Slide #1: Lecture 10 – Portfolio Expected Return, Variance, and Standard Deviation
Welcome to Lecture 10: The calculation of portfolio expected return, portfolio variance, and portfolio standard deviation.
A preamble for this lecture is necessary: This lecture is mainly concerned with the mechanics of calculating portfolio expected return, variance, and standard deviation. It is assumed that you are already familiar with the terms used in this lecture, such as states of the economy, probability of a state, asset return, portfolio return, variance, and standard deviation. If you do not know what these terms mean, please review the relevant materials in your textbook before proceeding with this lecture.
Slide #2: Numerical Example
This lecture will demonstrate the process of calculating portfolio expected return, variance, and standard deviation via the use of a numerical example. To be more specific, here is the example we shall be using:
State/Asset / X / Y / ProbabilityNormal / 0.25 / 0.13 / 0.4
Recession / 0.08 / 0.09 / 0.6
Amount invested / $5,000 / $3,000
The information that we will need to calculate portfolio expected return, variance, and standard deviation are these:
States of the economy: Normal or Recession
Probabilities: 0.4 for Normal state and 0.6 for Recession state
Assets: X and Y
Returns: 25% for X and 13% for Y in Normal State; 8% for X and 9% for Y in Recession State
Amount invested: $5000 in X and $3000 in Y
Slide #3: Seven-step process
The process for calculating portfolio expected return, variance, and standard deviation is divided, by me, into 7 steps:
Step 1: Identify the states and the probability associated with each state.
Step 2: Read the returns table across each state.
Step 3: Identify the assets and the weight associated with each asset.
Step 4: Calculate portfolio return in each state.
Step 5: Calculate the portfolio expected return.
Step 6: Calculate the portfolio variance.
Step 7: Calculate the portfolio standard deviation.
Slide #4: Step 1 – Identifying the states and probabilities
Let’s illustrate the 7-step process using our numerical example. The information table is reproduced here:
State/Asset / X / Y / ProbabilityNormal / 0.25 / 0.13 / 0.4
Recession / 0.08 / 0.09 / 0.6
Amount invested / $5,000 / $3,000
In Step 1, we want to identify the states and their associated probabilities. In this example, we have two possible states: Normal and Recession, and their associated probabilities are:
Probability of Normal state is: Prob(Normal) = 0.4.
Probability of Recession state is: Prob(Recession) = 0.6.
These are found in the last column in the table. Note that the total probabilities of all states must be equal to 1:
Prob(Normal) + Prob(Recession) = 0.4 + 0.6 = 1
Slide #5: Step 2 – Reading the table across each state
In Step 2, we want to read the returns table across each state and write out the returns.
In the Normal state, we haveReturn of X in Normal state is
RXNormal = 0.25
And Return of Y in Normal state is
RYNormal = 0.13
While in the Recession state, we have Return of X in Recession state is
RXRecession = 0.08
And Return of Y in Recession state is
RYRecession = 0.09
Slide #6: Step 3 - Identifying the assets and weights
In Step 3, we identify the assets in our portfolio and their associated weights.
We have two assets in this portfolio, X and Y.
The weight on the ith asset is equal to the amount invested in the ith asset, divided by the total amount invested in the portfolio:
Weight on Asset i = Amount invested in Asset i / Total amount invested
The total amount invested is simply equal to the sum of the amount invested in each of the assets. In this case, we add the $5000 and the $3000 invested in X and Y, respectively, to get a total of $8000 invested in the portfolio.
The weight on Asset X is, therefore:
Weight on X = 5000 / 8000 = 0.625
The weight on Asset Y is:
Weight on Y = 3000 / 8000 = 0.375
Notice that the sum of the weights on the assets must add up to 1.
Slide #7: Step 4 – Calculating portfolio return in each state
In Step 4, we calculate the portfolio return in each state. That is, the outcome in this step should be one return on the portfolio for each of the possible states. If there are two states, then we will have two portfolio returns. If there are three states, then we will have three portfolio returns, and so on and so forth.
Two sets of information are needed for completing this step:
One: The weights on the assets, which, in our numerical example, are:
Weight on X = 0.625
Weight on Y = 0.375
(These numbers were obtained in Step #3.)
Two: The returns on the assets in each state obtained in Step #2. These are shown in the second column in the table below:
State / Returns on assets / Portfolio ReturnNormal / RXNormal = 0.25
RYNormal = 0.13 / RpNormal
= WX(RXNormal) + WY(RYNormal)
= 0.625(0.25) + 0.375(0.13)
= 0.15625 + 0.04875
= 0.205
Recession / RXRecession = 0.08
RYRecession = 0.09 / RpRecession
= WX(RXRecession) + WY(RYRecession)
= 0.625(0.08) + 0.375(0.09)
= 0.05 + 0.03375
= 0.08375
The portfolio return in each state is calculated as the sum of the weight on each asset in State i, multiplied by their associated return in State i.
So, for the Normal State, reading across the row that says “Normal”, we see the portfolio return in Normal state is equal to the weight on Asset X multiplied by the return on Asset X in Normal state. Add to that the weight on Asset Y multiplied by the return on Asset Y in Normal state. Plugging in the weights on X and Y of 0.625 and 0.375, and plugging in the return on X and Y in Normal state of 0.25 and 0.13, we get a portfolio return in Normal state of 0.205 or 20.5%.
For the Recession State, reading across the row that says “Recession”, we see the portfolio return in Recession state is equal to the weight on Asset X multiplied by the return on Asset X in Recession state. Add to that the weight on Asset Y multiplied by the return on Asset Y in Recession state. Plugging in the weights on X and Y of 0.625 and 0.375, and plugging in the return on X and Y in Recession state of 0.08 and 0.09, we get a portfolio return in Recession state of 0.08375 or 8.375%.
Slide #8: Step 5 – Calculating portfolio expected return
Finally, in Step 5, we calculate the portfolio expected return.
Two sets of information are needed to complete Step 5: the probability of each state, and the portfolio return in each state.
One: The probability of each state was obtained in Step 1.
In our example, the probability of Normal state is 0.4, and the probability of Recession state is 0.6:
Prob(Normal) = 0.4
Prob(Recession) = 0.6
Two: The portfolio return in each state was obtained in Step 4.
In our example, the portfolio return in the Normal state is 0.205, and the portfolio return in the Recession state is 0.08375 :
RpNormal = 0.205
RpRecession = 0.08375
The portfolio expected return, as represented by E(Rp), is calculated as the sum of the product of the probability in each state and the portfolio return in each state:
E(Rp) = [Prob(Normal) x RpNormal] + [Prob(Recession) x RpRecession]
So, we have the probability of the Normal state multiplied by the portfolio return in the Normal state. Then, add to that the probability of the Recession state multiplied by the portfolio return in the Recession state.
Plugging in the probability of Normal state of 0.4, and the probability of Recession state of 0.6, and the portfolio return in Normal state of 0.205 and the portfolio return in Recession state of 0.08375, we get the portfolio expected return of 0.13225 or 13.225%:
E(Rp) = [0.4 x 0.205] + [0.6 x 0.08375] = 0.13225
We can also accomplish Step 5 by using a table such as this one.
State / Probability / Portfolio Return / Prob x RpNormal / 0.4 / 0.205 / 0.4 x 0.205 = 0.082
Recession / 0.6 / 0.08375 / 0.6 x 0.08375 = 0.05025
E(Rp) = / 0.082 + 0.05025 = 0.13225
We place the probabilities in one column, and the portfolio returns in the column next to it to the right. We then read across each row, and multiply the probability by the portfolio return. So, for example, in the “Normal” row, we have probability of 0.4, portfolio return of 0.205, and the probability multiplied with portfolio return of 0.4 x 0.205 to give us 0.082. Similarly for the “Recession” row.
We then add the two numbers in the last column to get us the expected return on the portfolio.
Slide #9: Step 6 - Calculating portfolio variance
In Step 6, we calculate the portfolio variance.
In this step, we will need three sets of information.
One: The probability of each state obtained in Step 1. So, we have the probability of Normal state equal to 0.4, and the probability of Recession state equal to 0.6:
Prob(Normal) = 0.4
Prob(Recession) = 0.6
Two: The portfolio return in each state obtained in Step 4. And so, we have the portfolio return in Normal state of 0.205, and the portfolio return in Recession state of 0.08375:
RpNormal = 0.205
RpRecession = 0.08375
Three: The portfolio expected return obtained from Step 5, which is equal to 0.13225:
E(Rp) = 0.13225
The portfolio variance (Var(p)) is calculated as the sum of the probability in each state, multiplied by the square of the difference between the portfolio return in each state and the expected return on the portfolio:
Var(p) = Sum[Prob(State i) x (RpStatei – E(Rp))2]
In our example, we then have the probability of the Normal state, multiplied by the square of the difference between portfolio return in Normal state and the expected portfolio return. Add to that the probability of the Recession state, multiplied by the square of the difference between the portfolio return in Recession state and the expected portfolio return. Our portfolio variance, Var(p), formula is then:
Var(p) = [Prob(Normal) x (RpNormal – E(Rp))2] + [Prob(Recession) x (RpRecession – E(Rp))2]
Plugging in the numbers, we get:
Var(p) = [0.4 x (0.205 – 0.13225)2] + [0.6 x (0.08375 – 0.13225)2]
= [0.4 x (0.07275)2] + [0.6 x (-0.0485)2]
= [0.4 x 0.005292563] + [0.6 x 0.00235225]
= 0.002117025 + 0.00141135
= 0.003528375
Therefore, the portfolio variance is 0.003528375.
Slide #10: Step 6 (cont.) – Calculating portfolio variance using a table
We can also use a table to accomplish Step 6 when calculating portfolio variance. For our example, we have the table below:
State / Prob / Portfolio Return / Rp in state i – E(Rp) / (Rp in state i – E(Rp))2 / Prob of state i x (Rp in state i – E(Rp))2Normal / 0.4 / 0.205 / 0.205 – 0.13225 = 0.07275 / 0.072752 = 0.005292563 / 0.4 x 0.005292563 = 0.002117025
Recession / 0.6 / 0.08375 / 0.08375 – 0.13225 = -0.0485 / -0.04852 = 0.00235225 / 0.6 x 0.00235225 = 0.00141135
Var(p) = / 0.002117025 + 0.00141135 = 0.003528375
We again work across each row. For the Normal state, we have probability of 0.4, portfolio return of 0.205, the difference of portfolio return in Normal state and expected portfolio return of 0.205 - 0.13225, which equals 0.07275. We move to the next cell to the right, which gives us the square of the difference of 0.07275, which gives us 0.005292563. We then move to the next cell to the right, which gives us the probability of the Normal state of 0.4, multiplied by the answer in the previous cell of 0.005292563, which gives us 0.002117025.
We do similar calculations for the “Recession” row, and obtain the answer of 0.00141135.
We then add up the numbers in the last column to get the variance of the portfolio, which is, again, 0.003528375.
Slide #11: Step 7 – Calculating portfolio standard deviation
In Step 7, we calculate the portfolio standard deviation. This part is easy.
The portfolio standard deviation is simply equal to the square root of the portfolio variance. In our example, this is
Portfolio standard deviation = = 0.059400126.
Slide #12: Practice problem
Since practice puts the “tise” in Expertise, try to do this question here by yourself to see if you can get the answer. The check answers are also provided on this slide.
State/Asset / A / B / C / Probability1 / 30% / 20% / -5% / 0.3
2 / 20% / 15% / 5% / 0.4
3 / 10% / -2% / 25% / 0.3
Amount invested / $2,500 / $2,500 / $2,500
Check answers:
Portfolio expected return = 0.131333333
Var(p) = 0.000242667
Portfolio standard deviation = 0.015577762
Slide #13: Hints for practice problem
Here are some hints for you. Have fun!
Step #1: States and probabilities
Prob(State 1) = 0.3
Prob(State 2) = 0.4
Prob(State 3) = 0.3
Step #2: States and returns
State 1: RA1 = 0.3; RB1 = 0.2; RC1 = -0.05
State 2: RA2 = 0.2; RB2 = 0.15; RC2 = 0.05
State 3: RA3 = 0.1; RB3 = -0.02; RC3 = 0.25
Step #3: Assets and weights
WA = 1/3
WB = 1/3
WC = 1/3
Step #4: Portfolio return in each state
RP1 = 0.15
RP2 = 0.1333333
RP3 = 0.11
Slide #14: End of lecture
Here ends the lecture on calculating portfolio expected return, variance, and standard deviation.