+ +2+  + +] and of the annu-alized log long horizon return 2¡1[+1+ +2+  + +]¢as a function of the horizon and the one-period variance 2(c) Find the standard deviation of the log long-horizon return  [+1+ +2+  + +] and ofthe annualized log long horizon return ¡1[+1+ +2+  + +]¢as a function of .Ifthestandard deviation of annual returns is 16%, what is the standard deviation of monthly anddaily returns?(d) A stock has mean and standard deviation of monthly returns equal to 1%. Express these onan annualized basis.(e) Find the “Sharpe ratio” (quotes since you’re using logs and not subtracting a risk free rate)(+1+  + +)(+1+  + +) as a function of horizon . Does the Sharpe ratiodepend on the units (annual vs. monthly return) in which you quote it?(f) The typical portfolio allocation form ula says stock allocation should depend on the ratio ofmean to variance of total return:stock share =1risk aversion×()2()For example, if mean returns are 8% (0.08), the standard deviation of returns is 20% so2()=004, and risk aversion is 2, then the investor should put all his money in stocks,share = 1. Now, how does this advice scale with horizon? Should an investor with a 10 y earhorizon allocate more to stocks than an investor with a 1 year horizon because “stoc ks aresafer in the long run” and he can “wait out market declines?”(g) Client: "No, you have it wrong. The mean return is independent of horizon, but standarddeviation of returns is lower at long horizons because of time-diversification. Long horizonaverage returns are very stable, and volatility of long-run average returns goes down withthe square root of horizon. I really can afford to take more equity risk


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Chicago Booth BUSF 35150 - Problem Set 1

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