 After going through the arbitrage pricing theory in its theoretical aspects, now let's see what how does it work with the help of a numerical example. This is the example one. We have two stocks X and Y and we have two factors model with the risk factor definition and sensitivities. The first is the changes in the inflation rate that is a factor one. The risk premium related to this factor is 2% for every 1% change in the rate. So 2% is the risk premium of this factor one of inflation. The second factor is the real GNP which says here that the average risk premium related to this factor is 3% for every 1% change in the rate. So the risk premium for the factor two, which is GNP is 3%, rate of return on zero systematic risk asset is 4%. So finally we have two factors are here, lambda naught is equal to 4%, lambda one or the factor one has risk premium of 2% and lambda two or the factor two has risk premium of 3%. So far as the stocks sensitivities to the common risk factors are concerned, beta x1 says that asset X responses to the changes in inflation factor is 0.5 for beta two, asset X response to change in GDP factor is 1.5. This one shows that any one unit change in beta two, the value of GDP factor will be changed by 1.5 times for beta y1, the asset y response to change in inflation factor is 2. For beta y2, asset y response to changes in GDP factor is 1.75. Now using this earlier data, we can determine the expected return for both of the assets X and Y. We know that we have an overall expected return equation, which says that expected return on asset I is equal to lambda naught plus lambda one into beta one plus lambda two into beta two. While putting the values into these symbols, we can see that the expected return on asset X is equal to 9.5% and the expected return on asset Y is equal to 13.25%. Now see what is the relationship between expected return and K risk factors as the APT's equation suggests a relationship that is in fact analogous to the SML associated with KPM. But instead of line connecting risk and expected return, the APT implies a security market plan with K plus dimensions, in which K risk factors and one additional dimension for the securities expected return on the right panel of the screen, we can see that we have multiple dimensions. This is the dimension of factor 2 beta. This is the dimension of factor 1 beta and here we have the expected return. So this is a dimensional diagram where we can see the values of our lambdas that is the values of risk premium values of betas for each of the stock beta 2 for stock Y beta 1 for stock X and beta 1 for stock Y here beta 2 for stock Y. So in this diagram we have for every stock we have the corresponding values of the betas for every stock we have corresponding values of the risk premium of the risk factors and the third value we have the expected returns on each of the two stocks. Now we have another example where we can see the security valuation for the riskless arbitrage strategy. It assumed that we have three stocks and two common systematic risk factors. Now stocks sensitivity to the common risk factor is for beta 1, A has value of 0.80 for beta 1, B has value of minus 2 0 and C has value of 1.8. So we have beta value for all of the three stocks. Similarly we have beta 2 value for the of the three stocks. We have two common risk factors that is lambda 1 which has a risk premium of 4 percent and lambda 2 which is having a risk premium of 5 percent. For 0 beta return we assume that it is 0. Now the first step is to determine the expected returns for each of the stocks using the equation of lambda 0 plus lambda 1 into beta 1 plus lambda 2 into beta 2. When we put the values in this equation we have the expected return for each of the stock like for stock A the expected return is 7.7 percent for stock B the expected return is 5.7 percent and for C the expected return has a value of 9.7 percent. So in the first step we have determined the stock's expected return. Now assume the current price for each stock is $35 with the expectation of no dividend in future. So what would be the next year's expected price? To determine this next year's expected price we simply multiply the current price of each individual stock with the corresponding expected return. So here we have the expected price at the end of year 1 which is 37.70 for stock A 37 for stock B and 38.40 for stock C. Now assume that after one year the actual prices of the stock are in another order like stock A has 37.20, stock B has 37.80 and stock C has 38.50. Now these are the prices we are assuming as being the actual price after one year. Next step we forecast that the stock A will not reach the price level after one year which is consistent with the investor's return expectation. This means that at the current share price of $35 each the stock A is overvalued, B is undervalued and stock C is slightly undervalued. This means that as a consequent investment strategy we can take the advantage of these price discrepancies. But how we can take this advantage? We need to buy stock B and C and we need to short sell for stock A. The question arises that how can one take advantage of market mispricing? The solution of this strategy is to go for riskless arbitrage which favors assembling a portfolio in an order that requires no net wealth invested initially that will bear no systematic or unsystematic risk but that portfolio will still earn a certain profit. Now let WI represent the proportional investment in security I. The condition that must be satisfied can be written formally as so we have now three equations in correspondence with our three conditions that we have earlier seen where in first equation this mission of WI is equal to zero which is no net wealth invested as a next in B part we have this mission of WI into beta I which is equal to zero that is for all of the K factors we have no systematic risk and WI is small for all assets that is the unsystematic risk is fully diversified. And for the third condition the mission of WI into RI is greater than zero this means that the actual portfolio return is positive meaning thereby we are earning a certain profit now sell we need to sell short the stock A which is overvalued and we need to buy stock B and C both are undervalued with the proceeds from the short selling of stock A in the lower half of the screen we can see that we have a proportion of negative one for A and point plus 0.5 for B and plus 0.5 for C so our total proportion is equal to zero if we sell the stock A we have a cash inflow of positive 70 with this positive 70 cash inflow we need to spend 35 each on stock B and stock C ultimately we have initial net investment equal to zero and that satisfies our first condition now let work for the second condition which is to estimate net exposure to the risk factors with the corresponding weights exposure from the stock we have factor one and factor two here are the weights and here are the beta values for the individual stocks the net risk exposure for each of the factor is equal to zero and that also satisfies our second condition now come to the third condition we assume that prices in one year actually rise to the levels as we have expected earlier then the net profit from covering our short position and liquidating the two long holdings would be equal to dollar 1.90 so this is the profit equation that we are earning loss at short sell of stock A but we are earning profit at acquiring the two stocks stock B and C the ultimate benefit we are gaining is net 1.90 so we are also satisfying the third condition the conclusion of this exercise is that the essence of arbitrage investing is that a portfolio with no net wealth invested and assuming no net risk has realized a positive profit and example of long short trading strategies which are generally used by hedge funds