Understanding Investment Concepts: Expected Returns, Minimum Variance Portfolios, and Utility
1. Below is given the log returns of asset X for 4 years. The standard deviation of the log returns is 5.39 %. Calculate the expected return of the asset (this means the arithmetic mean of the asset)- 1.15% 0.91% -0.04% 3.65% 1.77% 1.80% 3.36% -1.96% -0.12% -0.12% 3.50% 0.09% Step1- Calculate the average in this case = 0.0115,0.0091,(0.0004),0.0365,0.0177,0.0180,0.0336,(0.0196),(0.0012),(0.0012) ,0.0350,0.0009 add all and divide by 12 which = 0.0117 then + ½ (( 0.0539 )^2) = 0.0131 then convert to % which = 1.31% 2.)
Solution: First we need to calculate the average log return (α). This is also the geometric return of the asset. This is a simple average of the log returns. (-0.03+0.04+0.03+0.02 + 0.02 – 0.07 ) /6= 0.00166667. To get to the expected return we need to correct by the amount of ½ σ2.Expected return of an asset =Arithmetic Average= μ = 0.00167 + ½* (0.042^2) = 0.002552 = 0.2552%
3) In Question (2) what is the actual return over the entire 6 years? Sum of the log returns will give us the total log return for the entire period:
-0.03 + 0.04 + 0.03 + 0.02 + 0.02 – 0.07 = 0.01 (this is the geometric return) In order to find the actual return = EXP(0.01) = 1.01005. The exponential is PT/P0. in other words 1+ r. Therefore to get to the actual return over the period is 1.01005 -1 = 0.01005 or 1.005 %. 4) Given the prices below what is the total log return over the entire period? What is the actual return? Use log returns and show all your calculations.
P1= 23
P2 = 24
P3 = 22
P4 = 20
P5 = 26 Ln (26/20)(20/22)(22/24)(24/23) = ln (26/23) = 0.12256 (This is log return ; to find the actual return you need to find e0.12256= 1.1304 Remember this is 1 + r and we need to subtract 1 from this to get the actual return = 13.04 %
Alternatively= ln 26/20 + ln 20/22 + ln 22/24 + ln 24/23 =0.2623+ (-0.0953) + (-0.087) + (0.04256) = 0.12256 (this is the total log return over the entire period). To find the actual return e0.12256 = 1.1304. Again we need to subtract 1 to get the actual return = 13.04 %
5) Definition of the minimum variance portfolio-an investing method that helps you maximize returns and minimize risk. It involves diversifying your holdings to reduce volatility, or such that investments that may be risky on their own balance each other out when held together.
6)Given the below information calculate the expected return of the portfolio: α σi
Asset 1
Asset 2
Asset 3 σ_w = 23 % (σ_w for a portfolio of more than 2 assets will be given). R_w(T) = ∑i ω_i R_i(T) + ½ ∑i σ i ^2 T – ½ σ w^2 T. Can also be rewritten as the following: R_w(T) = (μω – ½ σ w2 )T.
First, you need to find the μω for each asset and then find the μω of the portfolio by using the following return formula μw = [ ∑i ωi μi ] and finally find the Rw(T) of the portfolio. EXAMPLE: Part A- Given the information below calculate the expected return of the portfolio:
w_i α_i σ_i σ_P SOLUTION: wi αi σi σP μi μP E(Rp)
Asset A 0.18 0.03 0.25 0.33 Asset A 0.18 0.03 0.25 0.33 0.061 0.2956 0.241
Asset B 0.2 -0.02 0.05 Asset B 0.2 -0.02 0.05 -0.019
Asset C 0.62 0.34 0.5 Asset C 0.62 0.34 0.5 0.465
Formulas: Calculations:
μi = αi + 1/2 * σi2 μA = αA + ½ * σA2
μP = ∑i μi wi 0.061 = 0.03 + ½ * (0.25^2) = 0.061
E(Rp) = μP – 1/2 * σP2 μB = -0.02 + ½ *(0.05^2) = 0.019
μC = 0.34 + ½ * (0.5)^2 = 0.465
μP = wA * μA + wB * μB + wB * μB + wc * μC
μP = (0.18 * 0.061) + ( 0.20 * -0.019) + ( 0.62*0.465) = 0.01098+ -0.0038 + 0.2883 = 0.2955
E(Rp) = μP – ½ * σP2
= 0.2955 – ½ (0.33^2) = 0.241
Part B – Given (the risk aversion level) λ of 0.7 of the investor calculate the MV utility of the investor. (μw – λ σ2w)T. = 0.2956 – 0.7 (0.33^2) = 0.219.
7) Difference between the utility of expected wealth U(W) and E (U(W))- The utility of expected wealth U(W) is about how happy you’d be with an average amount of money, while the expected utility of wealth
E(U(W)) is about how happy you’d typically be with all the different amounts of money you could possibly have.
Assume an investor with a risk aversion level of 0.5. The investor is faced with the following decision: A coin toss with a %10 probability of earning 400.000 or 26,667 with a 90 % probability
A guaranteed amount of 64.000 Notice that the expected return for both are 64.000! a) Would the investor prefer sure money or the bet? Investor will choose the option which is maximizing their expected utility of wealth.
Guaranteed money of 64.000 with 100 % probability: U(W) using the formula:U(W)=W^Y /Y Y
In order to find the expected utility of wealth we need to multiply the U(W) of each option with its respective probability. In this scenario there is only one probability therefore:
E(U(64,000)) = 100 % * 505.96 = 505.96
Let’s find the U(W) for 400,000 and 26,667 and then calculate expected utility of the bet:
U(400,000) = 1/0.5 (400,000^0.5) = 1264.91
U(26,667) = 1/0.5 * (26,667^0.5) = 326.60
E[U(W)] = 10 % * 1246.91 + 90 % * 326.60 = 420.43
The investor will compare the expected utility of the sure return versus the bet.
Expected Utility of sure return = 505.96
Expected Utility of bet = 420.43
Because the expected utility of wealth is higher with the sure bet at 505.96; the investor will choose the guaranteed money