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AI2135 - Midterm HT22, Answers and explanations

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The Market as a Rainforest

Key Theory in Midterm

What is a beta?

  • Beta (β), primarily used in the capital asset pricing model (CAPM), is a measure of the volatility–or systematic risk–of a security or portfolio compared to the market as a whole.
  • Beta data about an individual stock can only provide an investor with an approximation of how much risk the stock will add to a (presumably) diversified portfolio.
  • For beta to be meaningful, the stock should be related to the benchmark that is used in the calculation.
  • The S&P 500 has a beta of 1.0.
  • Stocks with betas above 1 will tend to move with more momentum than the S&P 500; stocks with betas less than 1 with less momentum.

What is Options?

There are four basic options positions:

  1. Buying a call option
  2. Selling a call option
  3. Buying a put option
  4. Selling a put option.

Risk when buying options

Buyers of call or put options are limited in their losses to the cost of the option (it's premium). Unhedged sellers of options face theoretically unlimited losses.

Capital Asset pricing model illustration to understand Beta
Capital Asset pricing model illustration to understand Beta

What are Call and Put options?

Call Options

With call options, the buyer is betting that the market price of an underlying asset will exceed a predetermined price, called the strike price, while the seller is betting it won't.

Put Options

With put options, the option buyer is betting the market price of an underlying asset will fall below the strike price, while the seller is betting it won't.

Example Midterm HT22 with Answers and Explanations

(A) The standard deviation of the return of an individual security is a measure of: (1p)

Firm specific risk of the security.
Market risk of the security.
The total risk of the security. ✅
None of the above.
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What is the standard deviation of returns a measure of?

Standard deviation of returns is a measure of volatility or risk.

(B) The beta of an individual security is a measure of:

Firm specific risk of the security.
Market risk of the security. ✅
The standard deviation of the return of the security.
None of the above.
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In the capital asset pricing model, the Y-axis is return, and the X-axis represents risk. The model basically states y = kx + m. k = beta. The beta is a measure of the volatility–or systematic risk–of a security or portfolio compared to the market as a whole.

(C) When you increase the number of assets in a portfolio: (1p)

The unsystematic risk decreases. ✅
The systematic risk decreases.
The expected portfolio return decreases.
The expected portfolio return increases.
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More stocks = Diversification → Decrease in firm specific risk = The unsystematic risk decreases.

(D) A multi-factor asset pricing model assumes that: (1p)

Firm-specific risk factors can have a risk premium.
There exist multiple systematic risk factors. ✅
All stocks must lie on the security market line (SML).
None of the above.
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What is a multi-factor asset pricing model? → That the expected return on an asset is a linear function of factor risk premiums and their associated factor sensitivities.

(E) The CAPM beta of a stock equals minus one. This indicates that: (1p)

The return on the stock is risk free.
The stock does not contribute to diversification when included in a portfolio.
The beta estimate is erroneous since no stock can have a negative beta.
The expected return on the stock is lower than the risk-free return. ✅
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y = -kx + m → The return of the stock is has a negative slope → Expected return is lower then risk free return.

(F) The CAPM beta of a stock equals zero. This indicates that: (1p)

The return on the stock is risk free.
The stock does not contribute to diversification when included in a portfolio.
The stock does not have any market risk. ✅
The stock does not have any firm-specific risk.
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A zero-beta portfolio is constructed to have no systematic risk, or a beta of zero, with performance not correlated to swings in the broader market. - Investopedia

(G) For a two-stock portfolio the maximum reduction in risk occurs when the correlation coefficient between the returns on the two stocks is: (1p)

+1
0
-1 ✅
-1,5
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To reduce the maximum amount of risk in a two-stock portfolio, you want one stock to go up when the other goes down. A correlation coefficient of -1 represents this exact phenomenon.

(H) Which of the following statements about options is correct: (1p)

Options are always less risky than the underlying asset. → (Only when buying Options)
The value of a call option is never affected by dividend payments of the underlying asset.
The value of a put option decreases when the value of the underlying asset increases. ✅
None of the above.
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Alt. 1 → Only buyers of call or put options are limited in their losses to the cost of the option. Alt. 2 → Options listed on stocks are affected by the payment of dividends, since holders of the underlying shares receive dividends but call and put holders do not receive these inflows. Alt. 3 → Put is betting on a the stock decreasing in value, if the asset increases in price your losing the bet. ✅

(I) Which of the following statements about options is correct: (1p)

In order to value a call option on a financial asset you need to know the required rate of return on the underlying asset.
The time value of an option must always be positive.
The time value of an option must always be negative.
None of the above. ✅
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Alt. 1 → Not true at all, see how we value options at the question 2. Alt. 2 → This is true for none-dividend paying stocks. Alt. 3 → This is not true any of the two types of options.

(J) The beta of a stock in a CAPM regression is 1.5. The risk-free rate of return is 3% and the expected return on the market portfolio equals 8%. According to the CAPM, the expected return on this stock is closest to: (1p)

0,03 + 1,5(0,08-0,03) = 0,105

8%
10% ✅
12%
14%
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CAPM-Expected return = B*(expected return - Risk free Return) + Risk Free Expected return: 1,5(0,08-0,03) + 0,03 = 0,105 (it’s closest to 10%)

(K) An efficient stock market implies that: (1p)

The net present value of buying or selling a security equals zero. ✅
Returns on financial assets are predictable.
You can never get a higher return on a security than what CAPM predicts.
None of the above.
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A security is a fungible, negotiable financial instrument that represents some type of financial value, usually in the form of a stock, bond, or option. Alt. 1 → The net present value of any security will be zero if securities markets are efficient. Inefficiencies get traded away. ”An immediate and direct implication of an efficient market is that no group of investors should be able to consistently beat the market using a common investment strategy.” - NYU Stern.

(L) The expected return on a portfolio is 8%. The risk-free rate of return is 2% and the Sharpe Ratio is 0.2. This information implies that the portfolio standard deviation is: (1p)´

30% ✅
20%
15%
10%
image
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Sharpe Ratio = Expected Return / (Standard Deviation) Expected Return → (Return on asset - Risk free return) Given: Asset Return = 0,08 Risk-Free = 0,02 Sharp Ratio = 0,2 Portfolio standard deviation = ?? → (0,08 - 0,02)/0,2 = 0,3 → 30%

QUESTION 2

Explain the following terms

  1. a) Diversification (2p)
    • Reducing risk by investing in multiple assets who are none-correlated or negatively or correlated. Ie. Usually by buying assets in none-correlated industries ex. Pharma and Oil
  2. b) Efficient frontier of a portfolio (2p)
    • Not possible to increase return without increasing risk.
  3. c) Market risk (2p)
    • Risk that affects all stocks. For example QE or Fed rate.

QUESTION 3

The risk free rate of return is 4%. The average return on large-firm stocks is expected to be 3% lower than the average return on small-firm stocks. The average return on firms with high book-to-market ratio is expected to be 2% higher than the average return on firms with low book-to-market ratio. The average return on the last 12 months worst performing stocks is expected to be 4% lower than average return on the last 12 months best performing stocks. The risk-premium on the market portfolio is expected to be 5%.

The Green Properties (GP) stock has the following factor sensitivities:

  • Firm size: 2.5
  • Book-to-market ratio: -2.3
  • Prior one-year momentum: 1.2
  • Risk-premium on the market portfolio: 0.5

Use the Fama-French-Carhart factor model to calculate the expected return on your investment in GP. (6p)

Fama-French-Carhart factor model

B(X-Y)
Term 1
Term 2
Term 3
Term 4
Term 5
B
1
Risk-premium on the market portfolio
Firm Size:
Book-to-market ratio
Prior one-year momentum: 1.2
X
Risk Free Rate of Return
Expected Risk-premium Market portfolio
Expected average return on large-firm stocks
Expected average return book to market
Expected to be 4% lower than average return on the last 12 months best performing stocks.
Y
0
Risk Free Rate of Return
0
0
0
1*0,04
0,5*(0,05-0,04)
2,5*0,03
-2,3*0,02
1,2*0,04

E(Rp) = 0,04 + 0,5*(0,05-0,04) + 2,5*0,03 + (-2,3*0,02) + 1,2*0,04

E(Rp) = 0,122

✅
Svar: The expected return according to the model is approximately 12,2%

QUESTION 4

The current price of the Delta stock is 80. In one period, the price will either rise by 15 or fall by 10. The one-period risk-free rate of return is 4%. Use risk-neutral probabilities and a one-period binomial model to value a call option on the Zero stock, with an exercise price of 85 and that expires in one period. (6p)

One Period Binomial Model

Current Stock Price: 80 USD

Strike Price: 85 USD

Risk free rate: 4%

Step 1) Make a table

Movement Assumptions
New Stock Price
New Price - Strike Price
Upper
15% (Increase)
92
Max(92-85) = 7
Lower
-10% (Decrease)
72
Max(72-85) = 0

Step 2) Calculate Delta 🔺

🔺 : (7 - 0) / (92-72) = 0,35

Step 3) Calculate B

B = 0 - 72*🔺 / (Risk free rate) → (-72*0,35) / 1,04 = -24,23

Step 4) Calculate Value of Call Option

Value of Call option → C = Stock Price * 🔺 + B

C = 80*0,35 - 24,23

C = 3,8

Risk-neutral probabilities

Current Stock Price: 80 USD

Strike Price: 85 USD

Risk free rate: 4%

Step 1) Make this table below

Movement Assumptions
New Stock Price
New Price - Strike Price
Upper
15% (Increase)
92
Upper: Max(92-85) = 7
Lower
-10% (Decrease)
72
Lower: Max(72-85) = 0

Step 2) Make probability equation with Stock Price, Max

Stock Price = ((max Upper)*p + (Max price)*(1-p)) / (risk free rate)

80 = (92*p + 72(1-p)) / 1,04 → p = 0,56

Step 3) Calculate the Value of the option

Value Call Option = (7*p + 0*(1-p))/ Bond → 7*0,56/1,04 = 3,76 → 3,8