Finance Numericals

Immunization of bond: It is a strategy used by the bond portfolio manager to reduce the interest risk which affects the bond price through ups and down in interest rate. There is an inverse relationship between interest rate and bond price. It means when interest rate fall then there is a rise in the price of a bond and vice versa.  So, to minimize the changes in interest rate during an investment period immunization is used. The changes in interest rate will affect the investors’ future obligations. The process of immunization is done by reinvesting the coupon amount in new bond which offers high interest rate due to which the price of a old bond price is decreases and reinvesting the coupon amount in a bond will offset the losses in bond price due to rise in interest rate. Immunization helps to ascertain the investment quantity in a security will provide the definite cash flows to meet the future obligations in time. Formula: Immunization = X 1 *d 1 + X 2 *d 2 = X 1 , X 2

Example: Find out the convexity of a 6 year bond whose face value is Rs. 10, 000. It bears 6% coupon rate p.a. The current market price of a bond is Rs. 9, 880. If the yield rate increases from 6% to 8% then the price will be Rs. 9, 860. And if the yield rate decreases from 6% to 4% then the price will increases up to Rs. 10, 100. Solution: P (interest decreases) + P (interest increases) – 2P 0 / 2* P 0 * ∆ y 2 = 10,100 + 9, 860 – 2*9, 880 / 2*9, 880*(0.02) 2 = 19, 960 – 19, 760 / 19, 760* 0.0004 = 200/ 7.904 = 25.30 Example: Find out which of these bonds are less risky with the help of convexity of bond.   Both bond A and B has 3 years of maturity and bears 7% coupon rate compounded annually. The face value of both the bond is Rs. 10, 000. The yield rate of Bond A is 8.5% and bond B is 10% compounded annually. Solution: Bond A: Period Cash flow (Coupon rate 7%) Discounted factor @8.2% Present value Pv*t Convexity (Pv*t(t+

  Convexity: Duration is a linear line which measures the changes in bond price in relation to changes in interest rate. But it does not determine the accurate bond price changes if there is large change in yield rate. On the other hand convexity is measure of the non-linear relationship of bond price in relation to changes in interest rate. It measures the changes in duration in relation to changes in interest rate to determine the accurate bond price. Formula: Convexity = 1 / P * (1+y) 2 Ʃ T t=1 [CF / (1+y) 2 (t 2 + t)] Where, P = bond price Y = yield to maturity CF = Cash flow T = maturity period Or Convexity = P (i decreases) + P (i increases) – 2P 0 / 2P 0 * ∆ Y 2 Convexity adjustment = Convexity*100* ∆ y 2 Change in bond price in percentage with the help of modified duration: ∆ p% = -D*p * ∆ y Change in bond price in percentage with the help of modified duration and convexity: ∆ p% = -D*p * ∆ y + 0.5*C* ∆ y 2 Where, ∆ p %= change on bo