so now with this words I'll start off that what we mean by a a demand and supply and what are the relevance of see for example demand Supply and what we mean by the concept of a price of of a particular product or a service so I'm using a very general word of a product or a service so generally product can be any any any thing which is being bought and sold in the market and and and it can be a stock it can be an option and services can be different type of financial services which you are giving but we will only restrict our discussions later on with only related to financial one but generally the first initial lectures would be very basic microeconomics which will give you that how demand and Supply can be considered in order to find out the optimum price and Optimum quantity based on which a product and service can be sold and bought in the market so if we if you see see the this this curve here you basically have the price along the y axis and you have basically quantity along the x-axis so when and and and if you go to the market buy anything you whether you want to buy some Goods whether it can be any vegetable it can be any product and it can be Financial instrument whatever is being bought and sold there would be a particular set of people who are interested to buy and particular set of people who are interested to sell now another reason which we which I think is very important to understand the the concept of demand and Supply is that because say for example if if if we are going to buy any product we are buying to buy basically going to the the market basically gives us a very good feel or regulates the price and the demand based on which that product and the service can be sold say for example I go to the bank and want to basically take a loan on the other hand they can be people who want to go to the bank and basically deposit some money so obviously the bank acts at an intermediate basically a space or a meeting place where different people with different demands and supplies for different products can go into the market and basically exchange such that apart from from perfect information which may not be true in the actual sense in the actual practical perfect information being available to all the buyers and suppliers the overall place where this this supply and demand takes place helps to regulate the price and demand at which value the overall product can be bought and sold so we'll we'll come to those those Concepts later on also so now if you see the the the coordinate where at where the this this price so at the price price and the quantity which is sold so basically what we know that is there's a quantity based on which that particular product is being bought and sold and there is a price so now consider that if you have baj share or say for example you have tataa shares or see for example you have some options so when you're are trying to find out your investment returns main concern comes is that what is the price based on which you can optimize your overall returns what is the price at you you can basically go for a derivative what is the price at which you can basically formulate your portfolio Point number one and point number two would be that what is the total Quantum of such Goods which you can buy and sell in order to basically optimize your portfolio so I'm using the word optimize optimize in a very general sense say for example if I a human being my main concern is only profit so I'll basically try to initialize My overall portfolio where I want to basically maximize or have the positive returns to the maximum sense another person can be where he or she wants to basically minimize the overall loss and for him or her profit is not important so for that person it will be more of a concern in order to basically try to see that the risk can be minimized to the maximum possible extent again I'm using the word risk in a very general sense there are different type of of quantification of risk and what are the quantification all of us B generally know in a very general sense it can be standard Devion it can be beta what is BET beta in the market sense we all know but there are different techniques which are come up where you use different terms in the concept of risk and try to analyze any different problems in order to mitigate the overall risk for persons who are more concerned about the losses so if you if you see the overall of the risk return profile as I were discussing so obviously the supplies can be of of different values and the demands can be of different values so for different type of of supplies and demands you will basically have different type of prices so one one will be this the second one will be this and the third one will be this so when you are trying to basically analyze any any any stock I'm using the stock in a very general sense it can be any Financial instrument what is more concerned is the price and what is the quantity which is being bought and sold this is basically small q now this price basically gives you the concept of returns so returns I'll use the word as a letter of small R so there are different type of books where they use the use of capital r also but those can be interchange depending on how you try to analyze the problem now our main concern is to find out the risk and return based on a particular set of returns which is small R and as we formulate the portfolio how different types of stocks can be combined giving their different type of values of R like for example if there are 10 type of 10 different stocks you can would basically have R1 till r10 which will give you the corresponding returns of that of these 10 number of stocks and obviously will have the corresponding risk so general terms let me denote the risk as Sigma 1 to Sigma 10 and all of you know basically Sigma means the standard deviation and obviously there are other different techniques or or or symbols or methodologist to trying to quantify the risk so our main concern is to see that given a products price and quantity I'm using the product as a very general term it's a financial product that how can you find out say for example the price of one particular stop is decreasing and you want to basically sell it so you want to find out that what is the other Financial product which can be substitute in your portfolio such that your overall risk mitigation is met at the maximum possible extent so how this this this relationship between risk and return can be done in the concept of financial instruments I'll come to that later on also so generally we know that for any investment or or or any concept of demand and Supply there is the concept of marginal revenues so marginal revenues in a very simple sense means the per unit rate of change of that particular Revenue now you may be thinking that why we are going to consider marginal revenues so later on we we we'll see that when you solve the concept of optimization problem they would be given a primal problem I'm using this word Primal for the first time but we'll see that later on that what that actual problem means if you have a primal problem or consider that in a very general sense as the primary problem there would be a counterpart of that which is known as the Dual problem so dual problem is basically consider that if you're standing in front of a mirror so if you are the Primal Primal problem or so called primary objective what what you want to solve and the mirror image which you see is basically the dwell so there are techniques depending on what are the conditions based on which the Primal problem have have been formulated you can convert that primary problem into a dual one and you'll see that the properties or the or the intrinsic properties of the Primal and the Dual problem as such there's a onetoone relationship between them and later on also we'll see that if you are able to solve the Primal problem with some techniques they would def to be methods that how you can solve the Dual problem also and there you will see the concept of marginal rates of substitution of products happening so how you can basically replace one product or One Financial instrument with the other can happen then we'll also see that what is the concept of Shadow prices so and what are the concept of Shadow prices which are used in the optimization sense can have very specific meaning in the area of of finance and optimal portfolios we will also see that later then we'll find out that as the prices or the quantities of the products change products means again Financial products how does sensitive in analysis help you to basically balance your portfolio balance your risk balance your returns we will also consider that later on now marginal revenues or return for any decision on financial instruments is basically the change in the revenue of the return associated with the unit change in the output so say for example your total output depending on the price change in the overall Market Falls by 100 rupees or Falls by 200 rupees or increases by 200 rupees So based on that you want to find out that what are the Readjustment you want to do in your portfolio such that you are able to mitigate the overall risk or you can basically try to maximize your return or whatever your concern is depending on how you have formulated the problem we'll see that the marginal revenues is zero add the quantity that generates the maximum total revenue when we consider the concept of maximization or minimization we will discuss now the concept of utility Theory and why utility theory is generally utilized in finance and how it can be utilized in the area of decision making and how quantitative techniques as a such is basically utilizing util Theory so what do we mean by util Theory so different investors have different type of Demands somebody would like to basically increase his or her returns somebody is interested to basically try to minimize the risk and others may be interested in basically trying to make a compromise between risk and return so risk and return go hand in hand but in opposite directions we'll see that later on also so each investor has in front of him or her a choice set Choice means a set of of decisions based on which he or she can take that that set of decisions so it is not single there are many different type of choices and and an investor or a person who's working in the area of Finance would like to basically take up the decision in such a way that he or she choose the optimum one and what is the optimum one we'll we'll discuss that later on also so so consider in a very simple sense there are outcomes and outcomes are with one3 probability so there are three and the outcome values are given by in the First Column if you see is 15 10 15 and in the other column it's 20 12 and 8 now consider the values of 20 12 8 which is in the second one and the first one which we discuss is 15 10 15 as given so we'll consider why those values are now if I ask you a question that which one decision would you take so obviously all of you would basically see that on an average what is the value of the outcome based on the probability of one/ thir for each an outcome which is there so if you basically analyze the two two different situations which is are A and B your obvious answer would be that if the both the values are coming out to the same value which is the expected value which is 13.33 obviously it means that you are indifferent or you are basically not willing to take a decision whether A and B so but there would be some some different type of of examples which can be formulated or they can be different type of people who are there in the market who would be interested to take either the a decision or the B decision so how that is done I'm going to come to that later now consider the situation is changed now you have the probabilities are half 1/4 and 1/4 so obviously the sum obviously always remains same for the second and the fourth column but now if you see the total expected value of the outcome for a is now 13.75 and for b b it is Still Remains 13.33 so if again I ask you the question now people who were now indifferent between A and B depending on the so-called expected value would like rather like to take the decision which is a because it is giving a higher value of 13.75 now consider that there are two teams and you want to B basically you want to rank them it's like there are two portfolios you want to rank them so if you see that the wins which you have in the First Column for both the cases A and B which is the first block and the second block you'll find out that the overall points based on which the outcome would be one or loss for the first case the points of outcome are if you win you get two points if you basically draw you get one point and for a loss it is zero but for the other one it is 510 so again if you are trying to compare so-called portfolio A and B which in our case is basically trying to rank team a a X and Y for a game you will find that the system of ranking changes so my question would be and obviously your question would immediately come out is that which means the ranking changes as the outcome changes so outcome doesn't mean only the probability it means the outcome in value sense now in case one if you consider team a has 100 Team B has 95 which means a is high ranked so what in in this case is the points basis like is a round robin League or whatever it is a knockout one you have the points based on which you take a decision which team is higher which team is lower who is first who is second so on and so forth in case two which you saw with the different points it will be team a has to 20 Team B has 230 so obviously it means B is much better than a so a was ranked higher which means that even when the situations are same if you give different points for the outcomes it means that the ranking immediately changes which means that if you change your your actual criteria based on which you're going to take a decision you will have different results for different type of investment purposes or financial decisions now in general utility is given by expected value in very simple terms so when you have expected value you have we'll see later two things to consider one is the random variables and what are the values which are known as the realized values and the second one is what is the probability distribution so probability distribution means the relative frequency which we all know in the general statistic sense now if a random value is a variable is there the probability distribution would basically be Ain to the how frequently a certain outcome is happening so does any type of probability distribution hold for the finance we'll consider that later on yes they do now here you have two things one is the expected value of the utility is considered multiplying two terms one is UW which is a utility function what is utility function we'll see that later and one the second term which is basically the ratio is the relative frequency is giving you the probability so our main task would be to find out what is the utility and also to find out the distribution of the utility function based on which you are going to take a decision of the ranking remember in general the utility values cannot be negative so obviously if I'm taking a decision if it is negative value obviously I would never take that decision but later on we'll see that in general economic sense utility values even if they are negative some decisions are taken but in the finance concept we'll consider such decisions are not taken and to make our life simple we'll basically have the rule that if there's some negative returns we would not be considering them but later on we'll consider also the cases if there are different type of negative returns how to basically choose the one which you gives you the the least of the overall wor scenario so we'll rank them accordingly so ranking would would happen based on the value of the utility even if they are negative now the question is that if the probability of the outcome changes as you saw in the game on the match between A and B there was a change in the ranking now the question comes that what if the utility function also changes that U which you have change is considering if it changes also obviously there's a change in the ranking so consider the first one which is the highlighted one which is UW is given by linear function and in the second case you have basically U which is given a quadratic function so we are first encountering the concept of quadratic utility function we will be considering later on in depth so if you have the out comes of as given in the First Column so these are just theoretical values hypothetical values and if you see the the W1 which is the utility function based on W1 and you have W2 which is the utility function based on utility function two the corresponding utility values are given in the third column and the second last column and the corresponding probabilities are next to that so if I find on the expected value of the utility using utility one you will have a value of 3.82 five and for the second one you'll have basically have a value of 12.69 so obviously it mean that your your ranking of of both the expected value would basically change depending on the utility you are using now consider we have two different utility functions and again the decisions are is like a no no go or no go that means if a decision is taken obviously you get some utility if decision is not taken obviously you will consider your utility value is net worth is zero so if you consider that for utility function U W1 the fin value comes out for the case of UA given the utility function one is 1.69 and for the case of U given the utility function is 1 it's two so obviously it means you will choose B because it is a value of two which is greater than 1.69 now what happens let us consider the utility function is now again if you see the value is basically a nonlinear one which is not a linear one then again if you see and and if again considering the same outcomes if you find out the expected value for both of them the value comes out to be 1.69 so now your indecisive between ranking of both the utilities utilities means the situations which you have in front of you consider a very simple case a venture capital is considering two possibilities the first alternative is buying government treasury bills so for the first time you are trying to encounter government treasury bills and they are the cases where the actual risk for the government Tre bills are zero in theoretical sense practical sense obviously we'll see that the values of a risk free interest rate which is a characteristics of the government bills or or t bills which we say are do change and we'll basically see what are the values of the T bills in the Indian market and you can find out information of them in the in the RBI side which is the Reserve Bank site and there is another another uh outcome which has three different nodes nodes means outcomes of particular sec cases one with the value of of 10 lakhs another is 5 lakhs and the third one is 1 lakh and the corresponding utility function which you're seeing is given by u w which is W to the^ half which is basically a square root of that now if you find out the value of the utility based on the expected one then the first alternative would have basically have a a net value of 776 while the second one which has three outcomes would basically have a value of 60 9 you can find it out very easily doing the simple calculations like in the first case if it is 6 lakhs you'll find out the square root of that and the second case if you basically have these three values and the corresponding expected returns uh value probabilities are given what you will do is that you will multiply the corresponding utility which is UW that means square root of this multiplied by half square root of this multiplied by4 square root of this multiped by point4 and you can find out the value of 68 so obviously your first government bond which one is the first decision gives you higher higher expected value you will take that now would the above problem give you the same result if you basically add any constant term answer is yes without going to the the the actual conceptual framework we'll consider there would be some decisions where if a constant is added how the value doesn't change now consider a very s very simple case a theoretical one but for the first time we'll try to basically bring a different type of utility which is basically based on the property of log and why log is coming for the first time we'll consider that very slowly in within another two or 3 minutes the First Column are the days second column are the prices third column are the log of the prices and any of the number outcomes and the corresponding probabilities are given so if you find out the expected value expected value comes out to be 6.91 if you use a different function which is given by P to the^ 1/4 then the value comes out to be 33. 63 now the general properties of utility which would be important for us later on are only two important characteristics one is that more I give to a person more he wants which is known as basically the property of nons sociation so the more wealth a person has more he or she wants in order to basically increase the overall expected value of the outcome that's Point number one and if you see the last line of this slide is means the rate of change of the util function is has to be neg positive which means the marginal rate is always positive so that has some significance later on next property we'll consider is that any human being can be classified under three categories that means he or she is a risk aversion person that means he or she is not interested in take a risk the thir second characteristics sub characteristic is basically risk neutral that means given two situations where one you have the determinist outcome and another has a prob outcome the person will always try to analyze the expected value of both the cases and take a decision such that he or she basically indifferent between them and the third case is basically a person who seeks risk that means in a way that he or she is taking the risk in such a way that he or she thinks that in the long run the outcome would be positive in a sense that this that would basically mitigate the risk which is being taken by that person and we'll consider these properties when when somebody takes a decision in the area of Finance so now Bas we will consider a very simple case which is known as a lottery so consider on the right hand side you have the cholle coin where if we toss it both sides are heads and your actual probabilities is one and the outcomes are also given in one in other case you invest one and your probabilities are simple coin you get a probability of of half outcome is two probability of half your outcome is zero so if you analyze both of them the expected value in both the cases are one and one now if I ask you a question without going to deta details that which one would you choose would you choose the right one where it is a deterministic case or would you choose the left one which is the probabilistic case so we'll see that they would be in general sense three categories of person person one who takes the problemistic case person two who is indifferent between them and person three who Tes basically takes the deterministic case so if I change the probabilities if I change the outcomes you will find out a human being can be categorized in any one of the category depending on what the outcome is so I may a maybe a person so if the value which is given as two and zero and one if those values changes in quantums so I may be tempted to change my decision in order to basically take such a case that means I'm basically being benefited by the overall decision