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Old 07-10-2018, 10:03 PM
eastla_student eastla_student is offline
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Default ASM IFM Exercises 6.5-6.6

The logic behind Exercise 6.5 ASM IFM manual is bugging me.

An investment of 10,000 in stock S has expected return E(R) = 0.10 and SD(R) = 0.30. Risk-free rate is r = 0.04.
You will borrow x at the risk-free rate and invest in more of stock S in order to increase your expected return to 0.12


Exercise 6.5 asks What is x?

The set up for the answer is 0.10(10,000 + x) -0.04x = .12(10,000).

I know I must be missing something easy, but I don't understand the right side of the equation. Why are we taking a percent of the original investment using the new expected return? Wouldn't the new expected return be applied to the new total investment of 10,000 + x, in other words, .12(10,000 + x) ?


As for 6.6, the same setup is used as above but they ask: What is the volatility of the portfolio after the loan is taken.

In the answer, they state the Sharpe ratio of a portfolio is not changed by a loan, and use that to solve. However, I'm not convinced. How do we derive that fact?
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Last edited by eastla_student; 07-10-2018 at 10:06 PM.. Reason: original was incomplete
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Old 07-10-2018, 10:35 PM
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I don't have the ASM manual, but I will answer based on the information that you provided.

Exercise 6.5
Originally you have $10,000 invested in Stock S. Now, you want to borrow $x and invest that money to buy more of Stock S. So, your portfolio has two things:

1. A short position in the risk-free asset. The value of this position is -x.
2. A long position in Stock S. The value of this position is +x+10,000.

Thus, the total value of the portfolio is -x+x+10,000=10,000.

The proportion weight on Stock S
= (Value of the investment in Stock S)/(Total value of the portfolio)
= (x+10,000)/10,000

The proportion weight on the risk-free asset
= 1 - (+x+10,000)/+10,000
= -x/10,000

Then, the portfolio's expected return is:

E[R_P]
=(Weight on Stock S)*(Expected Return on Stock S) + (Weight on the risk-free asset)*(Expected Return on the risk-free asset)
= (x+10,000)/10,000*0.10 + -x/10,000*0.04

Equating this to the given portfolio's expected return of 0.12, we have:

(x+10,000)/10,000*0.10 + -x/10,000*0.04 = 0.12

Finally, we have:

(x+10,000)*0.10 - x*0.04 = 0.12*10,000

which is the equation that you stated above.

Exercise 6.6

When you combine a risk-free asset with a risky asset, you will get a straight line (it's called the capital allocation line), and the slope of that line is the Sharpe ratio. Any combinations of the risk-free asset and the risky asset will lie on that line. Since all of them are on the same line, they will have the same slope, and thus they will have the same Sharpe ratio.

Hope this helps!
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Last edited by tkt; 07-10-2018 at 10:47 PM..
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Old 07-10-2018, 10:39 PM
Abraham Weishaus Abraham Weishaus is offline
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For 6.5, the problem is that the stock only pays 10%. Just because you want it to pay 12% doesn't mean it will. If you invest 10000+x in the stock and call up the company's president and ask him/her to reward you for investing more in their stock, they'll ignore you. The most you can do is take a loan and leverage the investment.


For 6.6, if the Sharpe ratio changed, you could create an arbitrage by buying the low-volatility one and selling the high-volatility one. For example, if A has 30% volatility and B has 40% volatility and A and B are perfectly correlated and both pay 4%, buy 4 units of A and sell 3 units of B. Since they're perfectly correlated, you can't lose anything, yet you'll earn 4%.
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Old 07-11-2018, 12:11 AM
eastla_student eastla_student is offline
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Thank you all very much, tkt and Abraham! I love this forum!
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Old 07-11-2018, 12:39 AM
eastla_student eastla_student is offline
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tkt, I'm am digesting your explanation and I having trouble with how the short position is playing into this problem. Isn't a short position when you borrow stock and sell it, in order to buy it back later at a lower price?

My understanding in 6.5 is that your are borrowing $x at the risk-free rate and buying $x of S, with the idea of selling it later at a higher price. How is that the same as occupying a short position? This is where my lack of knowledge about the world of investing is messing me up.
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Old 07-11-2018, 12:48 AM
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Quote:
Originally Posted by eastla_student View Post
tkt, I'm am digesting your explanation and I having trouble with how the short position is playing into this problem. Isn't a short position when you borrow stock and sell it, in order to buy it back later at a lower price?

My understanding in 6.5 is that your are borrowing $x at the risk-free rate and buying $x of S, with the idea of selling it later at a higher price. How is that the same as occupying a short position? This is where my lack of knowledge about the world of investing is messing me up.
Note that:
  • Borrowing money is like selling (i.e., short position) a zero-coupon bond. Both results in cash inflows now and cash outflows later.
  • Lending money is like buying (i.e., long position) a zero-coupon bond. Both results in cash outflows now and cash inflows later.

This is why borrowing x is the same thing as selling a zero-coupon bond for x, which is the same as a short position in the zero-coupon bond.
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Old 07-11-2018, 12:53 AM
eastla_student eastla_student is offline
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I see, thank you for generalizing the concepts of long and short.
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