Honor Code: You must do these problems on your own, and submit your answers no later than 11 PM on Sunday, October 21.
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The circus is back in town!
Bailey Barnum, the circus owner (and a grand-niece of P.T.), had planned to have two entrances set up on the north side of the circus grounds (near the main exit from the underground parking garage), and one on the south side. However, a city inspector ordered her to close one of the two northern entrances in order to simplify crowd control and security. She�d originally set up her staffing expecting the northeast entrance to process 7,000 circus-goers (with one standard-deviation�s-worth of uncertainty in her forecast being 850), and expecting the northwest entrance to process 5,000 (with a standard deviation of 600). From past experience, she expected the numbers of patrons passing through the two northern entrances to have a correlation of �0.35 (if more people choose one entrance, that leaves fewer to go through the other). Now, she needs to staff a single northern entrance dealing with all the northern (i.e., both northeastern and northwestern) arrivals.
What is the number of circus-goers expected to use the now-combined northern entrance? patrons
What is one standard-deviation�s-worth of uncertainty in the number who will use the northern entrance? patrons
Agilo the Juggler has $100,000 invested in his 401(K) retirement account. Currently, $55,000 is in shares of Bailey�s circus, and (to spread his risk a bit) $45,000 in shares of the circus� primary competitor (Cirque de la Lune). A recent analysts� report has predicted that the rate of return on shares in Bailey�s circus will be 16% over the next year, with one standard-deviation�s-worth of uncertainty in the prediction being 20%. The same report predicts a rate of return of only 10% for the shares in the competing circus, with a standard deviation of 13%.
What is the expected rate of return on his portfolio? %
Assume (somewhat unrealistically) that returns on the two investments vary independently.
What is one standard-deviation�s-worth of uncertainty in next year�s rate of return on his 401(K)? %
Agilo wants to limit one standard-deviation�s-worth of uncertainty in his portfolio�s rate of return to 15%. By rebalancing his portfolio (i.e., by shifting money between the two investments), what�s the highest expected rate of return he can achieve, consistent with his risk-limitation goal? %
Hint: Use Solver.
As the Christmas season approaches, you feel that consumer sentiment in your neighborhood is somewhat better than it is reported to be nationally, and you think this bodes well for regional retail sales. Indeed, you feel that shares in Webmart, a regional retail chain, are slightly undervalued. Over the next 20 trading days, you believe that the share price - currently $66/share - is likely to drift upwards. Specifically, you think that the share price will, each day, either drop $1 (a 20% chance), stay unchanged (a 45% chance), or move up by $1 (a 35% chance). (The day-to-day price changes will vary independently.)
What is the expected change in share price in the course of one day? $
What is one standard-deviation's-worth of uncertainty in the one-day price change? $
What is the expected stock price 20 days from now? $
What is one standard-deviation's-worth of uncertainty in the stock price 20 days from now? $
Hint: The price of a share 20 days from now will be $66 plus the sum of 20 single-day price changes.
You don't have enough free capital to buy 100 shares in Webmart right now. However, the November $70 call option, which expires on the third Friday in November - exactly 20 trading days from now - is selling for only a $0.50 premium.
(This means that, for $50, you can purchase an option to buy 100 shares in Webmart at a price of $70 on the expiration day. On the day of expiration, if the share price is below $70, your option will be valueless (and you'll lose your $50). However, if the share price is above $70, you'll be able to exercise the option, and immediately sell the 100 shares at the market price. For example, if the share price is $73 twenty days from now, the option will net you 100⋅($73-$70) - $50 = $250 in profit. To keep things simple, we'll ignore commissions and taxes.)
Using 50,000 simulation runs, estimate your expected profit from the purchase of a 100-share November $70 call option. $
Hint: Putting a RAND() in two places in the same formula will yield two distinct random values. To "freeze" a RAND() that will be used in more than one place, put it in a cell of its own, and then reference it several times. If you put =RAND() in cell A1, and =IF(A1<0.2,1,IF(A1<0.7,2,3)) in another cell, then the latter cell will contain the value 1 20% of the time, 2 50% of the time, and 3 30% of the time. Of course, you can also simulate a random variable directly from its cumulative probability distribution using the LOOKUP function, e.g., =LOOKUP(RAND(),{0,0.2,0.7;1,2,3}).
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