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In this video, I discuss standard deviation and expected return of portfolio. The Standard Deviation is a measure of how spread out numbers are. Its symbol is σ (the greek letter sigma). The basic expected return formula involves multiplying each asset’s weight in the portfolio by its expected return, then adding all those figures together.

Hello and welcome to this session this is professor forehead in which we would look at risk and risk premium and specifically to measure risk and risk premium we’re going to look at the variance and standard deviation these topics are covered on the cpa exam as well as the cfa exam also if you’re taking an essentials or principles of investment graduate or undergraduate

As always i’m going to remind you to connect with me on linkedin if you haven’t done so youtube is where you would need to subscribe i have 1700 plus accounting auditing tax finance as well as excel tutorial if you like my lectures please like them share them put them in playlist if they benefit you it means they might benefit other people and connect with me on

Instagram on my website foreheadlectures.com you will find additional resources to complement and supplement your accounting courses your finance courses as well as your professional certifications any investment you undertake comes with risk comes with uncertainty about the future holding period return what does it mean future holding period it means your future

Rate of return is uncertain and there are many reasons why that’s uncertain it could be from macroeconomics fluctuation the economy is not doing well you have no control over that maybe changes in your industries maybe changes that are firm specific unexpected development at your company maybe a competitor came with a new product there’s always risk when you make

An investment also there are unknown reasons simply put you undertaken investments you don’t do well sometimes there’s no specific explanation for that but from a finance perspective what we can do we can try to measure risk by using scenario analysis and probability distribution so how to measure risk this is basically the idea of what we are doing today so we

Attempt to measure risk by asking ourselves to couple questions basically what is the holding period return for this investment so basically what is the holding period it means how much returns can we earn on this investment and how likely are they so basically we have to kind of guess different type of returns under different scenarios so basically what we have

To do is we have to list the possible economic outcome or scenarios and specify the probability for each scenario and the holding period return for the asset assuming given this scenario so this approach called the scenario analysis so how how does it work it’s pretty straightforward for example here let’s take a look at this example we’re going to be assuming

Four scenarios we’re gonna have a severe recession we’re gonna have a mild recession we’re gonna have a normal growth and we’re gonna have a boom what are the probability of those events happening having a severe recession is five percent mild recession 25 percent normal growth is 40 and a boom is 30 always add up your probabilities your probabilities should

Always add up to 100 percent now we have to guess or estimate our holding period that are possible giving the giving the different scenarios in a severe recession we might have a holding period return of negative 37 percent if you don’t know how to compute the holding period return please go to the prior session because i did show you how to compute this um in

A mild recession we could have 11 negative 11 percent normal growth we could have a positive 14 in a boom if everything goes well and the economy is booming our investment should earn 30 so basically what we did is we answered those two questions how likely are they and what’s the holding period now once we have this information the probability we’re going to

Have a probability distribution that’s going to let us derive find out the measurement for both the reward and risk of the investment basically the reward and risk of the investments are called especially the reward is called the expected return basically how much are we expected to earn from this investment giving the probabilities this is what we’re looking

For the expected return is also called the mean of the distribution of the holding period rate and it’s often referred to as the mean return so i’m going to be referring referring referring to it as the mean return and how do we compute the mean return this is the formula basically it’s the sum of the probability times the return what does that mean we have the

Probabilities 5 percent we have to return the holding period return negative 37 so if that happened if we multiply the probability by the holding period return we’re going to get negative 1.85 under the mild recession we’ll do the same thing 25 times the return two point negative two point seven five normal growth forty percent probability times fourteen percent

Five point six zero and a boom 30 probability and we’re going to get 30 return expected return 9 now when we add up all these returns we’re going to get when we net them out we’re going to get what’s called the expected return or the mean return so the mean return basically what does that mean i mean we expect given the different probabilities we expect to

Earn 10 is this accurate absolutely not we need to learn a little bit more about this expected return what do we want to learn about we want to learn about the variability of it because there’s going to be well what does that mean it means if we have a boom we could go up to 30 you know what’s the difference between 30 and 10 that’s that’s a variance of 20 20

Percentage so what we need to do is because there’s a risk of the invest to the investment the actual return may be a lot more or a lot less so we don’t know we expect the 10 percent so if the boom materializes the return will be better which is 30 but in a severe recession the return will be negative 37. now what we need to know we need to know we need to learn

A little bit more we need to quantify this uncertainty this spread between from from the mean to the other returns giving the probabilities so there’s this is what’s called the surprise return in any scenario is the difference between the actual return and the expected return so the surprise return is we expected 10 what was the actual okay so this to summarize

Uncertainty with a single number we’re going to compute something called the variance and the variance it’s going to help us measure this uncertainty and we’re going to end the variance as the expected value of the square deviation from the mean so the expected square dv the expected square surprise across different scenarios how do we compute the variance let’s

Look at the formula first and we’re going to compute it it’s pretty straightforward it’s the sum of uh the probabilities times the the return given a scenario minus the expected return the expected return for us is 10 raised to the second power okay in the sum of all of those probabilities simply put let’s take a look at let’s take a look at it here we already did

We already looked at the probabilities and the holding period return at the expected return now what we need to find out is the deviation and what is the deviation the deviation is the range is the range from the mean return let’s let me show you what do i mean by what by the deviation so the expected return is 10 this is the expected return notice here in a

Severe recession we could go down to negative 37 so what is the deviation well the deviation is negative 47 and this is what we mean by the deviation negative 47. under a mild recession we could be standing at negative 11 negative 11. so what’s the difference between 10 and negative 11 that’s negative 21 that’s negative 21. also the normal growth again we’ll

Do the same thing it’s 14 that’s 4 the deviation is 4 and under the boom we’re gonna have 30 percent and that’s the deviation is 20. so i want you to see how what what do i mean by the deviation so what you do is you compute the deviation or or the range the range how much does it does it differ from the expected return basically then we compute the um the

Uh the squared or the variance the variance we’re going to compute the variance and again how do we compute the variance using this formula right here so let’s go ahead and look at the first example using this formula to see how we compute this so let’s take a look at it okay so we’re going to have the probability well we’re going to have what are we going to

Have we’re going to have the probability which is for the first one 0.05 and we’re going to multiply this by the difference between the holding period return 37 minus the expected return the expected return is 10 again what are we going to get we’re going to get negative 47 and we’re going to square this and we’re going to get 110.45 and we’ll do the same thing

For the mild recession the same thing for the normal and the boom and what we did is we add up all the variances and why did we do so we do so because we want to eliminate the negatives once we squ once we raise to the second power we eliminate the negative because when you multiply negative by negative negative 47 times multiply by negative 47 it’s going to

Give you a positive number so that’s the reason we compute the uh the variance is to get rid of the negative because we’re really concerned with this with the range with the spread okay so to give measure of risk this risk the same dimension as expected return percentage we use the standard deviation now how do we compute the standard deviation because we looked

At the variance the standard deviation is simply put as the square root of the variance so this is the variance all we’re going to do is going to we’re going to find the square root and this is sigma so the standard deviation sigma is the square root of the uh of the of the of the variance and basically what we did because we raised those to the second power you

Know what we do now is we reverse this we take the square root of it the square root of it so basically the standard deviation which is the measure measure of disbursement is 18.63 now we’re going to talk a few things about the standard deviation generally speaking giving the same expected return the largest the standard deviation the more risk there is for the

Portfolio or for that investment standard deviation means there’s more variances in the data the more variances in the data the riskier the riskier is the problem let me give you a simple example like is an example that you can maybe you can maybe relate to to understand this concept let’s assume i have two classes i have two classes class a and class b and the

Average score the average score the average score for class a is 75 and the average for class b is 75. if that’s all the information that you know you would say this both classes have the same average now let’s assume i ask you to do the following um or let’s yeah let’s assume i ask you to do the following so i pulled i pulled the first first exam from class a

And the score was 70. i pulled the first test randomly and the score was 70. i pulled the second test from class a the score was 80. i pulled the third test from class a the score was 73 i pulled the fourth exam the score was 86. i pulled the fifth exam the score was 72. i pulled the sixth exam and the score was the score was let’s assume 88. now if i ask you to

Pull the seventh exam or if i ask you to guess what the seventh exam most likely be in class a and i bet you’re gonna say something between 70 to 80 percent because most of them they’re between you know do we have this 86 but we have 70. so it’s between 17 86 88 so it’s ranging in that it’s in that in that ballpark okay if you want the guess if i ask you to guess

Let me go to class b i’m going to do the same thing for class b so randomly i pulled the first test from class b and the score was 20. i pulled the second test from class b the score was 100 i pulled the third test randomly from class b and the score was 60. i pulled the fourth test it was 90. the fifth test was 75. now let’s that’s five let’s pull one more it

Was 40. if i ask you to pull the seventh or guess the seventh test in class b how are you are you more comfortable guessing class a the seventh test or class b and hopefully you are more comfortable guessing the seventh grade for class a why because it seems based on what we have based on the variance and the grade the grade are centered around 75 the average

Is 75 but the grades are centered around 75. for class for class v the average is the same at 75 however some students are getting 20 some students are getting 100 so there’s wide dispersion so the to guess class b exam you’re going to have a few if we compute the standard deviation the standard if we compute the standard deviation for class b if we compute the

Standard deviation i could assure you the standard deviation is higher than the standard deviation in class a so class a will be lower therefore guessing the seventh grade will be riskier for class b then class a although they both have the same average so that’s the point of computing the standard deviation is the the larger the standard deviation generally

Speaking the more risk there is again the standard deviation works best when we have the same expected expected return which is 10 the same average grade what i was talking about 75. now why are we doing all of this the reason we’re doing all of this because in the next session we’re going to be looking at we’re going to be using the normal distribution bell

Curve to explain a little bit more about the average return the mean and the standard deviation but first i wanted to make sure you understand how we came up with the standard deviation and what does it mean it measures the variability in the data and more variable more variability there is if a stock goes up up and down swing widely from day to day it’s going

To be riskier than a stock that doesn’t move too much but this is what we’re going to be talking about next session but i want to make sure you know what how we compute the standard deviation and what the standard deviation is in the next session we would look we would measure risk looking at the normal distribution as always i’m going to ask you to like this

Recording and share it if you do like it and share it and obviously if you’re listening that far you must have liked what i said please share it put it in playlist and don’t forget to visit my website foreheadlectures.com for additional resources in order to succeed in your education professional certification and exams good luck and study hall

Transcribed from video

Standard Deviation and expected Return. Scenario Analysis Measure Risk Investment Course. CFA exam. By Farhat’s Accounting Lectures