Question

The five largest tech companies (Google, Apple, Facebook, Amazon, Microsoft or GAFAM)

now make up 20% of the S&P 500 (the market portfolio). This problem aims to quantify

the consequence of this increasing concentration for investors.

We assume that the entire security market is made of only three types of assets: a risk-free

asset (rf = 3%) and two risky securities G (a portfolio composed by investing in the ve

GAFAM stocks) and B (a portfolio composed by investing in the rest of the S&P 500,

i.e., it is composed of 495 dierent stocks). You can think of G and B as stocks.

There are 500 shares of G is worth $10 per share, and 10,000 shares of B worth $2 per

share. G and B both generate expected returns equal to 7%. The volatility of G is equal

to 22.6% and the volatility of B is equal to 5.9%. The correlation of the returns of G and

B is equal to 0.76.

1. What is the composition of the market portfolio (portfolio of risky assets)? (2.5

points)

2. Intuitively, based on what G and B represent, why do you think it is legitimate to

model stock G as riskier than B? (2.5 points)

3. Compute the return and the volatility and the return-to-volatility ratio ((return of

the portfolio - risk free rate)/volatility) of the market portfolio. What would have

been the return-to-volatility ratio if G was worth $2 per share? (5 points)

Answer 1

Given,

Stock G represents GAFAM stocks

Stock B represents 495 dierent stocks

Expected return of risky portfolio = 7%

Volatility of G = 22.6%

Volatility of B = 5.9%

Correlation coefficient between G and B = 0.76

(1) Composition of Market portfolio:

Stock G = 500×10 = $5,000

Stock B = 10,000×2 = $20,000

Total value of portfolio = $25,000

Composition of stock B which represents 495 dierent stocks = (20,000÷25,000)×100 = 80%

Composition of stock G which represents GAFAM stocks = 20%

(2) Stock G represents GAFAM stocks

Stock B represents dierent stocks

Stock G is riskier than stock B as the volatility for stock G is higher compared to stock B

(3) Return and Volatility of Market portfolio:

Expected return of portfolio = 7%

Volatility of portfolio:

= √(probability of stock G)^2 × (volatility of stock G)^2 + (probability of stock B)^2 × ( volatility of stock B)^2 + 2 × probably of stock G × probability of stock B × volatility of stock G × volatility of stock B × correlation coefficient of stock G and B

= √( 0.20)^2 × (22.6)^2 + (0.80)^2 × (5.9)^2 + 2 × 0.20 × 0.80 × 22.6 × 5.9 × 0.76

= √ 0.04 × 510.76 + 0.64 × 34.81 + 32.43

= √ 20.4304 + 22.2784 + 32.43

= √ 75.1388

= 8.67%

Return to volatility ratio:

Return of volatility ratio

=[( Return on portfolio - Risk free rate) ÷ volatility]×100

=[( 0.07 - 0.03) ÷ 0.0867 ]×100

= [0.04 ÷ 0.0867]×100

= 46.136%

Return to volatility ratio if stock G was worth $2 per share:

Stock of G = 500×2 = $1,000

Stock of B = $20,000

Total value = $21,000

Composition of stock G = 1,000÷21,000 = 4.76%

Composition of stock B = 100 - 4.76 = 95.24%

Volatility of risky portfolio:

= √( 0.0476)^2 × (22.6)^2 + (0.9524)^2 × (5.9)^2 + 2 × 0.0476 × 0.9524 × 22.6 × 5.9 × 0.76

= √ 0.00226 × 510.76 + 0.9071 × 34.81 + 9.1882

= √ 1.1543176 + 31.576151 + 9.1882

= √ 41.9187

= 6.47%

Return to volatility ratio

= [(0.07 - 0.03) ÷ 0.0647]×100

= (0.04 ÷ 0.0647)×100

= 61.82%