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Hello,
please read the attached file. Only question 1,2,3,4 needs to be done.
I will give a tip if everything is in order :)
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Explanation & Answer
Finally done xDI have also attached it in PDF incase some symbols or boxes got moved
Question 1
The returns per annum on two securities are denoted by 𝑅1 and 𝑅2 , and are modelled as follows:
𝑅1 ~𝑁(0.1,0.22 ), 𝑅2 ~𝑁(0.05,0.042 ), 𝐶𝑜𝑟𝑟(𝑅1 , 𝑅2 ) = 0.5
where 𝐶𝑜𝑟𝑟(𝑅1 , 𝑅2 ) denotes the correlation between 𝑅1 and 𝑅2
(i)
Suppose that $1,000 are invested in the minimum variance portfolio formed from the two
securities. Calculate the composition of the minimum variance portfolio and state how
much is invested in each security.
From the above information we have the following parameters:
𝐸(𝑅1 ) = 0.1, 𝜎12 = 0.22 , 𝐸(𝑅2 ) = 0.05, 𝜎22 = 0.042 , 𝐶𝑜𝑟𝑟(𝑅1 , 𝑅2 ) = 0.5
Let us define 𝑥1 as the weight of security 1, 𝑥1 that minimizes the variance is as follows:
𝑥1 𝑚𝑖𝑛 =
𝜎22 − 𝐶𝑜𝑟𝑟(𝑅1 , 𝑅2 )
0.042 − 0.5
=
= 0.5191
𝜎12 + 𝜎22 − 2𝐶𝑜𝑟𝑟(𝑅1 , 𝑅2 ) 0.22 + 0.042 − 2(0.5)
Thus, the weight of security 2, 𝑥2 = 1 − 0.5191 = 0.4809
Therefore, the composition of minimum variance portfolio is 51.91% of the investment is in security 1
and 51.91% of the investment is in security 2.
Finally, we find how much money is invested in each security,
In security 1: $ 1,000(0.5191) = $ 519.10
In security 2: $ 1,000(0.4809) = $ 480.90
Next, suppose that, at the beginning of the year, the balance on an investment account is $1,000. The
account is invested in securities 1 and 2 only, and short selling is permitted.
Suppose also that, at the end of the year, a random sum of $1000 × (1 + 𝑄) is withdrawn from the
investment account. The random variable Q is modelled as follows:
𝑄~𝑁(0.08,0.12 ), 𝐶𝑜𝑟𝑟(𝑄, 𝑅1 ) = 0.8,
𝐶𝑜𝑟𝑟(𝑄, 𝑅2 ) = 0.2
If x is the portfolio weight for security 1, then the end-of-year account balance is:
𝑊1 = 1000(1 + 𝑥𝑅1 + (1 − 𝑥)𝑅2 ) − 1000(1 + 𝑄)
(ii)
Calculate the composition of the portfolio such that the end-of-year balance on the
investment account has the least variance.
The variance of 𝑊1 is
𝑣(𝑥) = 24,000𝑥 2 − 84,000
The calculation of this is in part (iii)
We determine the derivative
𝑣 ′ (𝑥) = 48,000𝑥
Note that the value that minimizes this is x=0
Therefore, for of end-of-year balance to have the least variance, we must invest all the money
in security 2
(iii)
Let 𝑚(𝑥) = 𝔼[W1 ] and 𝑣(𝑥) = 𝑉𝑎𝑟[𝑊1 ]
(a) Derive expressions for m(x) and v(x)
First,
𝑊1 = 1000 − 1000𝑅1 𝑥 + 1000(1 − 𝑥)𝑅2 − 1000 − 1000𝑄
𝑊1 = 1000𝑅1 𝑥 + 1000(1 − 𝑥)𝑅2 − 1000𝑄
We take the expected value
𝑚(𝑥) = 𝔼[W1 ] = 𝔼[1000𝑅1 𝑥 + 1000(1 − 𝑥)𝑅2 − 1000𝑄]
= 100𝑥𝔼[R1 ] + 1000(1 − X)𝔼[𝑅2 ] − 1000𝔼[Q]
= 100𝑥(0.1) + 100(1 − 𝑥)(0.05) − 1000(0.08)
= 10𝑥 + 5(1 − 𝑥) − 80
= 5𝑥 − 75
•
We have 𝑚(𝑥) = 5𝑥 − 75
Next, we take the variance
𝑣(𝑥) = 𝑉𝑎𝑟(𝑊1 ) = 𝑉𝑎𝑟(1000𝑅1 𝑥) + 𝑉𝑎𝑟(1000(1 − 𝑥)𝑅2 ) − 𝑉𝑎𝑟(1000𝑄)
= 10002 𝑥 2 𝑉𝑎𝑟(𝑅1 ) + 10002 (1 − 𝑥)2 𝑉𝑎𝑟(𝑅2 ) − 10002 𝑉𝑎𝑟(𝑄)
= 10002 𝑥 2 (0.22 ) + 10002 (1 − 𝑥)2 (0.042 ) − 10002 (0.1)
= 40,000𝑥 2 + 16,000(1 − 𝑥)2 − 100,000)
= 24,000𝑥 2 − 84,000
Therefore,
•
𝑣(𝑥) = 24,000𝑥 2 − 84,000
(b) Show that 𝑚(𝑥 ∗ )𝑣 ′ (𝑥 ∗ ) = 2𝑚′ (𝑥 ∗ )𝑣(𝑥 ∗ )
We know that 𝑚(𝑥) = 5𝑥 − 75 and therefore 𝑚′(𝑥) = 5
𝑣(𝑥) = 24,000𝑥 2 − 84,000 and therefore 𝑣′(𝑥) = 48,000𝑥
Now, 𝑚(𝑥 ∗ )𝑣 ′ (𝑥 ∗ ) = (5𝑥 ∗ − 75)48,000𝑥 ∗ = 240,000𝑥 ∗2 − 3,600,000𝑥 ∗
2𝑚′ (𝑥 ∗ )𝑣(𝑥 ∗ ) = 2(5)( 24,000𝑥 ∗2 − 84,0002∗ ) =240,000𝑥 ∗2 − 3,600,000𝑥 ∗
Thus, 𝑚(𝑥 ∗ )𝑣 ′ (𝑥 ∗ ) = 2𝑚′ (𝑥 ∗ )𝑣(𝑥 ∗ ) are both equal to240,000𝑥 ∗2 − 3,600,000𝑥 ∗
Question 2
(i)
(a) State the no-arbitrage principle
In a financial market, there must not be any free risk profits. When arbitrage opportunities
exist, an investor can earn riskless profits and still make a net investment. Market prices will
move in such a way to rule out arbitrage opportunities.
(b) Explain why only a few arbitrageurs are required in a market to enforce the no-arbitrage
condition.
When there are arbitrage opportunities, investors will want to take a large a position as
possible to increase their earnings. Only a few arbitrageurs are required to induce the price
pressures that are needed to restore equilibrium.
(ii)
The returns on securities in a market are modelled using a 2-factor model:
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖1 𝐼1 + 𝑏𝑖2 𝐼2 + 𝑐𝑖
The following data, concerning three securities are known:
Security i
1
2
3
𝑅̅𝑖
12%
8%
9%
𝑏𝑖1
0.5
1.0
0.7
𝑏𝑖2
1.2
0.8
-1.0
(a) Derive an equation for the arbitrage pricing plane in 𝑅̅𝑖 − 𝑏𝑖1 − 𝑏12 space.
We have the three equations:
12 = 𝑎𝑖 − 0.5𝑏𝑖1 − 1.2𝑏𝑖2
8 = 𝑎𝑖 − 𝑏𝑖1 − 0.8𝑏𝑖2
9 = 𝑎𝑖 − 0.7𝑏𝑖1 + 1.2𝑏𝑖2
These are three equations with three unknowns: 𝑎𝑖 , 𝑏𝑖1 , 𝑏𝑖2
Solving the system of equations, we get the solutions:
𝑎𝑖 = 14.90,
𝑏𝑖1 = 7.45, 𝑏𝑖2 = −0.69
Thus, the equation for the arbitrage pricing plane in the 𝑅̅𝑖 − 𝑏𝑖1 − 𝑏12 space is
𝑅̅𝑖 = 14.90 + 7.45𝑏𝑖1 − 0.69𝑏12
(b) A fourth security exists with
𝑅̅4 = 10%, 𝑏41 = 0.9, 𝑏42 = −0.2
Explain carefully why an opportunity for arbitrage arises.
The new security is not the arbitrage pricing plane calculated above,
Now, let’s see if we can find weights such that:
The weighted average of the sensitivity 𝑏𝑖1 is
𝑤1 (0.5) + 𝑤1 (1) + 𝑤2 (0.7) = 0.9
The weighted average of the sensitivity 𝑏𝑖2 is
𝑤1 (1.2) + 𝑤2 (0.8) + 𝑤3 (−1) = −0.2
And we must have
𝑤1 + 𝑤2 + 𝑤3 = 1
This gives us another system of three equations with three unknowns and gives us the
solution
𝑤1 = 0
𝑤2 = 0.67
𝑤3 = 0.33
This portfolio is on the arbitrage pricing plane
Therefore, an opportunity for arbitrage arises since an investor and choose a security on the
plane that has the same risk as the fourth security with a better return.
(c) Construct an arbitrage portfolio from securities 1, 2, 3 and 4 (stating relevant proportions) to
exploit this arbitrage portfolio.
Security i
1
2
3
4
𝑅̅𝑖
12%
8%
9%
10%
𝑏𝑖1
0.5
1.0
0.7
0.9
𝑏𝑖2
1.2
0.8
-1.0
-0.2
Portfolio 4 is underpriced, and we would benefit form weighting security 4 heavily.
We can construct the portfolio such that we have the following weights (proportions)
Security i
1
2
3
4
Weight(proportions)
1/4
1/4
1/4
2/4
Question 3
(i)
A model for the price 𝑃𝑡 of a share is:
𝑃𝑡 = 𝑃0 exp {𝑎𝑡 + 𝑏𝐵𝑡 }
where 𝐵𝑡 denotes the standard Brownian motion and a, b are constants.
(a) Calculate the expected share price at time 𝑡 given that its price is 115 pence at time 2.
We have,
𝑃𝑡 = 𝑃0 exp {0.06𝑡 + 0.15𝐵...