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FINS2624 PROBLEM SET 5 SOLUTIONS Question 1. a)

ℎ  →

( ) −  

Sharpe Ratio is a measure for calculating risk-adjusted return. It is the average return earned in excess of the risk-free rate per unit of volatility or total risk Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.  =  =

( ) −  0.08 − 0.04 = 0.2 = 0.2 

( ) −  0.16 − 0.04 = = 0.4  0.3

b) A combination of asset Z + the risk-free asset dominates asset X if: - E[Combination] > E[X] AND  =  ; or - E[Combination] = E[X] AND  < 

Below, we consider the first case:

We want to create a portfolio C, consisting of (weights) wZ in Asset Z and wrf = 1- wZ in the rf asset, such that:  = 

( ) > ( )

and

( ) =  ( ) + (1 −  ) ( ) =  +  ( ) −  

 + 2 (1 −  ) ,    =   + (1 −  )   ∴  =   ,   =  ,   = 0

∴  =     =  ,

  = 0.2 →  =

∴ ( ) = 0.04 +

0.2 2 = 0.3 3

2 (0.16 − 0.04) = 0.12 > 0.08 (= [ ]) 3

∴   

c)

  →  =  = 0.5 ( ) =  +  ( ) −  

( ) = 0.04 +

1 × 0.04 = 0.06 2

 =   = 0.5 × 0.2 = 0.1

  →       ℎ  =  = 0.1  =   = 0.1 →  =

0.1 1 = 0.3 3

( ) =  +  (( ) −  )

( ) = 0.04 +

1 × 0.12 = 0.08 > ( ) 3

∴   

d) Need to show: d > 0, where

 →  ( ) =  +  ( ) −  ,

( =     )

 =   →  =

 

( ) =  + 󰇩

 ( ) =  +  ( ) −   

( ) − 

Similarly, ( ) =  + 󰇩



( ) −  

󰇪  ,

ℎ

( ) −  =  , ℎ ℎ    

󰇪  ,

ℎ

( ) −  =  , ℎ ℎ    

(CZ = representative portfolio on CAL Z)   =  , ℎ

( ) =  + 󰇩

( ) −  󰇪  

∴  = ( ) − ( ) =  +   −  +   

 = ( −  ) ≥ 0

(  >    ≥ 0)

∴   

Question 1d, Alternative Proof: Return of any complete portfolio of asset X and risk-free asset is:  =  +   −   = 0.04 +  ( − 0.04)   󰇱 [ ] = 0.04 +  × 0.04 … … … … … (1) [ ] =  ×  =  × 0.2 … … … … (2)

Similarly, any complete portfolio of asset Z and risk-free asset is:

 =  +   −   = 0.04 +  ( − 0.04)   󰇱 [ ] = 0.04 +  × 0.12 … … … … … (3) [ ] =  ×  =  × 0.3 … … … … (4) To make sure the portfolio CZ dominates the portfolio CX, we only need: [ ] = [ ] … … … … (5)

[ ] > [ ] … … … … (6)

For (5) to hold, equating (2) and (4), we get:  =

2 ×  … … … … (7) 3

Now, let’s check whether (6) holds given (7) ( ) = 0.04 +  × 0.12 = 0.04 +

End of proof

2 ×  × 0.12 3

( ) = 0.04 + 0.08 ×  > ( ) = 0.04 + 0.04 × 

Question 2.

 = 3% (∗) = 15% ∗ = 20%

1  = () −   2 A complete portfolio C with the fraction y invested in P* & (1-y) in risk-free asset has the following return:  =  + ∗ −   ( ) =  + (∗) −    =  × ∗

a) We want to maximise the investor’s utility function. The utility function is given by: 1  = ( ) −  2 1  =  + (∗) −   − ( × ∗ ) 2 Thus, set the first derivative (with respect to y) equal to 0, and solve for the optimal weight, y*:   = (∗) −  − ∗ =0  → ∗ =

(∗) −  1 0.15 − 0.03 1 ×󰇧 󰇨= × = 1.5   0.2 2 ∗

So, (1-y*) = -0.5 This means the investor should borrow 50% of her wealth and then invest the total amount of money she has, 150% of her wealth, in the optimal risky portfolio P*.

CAL

Efficient frontier of risky assets P*

rf

b) & c) We know the optimal fraction in P*, ∗ =

1 (∗ ) −  ×   ∗

So, a more risk-averse investor (i.e. with a higher A) should invest less in P*.

And if the portfolio P* offers a higher E(rP*), but same ∗ , then the risk-reward ratio, (∗ ) , ∗

is higher, so a higher y*.

The investors should invest more.

d)

New P* New rf

P*

rf = 3%

The optimal risky portfolio is the tangent portfolio of the linear line from rf to the efficient frontier. From the graph above, you can see the new P* should have a higher expected return and a higher volatility.

e) From a), we know the optimal fraction in P* is given by: ∗ =

1 (∗) −  ×   ∗

1. The investor can invest at  = 3%, where  is the mean Lending rate. So, the optimal fraction in P* should be: ∗ =

1 (∗) −  × ,  ℎ      100%   ∗ 

Why not above 100%? Because she can only lend (not borrow) at  = 3%

2. The investor can borrow at  = 5%, where  is the mean Borrowing rate. So, the optimal fraction in P* should be: ∗ =

1 (∗) −  × ,  ∗ > 1   ∗

3. Let’s put numbers in: 1 0.15 − 0.03 ∗ = × = 1.5 > 100%,    2 0.2

1 0.15 − 0.05 ∗ = × = 1.25 > 1 → ℎ    ℎ      5% 2 0.2 →  = 1 − ∗ = 1 − 1.25 = −0.25

Hence, the optimal portfolio is to borrow 25% of her wealth at 5% and invest 125% of her wealth in the optimal risky portfolio. With this higher borrowing rate, allocation of wealth shifts towards less leverage (compared to part a).

f)

 =  (1 − ∗ ) + ∗ ∗

( ) =  + ∗ (∗) −   ( ) = 0.05 + 1.25 × (0.15 − 0.05) = 17.5%  = ∗ ∗ = 1.25 × 0.2 = 0.25...