 Although the duration is a useful and effective tool in measuring the interest rate changes. However, it is just an approximation to check the impact of interest rate changes on the bond prices. What we need here is to determine convexity of the bond in order to further determine the changes in the bond's yields. We know that the direct proportional relationship between percentage changes in prices and the change in yield may give a straight line graph to reflect this relationship. But in real, the relationship is not the straight line because it is in fact non-linear. This means that the line is not straight but there is some curve in the line. Now the degree of this curve in relationship between bond prices and the bond yield is termed as the convexity. But duration rule is, in fact basically it is a good approximation for only small changes in the bond's yield but it does not work for the larger changes. The bonds with greater convexity means that more curvature in the price yield relationship. On the screen we see in the right panel a graphic relationship between percentage change in bond prices and change in the bond's YTM. We see that there is a curved line colored in blue and that curved line shows the percentage price change for a 30 year 8% coupon bond selling at initial YTM of 8%. Then there is a straight line and that straight line is showing the percentage price changes predicted by the duration rule. Now the slope of this straight line and that is this straight line, the slope of this straight line is the modified duration of this bond at its initial YTM and that modified duration is 11.26 years. Now we see that for small changes in the bond's YTM, the duration rule is quite accurate as we have earlier seen. But for larger changes there is progressively more daylight between the two plots that we can see a difference between these two lines here. These demonstrate that duration rule becomes progressively less accurate. We also see that these two plots that is the blue and black dotted line, these two plots are tangent at the initial yield which is 0 here in the graph. Duration approximation always understates the bond's value because this understate estimate increase in the bond price when yield falls and it overestimates the decline in the prices when the yield rises. This is due to the curvature of the true price yield relationship and these curves with shapes like that of the price yield relationship are said to be the more convex. The curvature of the price yield curve is called the convexity of the bond as we have earlier seen that bond is a curvy and the degree of this curve is termed as the convexity. Now why investors do like convexity for their bonds? Because the bonds with greater curvature mean there is gain more price when the yield falls then they lose when yield riles. So the degree of gain is much higher than the degree of loss. The second reason is that this convexity tend to have higher prices end or lower yields all else equal but the desire for having convexity is not free because investors will have to pay higher prices and they expect lower white Tm on the bonds with more volatility. So how to determine the duration and convexity on callable bonds? You see on the screens bottom panel at the right side in the graph that at higher rate of 10% the curve is convex and this price yield curve lies above its tangency line whereas at the rate at 5% as it falls from 10% to 5% there is a ceiling on the bonds market price which cannot rise above the call price. This means that the price yield curve lies below this tangency line and the curve is not set to have a negative convexity. This price yield curve shows unattractive asymmetry. This means that the issuer has retained an option to call back the bond and if rates rise the bond holder loses and if rates fall the rather than reaping a larger capital gain the bond holder may have the bond called back from her by the bond issuer. Now to determine the duration on such bonds we need to compute effective duration of the bonds with embedded options. By effective duration we mean the proportional change in bonds price per unit change in the market rate of interest. This means that effective duration is the relationship between changes in price and the changes in interest rate. This particular equation is different from the modified duration model in the sense that it does not compute effective duration relative to change in the bonds on YTM or yield to maturity. Second, the effective duration formula relies on the pricing model that accounts for embedded option and that pricing we see in the equation in terms of changes in price that is delta P over P. So how to determine now the convexity of mortgage-backed securities or MBS? How these mortgage-backed securities work let's talk on first. In this case the lenders originating mortgage loans commonly sell these bonds, sell these securities to the other financial agencies. The original borrowers continue to make repayments to the lenders. The lenders pass these payments along the loans to the buyer agencies. Then the agencies combine many mortgages into pool called a mortgage-backed security and that mortgage-backed security is then sold in the fixed income market by the buyer. The borrower has the right to prepay the loan at any time. These MBS or the mortgage-backed securities are subject to same negative convexity as the other callable bonds. In fact, when rates fall and the borrowers prepay their mortgages, the repayment of the principal is passed through to the investors. Now rather than enjoying the capital gain on their investment, these investors simply receive the outstanding principal balance on the loan. The value of the MBS as a function of interest rate is somewhat different because this value of MBS as a function of interest rate is much like the plot for our callable bonds that we have just seen in the graph of the callable bonds earlier. The implicit call price or the principal balance on the loan is not a firm ceiling on its value and that we have seen earlier in the case of callable bonds. Here, a term is used as trenches. What these trenches are? These are underlying mortgage pools divided into a set of derivative securities. These trenches are made to allocate interest rate risk to the investors who most willing to bear that particular risk.