University of Toronto (STG), Department of …...ECO 204 Microeconomic Theory for Commerce 2017-2018 Ajaz’s Sections Test 2 Solutions [Detailed step-by-step solutions are provided

1 © ECO 204 2017-2018 TEST 2 SOLUTIONS

University of Toronto (STG), Department of Economics ECO 204 Microeconomic Theory for Commerce 2017-2018

Ajaz’s Sections Test 2 Solutions

[Detailed step-by-step solutions are provided to help students understand the concepts, ideas, and methods. Such detailed solutions are not required in the actual test]

Due to rounding errors and how you carry out calculations, your numerical answers will probably differ from these solutions

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TEST DETAILS

Duration: Two hours (5 – 7 pm)

Total number of questions: 2

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Maximum number of points: 175

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QUESTION 1 [TOTAL 90 Points] A French supermarket chain with seven stores sells, amongst other items, two brands of soda: “Brand X” and “Brand Y”. Each brand of soda comes in two varieties “regular” and “lite”1 (i.e. the chain carries “Brand X-regular”, “Brand X-lite”, “Brand Y-regular”, and “Brand Y-lite”). You’re given the Below is a description of the variables in the “point of sale” data set for this French supermarket chain (data is across all seven stores (stores “62” through “68”) over 52 weeks (week “101” through “152”)):

Variable Description of Variable

STORE Store # ("62" through "68")

Hval_150 % of Real estate in "Trading area" surrounding store that is => $150,000

WEEK Week # ("151" through "152")

deal_X = 𝐷𝑑𝑒𝑎𝑙 𝑋 Dummy variable. If Brand X product had a price promotion: 0 = No, 1 = yes

promo_X = 𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 Dummy variable. If Brand X product had an in-store promotion: 0 = No, 1 = yes

oz_X = 𝑞𝑋 Brand X product: # of liquid ozs. sold in store that week

deal_Y = 𝐷𝑑𝑒𝑎𝑙 𝑌 Dummy variable. If Brand Y product had a price promotion: 0 = No, 1 = yes

promo_Y= 𝐷𝑝𝑟𝑜𝑚𝑜 𝑌 Dummy variable. If Brand Y product had an in-store display/ad promotion: 0 = No, 1 = yes

oz_Y= 𝑞𝑌 Brand Y product # of liquid ozs. sold in store that week

Price_X = 𝑃𝑋 Price/liquid oz. of brand X product

Price_Y= 𝑃𝑌 Price/liquid oz. of brand Y product

MC_X = 𝑀𝐶𝑥 Retailer's Marginal Cost/liquid oz. of brand X product

MC_Y = 𝑀𝐶𝑌 Retailer’s Marginal Cost/liquid oz. of brand Y product

Dummy Class = 𝐷𝐶𝑙𝑎𝑠𝑠 Dummy variable. If product is regular or lite: 0 = Lite, 1 = Regular

Here are the average, minimum, and maximum values of the variables in the data set (𝑁 = 728 observations):

ST

OR

E

Hva

l_15

0

WE

EK

oz_X

oz_Y

Pric

e_X

Pric

e_Y

MC

_X

MC

_Y

Cla

ss

Du

mm

y

deal_

X

pro

mo

_X

deal_

Y

pro

mo

_Y

Mean 65.00 0.21 126.50 7078.95 9504.79 0.03 0.03 0.02 0.02 0.50 0.20 0.29 0.20 0.26

Minimum 62.00 0.09 101.00 936.00 144.00 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00 0.00

Maximum 68.00 0.35 152.00 50760.00 160992.00 0.04 0.04 0.03 0.03 1.00 1.00 1.00 1.00 1.00

1 Technically, the “lite” variety is “lit” in French.


(a) [12 Points] Ceteris paribus, what was the frequency of each of the following types of marketing campaigns:

Price promotion campaign for Brand X?

In-store promotion campaign for Brand X?

Price promotion campaign for Brand Y?

In-store promotion campaign for Brand Y? Explain your answer in one-two sentences: Answer Incidents of price promotion campaigns and in-store promotion campaigns are recorded by a dummy variable:

𝐷𝑐𝑎𝑚𝑝𝑎𝑖𝑔𝑛 = {1 = if there is a campaign

0 = if there is no campaign

As such, the average values of the following dummy variables tell us the frequency of the corresponding campaigns:

𝐷Price promotion campaign for Brand X̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = 0.20 ⟹ there was a Price promotion campaign for Brand X 20% of the time

𝐷In store promotion campaign for Brand X̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = 0.29 ⟹ there was an in store promotion campaign for Brand X 29% of the

time

𝐷Price promotion campaign for Brand Y̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = 0.20 ⟹ there was a Price promotion campaign for Brand Y 20% of the time

𝐷In store promotion campaign for Brand Y̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = 0.26 ⟹ there was an in store promotion campaign for Brand Y 26% of the

time It’s interesting that both brands X and Y reduced prices 20% of the time.

(b) [2 Points] Which variable in the data set is a proxy for customers’ income level? Explain your answer in one-two sentences: Answer

Hval 150, the % of surrounding real estate ≥ $150,000 is a reasonable proxy for income levels.


(c) [5 Points] Below, you’ll be asked to calculate a common (uniform) revenue-maximizing price point for Brand X-regular under certain conditions (to be specified below). In order for you to determine the revenue-maximizing price, you need to know the demand function for Brand X-regular. You ask your intern, a Commerce student at “Vestern University”, to estimate a demand function for Brand X regular. The intern proposes using regression analysis to estimate the following demand function:

𝑞𝑥 = 𝛽0 + 𝛽1𝐻𝑣𝑎𝑙150 + 𝛽2𝑃𝑥 + 𝛽3𝑃𝑦 + 𝛽4𝐷𝑐𝑙𝑎𝑠𝑠 + 𝛽5𝐷𝑑𝑒𝑎𝑙 𝑋 + 𝛽6𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 + 𝛽7𝐷𝑑𝑒𝑎𝑙 𝑌 + 𝛽8𝐷𝑝𝑟𝑜𝑚𝑜 𝑌 + 𝛽9𝑀𝐶𝑥

+ 𝛽10𝑀𝐶𝑌 What’s wrong with this proposed demand function model? Explain your answer in one-two sentences: Answer As a consumer, what are some of the factors that determine how much soda you buy? Surely, some of these factors are: income, prices, whether there is a price cut, and ads. What about marginal cost? Do you care how much the supermarket paid

Brands X and Y? No. As such, 𝑀𝐶𝑥 and 𝑀𝐶𝑦 should not be in the demand function.


(d) [16 Points] In order for you to determine the revenue-maximizing price, you need to know the demand function for

Brand X-regular. This time, you decide to do everything yourself (proof that Rotman > Vesturn). You propose the following – “correct” – demand function:

𝑞𝑥 = 𝛽0 + 𝛽1𝐻𝑣𝑎𝑙150 + 𝛽2𝑃𝑥 + 𝛽3𝑃𝑦 + 𝛽4𝐷𝑐𝑙𝑎𝑠𝑠 + 𝛽5𝐷𝑑𝑒𝑎𝑙 𝑋 + 𝛽6𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 + 𝛽7𝐷𝑑𝑒𝑎𝑙 𝑌 + 𝛽8𝐷𝑝𝑟𝑜𝑚𝑜 𝑌

You use regression analysis to estimate the coefficients of your demand model:

Dependent Variable: 𝒒𝒙

Independent Variable Coefficient

Constant 12768.5

Hval_150 -1002.42

Price_X = 𝑃𝑋 -272917.07

Price_Y= 𝑃𝑌 -70726.52

Dummy Class = 𝐷𝐶𝑙𝑎𝑠𝑠 -106.31

deal_X = 𝐷𝑑𝑒𝑎𝑙 𝑋 18069.7

promo_X = 𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 3590.16

deal_Y = 𝐷𝑑𝑒𝑎𝑙 𝑌 -1437.49

promo_Y= 𝐷𝑝𝑟𝑜𝑚𝑜 𝑌 -180.75

𝑁 = 728. 𝑅2 = 0.96

Interpret each coefficient in economic terms (each interpretation should be at most one sentence). Answer

Ceteris paribus, a 1 unit increase in the % of surrounding area that is ≥ $150,000 reduces demand for Brand X by 1002.42 ozs per store per week Ceteris paribus, a 1 unit increase in the price of Brand X, reduces demand for Brand X by 272917.07 ozs/store/week. Ceteris paribus, a 1 unit increase in the price of Brand Y, reduces demand for Brand X by 70726.52 ozs/store/week. Ceteris paribus, sales of Brand X “lit” soda are 106.31oz/week/store lower than the sales of Brand X “reg” soda Ceteris paribus, a price promotion on Brand X boosts sales of Brand X by 18069.7 ozs/store/week. Ceteris paribus, an in store promotion on Brand X boosts sales of Brand X by 3590.16 ozs/store/week. Ceteris paribus, a price promotion on Brand Y reduces sales of Brand X by 1437.49 ozs/store/week. Ceteris paribus, an in store promotion on Brand Y reduces sales of Brand X by 180.75 ozs/store/week.


(e) [25 Points] Use the demand model in part (d) to calculate the common (uniform) revenue-maximizing price point for Brand X-regular given that currently:

There is a Price promotion campaign for Brand X

There is an In-store promotion campaign for Brand X

There is no Price promotion campaign for Brand Y

There is an In-store promotion campaign for Brand Y For your convenience:

ST

OR

E

Hva

l_15

0

WE

EK

oz_

X

oz_

Y

Pric

e_X

Pric

e_Y

MC

_X

MC

_Y

𝑫𝑪

𝒍𝒂𝒔

𝒔

deal_

X

pro

mo

_X

deal_

Y

pro

mo

_Y

Mean 65 0.21 126.50 7078.95 9504.79 0.03 0.03 0.02 0.02 0.50 0.20 0.29 0.20 0.26

Min 62 0.09 101.00 936.00 144.00 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00 0.00

Max 68 0.35 152.00 50760.00 160992.00 0.04 0.04 0.03 0.03 1.00 1.00 1.00 1.00 1.00

Answer The demand equation is:

𝑞𝑥 = 𝛽0 + 𝛽1𝐻𝑣𝑎𝑙150 + 𝛽2𝑃𝑥 + 𝛽3𝑃𝑦 + 𝛽4𝐷𝑐𝑙𝑎𝑠𝑠 + 𝛽5𝐷𝑑𝑒𝑎𝑙 𝑋 + 𝛽6𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 + 𝛽7𝐷𝑑𝑒𝑎𝑙 𝑌 + 𝛽8𝐷𝑝𝑟𝑜𝑚𝑜 𝑌

𝑞𝑥 = 12768.5 − 1002.42𝐻𝑣𝑎𝑙150 − 272917.07𝑃𝑥 − 70726.52𝑃𝑦 − 106.31𝐷𝑐𝑙𝑎𝑠𝑠 + 18069.7𝐷𝑑𝑒𝑎𝑙 𝑋

+ 3590.16𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 − 1437.49𝐷𝑑𝑒𝑎𝑙 𝑌 − 180.75𝐷𝑝𝑟𝑜𝑚𝑜 𝑌

Use average values for 𝐻𝑣𝑎𝑙150 and 𝑃𝑦 and the following values for the dummy variables:

𝐷𝑐𝑙𝑎𝑠𝑠 = 1 (𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑣𝑎𝑟𝑖𝑒𝑡𝑦)

𝐷𝑑𝑒𝑎𝑙 𝑋 = 1 (𝑎 𝑝𝑟𝑖𝑐𝑒 𝑝𝑟𝑜𝑚𝑜𝑡𝑖𝑜𝑛 𝑔𝑜𝑖𝑛𝑔 𝑜𝑛)

𝐷𝑝𝑟𝑜𝑚𝑜 𝑋 = 1 (𝑎𝑛 𝑖𝑛𝑠𝑡𝑜𝑟𝑒 𝑝𝑟𝑜𝑚𝑜𝑡𝑖𝑜𝑛 𝑔𝑜𝑖𝑛𝑔 𝑜𝑛)

𝐷𝑑𝑒𝑎𝑙 𝑌 = 0 (𝑛𝑜 𝑝𝑟𝑖𝑐𝑒 𝑝𝑟𝑜𝑚𝑜𝑡𝑖𝑜𝑛 𝑔𝑜𝑖𝑛𝑔 𝑜𝑛)

𝐷𝑝𝑟𝑜𝑚𝑜 𝑌 = 1 (𝑎𝑛 𝑖𝑛𝑠𝑡𝑜𝑟𝑒 𝑝𝑟𝑜𝑚𝑜𝑡𝑖𝑜𝑛 𝑔𝑜𝑖𝑛𝑔 𝑜𝑛)

𝑞𝑥 = 12768.5 − 1002.42(0.21) − 272917.07𝑃𝑥 − 70726.52(0.03) − 106.31(1) + 18069.7(1) + 3590.16(1)− 1437.49(0) − 180.75(1)

𝑞𝑥 = 31809 − 272917.07𝑃𝑥 From this:

𝑃𝑥 =31809

272917.07 −

1

272917.07 𝑞𝑥

Revenues are:


𝑅𝑥 = 𝑃𝑥𝑞𝑥 = (31809

272917.07 −

1

272917.07 𝑞𝑥) 𝑞𝑥

Setting 𝑀𝑅 =𝑑𝑅𝑥

𝑑𝑞𝑥= 0 yields:

𝑞𝑥∗ =

31809272917.07

2 ∗1

272917.07 =

31809

2≈ 15,904.5

𝑃𝑥∗ =

31809

272917.07 −

1

272917.07 𝑞𝑥 =

31809

272917.07 −

1

272917.07 31809

2=

1

272917.07 31809

2≈ 0.06


(f) [10 Points] What is the impact on Brand-X-regular customers’ utility due to a 0.5% “fat tax” on Brand-X regular? Show all calculations. Answer Assume that the representative Brand-X-reg customer has a quasi linear utility function where good 1 = Brand-X-regular and good 2 = all other goods (also the base good) and that the consumer is buying both goods 1 and 2 , then, we know that (proof not required):

𝐶𝑆 𝑓𝑟𝑜𝑚 𝑑𝑒𝑚𝑎𝑛𝑑 𝑐𝑢𝑟𝑣𝑒 𝑜𝑓 𝑔𝑜𝑜𝑑 1 = 𝑇𝑜𝑡𝑎𝑙 𝑈𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑣𝑒𝑟 𝑎𝑙𝑙 𝑔𝑜𝑜𝑑𝑠 − 𝐼𝑛𝑐𝑜𝑚𝑒 Since we have a linear demand curve:

𝐶𝑆 =1

2(ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑝𝑟𝑖𝑐𝑒 𝑡ℎ𝑎𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑐ℎ𝑎𝑟𝑔𝑒𝑑 − 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑟𝑖𝑐𝑒) 𝐶𝑢𝑟𝑒𝑛𝑡 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦

Thus, before the excise tax:

𝐶𝑆𝑏𝑒𝑓𝑜𝑟𝑒 𝑓𝑎𝑡 𝑡𝑎𝑥 =1

2(

31809

272917.07−

1

272917.07 31809

2)

31809

2≈ $463.43

After the excise tax:

𝐶𝑆𝑎𝑓𝑡𝑒𝑟 𝑓𝑎𝑡 𝑡𝑎𝑥 =1

2(

31809

272917.07−

1.05

272917.07 31809

2) (31809 − 272917.07

1.05

272917.07 31809

2) ≈ $418.24

Thus:

𝐶𝑆𝑎𝑓𝑡𝑒𝑟 𝑓𝑎𝑡 𝑡𝑎𝑥 − 𝐶𝑆𝑏𝑒𝑓𝑜𝑟𝑒 𝑓𝑎𝑡 𝑡𝑎𝑥 = (𝑈𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 − 𝑌) − (𝑈𝑏𝑒𝑓𝑜𝑟𝑒 𝑡𝑎𝑥 − 𝑌) = (𝑈𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 − 𝑈𝑏𝑒𝑓𝑜𝑟𝑒 𝑡𝑎𝑥)

𝐶𝑆𝑎𝑓𝑡𝑒𝑟 𝑓𝑎𝑡 𝑡𝑎𝑥 − 𝐶𝑆𝑏𝑒𝑓𝑜𝑟𝑒 𝑓𝑎𝑡 𝑡𝑎𝑥 = (𝑈𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 − 𝑈𝑏𝑒𝑓𝑜𝑟𝑒 𝑡𝑎𝑥) = $418.24 − $463.43 ≈ − $45.19 The fat tax reduces Brand-X-regular customers’ utility by about $45.


(g) [10 Points] How would answer part (f) if you were informed that the “fat tax” and a reduction in Brand-X-regular customers’ income occurred simultaneously? State salient assumptions and show essential calculations. Answer We assumed that Brand-X-reg customer have a quasi linear utility function where good 1 = Brand-X-regular and good 2 = all other goods (also the base good) and that the consumer is buying both goods 1 and 2. As such:

𝑈 = 𝑓(𝑞1) + 𝑞2

|𝑀𝑅𝑆| =𝑃1

𝑃2

𝑓′(𝑞1)

1=

𝑃1

1

Thus:

𝑓′(𝑞1) = 𝑃1 This implies that so long as the customer is buying both goods 1 and 2, the demand for good 1 will not depend on income levels. As such, our answer to part (f) is valid even if the customer’s income changed at the same time that the excise tax was imposed (so long as the customer purchases both goods 1 and 2).


(h) [10 Points] Derive the brand-X-regular customer’s utility function over Brand-X-regular and “all other goods”. State salient assumptions and show essential calculations. Answer From above: we assumed that Brand-X-reg customer have a quasi linear utility function where good 1 = Brand-X-regular and good 2 = all other goods (also the base good) and that the consumer is buying both goods 1 and 2. As such:

𝑈 = 𝑓(𝑞1) + 𝑞2

|𝑀𝑅𝑆| =𝑃1

𝑃2

𝑓′(𝑞1)

1=

𝑃1

1

Thus:

𝑓′(𝑞1) = 𝑃1

𝑑𝑓(𝑞1)

𝑑𝑞1= 𝑃1

𝑑𝑓(𝑞1) = 𝑃1 𝑑𝑞1

∫ 𝑑𝑓(𝑞1) = ∫ 𝑃1 𝑑𝑞1

𝑞1

0

𝑞1

0

𝑓(𝑞1) = ∫ {31809

272917.07 −

1

272917.07 𝑞𝑥} 𝑑𝑞1

𝑞1

0

= 31809

272917.07𝑞𝑥 −

1

2(272917.07) 𝑞𝑥

2

The utility function of the representative Brand-X-regular customer is:

𝑈 = 𝑓(𝑞𝑥) + 𝑞2 =31809

272917.07𝑞𝑥 −

1

2(272917.07) 𝑞𝑥

2 + 𝑞2


QUESTION 2 [TOTAL 85 Points] The following table contains summary stats for certain financial metrics for Walmart (WMT), Amazon (AMZN), the S&P 500 Index (SP 500) and 90 Day US T-Bills:

RET WMT

RETX WMT

RET AMZN

RETX AMZN

RET SP 500

US 90 DAY T-BILL ANNUAL RATE %

Mean 0.0098 0.0085 0.0415 0.0415 0.0051 2.3240

Std. Dev. 0.0632 0.0633 0.1860 0.1860 0.0440 2.1981

Note: Data for WMT, AMZN, and SP 500 on a monthly basis expressed in decimal form out of 100

Source: CRSP (accessed through U of T CHASS). Data from 6/30/97 – 12/31/16

(a) [10 Points] What is the average monthly dividend yield on Walmart and Amazon stocks? Show key calculations. Answer By definition:

𝑅𝑒𝑡𝑢𝑟𝑛𝑠 = 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛𝑠 + 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 Using CRSP’s nomenclature:

𝑅𝐸𝑇 = 𝑅𝐸𝑇𝑋 + 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑

𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 = 𝑅𝐸𝑇 − 𝑅𝐸𝑇𝑋 Thus:

𝑊𝑎𝑙𝑚𝑎𝑟𝑡 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 = 𝑊𝑎𝑙𝑚𝑎𝑟𝑡 𝑅𝐸𝑇 − 𝑊𝑎𝑙𝑚𝑎𝑟𝑡 𝑅𝐸𝑇𝑋 = 0.0098 − 0.0085 = 0.0013

𝐴𝑚𝑎𝑧𝑜𝑛 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 = 𝐴𝑚𝑎𝑧𝑜𝑛 𝑅𝐸𝑇 − 𝐴𝑚𝑎𝑧𝑜𝑛 𝑅𝐸𝑇𝑋 = 0.0415 − 0.0415 = 0 That is, Amazon has never issued a dividend.


(b) [10 Points] A “Finance Specialist” from Princess University in Queenston ON insists there’s something wrong with the data used to compile the summary stats in part (a): he points out that the return on each of the three risky assets in that table are less than the return on US 90-day T-Bills. The “Princess Finance Specialist” starts making fun of U of T. Well, we all know that U of T is #1 (and always right). How would you respond to this petulant “Princess Finance Specialist” and set him straight (and have him scurry back to Queenston)? Show key calculations. Answer The Princess Finance Specialist is (of course) wrong – the issue is that CRSP reports returns on a monthly basis out of 100 but the 90 day US T-Bill rate is reported on an annual basis in % terms. To correctly compute risk premium, we need the 90 day US T-Bill rate on a monthly basis out of 100:

2.3240

12 × 100≈ 0.0019

Now, we see that the returns of all risky assets are indeed greater than risk free returns:

RET WMT

RETX WMT

RET AMZN

RETX AMZN

RET SP 500

US 90 DAY T-BILL RATE

Mean 0.0098 0.0085 0.0415 0.0415 0.0051 0.0019

Note: All data on a monthly basis expressed in decimal form out of 100


(c) [10 Points] Calculate and interpret Walmart’s, Amazon’s, and the S&P 500 Index’s Sharpe Ratios. Show key calculations and state salient assumptions. Answer The Sharpe Ratio is:

𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚

𝑅𝑖𝑠𝑘=

𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝐴𝑠𝑠𝑒𝑡 𝑅𝑒𝑡𝑢𝑟𝑛

𝑅𝑖𝑠𝑘

As such:

𝑊𝑀𝑇 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝑊𝑀𝑇 𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝐴𝑠𝑠𝑒𝑡 𝑅𝑒𝑡𝑢𝑟𝑛

𝑊𝑀𝑇 𝑅𝑖𝑠𝑘=

0.0098 − 0.0019

0.0632= 0.125

That is, Walmart has a 12.5% risk premium per unit of risk.

𝐴𝑀𝑍𝑁 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝐴𝑀𝑍𝑁 𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝐴𝑠𝑠𝑒𝑡 𝑅𝑒𝑡𝑢𝑟𝑛

𝐴𝑀𝑍𝑁 𝑅𝑖𝑠𝑘=

0.0415 − 0.0019

0.1860≈ 0.21

That is, Amazon has a 21% risk premium per unit of risk.

𝑆𝑃500 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝑆𝑃500 𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑖𝑠𝑘 𝐹𝑟𝑒𝑒 𝐴𝑠𝑠𝑒𝑡 𝑅𝑒𝑡𝑢𝑟𝑛

𝑆𝑃500 𝑅𝑖𝑠𝑘=

0.0051 − 0.0019

0.0440≈ 0.072

That is, the S&P 500 has a 7.2% risk premium per unit of risk.


(d) [25 Points] Suppose an investor wants to construct a leveraged portfolio (with 5% borrowed funds) consisting of a risk free asset and one of the three risky assets in the following table:

RET WMT

RETX WMT

RET AMZN

RETX AMZN

RET SP 500


Mean 0.0098 0.0085 0.0415 0.0415 0.0051 2.3240

Std. Dev. 0.0632 0.0633 0.1860 0.1860 0.0440 2.1981



What should be that one “risky” asset in this investor’s portfolio? What is the expected return and risk of the portfolio? What is the investor’s “degree of risk aversion”? Show key calculations and state salient assumptions. Answer We saw that:

𝑊𝑀𝑇 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = 0.125

𝐴𝑀𝑍𝑁 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 ≈ 0.21

𝑆𝑃500 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 ≈ 0.072 If we had to choose one of these risky assets for this portfolio, we’d choose Amazon because it has the highest Sharpe ratio. Now, we’d choose the fraction of the portfolio invested in Amazon by solving the UMP:

max𝛽

𝑈 = 𝐸[𝑟𝑝] − 𝑐𝜎𝑝2

Now:

𝐸[𝑟𝑝] = (1 − 𝛽)𝑟𝑓 + 𝛽𝑟𝑟

𝜎𝑝2 = 𝑉 ((1 − 𝛽)𝑟𝑓 + 𝛽𝑟𝑟) = (1 − 𝛽)2𝜎𝑓

2 + 𝛽2𝜎𝑟2 + 2 𝛽(1 − 𝛽)𝜎𝑟𝑓 = 𝛽2𝜎𝑟

2

Thus:

max𝛽

𝑈 = 𝐸[𝑟𝑝] − 𝑐𝜎𝑝2 = (1 − 𝛽)𝑟𝑓 + 𝛽𝑟𝑟 − 𝑐𝛽2𝜎𝑟

2

𝑑𝑈

𝑑𝛽= −𝑟𝑓 + 𝑟𝑟 − 2𝑐𝛽𝜎𝑟

2 = 0

𝛽 =𝑟𝑟 − 𝑟𝑓

2𝑐 𝜎𝑟2 =

0.0415 − 0.0019

2 𝑐 0.18602

The investor wants a leverage portfolio with 5% borrowed funds. Thus:

𝛽 =0.0415 − 0.0019

2 𝑐 0.18602= 1.05

𝑐 = 0.0415 − 0.0019

2 (1.05)0.18602≈ 0.55

Thus, the investor’s degree of risk aversion parameter is 0.55. The portfolio return and risk are:


𝐸[𝑟𝑝] = (1 − 𝛽)𝑟𝑓 + 𝛽𝑟𝑟 = −0.05(0.0019) + 1.05(0.0415) ≈ 0.043

𝜎𝑝2 = (1.05)20.18602 ≈ 0.038


(e) [30 Points] Now suppose another investor wants to construct a leveraged portfolio (with 5% borrowed funds) consisting of a risk free asset and a synthetic risky asset (where the synthetic risky asset consists of the “top two” risky assets listed in the following table):

RET WMT

RETX WMT

RET AMZN

RETX AMZN

RET SP 500


Mean 0.0098 0.0085 0.0415 0.0415 0.0051 2.3240

Std. Dev. 0.0632 0.0633 0.1860 0.1860 0.0440 2.1981



Covariance Table

RET WMT

RET AMZN

RET SP 500

RET WMT 0.00400

RET AMZN 0.00231 0.03461

RET SP 500 0.00110 0.00377 0.00194

Note: Data for WMT, AMZN, and SP 500 on a monthly basis expressed in decimal form out of 100. Source: CRSP (accessed through U of T CHASS). Data from 6/30/97 – 12/31/16

Which two risky assets should be used to form the synthetic risky asset? What is the expected return and risk of the synthetic risky asset? What is the expected return and risk of the portfolio? What is this investor’s “degree of risk aversion”? Show key calculations and state salient assumptions. If needed, you may use:

𝑓∗ = 1 −(𝑟�̅� − 𝑟𝑓)𝑠𝐵

2 − (𝑟�̅� − 𝑟𝑓)𝑠𝐴𝐵

(𝑟�̅� − 𝑟𝑓)𝑠𝐵 2 + (𝑟�̅� − 𝑟𝑓)𝑠𝐴

2 − [𝑟�̅� − 𝑟𝑓 + 𝑟�̅� − 𝑟𝑓]𝑠𝐴𝐵

Answer

𝑊𝑀𝑇 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = 0.125

𝐴𝑀𝑍𝑁 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 ≈ 0.21

𝑆𝑃500 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 ≈ 0.072 If we had to choose two of these risky assets for our 2-asset mutual fund, we’d choose Walmart and Amazon because these have the top two Sharpe ratios. We’d label Amazon “asset B” and label Walmart “asset A” where the fraction of the mutual fund consisting of Amazon is:

𝑓∗ = 1 −(𝑟�̅� − 𝑟𝑓)𝑠𝐵

2 − (𝑟�̅� − 𝑟𝑓)𝑠𝐴𝐵

(𝑟�̅� − 𝑟𝑓)𝑠𝐵 2 + (𝑟�̅� − 𝑟𝑓)𝑠𝐴

2 − [𝑟�̅� − 𝑟𝑓 + 𝑟�̅� − 𝑟𝑓]𝑠𝐴𝐵

Now:

𝑓∗ = 1 −(0.0098 − 0.0019)0.18602 − (0.0415 − 0.0019)0.00231

(0.0098 − 0.001)0.18602 + (0.0415 − 0.0019)0.06322 − [0.0098 − 0.001 + 0.0415 − 0.0019]0.00231

𝑓∗ = 1 −0.0001818324

0.000350812704≈ 0.48


48% of the synthetic asset consists of Amazon stocks. Now, the expected return and risk of this synthetic risky asset is:

𝐸[𝑟𝑟] = (1 − 𝑓)𝐸[𝑟𝐴] + 𝑓 𝐸[𝑟𝐵] = (1 − 0.48)0.0098 + 0.48(0.0415) ≈ 0.025

𝜎𝑟2 = 𝑉((1 − 𝑓)𝑟𝐴 + 𝑓𝑟𝐵) = (1 − 𝑓)2𝜎𝐴

2 + 𝑓2𝜎𝐵2 + 2 𝑓(1 − 𝑓)𝜎𝐴𝐵

𝜎𝑟2 = (1 − 0.48)20.06322 + 0.4820.18602 + 2 0.48(1 − 0.48)0.00231 ≈ 0.01

Previously, we showed that the fraction of the portfolio invested in a risky asset is found by solving the UMP:

max𝛽

𝑈 = 𝐸[𝑟𝑝] − 𝑐𝜎𝑝2


2𝑐 𝜎𝑟2

Here, the risky asset is the synthetic risky asset so that:


2𝑐 𝜎𝑟2 =

0.025 − 0.0019

2 𝑐 0.01

The investor wants a leveraged portfolio with 5% borrowed funds. Thus:

𝛽 =0.025 − 0.0019

2 𝑐 0.01= 1.05

𝑐 = 0.025 − 0.0019

2 (1.05) 0.01≈ 1.1

Thus, the investor’s degree of risk aversion parameter is 1.1. The portfolio return and risk are:

𝐸[𝑟𝑝] = (1 − 𝛽)𝑟𝑓 + 𝛽𝑟𝑟 = −0.05(0.0019) + 1.05(0.025 ) ≈ 0.026

𝜎𝑝2 = (1.05)20.01 ≈ 0.011

Documents

University of Toronto (STG), Department of …...ECO 204 Microeconomic Theory for Commerce 2017-2018 Ajaz’s Sections Test 2 Solutions [Detailed step-by-step solutions are provided