Two common factors, F1 and F2, drive stock returns. We have the following factor

Question

Two common factors, F1 and F2, drive stock returns. We have the following factor equations for stocks 1, 2, and 3:

r˜1 = 0.07 + 1F˜ 1 1F˜ 2 + ˜1

r˜1 = 0.11 + 1F˜ 1 + 1F˜ 2 + ˜2

r˜1 = 0.16 + 2F˜ 1 + 1F˜ 2 + ˜3

Assume that these stocks are priced correctly.

• What are the weights of the first pure factor portfolio, that is, a portfolio that has loadings of 1 and 0 on the two factors?

• What is the riskless rate?

• What are the risk premiums of the two factors?

• We have another stock with loadings of 1 = 1 and 2 = 2. This stock’s expected return is 10%. Is this stock mispriced relative to the other stocks? Explain.

Explanation / Answer

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Subject: Allocating Assets - in theory and in practice - a BRIEF glance:

Citation: Lecturer notes from Booth School of Business, University of Chicago;

Let r = The stock returns

Beta = ?

Part 1) Weights of the 1st Pure factor portfolio

Weight 1 + 3 * Weight 2 + 4 * (1 - Weight 1 - Weight 2) = zero

- 2 * weight 1 + 2 * weight 2 = zero

Once we get a solution for weights, we get, weight 1 = one, weight 2 = one, and weight 3 = minus one;

Part 2)

Riskless rate:

E [ r rf ] = rf = 1 * 0.02 + 1 * 0.10 - 1 * 0.10 = 0.02 = 2 %

Part 3)

risk premium of factor 1:

Lambda 1 = E [ r p factor 1] - rf = 0.04 - 0.02 = 0.02 = 2%

risk premium of factor 2:

Lambda 2 = E [ r p factor 2] - rf = 0.03 - 0.02 = 0.01 = 1%

Part 4)

Next stock

Loaded as Beta1 = 1, Beta 2 = 2

r = 0.01 = 10%

Is the pricing correct or mispriced?

Remember that Beta is also known as loadings in the cases of multi factor portfolios

If the pricing were correct:

If the pricing were mispriced:

Hence the pricing is assessed to be:

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-- the following are additional data - not directly related to this question, though it is the same subject of Finance, but just for extra knowledge! --

Citation: University of Manitoba, Canada on Advanced topics in Actuarial science

European Put: (On the Expiry date only NOT before)

(Market) Stock price = So > 0.1

(Pre Agreed) Strike = K > 0.1 e ^( r * (T- to) )

Strike is a pre agreed price, stock is the fluctuating market price; Say, John, enters a contract that he will strike put (sell) at $K (say $100) in a time period t (say 6 months); Say the market stock price falls to $90 - as John agreed he will sell for $100 and make

a profit of 100-90 = $10;

But, unfortunately, say the market stock price fluctuates to $140, alas, poor john can sell just for $100 only and he will suffer the loss of 140-100 = $40;

The above is the basic style of put option trading;

PV(Strike Price) - 0.1

Trade strategy:

a) Buy the Put (Sell)

b) Borrow money from bank (loan, interest involved in repayment )

Portfolio at start = zero = 0

(opening balance = 0)

Writer of the put = seller;

Profit in Put:

As already illustrated in the example of John, the option gives you a

profit when the market price falls below the strike price;

strike < market leads to profit;

Loss in Put:

Just opposite of above:

As already illustrated in the example of John, the option gives you a

loss when the market price rises above the strike price;

strike > market leads to loss;

The difference between the put options of American and European continent lies predominantly in the duration; i.e. when you are at liberty to exercise the option;

European option is exercisable iff after the option expires, but the American option is exercisable on or before the expiry date - so premature closures, pre closures are allowed for American options including put option;

The continuous compounding risk free interest rate for an option that is not paying any dividend

The formula to find the parity of put call is given below (this applies to European put options with an equal strike price and a pre determined or preset maturity option)

Call - Put = F 0,T ^ P (Stock) - F 0,T ^ P (Strike)

= F 0,T ^ P (Stock) - PV 0,T (Strike)

= F 0,T ^ P (Stock) - Strike * e ^ ( -r * T )

= S 0 - Strike * e ^ ( -r * T )

we can computer r when we have the difference between call and put

Say the call option is selling for 0.25 above the option of put.

Call = C; Put = P; Strike = K; Stock = S;

Hence C-P = 0.25

Let S0 = 600; (is > 0.1 hence valid as per the question)

Let K = 700; (is also valid as K > 0.1 * e ^(r*(T-t0)) as per the question)

Let T = the tenure or time limit to exercise the options be = 6 years; which means we must deliver it to the writer of the put at the completion of the 6th year;

Now r will be approximately around 4%

American Put: (on or before expiry date)

Profit:

This scenario happens when the Strike price is less than the Stock or market price;

i.e. Strike < market leads to profit on or before the date of exercise tenure;

K < So;

Put:

strike > market leads to loss on or before he expiry date;

i.e. K > So will lead to loss;

Let T = the tenure or time limit to exercise the options be = 7 years; which means we can deliver it to the writer of the put at the completion of the 7th year or earlier;

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