Lecture on Probability Trees and Conditional Expectations

hey it's Jim and this is level one of the CFA program the topic on quantitative methods and the learning module on probability trees and conditional expectations let me read the first sentence in this learning module to you at least the first part of it investment decisions are made under uncertainty wow what a mouthful that is let me quickly remind you that in the previous learning module we had great conversations about lots of things but in particular standard deviation skewness and curtosis these are great measures of dispersion and consequently they become great measures of risk so what we did in that previous learning module is we laid a really good foundation now we're just adding to it we're going to add the essence of looking to the Future and estimating probability of outcomes and how does that contribute to our understanding of dispersion and here look at the first loss we're going to go ahead and do expected value variance and standard deviation just like we did in that previous learning module but we're going to add the Spectre of probabilities second loss we'll look at a probability tree and then the third losos we'll look at the Reverend Thomas Bay's formula what these three loss's are going to do is simply add to our base of knowledge contributing to our understanding of risk management let me read that first sentence to you again investment decisions are made under uncertainty boy we've got lots and lots to talk about it's going to take us all the way through level three to kind of dissect and digest that super simple sentence all right that first losos we've already done this but we haven't done it with probabilities so all we're going to do is we're going to say all right we have a potential outcome in the future that we can assign a probability to then we have another potential outcome maybe it's just a simple uh binomial distribution where the firm can either increase its dividend or decrease its dividend and then we attach probabilities to those so look on the blue box on the far right of the equal sign there's a summation sign all we're going to do is we're going to take each individual possible outcome and we're going to weit it by its probability so look at that first diamond point up there what's the definition that we've given you uh is the probability weighted average of the possible outcomes of a random variable like stock returns or maybe even dividend yields so remember what I said to you back in that previous learning module that the expected value also known as the mean also known as the point of central tendency also known as as the average that's the first moment of the distribution which implies that there is a second moment of the distribution and that of course is the variance of that random variable so the question then becomes how do we take the computations that we did in the previous learning module and add them to this probability weighted well we're going to do the exact same thing that we did before and we're going to couple it with the blue box so look down at the Orange Box we're going to take each individual outcome notice there's an X1 and an X2 and we're going to subtract the expected value of that outcome then we're going to square it and then notice we'll multiply it by the probability so this is a probability weighted average again now I always ask my students why do we Square the difference between those two now I probably said this back in the previous learning module but I think it Bears repeating here's the example that I give my students in my class now I teach in the finance trading lab and so we have I have the board behind me and my students are sitting in two rows on either side so I have an aisle so I walk up and down the aisle um and we have we have monitors on that side so I can walk up and down and I look this way and I look that way so I say to them I say hey look let's play a game let's play a game called count the number of steps that Jim takes if I stand in front of them and I go one two three four so I look at him and I say how many steps did Jim take and they of course look at me like I'm some kind of a cartoon character and they go all right Jim we'll play along because we love finance and we love the statistical application uh that you can use in finance they say four so I go back to the front of the room and I say all right let's play the same game but what I want you to do now is I want you to close your eyes and close your ears and of course most of them don't play along like that so I say all right ready and I tiptoe I go one two and I turn around and I go one two then I turn around and face the class and I say how many steps did Jim take and then I say how many steps does it look like Jim took right and and I said for those of you who played along it looks like I didn't go anywhere you want to say zero but everybody knows that I took four steps I just put them in One Direction and I turned around and came right back from that same direction so the reason that we Square the difference between each possible outcome and its mean is to get rid of the negative but also to be able to accurately count the number of steps that Jim takes you see statistics is your friend if you understand this quantitative method stuff you will be a much better financial analyst and you'll have a higher probability of passing passing the exam all right so thank you for bearing with me there now my wife never asks me to go to the food store and get 12 square pounds of potatoes so the variance is in a squared term a squared unit so we need to take the square root of that just like we did in the previous learning module so that's the standard deviation look at the uh look at that embedded Diamond point there if the variance is zero there is no dispersion which means that there is no risk this is super super important now when the variance is positive which it will be for almost all financial securities out there with the exception of a treasury security of course then it signifies a dispersion of outcom so the higher the variance the higher the standard deviation that means the greater is the risk and that's super important as we move throughout uh the CFA program let's take a quick example here we've got a distribution probability distribution of a company's sales so looking that far right column we don't know what those sales are going to be during this course of the next year they might be 50 they might be 40 they might be 30 so what we have to do is say to ourselves oh let's just hope for the best and hope that we sell 50 million in sales next year no of course you can't do that what you must do as a good financial analyst is assign probabilities to those potential sales so I always tell my students the following I say look rank these from high to low which we've done in the far right column and just think of them as best case scenario 50 worst case scenario 30 and then anything in between in this case we only have one one middle sales but you could have 10 or 20 or 80 of them think about putting this together in an Excel spreadsheet so we assign those probabilities based on things like well what happens to the economy right if the economy expands then we'll probably be able to generate more sales if the economy contracts then we'll probably be able to generate fewer sales so probabilities can be based on things like GDP they can be based on things like our marketing campaign quality of our board of directors quality of our Salesforce quality of our supply chain quality qualities so you get the sense that those probabilities are tied to the potential outcomes so that far right column we have a distribution of sales when you combine the two columns we have a probability distribution of sales so all we're trying to do is find the midpoint can you guys see that if I point if I point at my screen I guess it looks like I'm pointing at you but I don't I don't mean to point at you all we're going to do is take the average and what did we say in that previous uh slide it's a probability weighted so all we're going to do is we're going to say 20% of 50 30% of 40 50% of 30 and we're just going to sum though so our expected sales is 37 million think about what this means that's our average right that's our point of central tendency it's our expected value so we need to make decisions today based on the expected sales of 37 million now we know this is a distribution we know that when we predict when we estimate that 37 million we know we're going to do it with an error we know we're not going to be perfectly 100% accurate every single time so what do we do we compute the variance and the standard deviation this is a measure of the degree of error how wrong could we be I mean think about it if we expect sales of 37 million for this coming year and next year we have 1,000 million of sales we were super wrong I mean we're happy that we had generated all those sales but we were super wrong that's going to translate into a high standard deviation all right so how do we do this well the same way that we did it back in the previous learning module and the same way that we did it here on that previous slide we're just going to probability weight it all right so look down there what are we doing the probability that we get 50 times the difference between 50 and the expected sales so that expected sales figure in there that's the 37 from the previous slide and then we Square it so we do all the math and I encourage you to go ahead and pause the recording and do the math so that you can get that $61 million in variance and then uh hit your square root button and you get what's that about 8 million so what are we doing here we're saying something like let me go back here we're expecting 37 million but on average on average we're going to be wrong by about 8 million so think about how that contributes to our planning for the next year in terms of putting together not just an income statement but a balance sheet and a cash flow statement we expect 37 but on average we're going to be wrong by 8 million on average that doesn't mean we'll be wrong by 8 million during this coming year but if we do this a million times on average we'll be wrong by 8 million now compare that if for some reason we would have gotten a standard deviation of 2 million oh my gosh now we're now we're pretty accurate right a lower standard deviation means that we expect 37 and we're pretty confident that we're going to get around 37 but if our standard deviation is let's say 188 million boy then we expect 37 but we have no idea what's going to happen right so high standard deviations imply uh uh a super unknowledge of the future not quite quite sure if unknowledge is a word but you uh you get the point all right moving to a probability tree so look down on the left side of the Box notice we start at time period zero and then we can either move up or down and then after we've moved up or down we can either move up or down again this is called uh a decision tree and it has tremendous applications especially for valuing options and valuing bonds that have embedded options so you are going to see this multiple times in level two and in level three I'm guessing that the Institute is very likely to test you on this at those later levels under derivative Securities uh than it might in this particular example but what this does is this gives us an additional understanding of risk so here's the traditional example that textbooks and professors use all throughout the world when we're starting to explain a decision treade suppose we're C tossing a coin right you toss it it can come up heads it can come up tails and it has a 50% chance because there's only two possible chances right so notice at first toss we have a 50% chance of getting heads and a 50% chance of getting Tails then we have a second toss and that second toss we also have a 50% chance of getting heads and a 50% chance of getting Tails but here's what this visual representation of all future outcomes this tree diagram tells us that we can ask ourselves the question what's the probability of tossing two heads together well you just multiply 0.5 time .5 look at the middle column out there there's 25% so heads heads Heads Tails Tails heads Tails taals they each have a 25% probability of occurring so think of about what's happening here you have a first toss and then a second toss after that second toss you have a whole bunch of outcomes and you can assign a known probability of those outcomes now what does this mean for our previous examples you know what did I say earlier dividend yields or stock Returns what we can do is we can take a share of stock like Proctor and Gamble and we can say something like today the stock price is let's say it's 100 it could either go up or it could go down what's the probability of it going up well maybe it's 50% maybe it's some other percent we'll talk at length about that especially when we get into level two but then suppose that the stock price goes up during that first day what about the second day so what's the probability that Proctor and Gamble will rise to consecutive days well we use this decision tree to visualize those kinds of outcomes now each of the tree has a branch and so you can assign a probability to each of those branches and so we get into really really cool conversations about things that sound like this Proctor and Gamble let's suppose has a 50% chance of going up or down today so so at the end of today it goes up but then overnight Proctor and Gamble announces that it it has just introduced a new product line now those of you who have sons out there you will know that once they hit a certain age even after they take a shower they still come out smelling so Proctor and Gamble has developed a bar of soap that's going to uh defunkify teenage boys so that they don't smell so everyone says wait a minute I have sons who smell I'm going to go buy this product line so so overnight well what happens well that probability is no longer it's no longer 50% because people are going to flock to Target and Walmart and all the retail stores and buy this product line so maybe it has a 90% chance so you see how cool this can get this super simple example where we just CAU toss a coin can lead into much more intricate much more sophisticated and much more meaningful as Financial analysts so that we can make better decisions for our client right we're never going to toss a coin when our client says here's a bunch of money go ahead toss the coin oh heads we invest over here Tales we invest over here that's not why we're in the CFA program all right so let's go ahead and work through an example what the learning module does for you is that it pro provides a series of super complex equations and uh I'm going to work through an example and we'll show you just how how simple this is but let's go ahead and read through some of these things what does a conditional expectation mean it really just means something like the expected value of a certain kind of an investment like like Proctor and Gamble given a certain set of real world events you know a new product line with this soap so those expected values I was saying this earlier right actions of GDP actions of governments whether it's changes in money supply or fiscal policy changes in financial institutions go back to the beginning of 2023 we had the collapse of SV Bank what does that do well that puts all kind of other Regional banks on notice so if we go back here and we try to do this decision tree we're trying to predict whether Regional banks are going to increase or decrease in price during the course of a day or a week or a month well that that uh SV Bank debacle that happened on a certain day well that changes stuff right that changes these probabilities so look down at the bottom all we're going to do is we're going to say something like we're going to sum those probabilities right and what we're going to do is these probabilities now notice that there's a slash sign so the probability of X dividend yields stock returns whatever that is given some kind an event or a change in scenario that's what the notation for S is so the probability of X giv some scenario maybe that scenario is SV Bank failing maybe that scenario is Spike in inflation maybe maybe yeah so we can do all that stuff and we can incorporate oh so you should be thinking wait a minute Jim how do I process all of the macroeconomic variables out there into a simple decision tree and well that's a that's a challenge all right so let's take a look at a quick example here uh we want to predict recovering principle for a defaulted bond issue and that defaulted bond issue depends on two economic scenarios all right so notice that we have scenario one with a probability of 60% we put that in red scenario 2 with a probability of 40% so let's not worry too much about what scenario 1 and scenario 2 are but maybe they are things like changes in GDP maybe they are things like SV Bank default maybe they are what whatever we come up with notice that 60 and 40 they add to 100% right so scenario one and scenario to have a 100% chance of one or the other occurring now once we hit scenario one then then we have another branch on the decision tree we might get 80 cents per $1 and we might get. 70 cents per $1 so we assign probabilities 35% to the 80 and 65% to the 70 all right you see how this decision tree is forming scenario two well we might get 60% I'm sorry we might get 60 cents or we might get 30 cents all right so scenario one is obviously better for us if we're uh one of these defaulted Bond holders scenario 2 is worse for us so we have a 70% chance of 60 and a 30% chance of the 30 cents right so what is the condition expected recovery so what we're trying to do here in this case here let me go back real quick what we're trying to do is come up with the column of HH HT th and TT all right so all we're going to do we're going to use that equation down to the bottom of the blue box are you ready for this so there's our decision tree all we're going to do is we're going to say we have a 35% chance of getting 80 we have a 65% chance of getting 70 cents so there under scenario one our recovery is 73.5 cents so look on the left side of that first uh equation the top of the gray box the expected recovery given scenario one is 73.5 cents the expected recovery given scenario 2 is 51 cents so all we're doing is Computing kind of like a future value it's not really a future value but it's kind of like a future value then what do we well then we go ahead and say let's keep coming backwards and we call these unconditional expected values there's the formula down there all we're going to do is here's the same thing that we have before the unconditional expected recovery is closest to well now we're just going to say we have a 60% chance of scenario one under which we'll recover 73 and a half cents we have a 40% chance under scenario 2 of recovering 51 cents so you just are you ready for this probability wait those future outcomes so our recovery is 645 cents that that is the very likely question on the exam not the compute the 73 and A2 and the 51 you'll have to compute the 73 and2 and the 51 in order to get to that 64 this is a great this is a great exam question what does this do this gives us an expected value right so we expect to recover 64.5 cents so we can make decisions today based on what we expect to happen in the future given our estimates of probabilities and our understanding of scenario one and scenario 2 let's go ahead and look at this last losos in which we ask the following question let's suppose that during the course of our decision tree that the information set changes think about how valuable that new information is so this is this idea of uh of Thomas Bay notice what we have there in that very first uh Point updated or posterior probability given a set of Prior probabilities it allows us to update our decisions so think about our decision Tree in that last example with Proctor and Gamble and then Proctor and Gamble decides to do something else and then the economy changes and then and then so we have all we have all of these changes and how can we explicitly explicitly measure those updated probabilities the example that I've been giving to my class uh for the last oh what has it been now two years or so is the following you guys should know that I'm a huge James Bond fan I saw my first movie my dad took me when I was 5 years old 1966 anyway when Daniel Craig announced that he was going to this last movie was going to be his his very final episode his very final role you know of course we're trying to figure out who is going to be the new James Bond well you can use this base formula to kind of figure this out to estimate the probability that an individ ual actor will get that role you know just just take a simple example Suppose there are five dudes out there who are in the running so you could just say something like well each has a 20% chance of getting the role of James Bond but but then over time what happened you hear all these rumors so maybe maybe one of these actors um Mary's a famous country music star and that changes his probability maybe the producers of James Bond maybe they want their James Bond actor to be married to country music star or maybe they don't so that might increase or decrease his probability uh heaven forbid remember what happened to Luke Skywalker in between the first and second movies he was in an accident he need to have some plastic surgery you know maybe one of these five dudes has an accident heaven forbid and you know he he can't uh perform all of the running and jumping and all the cool stuff that James Bond has to do because you know broken arm broken leg whatever whatever that is or maybe or maybe or maybe I mean you have all these probabilities and so as the James Bond decisionmaking process evolves over time you can update these probabilities now um I'm always fascinated by the way Finance textbooks and finance people try to explain the statistics behind um all of these kinds of models and The Institute really is no exception and I'm not being critical at all there's really no way around this excuse me what what you can do is you can take a look at look at the very bottom blue box here's our summary of this uh this base formula we're going to try to estimate the probability of an event given some information and we can adjust some things over on the far right hand side of the equal sign but notice we have probabilities in there in the numerator and the denominator what that means is that we need to go back to our quantitative methods beginning so look at these equations here I'm not going to suggest that you memorize these things remember we've got things of like Vin diagrams and unions and intersections and that's what all of this stuff is using conditional probabilities which we just you uh used it in a previous example the total probability rule which we did before and so uh here are the variables that are super important uh the probability of B subi these are known prior probabilities the event a is some event known to have occurred and then the probability of B given that event is known as the posterior probability so keep in mind that these are all good equations these are all good mathematical stuff for you to know here's the good orange and blue box for you to remember kind of General formulations but let's work through an example and you'll see how simple this is now here's my advice this example and the example in the body of this learning module both ask a similar question here notice at the bottom what is the probability that uh there's a negative return on the New York Stock Exchange this is a very similar question as in the body of learning this learning module so it's very likely that if the Institute asked you this question on Bay it's going to be in this context now I will also say to you that when you look at the problems at the end of this learning module there's only two one ask you for the mean one ask you for the standard deviation so there are no problems on any of this stuff that we've done over the last 10 minutes or so I'll let you decide whether not you want to spend extra time on these things my advice is to spend the extra time because as I say regularly throughout all of my recordings is that every losos is a potential exam question but we can also infer from the problems at the end of each learning module what kind of importance which topics are important uh to The Institute all right so enough of my babbling there about what can show up and what can't show up on on the exam all right so we have uh an analyst investigating the performance of stocks on different exchanges all right so 50% come from the New York Stock Exchange 30% from the London Stock Exchange 20% from the Tokyo Stock Exchange now when you get to a problem like this I think it's easy and it might be just the way I think but I will visualize that in this number there are 100 so 50 from the New York Stock Exchange 30 from London 20 from the Tokyo all right so what are we doing here we've got 100 stocks think about think about them as a 100 little pingpong balls in in a swimming pool so we're going to reach in and we're going to grab one of them out all right now here's this updated information the probability of a stock posting a negative return is 40 35 and 25% on each of those exchanges so what we want to do is we want the probability that when we pick out a pingpong ball from the swimming pool that it's going to have a negative return ah so wait a minute if we were to say something like look look if we knew that the pingpong ball was New York Stock Exchange and we pick that out well then we would know that that probability is 40% similarly London 35% Tokyo 25% but here's the deal ready look at the question picks a stock at random so the pingpong balls in the pool are not colored by the type of exchange they're all white or yellow or pink or blue right we don't know which ones are New York which ones are London and which ones are Tokyo so how do we figure out that probability that that randomly selected pingpong ball is going to have a negative return so let me just go back here quickly all right so look down at well let's go back to the uh The Orange Box so probability of the new information given the event that's going to be in our numerator so there's probability of the information slash given that event all right so here we look at the bottom here probability of A/B subi all right you ready for this so this is super simple so what do we know we know we have a let's go back here we know we have a 50% chance of picking a New York Stock Exchange pingpong ball which tells us that then we're going to have a 40% chance of posting a negative return well there's our numerator right 50% times that 40% % chance right that's the numerator then all we're going to do is divide by all of the other possibilities right we're trying to do a probability so in the denominator we take the 50% times the 40 That's New York Stock Exchange 30% times the 35 uh percent that's the London Stock Exchange and then 20% times the 25% that's the uh Tokyo Stock Exchange so do you see how this Baye formula it kind of takes away from what we've done during this entire slide deck but it puts it in a numerator and a denominator in a form so that we have this updated information so there you go 56.3% and then of course my advice for you is to go back and change the question say probability that it has a negative return on London do that one probability of negative negative return on Tokyo do that one so you get the sense of what's going on here then what I want you to do is I want you to go back to the reading and work through the example inside the middle of that learning module so I believe that takes us to the end of these three learning outcome statements this is super fun what did I say somewhere in there statistics and quantitative methods is your friend so get your arms out wrap your arms around this material embrace it and love it and and hug it and do all those really really good things because look I can warn you you now but you'll see as we move throughout the program you're going to use this material and it's going to be applied to lots and lots of other different kinds of topics so what do I want you to do now I want you to go to the middle of this learning module and work through that base formula then I want you to go and compute the expected value and standard deviation and the problems at the end of this learning module so hey that was fun for me I hope it was fun for you thanks for watching and good luck studying

hey it&#39;s Jim and this is level one of the CFA program the topic on quantitative methods and the learning module on probability trees and conditional expectations let me read the first sentence in this learning module to you at least the first part of it investment decisions are made under uncertainty wow what a mouthful that is let me quickly remind you that in the previous learning module we had great conversations about lots of things but in particular standard deviation skewness and curtosis these are great measures of dispersion and consequently they become great measures of risk so what we did in that previous learning module is we laid a really good foundation now we&#39;re just adding to it we&#39;re going to add the essence of looking to the Future and estimating probability of outcomes and how does that contribute to our understanding of dispersion and here look at the first loss we&#39;re going to go ahead and do expected value variance and standard deviation just like we did in that previous learning module but we&#39;re going to add the Spectre of probabilities second loss we&#39;ll look at a probability tree and then the third losos we&#39;ll look at the Reverend Thomas Bay&#39;s formula what these three loss&#39;s are going to do is simply add to our base of knowledge contributing to our understanding of risk management let me read that first sentence to you again investment decisions are made under uncertainty boy we&#39;ve got lots and lots to talk about it&#39;s going to take us all the way through level three to kind of dissect and digest that super simple sentence all right that first losos we&#39;ve already done this but we haven&#39;t done it with probabilities so all we&#39;re going to do is we&#39;re going to say all right we have a potential outcome in the future that we can assign a probability to then we have another potential outcome maybe it&#39;s just a simple uh binomial distribution where the firm can either increase its dividend or decrease its dividend and then we attach probabilities to those so look on the blue box on the far right of the equal sign there&#39;s a summation sign all we&#39;re going to do is we&#39;re going to take each individual possible outcome and we&#39;re going to weit it by its probability so look at that first diamond point up there what&#39;s the definition that we&#39;ve given you uh is the probability weighted average of the possible outcomes of a random variable like stock returns or maybe even dividend yields so remember what I said to you back in that previous learning module that the expected value also known as the mean also known as the point of central tendency also known as as the average that&#39;s the first moment of the distribution which implies that there is a second moment of the distribution and that of course is the variance of that random variable so the question then becomes how do we take the computations that we did in the previous learning module and add them to this probability weighted well we&#39;re going to do the exact same thing that we did before and we&#39;re going to couple it with the blue box so look down at the Orange Box we&#39;re going to take each individual outcome notice there&#39;s an X1 and an X2 and we&#39;re going to subtract the expected value of that outcome then we&#39;re going to square it and then notice we&#39;ll multiply it by the probability so this is a probability weighted average again now I always ask my students why do we Square the difference between those two now I probably said this back in the previous learning module but I think it Bears repeating here&#39;s the example that I give my students in my class now I teach in the finance trading lab and so we have I have the board behind me and my students are sitting in two rows on either side so I have an aisle so I walk up and down the aisle um and we have we have monitors on that side so I can walk up and down and I look this way and I look that way so I say to them I say hey look let&#39;s play a game let&#39;s play a game called count the number of steps that Jim takes if I stand in front of them and I go one two three four so I look at him and I say how many steps did Jim take and they of course look at me like I&#39;m some kind of a cartoon character and they go all right Jim we&#39;ll play along because we love finance and we love the statistical application uh that you can use in finance they say four so I go back to the front of the room and I say all right let&#39;s play the same game but what I want you to do now is I want you to close your eyes and close your ears and of course most of them don&#39;t play along like that so I say all right ready and I tiptoe I go one two and I turn around and I go one two then I turn around and face the class and I say how many steps did Jim take and then I say how many steps does it look like Jim took right and and I said for those of you who played along it looks like I didn&#39;t go anywhere you want to say zero but everybody knows that I took four steps I just put them in One Direction and I turned around and came right back from that same direction so the reason that we Square the difference between each possible outcome and its mean is to get rid of the negative but also to be able to accurately count the number of steps that Jim takes you see statistics is your friend if you understand this quantitative method stuff you will be a much better financial analyst and you&#39;ll have a higher probability of passing passing the exam all right so thank you for bearing with me there now my wife never asks me to go to the food store and get 12 square pounds of potatoes so the variance is in a squared term a squared unit so we need to take the square root of that just like we did in the previous learning module so that&#39;s the standard deviation look at the uh look at that embedded Diamond point there if the variance is zero there is no dispersion which means that there is no risk this is super super important now when the variance is positive which it will be for almost all financial securities out there with the exception of a treasury security of course then it signifies a dispersion of outcom so the higher the variance the higher the standard deviation that means the greater is the risk and that&#39;s super important as we move throughout uh the CFA program let&#39;s take a quick example here we&#39;ve got a distribution probability distribution of a company&#39;s sales so looking that far right column we don&#39;t know what those sales are going to be during this course of the next year they might be 50 they might be 40 they might be 30 so what we have to do is say to ourselves oh let&#39;s just hope for the best and hope that we sell 50 million in sales next year no of course you can&#39;t do that what you must do as a good financial analyst is assign probabilities to those potential sales so I always tell my students the following I say look rank these from high to low which we&#39;ve done in the far right column and just think of them as best case scenario 50 worst case scenario 30 and then anything in between in this case we only have one one middle sales but you could have 10 or 20 or 80 of them think about putting this together in an Excel spreadsheet so we assign those probabilities based on things like well what happens to the economy right if the economy expands then we&#39;ll probably be able to generate more sales if the economy contracts then we&#39;ll probably be able to generate fewer sales so probabilities can be based on things like GDP they can be based on things like our marketing campaign quality of our board of directors quality of our Salesforce quality of our supply chain quality qualities so you get the sense that those probabilities are tied to the potential outcomes so that far right column we have a distribution of sales when you combine the two columns we have a probability distribution of sales so all we&#39;re trying to do is find the midpoint can you guys see that if I point if I point at my screen I guess it looks like I&#39;m pointing at you but I don&#39;t I don&#39;t mean to point at you all we&#39;re going to do is take the average and what did we say in that previous uh slide it&#39;s a probability weighted so all we&#39;re going to do is we&#39;re going to say 20% of 50 30% of 40 50% of 30 and we&#39;re just going to sum though so our expected sales is 37 million think about what this means that&#39;s our average right that&#39;s our point of central tendency it&#39;s our expected value so we need to make decisions today based on the expected sales of 37 million now we know this is a distribution we know that when we predict when we estimate that 37 million we know we&#39;re going to do it with an error we know we&#39;re not going to be perfectly 100% accurate every single time so what do we do we compute the variance and the standard deviation this is a measure of the degree of error how wrong could we be I mean think about it if we expect sales of 37 million for this coming year and next year we have 1,000 million of sales we were super wrong I mean we&#39;re happy that we had generated all those sales but we were super wrong that&#39;s going to translate into a high standard deviation all right so how do we do this well the same way that we did it back in the previous learning module and the same way that we did it here on that previous slide we&#39;re just going to probability weight it all right so look down there what are we doing the probability that we get 50 times the difference between 50 and the expected sales so that expected sales figure in there that&#39;s the 37 from the previous slide and then we Square it so we do all the math and I encourage you to go ahead and pause the recording and do the math so that you can get that $61 million in variance and then uh hit your square root button and you get what&#39;s that about 8 million so what are we doing here we&#39;re saying something like let me go back here we&#39;re expecting 37 million but on average on average we&#39;re going to be wrong by about 8 million so think about how that contributes to our planning for the next year in terms of putting together not just an income statement but a balance sheet and a cash flow statement we expect 37 but on average we&#39;re going to be wrong by 8 million on average that doesn&#39;t mean we&#39;ll be wrong by 8 million during this coming year but if we do this a million times on average we&#39;ll be wrong by 8 million now compare that if for some reason we would have gotten a standard deviation of 2 million oh my gosh now we&#39;re now we&#39;re pretty accurate right a lower standard deviation means that we expect 37 and we&#39;re pretty confident that we&#39;re going to get around 37 but if our standard deviation is let&#39;s say 188 million boy then we expect 37 but we have no idea what&#39;s going to happen right so high standard deviations imply uh uh a super unknowledge of the future not quite quite sure if unknowledge is a word but you uh you get the point all right moving to a probability tree so look down on the left side of the Box notice we start at time period zero and then we can either move up or down and then after we&#39;ve moved up or down we can either move up or down again this is called uh a decision tree and it has tremendous applications especially for valuing options and valuing bonds that have embedded options so you are going to see this multiple times in level two and in level three I&#39;m guessing that the Institute is very likely to test you on this at those later levels under derivative Securities uh than it might in this particular example but what this does is this gives us an additional understanding of risk so here&#39;s the traditional example that textbooks and professors use all throughout the world when we&#39;re starting to explain a decision treade suppose we&#39;re C tossing a coin right you toss it it can come up heads it can come up tails and it has a 50% chance because there&#39;s only two possible chances right so notice at first toss we have a 50% chance of getting heads and a 50% chance of getting Tails then we have a second toss and that second toss we also have a 50% chance of getting heads and a 50% chance of getting Tails but here&#39;s what this visual representation of all future outcomes this tree diagram tells us that we can ask ourselves the question what&#39;s the probability of tossing two heads together well you just multiply 0.5 time .5 look at the middle column out there there&#39;s 25% so heads heads Heads Tails Tails heads Tails taals they each have a 25% probability of occurring so think of about what&#39;s happening here you have a first toss and then a second toss after that second toss you have a whole bunch of outcomes and you can assign a known probability of those outcomes now what does this mean for our previous examples you know what did I say earlier dividend yields or stock Returns what we can do is we can take a share of stock like Proctor and Gamble and we can say something like today the stock price is let&#39;s say it&#39;s 100 it could either go up or it could go down what&#39;s the probability of it going up well maybe it&#39;s 50% maybe it&#39;s some other percent we&#39;ll talk at length about that especially when we get into level two but then suppose that the stock price goes up during that first day what about the second day so what&#39;s the probability that Proctor and Gamble will rise to consecutive days well we use this decision tree to visualize those kinds of outcomes now each of the tree has a branch and so you can assign a probability to each of those branches and so we get into really really cool conversations about things that sound like this Proctor and Gamble let&#39;s suppose has a 50% chance of going up or down today so so at the end of today it goes up but then overnight Proctor and Gamble announces that it it has just introduced a new product line now those of you who have sons out there you will know that once they hit a certain age even after they take a shower they still come out smelling so Proctor and Gamble has developed a bar of soap that&#39;s going to uh defunkify teenage boys so that they don&#39;t smell so everyone says wait a minute I have sons who smell I&#39;m going to go buy this product line so so overnight well what happens well that probability is no longer it&#39;s no longer 50% because people are going to flock to Target and Walmart and all the retail stores and buy this product line so maybe it has a 90% chance so you see how cool this can get this super simple example where we just CAU toss a coin can lead into much more intricate much more sophisticated and much more meaningful as Financial analysts so that we can make better decisions for our client right we&#39;re never going to toss a coin when our client says here&#39;s a bunch of money go ahead toss the coin oh heads we invest over here Tales we invest over here that&#39;s not why we&#39;re in the CFA program all right so let&#39;s go ahead and work through an example what the learning module does for you is that it pro provides a series of super complex equations and uh I&#39;m going to work through an example and we&#39;ll show you just how how simple this is but let&#39;s go ahead and read through some of these things what does a conditional expectation mean it really just means something like the expected value of a certain kind of an investment like like Proctor and Gamble given a certain set of real world events you know a new product line with this soap so those expected values I was saying this earlier right actions of GDP actions of governments whether it&#39;s changes in money supply or fiscal policy changes in financial institutions go back to the beginning of 2023 we had the collapse of SV Bank what does that do well that puts all kind of other Regional banks on notice so if we go back here and we try to do this decision tree we&#39;re trying to predict whether Regional banks are going to increase or decrease in price during the course of a day or a week or a month well that that uh SV Bank debacle that happened on a certain day well that changes stuff right that changes these probabilities so look down at the bottom all we&#39;re going to do is we&#39;re going to say something like we&#39;re going to sum those probabilities right and what we&#39;re going to do is these probabilities now notice that there&#39;s a slash sign so the probability of X dividend yields stock returns whatever that is given some kind an event or a change in scenario that&#39;s what the notation for S is so the probability of X giv some scenario maybe that scenario is SV Bank failing maybe that scenario is Spike in inflation maybe maybe yeah so we can do all that stuff and we can incorporate oh so you should be thinking wait a minute Jim how do I process all of the macroeconomic variables out there into a simple decision tree and well that&#39;s a that&#39;s a challenge all right so let&#39;s take a look at a quick example here uh we want to predict recovering principle for a defaulted bond issue and that defaulted bond issue depends on two economic scenarios all right so notice that we have scenario one with a probability of 60% we put that in red scenario 2 with a probability of 40% so let&#39;s not worry too much about what scenario 1 and scenario 2 are but maybe they are things like changes in GDP maybe they are things like SV Bank default maybe they are what whatever we come up with notice that 60 and 40 they add to 100% right so scenario one and scenario to have a 100% chance of one or the other occurring now once we hit scenario one then then we have another branch on the decision tree we might get 80 cents per $1 and we might get. 70 cents per $1 so we assign probabilities 35% to the 80 and 65% to the 70 all right you see how this decision tree is forming scenario two well we might get 60% I&#39;m sorry we might get 60 cents or we might get 30 cents all right so scenario one is obviously better for us if we&#39;re uh one of these defaulted Bond holders scenario 2 is worse for us so we have a 70% chance of 60 and a 30% chance of the 30 cents right so what is the condition expected recovery so what we&#39;re trying to do here in this case here let me go back real quick what we&#39;re trying to do is come up with the column of HH HT th and TT all right so all we&#39;re going to do we&#39;re going to use that equation down to the bottom of the blue box are you ready for this so there&#39;s our decision tree all we&#39;re going to do is we&#39;re going to say we have a 35% chance of getting 80 we have a 65% chance of getting 70 cents so there under scenario one our recovery is 73.5 cents so look on the left side of that first uh equation the top of the gray box the expected recovery given scenario one is 73.5 cents the expected recovery given scenario 2 is 51 cents so all we&#39;re doing is Computing kind of like a future value it&#39;s not really a future value but it&#39;s kind of like a future value then what do we well then we go ahead and say let&#39;s keep coming backwards and we call these unconditional expected values there&#39;s the formula down there all we&#39;re going to do is here&#39;s the same thing that we have before the unconditional expected recovery is closest to well now we&#39;re just going to say we have a 60% chance of scenario one under which we&#39;ll recover 73 and a half cents we have a 40% chance under scenario 2 of recovering 51 cents so you just are you ready for this probability wait those future outcomes so our recovery is 645 cents that that is the very likely question on the exam not the compute the 73 and A2 and the 51 you&#39;ll have to compute the 73 and2 and the 51 in order to get to that 64 this is a great this is a great exam question what does this do this gives us an expected value right so we expect to recover 64.5 cents so we can make decisions today based on what we expect to happen in the future given our estimates of probabilities and our understanding of scenario one and scenario 2 let&#39;s go ahead and look at this last losos in which we ask the following question let&#39;s suppose that during the course of our decision tree that the information set changes think about how valuable that new information is so this is this idea of uh of Thomas Bay notice what we have there in that very first uh Point updated or posterior probability given a set of Prior probabilities it allows us to update our decisions so think about our decision Tree in that last example with Proctor and Gamble and then Proctor and Gamble decides to do something else and then the economy changes and then and then so we have all we have all of these changes and how can we explicitly explicitly measure those updated probabilities the example that I&#39;ve been giving to my class uh for the last oh what has it been now two years or so is the following you guys should know that I&#39;m a huge James Bond fan I saw my first movie my dad took me when I was 5 years old 1966 anyway when Daniel Craig announced that he was going to this last movie was going to be his his very final episode his very final role you know of course we&#39;re trying to figure out who is going to be the new James Bond well you can use this base formula to kind of figure this out to estimate the probability that an individ ual actor will get that role you know just just take a simple example Suppose there are five dudes out there who are in the running so you could just say something like well each has a 20% chance of getting the role of James Bond but but then over time what happened you hear all these rumors so maybe maybe one of these actors um Mary&#39;s a famous country music star and that changes his probability maybe the producers of James Bond maybe they want their James Bond actor to be married to country music star or maybe they don&#39;t so that might increase or decrease his probability uh heaven forbid remember what happened to Luke Skywalker in between the first and second movies he was in an accident he need to have some plastic surgery you know maybe one of these five dudes has an accident heaven forbid and you know he he can&#39;t uh perform all of the running and jumping and all the cool stuff that James Bond has to do because you know broken arm broken leg whatever whatever that is or maybe or maybe or maybe I mean you have all these probabilities and so as the James Bond decisionmaking process evolves over time you can update these probabilities now um I&#39;m always fascinated by the way Finance textbooks and finance people try to explain the statistics behind um all of these kinds of models and The Institute really is no exception and I&#39;m not being critical at all there&#39;s really no way around this excuse me what what you can do is you can take a look at look at the very bottom blue box here&#39;s our summary of this uh this base formula we&#39;re going to try to estimate the probability of an event given some information and we can adjust some things over on the far right hand side of the equal sign but notice we have probabilities in there in the numerator and the denominator what that means is that we need to go back to our quantitative methods beginning so look at these equations here I&#39;m not going to suggest that you memorize these things remember we&#39;ve got things of like Vin diagrams and unions and intersections and that&#39;s what all of this stuff is using conditional probabilities which we just you uh used it in a previous example the total probability rule which we did before and so uh here are the variables that are super important uh the probability of B subi these are known prior probabilities the event a is some event known to have occurred and then the probability of B given that event is known as the posterior probability so keep in mind that these are all good equations these are all good mathematical stuff for you to know here&#39;s the good orange and blue box for you to remember kind of General formulations but let&#39;s work through an example and you&#39;ll see how simple this is now here&#39;s my advice this example and the example in the body of this learning module both ask a similar question here notice at the bottom what is the probability that uh there&#39;s a negative return on the New York Stock Exchange this is a very similar question as in the body of learning this learning module so it&#39;s very likely that if the Institute asked you this question on Bay it&#39;s going to be in this context now I will also say to you that when you look at the problems at the end of this learning module there&#39;s only two one ask you for the mean one ask you for the standard deviation so there are no problems on any of this stuff that we&#39;ve done over the last 10 minutes or so I&#39;ll let you decide whether not you want to spend extra time on these things my advice is to spend the extra time because as I say regularly throughout all of my recordings is that every losos is a potential exam question but we can also infer from the problems at the end of each learning module what kind of importance which topics are important uh to The Institute all right so enough of my babbling there about what can show up and what can&#39;t show up on on the exam all right so we have uh an analyst investigating the performance of stocks on different exchanges all right so 50% come from the New York Stock Exchange 30% from the London Stock Exchange 20% from the Tokyo Stock Exchange now when you get to a problem like this I think it&#39;s easy and it might be just the way I think but I will visualize that in this number there are 100 so 50 from the New York Stock Exchange 30 from London 20 from the Tokyo all right so what are we doing here we&#39;ve got 100 stocks think about think about them as a 100 little pingpong balls in in a swimming pool so we&#39;re going to reach in and we&#39;re going to grab one of them out all right now here&#39;s this updated information the probability of a stock posting a negative return is 40 35 and 25% on each of those exchanges so what we want to do is we want the probability that when we pick out a pingpong ball from the swimming pool that it&#39;s going to have a negative return ah so wait a minute if we were to say something like look look if we knew that the pingpong ball was New York Stock Exchange and we pick that out well then we would know that that probability is 40% similarly London 35% Tokyo 25% but here&#39;s the deal ready look at the question picks a stock at random so the pingpong balls in the pool are not colored by the type of exchange they&#39;re all white or yellow or pink or blue right we don&#39;t know which ones are New York which ones are London and which ones are Tokyo so how do we figure out that probability that that randomly selected pingpong ball is going to have a negative return so let me just go back here quickly all right so look down at well let&#39;s go back to the uh The Orange Box so probability of the new information given the event that&#39;s going to be in our numerator so there&#39;s probability of the information slash given that event all right so here we look at the bottom here probability of A/B subi all right you ready for this so this is super simple so what do we know we know we have a let&#39;s go back here we know we have a 50% chance of picking a New York Stock Exchange pingpong ball which tells us that then we&#39;re going to have a 40% chance of posting a negative return well there&#39;s our numerator right 50% times that 40% % chance right that&#39;s the numerator then all we&#39;re going to do is divide by all of the other possibilities right we&#39;re trying to do a probability so in the denominator we take the 50% times the 40 That&#39;s New York Stock Exchange 30% times the 35 uh percent that&#39;s the London Stock Exchange and then 20% times the 25% that&#39;s the uh Tokyo Stock Exchange so do you see how this Baye formula it kind of takes away from what we&#39;ve done during this entire slide deck but it puts it in a numerator and a denominator in a form so that we have this updated information so there you go 56.3% and then of course my advice for you is to go back and change the question say probability that it has a negative return on London do that one probability of negative negative return on Tokyo do that one so you get the sense of what&#39;s going on here then what I want you to do is I want you to go back to the reading and work through the example inside the middle of that learning module so I believe that takes us to the end of these three learning outcome statements this is super fun what did I say somewhere in there statistics and quantitative methods is your friend so get your arms out wrap your arms around this material embrace it and love it and and hug it and do all those really really good things because look I can warn you you now but you&#39;ll see as we move throughout the program you&#39;re going to use this material and it&#39;s going to be applied to lots and lots of other different kinds of topics so what do I want you to do now I want you to go to the middle of this learning module and work through that base formula then I want you to go and compute the expected value and standard deviation and the problems at the end of this learning module so hey that was fun for me I hope it was fun for you thanks for watching and good luck studying

Transcript for:Lecture on Probability Trees and Conditional Expectations

Transcript for:
Lecture on Probability Trees and Conditional Expectations