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Sensitivity of Bond Prices: Duration and Convexity

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Sensitivity of Bond Prices: Duration and Convexity BOND PORTFOLIO MANAGEMENT Protecting Against Term Structure Shifts: • Shifts in the term structure are viewed by most managers as the sources of risk to bond portfolios. • Two techniques to insulate a portfolio from shifts in the term structure: • Exact matching and immunization. Exact Matching • Assume it is necessary to meet flows of Rs.100, Rs.1000 and Rs.2000 over next three years for pension payments. • An exact matching programme would determine a bond portfolio of one, two and three year bonds so that the coupon plus principal match the three flows mentioned. Cash Flow Matched Portfolios Year 1 2 3 Liability Rs100 Rs1000 Rs2,000 Portfolio A Rs100 Rs1000 Rs2,000 Portfolio B Rs195 Rs900 Rs2,000 Immunization • Immunization theory attempts to eliminate sensitivity to shifts in the term structure by matching the duration of the assets to the duration of the liabilities. • [DA = DL] • Suppose we had a single payment liability at period 4. Since, it is a single payment liability it has the duration of 4 i.e., D = M = 4. • To immunize we can purchase bonds or portfolios of bonds with 4 years Duration. Assume that maturity price of a bond is Rs.100, maturity period is 5 years, annual coupon payment is 13.52%. If the yield rate is 11% (Interest rate for all maturity is 11%), find out the Duration. Time Ct PV W Wt 1 13.52 12.18018 0.111424 0.111424 2 13.52 10.97314 0.100382 0.200764 3 13.52 9.88570 0.090434 0.271303 4 13.52 8.90604 0.081472 0.325889 5 113.52 67.36859 0.616287 3.081435 ________ ________ P =109.31 3.9908 The Value of Bond with Changing Interest Rates: Time CF Value at 11% Value at 10% Value at 12% 1 13.52 13.52 (1.11)3 = 18.49 13.52(1.1)3=17.99 13.52(1.12)3=18.99 2 13.52 13.52(1.11)2 = 16.66 13.52(1.1)2=16.36 13.52(1.12)2=16.96 3 13.52 13.52(1.11) = 15.00 13.52(1.1) =14.87 13.52(1.12) =15.14 4 13.52 13.52 = 13.52 13.52 =13.52 13.52 =13.52 _______ ______ ______ 63.67 62.75 64.61 5 113.52 113.52(1.11)-1 = 102.27 113.52(1.1)-1= 103.2 113.52(1.12)-1 =101.35 Total 165.94 165.95 165.96 • If we had a liability at period 4, we could purchase a sufficient quantity of the bond to just meet the liability. • For example, a Rs.1659.5 liability could be meet with 10 bonds. • If Interest rate is 11%, PV of the liability is [Rs.1659.5/(1.11)4] = Rs.1093.16 • Present value of one bond was found to be Rs.109.314. • Thus, PV of 10 Bonds = Rs.1093.16 • i.e., the PV of the Liability of Rs.1659.5 • The duration of a portfolio of bonds is equal to the weighted average of the durations of the individual bonds in the portfolio. • We have seen, dP d(1 Y) R DY D u P (1 Y) 1 dP 1 or D P d(1 Y) (1 Y) • Let, P Pi Where, P Priceof Portfolio Pi Priceof IndividualBondsof the Portfolio dP dP i d(1 Y ) d(1 Y ) 1 dP 1 dP or i P d(1 Y ) P d(1 Y ) P 1 dP i i P Pi d(1 Y ) D P D or i i (1 Y ) P (1 Y ) P or D i D P i • Duration of a portfolio of bonds = Weighted average of the durations of individual bonds in the portfolio. • Thus, if a portfolio has 1/3 of its funds invested in a bond with 6 years duration and 2/3 of its funds invested in a bond with 3 years duration, then the duration of the portfolio is • D = ∑[(Pi/P) x Di] = 1/3 x 6 + 2/3 x 3 = 4 Years. • Example II: • Consider a situation in which a portfolio manager has one and only one cash outflow to make from a portfolio: an amount equal to Rs.1,00,000 which is to be paid in two years. Because there is one cash outflow, its duration is simply two years (D = 2 Years). • Now two bonds are available to invest. • First bond is with annual coupon payments of Rs.80, maturity of 3 years, and per value of Rs.1000. Yield to maturity is 10% (Y = 0.10). • The second bond matures in one year, providing the holder of the bond with a single payment of Rs.1070. Yield to maturity is 10% hence the bond is currently selling for Rs.1070/(1.1) = Rs.972.73. Calculation of Duration for the 1st Bond (Y = 10%) t Ct PV of Ct PV of Ct x t 1 80 72.72 72.73 2 80 66.12 132.23 3 1080 811.40 2434.21 Rs950.25 Rs2639.17 • Duration = 2639.17/950.25 = 2.78 Years • The manager has to invest a part of the portfolios fund in the one year bonds and rest in the three year bonds. • How much should be placed in each bond? If immunization is to be used, the solution can be found by solving simultaneously a set of two equations involving two unknown. • W1 + W2 = 1 ……….. (1) • (W1 x 1) + (W2 x 2.78) = 2 ………… (2) • From Equation (1), • W1 = 1 – W2 • Substituting the value of W1 in Equation (2), we get, • (1 – W2) x 1 + 2.78 W2 = 2 • 1 – W2 (1 – 2.78) = 2 • – W2 (– 1.78) = 2 – 1 = 1 • W2 = 1/1.78 = 0.5618 • W1 = 0.4382 • Since the manager has to pay Rs.1,00,000 after 2 years. It’s present value at 10% yield rate is Rs.1,00,000/(1.1)2 = Rs.82,645. • Thus the manager would need Rs.82,648 in order to purchase bonds that would create a fully immunized portfolio. • He invests Rs.82,645(0.4382) = Rs.36,215 in one year bond and Rs.82,645(0.5618) = Rs.46,430 in three year bond. • Because, the current market prices of one year and three year bonds are Rs.972.73 and Rs.950.25 respectively. • He would buy • [36,215/972.73] = 37.23 one year bond, and [46,430/950.25] = 48.86 three years bond. • The following table shows what would happen to the portfolio if yield rate remains the same or increases or decreases. The Value of the Portfolio with Changing Interest Rate at the End of Year 2 S. Descriptions YTM 9% YTM 10% YTM 11% N 1 Value at t = 2 from reinvesting one year 43,421.3 43,819.7 44,218.1 bond proceeds 1070 x 37.23 x (1+Y) = 2 Value at t = 2 of three year bonds: (A)Value from reinvesting of (i) coupons received at t = 1 [Rs.80 x 4,260.6 4,299.7 4,338.8 48.86 x (1+Y)] (ii) Coupon received at t = 2 [80 x 3,908.8 3,908.8 3,908.8 48.86] (B) Selling price at t = 2, [1080 x 48,411.7 47,971.6 47,539.5 48.86/(1+Y)] Aggregate Portfolio Value at t = 2 1,00,002.4 99,999.8 1,00,003.2 • Example III: Another Case of a Single-Payment Liability: • Suppose you have a liability where you must make a single payment of Rs1,931.00 in 10 years. The rate of interest is currently 10%, and the term structure is flat. • The present value of the liability is Rs 745.00, as given by; • Present Value of Liability = Rs1,931.00/(1+Y)10 • Or Rs745.00 = Rs1,931.00/(1.10)10 • Since, it is a single payment, the liability has a Macaulay duration equal to its maturity, 10 years. • To immunize you must invest Rs745 in a bond that also has a duration of 10 years. • At a 10% interest rate, a 20-years bond with maturity value of Rs1000, carrying a Rs70.00 annual interest payment has a current market value of Rs.745.00 and a duration of 10 years. • That is, • Price • = Rs745 = [Rs70.00/1.101] + [Rs70.00/1.102] + ……… + [(Rs70.00 + Rs1,000.00)/1.1020] • Duration • 10 = 1 x [(Rs70.00/1.101)/Rs745.00] + 2 x [(Rs70.00/1.102)/Rs745.00] + …… • + 20 x [{(Rs70.00 + Rs1,000.00)/1.1020}/Rs745.00] Effect of Interest Rate Changes on Terminal Values Rates Stay at 10% Rates Fall to 4% Rates Rise to 16% Accumulated value of 70x1.109=165 70x1.049=100 70x1.169=266 interest payments 70x1.108=150 70x1.048=96 70x1.168=229 received and 70x1.107=136 70x1.047=92 70x1.167=198 reinvested at indicated 6 6 6 interest rates 70x1.10 =124 70x1.04 =89 70x1.16 =171 70x1.105=113 70x1.045=85 70x1.165=147 70x1.104=102 70x1.044=82 70x1.164=127 70x1.103=93 70x1.043=79 70x1.163=109 70x1.102=85 70x1.042=76 70x1.162=94 70x1.101=77 70x1.041=73 70x1.161=81 70x1=70 70x1=70 70x1=70 Total = 1,115 Total= 842 Total = 1,492 Market value of bond in 816 1243 565 the 10th year at indicated interest rate Grand Total Rs1931 Rs2085 Rs2057 Less Required Payment Rs1931 Rs1931 Rs1931 Surplus 0 Rs154 Rs126 • Market value of bond in 10th year • = [70/(1+Y)1] + [70/(1+Y)2] + …….
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