 Maturity of a bond may be considered as nominal time period as stated by the issuer of the bond. But with reference to the bond holder, this definition may not be as much meaningful. In fact, duration is the effective time period covered by the promised cash flow of a bond. With reference to the changing interest rate environment, what duration is? It measures the average maturity of the bond's promised cash flows. It is a guide to sensitivity of a bond to interest rate changes. It is the weighted average of time on the bond's cash flows and these weights are the proportional of the present value of the individual payments of the bond as a whole which is termed as the bond's fair market price. Duration is equal to the maturity for zero coupon bonds because there is only one cash flow occurred at the end of the bonds. When zero coupon bonds duration is shorter than the maturity of all the bonds. The weights that is WT associated with the cash flow made at time t. In fact, this is the present value of all the cash flows divided by the bond's fair market price. Now to determine the weight of the individual cash flow, we need to divide the present value of the individual cash flow or the coupon value over the whole price of the bond. And that whole price is basically the fair market price of the bond. In this equation that we are seeing on the screen, why is the bonds YTM then the numerator is basically the present value of the cash flows accounted at the time t. The denominator is the value of all the bonds payments and that is the fair market price of the bond. All cash flows weights are equal to one or the sum of the weights is equal to one as sum of the cash flows discounted at YTM equals to the bonds market price. Now using all these values that we have earlier discussed, we get Macaulay's duration formula and their duration formula states that the cash flows at t times are multiplied with the individual weight of the individual cash flows. And their overall sum is termed as the bond's duration. Now duration leads to a key relationship between the change in the yield on the bond and the change in its price. So we can say that the change in price or the change in yield is basically the function of D and the change in yield. So price change is the function of D steric and the change in yield. And this D steric is termed as the modified or the Macaulay's duration. To understand this duration, we have an example here on the screen, we see that the duration of 8% coupon and 0 coupon bond each with two years to maturity. So we have two bonds computations here. The first half of the computation is about the 8% coupon bond and the second half is about the 0 coupon bond. We have two years semi-annual compounding. So we have four periods in all for the both bonds. We have the time until the last payment. It starts from six year, six a half year. Then it is one year goes to one and a half year and ends on two year. We have cash flows and of $40 for each half year period. And in the last period, we have two cash flows, the principal price, face value of the bond and the half year coupon payment. Then we are discounting these individual cash flows at YTM, which is 5% at the half year rate. Then we have the individual rates to determine this rate, we need to divide the individual cash flows present value over the sum of the present values. Then we have the sum of these rates and when we multiply the individual rate with the individual time of the periodic payment, we determine the overall duration of the individual bonds. For bond A, we have duration of 1.8852 years and for bond B, we have the duration of exactly the two, which is its nominal duration. So we see that the duration of the zero coupon bond is exactly equal to its time to maturity and the duration of two years bond is a shorter duration, which is 1.8852 years. So what is the importance of duration? Duration is a key concept in fixed income portfolio management because of three things. Number one, it is simply a summary static of the average effective maturity of the portfolio. Then it is an essential tool used in immunizing the portfolio from interest rate risk and third, it is a measure of interest rate sensitivity of a portfolio. Apart from these, the duration quantifies the relationship between bond prices, sensitivity to the interest rate changes and maturity. We can also see that the bond with equal duration will have equal interest rate sensitivity. For this, we have two bonds, bond A and bond B, both have maturity of two years. Bond A has a coupon interest rate of 8%, whereas the bond B is the zero coupon bond. The current price of bond A is 964.540 whereas the current price of bond B is 831.970. Both bonds have the YTM of 10% and we have already seen that the duration of these two bonds is equal, which is 1.8852 years. Now if we treat one period as a half year, then each bond's duration is 3.7704 SEMI annual periods and to determine the modified duration with 5% or the SEMI annual rate, we need to divide this nominal duration over 1.05 and both the bonds have SEMI annual duration of 3.591 period. Now suppose that SEMI annual interest rate increases by 1%, then what will be the effect on the bond prices? Certainly there would be the fall in the price. Then putting these value into this McCullis modified duration formula, we get our negative 0.03591% as a decline in the bond prices. If we want to confirm this decline, we have certain values like we have initial selling price that we have earlier seen, we have new selling price for bond A which is 964.1942 for bond B, we have new selling price of 831.6717. If we see the changes in the prices, we see that both the bonds have same decline in their prices and that decline we have just seen which is negative 0.0359%. Now what determine duration? We have certain rules here that determine the duration. These rules are in line with the diagram that we can see on the upper right column of the screen. The first rule is that the duration on a 0 copen bond equals its time to maturity. Holding maturity constant duration is higher when the copen rate is lower and that we can see for the bond having 15% copen rate and the bond having 3% copen rate. Then holding copen rate constant duration generally increases with time to maturity that we can see in the case of bond with 6% yield and the bond with 15% yield and copen rate of 3%. We also see that holding others factor constant duration is higher or longer when YTM is lower and that we can see in our diagram where the YTM is 6% and the duration is much higher over the other two bonds. Then we see that the minute or time just after a copen is paid duration jumps as the cash flows disappear and that we see in the diagram as well. Next we see that duration always increases with maturity for bonds selling at power or at a premium to power and final rule is that the duration of a level perpetuity is 1 plus Y divided by Y.