Suppose that you recently purchased a $500,000 house and have a $400,000 mortgage

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Suppose that you recently purchased a $500,000 house and have a $400,000 mortgage. You alsohave $100,000 invested in the S&P 500. Take your portfolio as the investment in the house,mortgage, and stock market and consider the following statement:If house prices go down by 10% and the stock market and interest rates don’t change, yourportfolio will decrease in value by 5%.Is this statement true or false?(b) Consider the cumulative return for holding a stock over a two-week period. We can computethis return by compounded each weeks return:1 + return2wk=(1 + returnwk1)×(1 + returnwk2).Because of the compounding, we have a cross term:1 + return2wk= 1 + returnwk1+ returnwk2+ returnwk1×returnwk2????cross term.For example, if we had a 1% return over wk1 and a 2% return over wk2, the cumulative returnwould be 3.02% since1.01×1.02 = 1.0302.Notice it is slightly bigger than 1% + 2% because of the compounding.Consider the facts:(i) The cross term will be a lot smaller usually and we can approximate it as zero. Based onthis logic, if we ignore this cross term:return2wk?returnwk1+ returnwk2(ii) The correlation of returns between each week is very low and we can approximate as zero.3That is,corr(returnwk1,returnwk2)?0.(iii) The volatility is about the same each week so?(returnwk2)??(returnwk1)Your friend from UCLA reasons that based on these three assumptions,?(return2wk)?2×?(returnwk1)Is your friend’s conclusion true? Hint: if you are not sure, you could try looking at the SPYhistorical data for historical weekly returns and check your answer.

USC Marshall School of Business

Financial Markets             GSBA 539

Scott Joslin          Summer 2021

Problem Set 2: Risk and Return

Due: Monday July 26 at 5:59 PM

Show your work cleanly and write out the formulas that you used to solve the problems. You are encouraged to work together on the problems, but you must write up the solutions on your own. You also need to acknowledge any help you received (beyond class MarshallTalk discussions) on the first page of your work.

All problems may be done with Excel.

I will hold office hours on Zoom on Thursday July 22 from 7–8 PM and Saturday July 24 from 10 – 11 AM.

Problem 1

(40 points) SPY is an exchange-traded fund that tracks the S&P 500. Go to Yahoo! finance and download the monthly adjusted close price for SPY and DIS for the period March 1, 2011 to March 1, 2021.

(a)          Compute the following estimate for µ, the average monthly return for each price series:

 .

Remember that to compute the return you can use the formula

adjusted close t+1

Rt,t+1 =                − 1,

 

adjusted closet

so it will be useful to add a column in Excel for this return. Here T is the number of months that we have data for the price; so T − 1 is the number of monthly returns that we have.

(b)          Make another column for the surprise and (surprise)  in each return relative to the expectation you computed above in ( a ): surpriset+1 = returnt,t+1 − µ.ˆ

Use this to compute estimates of the monthly volatility of the returns:

v

 

u             T−1

 surprise

You can use stdev.p function in Excel to check your answer.2

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(c)           Repeat parts (a) and (b), except use only the data since March 1, 2020 (around the start of the pandemic).

(d)          If you wanted to estimate the volatility going forward, which of the answers do you think is a better measure?

Problem 2

(10 points) True or False. There are 2 questions. Each question you answer correctly is worth 7 points to a maximum of 10.

(a)          Suppose that you recently purchased a $500,000 house and have a $400,000 mortgage. You also have $100,000 invested in the S&P 500. Take your portfolio as the investment in the house, mortgage, and stock market and consider the following statement:

If house prices go down by 10% and the stock market and interest rates don’t change, your portfolio will decrease in value by 5 %.

Is this statement true or false?

(b)          Consider the cumulative return for holding a stock over a two-week period. We can compute this return by compounded each weeks return:

1 + return2wk = 1 + return 1 + return .

Because of the compounding, we have a cross term:

1 + return2wk = 1 + returnwk1 + returnwk2 + returnwk1 × returnwk2 .

 

|              {z            }

cross term

For example, if we had a 1% return over wk1 and a 2% return over wk2, the cumulative return would be 3.02% since

1.01 × 1.02 = 1.0302.

Notice it is slightly bigger than 1% + 2% because of the compounding.

Consider the facts:

(i)            The cross term will be a lot smaller usually and we can approximate it as zero. Based on this logic, if we ignore this cross term:

return2wk ≈ returnwk1 + returnwk2

(ii)           The correlation of returns between each week is very low and we can approximate as zero. That is, corr(returnwk1,returnwk2) ≈ 0.

(iii)          The volatility is about the same each week so

σ(returnwk2) ≈ σ(returnwk1)

Your friend from UCLA reasons that based on these three assumptions,

σ(return2wk) ≈ 2 × σ(returnwk1)

Is your friend’s conclusion true? Hint: if you are not sure, you could try looking at the SPY historical data for historical weekly returns and check your answer.

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