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] + ……. + [{70/(1+Y)10} + {1,000/(1+Y)10}] • You are immunized if interest rates rise or fall in the first year and then stay there.
• But what if rates go down to 4%, stay there until the tenth year, and then rise to 16% just before you have to sell the bond. • When interest rate falls to 4% do you stay immunized? • What is to be done to get immunized? Example IV
• An insurance company must make a payment of Rs19,487 in 7 years. The market interest rate is 10%, so the present value of the obligation is Rs10,000. The company’s portfolio manager wishes to fund the obligation using 3-year zero- coupon bonds and perpetuities paying annual coupons. How can the manager immunize the obligation?
• Immunizations requires that the duration of the portfolio of assets equal the duration of the liability. We can proceed in four steps: • Step 1: Calculate the duration of the liability. In this case, the liability duration is simple to compute. It is a single payment obligation with duration of 7 years.
• Step 2: Calculate the duration of the asset portfolio. • The portfolio duration is the weighted average of duration of each component asset, with weights proportional to the funds placed in each asset. • The duration of zero-coupon bond is simply its maturity, 3 years. • The duration of the perpetuity = (1+y)/y =1.10/.10 • =11 years • If the fraction of the portfolio invested in the zero coupon bond is called w and the fraction invested in the perpetuity is (1-w), the portfolio duration will be • Asset duration = w x 3 years + (1-w)x 11 years
• Step 3: Find the asset mix that sets the duration of assets equal to the 7-year duration of liabilities. • This requires us to solve for w in the following equation: • w x 3 years + (1-w)x 11 years = 7 years • Solving the equation we get w = ½. The manager should invest half the portfolio in the zero coupon bond and the half in the perpetuity. This will result in an asset duration of 7 years.
• Step 4: Fully fund the obligation. Since the obligation has a present value of Rs10,000, and the fund will be invested equally in the zero and the perpetuity, the manager must purchase Rs5,000 of the zero coupon bond and Rs5,000 of the perpetuity
Rebalancing
• Suppose that I year has passed, and the interest rate remains at 10%. The portfolio manager needs to reexamine her position. Is the position still fully funded? Is it still immunized? If not, what actions are required?
• First examine funding. The present value of the obligation will have grown to Rs11,000, as it is 1 year closer to maturity. The manager’s funds also have grown to Rs11,000: The zero coupon bonds have increased in value from Rs5,000 to Rs5,500 with the passage of time, while the perpetuity has paid its annual Rs500 coupons and remains worth Rs5000. Therefore, the obligation is still fully funded.
• The portfolio weights must be changed, however. The zero-coupon bond now will have a duration of 2 years, while the perpetuity duration remains at 11 years. The obligation is now due in 6 years. The weights must now satisfy the equation • w x 2 + (1-w) x 11 = 6 • Or w = 5/9. To rebalance the portfolio and maintain the duration match, the manager now must invest a total of Rs11,000 (5/9) = Rs6,111.11 in the zero coupon bond. This requires that the entire Rs500 coupon payment be invested in the zero, with an additional Rs111.11 of the perpetuity sold and invested in the zero-coupon bond.
• Computing the Macaulay Duration and Internal Yield of a Bond Portfolio:
• Consider two pure discount bonds, both of which pay no interest until maturity.
• Bond A pays Rs1,100 and matures in 1 year. Bond B pays Rs1,407 and matures in 7 years. Both bonds are selling for the same market price, Rs1,000.
• The yields on each bond can be computed as:
• YA = 10% = [(Rs1,100/Rs1,000) – 1.00] 1/7 • YB = 5% = [(Rs1,407/Rs1,000) – 1.00]
• Suppose you bought both bonds, investing half your money in each bond. Would the yield on your portfolio be 7.5%, the weighted average of the two yields?
Cash flows of the portfolio
t0 t1 t7
Bond 1 -1,000 1,100
Bond 2 -1,000 1,407
Total -2,000 1,100 1,407 • The internal rate of return is the rate of discount that will discount the stream of payments associated with the portfolio to a present value equal to its market value. In the case of this portfolio, the internal yield is 5.634%:
• Rs2,000.00 = [Rs1,100.00/(1+Y)1 + (Rs1,407.00/(1+Y)7)] • Solving this we get, • Y = 0.05634
Calculation of Duration (Y = 0.5634)
t Ct PV (PV)xt 1 1,100 1,041.33 1041.33 7 1,407 958.67 6710.67 P=2000 Σ=7752
• D = 7752/2000= 3.876
• But weighted average of the two individual bond = 1(1/2) + 7(1/2) = 4
• Suppose you have a portfolio of two bonds with following characteristics. Face value of each bond is Rs1000.
• Bond Yield rate face value maturity period coupon payment • 1 10% Rs1,000 2 years Rs100 • 2 20% Rs1,000 4 years Rs200
• Find out he yield rate and duration of the portfolio.
COMBINATION LINES FOR INTERNAL YIELD AND DURATION • A B •
• C
• yield Internal D
• E • Duration Suppose you purchase one of each of the following bonds which are traded at par:
• Bond Annual Face Year to coupon value maturity 1 Rs 100 Rs 1000 2 2 Rs 200 Rs 1000 3 3 Rs 300 Rs 1000 4 • If portfolio yield rate changes from y % to (y + 5) %, find out the unanticipated price changes of your portfolio based on Macaulay’s Duration and Convexity.
ACTIVE BOND MANAGEMENT
• Horizon Analysis • Contingent Immunization • Interest rate swaps • Riding the Yield Curve Horizon Analysis
• One form of interest rate forecasting is called horizon analysis. • The analyst using this approach selects a particular holding period and predicts the yield curve at the end of that period. • Given a bond’s time to maturity at the end of the holding period, its yield can be read from the predicted yield curve and its end-of-period price calculated. • Then the analyst adds the coupon income and prospective capital gain of the bond to obtain the total return on the bond over the holding period. Example I • A 20-year maturity bond with a 10% coupon rate (paid annually) currently sells at a yield to maturity of 9%. • A portfolio manager with a 2-year horizon needs to forecast the total return on the bond over the coming 2 years. • In 2 years, the bond will have an 18-year maturity. • The analyst forecasts that 2 years from now, 18- year bonds will sell at yields to maturity of 8%, and that coupon payments can be reinvested in short-term securities over the coming 2 years at a rate of 7%. • To calculate the 2-year return on the bond, the analyst would perform the following calculations:
• 1. Current price=Rs100 x Annuity factor (9%, 20 years) + • Rs1000 x PV factor (9%, 20 years) • = Rs1091.29 • 2. Forecast price= Rs100 x Annuity factor (8%, 18 years) • Rs1000 x PV factor (8%, 18 years) • = Rs1187.44
• 3. The future value of reinvested coupons will be (Rs100x1.07) + Rs100 = Rs207 • 4. The 2-year return is • (Rs 207 + Rs1187.44 - Rs1091.29) / Rs1091.29 • = 0.278 or 27.8% • The annualized rate of return over the 2-year period would then be (1.278)1/2 -1 = 0.13, or 13%.
Example II
• Consider a 4% bond with ten years remaining to maturity (maturity price Rs100). • Coupon payment is semiannual. • Current Annual YTM is 9%. • Five years into future the bond’s term to maturity will decrease to 10-5 = 5 years. Horizon period is 5 years. • At the horizon (five years hence), annual YTM is predicted to be 8% (or 4% semiannually) YTM 10 Yrs 9 Yrs 5 Yrs 1 Yrs 0 Yrs
7.00 78.68 80.22 87.53 97.15 100.00
7.50 75.68 77.39 85.63 96.69 100.00
8.00 72.82 74.68 83.78 96.23 100.00
8.50 70.09 72.09 81.98 95.77 100.00
9.00 67.48 69.60 80.22 95.32 100.00
9.50 64.99 67.22 78.51 94.87 100.00
10.00 62.61 64.92 76.83 94.42 100.00 • Current price of the bond is Rs67.48 • As time passes, the bond might follow a path through the table such as that shown by the dashed line. • If so, it would end up at a price of Rs83.78 at the horizon, with an 8% promised annual YTM. • Over any holding period, a bond’s return typically will be affected by both the passage of time and a change in yields. • Horizon analysis breaks this into two parts: one due solely to passage of time, with no change in yields, and the other due solely to a change in yield, with no passage of time. • Thus total price change from Rs67.48 to Rs83.78 (Rs16.30) is broken into a change from Rs67.48 to Rs80.22 (or Rs12.74), followed by an instantaneous change from Rs80.22 to Rs83.78 (or Rs3.56).
• The intermediate value of Rs80.22 is the price the bond would command at the horizon if its promised YTM had remained unchanged at its initial level of 9%. • The actual price is that which it commands at its actual YTM of 8%. • In summary the total price change can be broken into two parts: • Price change = time effect + yield change effect
If the semiannual coupon of Rs2 is reinvested @ 4.25% per six months
• The value at the end of 5 years is • 2(1+.0425)9 = 2.908 • 2(1+.0425)8 =2.790 • 2(1+.0425)7 =2.676 • 2(1+.0425)6 =2.567 • 2(1+.0425)5 =2.463 • 2(1+.0425)4 =2.362 • 2(1+.0425)3 =2.267 • 2(1+.0425)2 =2.173 • 2(1+.0425)1 =2.085 • 2(1+.0425)0 = 2.00 • ______• Total 20+4.25= 24.25. Of this amount Rs20 can be considered interest and Rs4.25 is the interest on interest. • In summary, a bond’s overall return has four components – the time effect, the yield effect, the coupons, and the interest from reinvesting the coupons • Overall return = time effect+ yield effect+ coupons + interest on coupons • = Rs12.74 +Rs3.56 +Rs20+Rs4.29 = 40.59 • This is the 5 years holding period return. • Holding period rate of return is (40.59/67.48)x100=60.15 • Annualized rate of return = (1.6015)1/5 - 1= 9.87%
CONTINGENT IMMUNIZATION
• Assume that your desired ending –wealth value is Rs161.05. • Assume a 5 year horizon and 10 per cent return. • The PV factor at 10% for a lum sum of Re 1 at 5th year is 0.62092. • The PV factor of 0.62092 times Rs 161.05 ending value equals Rs 100 – that is, this is the required initial investment under these assumptions to attain the desired value. Assuming the 5 year horizon, we can do it for other interest rates as follows:
Percent PV factor Required investment to get Rs161.05 after 5 years 6% 0.747 120.34 8% 0.681 109.60 9% 0.650 104.68 10% 0.621 100.00 10.8% 0.598826 96.44 11% 0.592 95.50 12% 0.567 91.38 13% 0.543 87.45 14% 0.519 83.58
• Required Assets
• 120.34
• 109.60
• 100
• 91.38
• 83.64
• • 6 10 14 • Yield rate • Contingent immunization requires that the client will be willing to accept a potential return (9%) below the current market return (10%). • The difference between the current market return and the floor rate is referred to as cushion spread. • The cushion spread in required yield provides flexibility for the portfolio manager to engage in active portfolio strategies. • If we assume that the client initiated the fund with Rs 100, the acceptance of this lower rate will mean that the portfolio manager does not have the same ending-asset requirements. • Specifically, at 9% the required ending-wealth value would be Rs153.86 {=100(1.09)5} compared to Rs 161.05 {=100(1.10)5} at 10 per cent. Price behaviour required for floor return
Percent PV factor Required investment to Required investment to get get Rs 153.86 after 5 years Rs161.05 after 5 years
6% 0.747 114.93 120.34 8% 0.681 104.77 109.60 9% 0.650 100 104.68 10% 0.621 95.54 100.00 10.8% 0.598826 92.1354 96.44 11% 0.592 91.24 95.50 12% 0.567 87.238 91.38 13% 0.543 83.55 87.45 14% 0.519 79.85 83.58 Price behaviour required for floor return
• Required Assets
• 120.34 • 114.93 • 109.60
• 100 • 95.54 • 91.38
• 83.64
• • 6 9 10 14 • Yield rate
• The above figure (the dotted line) shows the value of assets that are required at the beginning assuming a 9% required return and the implied ending-wealth value of Rs153.86. • Assuming current market rates of 10%, the required amount to be invested is Rs 95.54 to get Rs153.86 after 5 years. • The difference between the client’s initial fund of Rs 100 (actual investment) and the required investment of Rs95.54 is the cushion available to the portfolio manager.
Active portfolio management strategies
• If the portfolio manager believes that interest rate will come down, he can consider to acquire a bond of 10 year duration (but the required duration is 5 year as the time horizon is 5 years and one time liability).
• Now if interest rate declines to 6%, portfolio value will
increase by Ru = DΔy. • Δy = (1.10 -1.06)/ 1.10 = 0.0363636 • DΔy = 10 (0.0363636) = 0.363636 • Thus the value of the long duration portfolio will be Rs136.36.
• Now if interest rate declines to 8 %, portfolio value will increase by Ru = DΔy. • Δy = (1.10 -1.08)/ 1.10 = 0.0181818 • DΔy = 10 (0.0181818) = 0.181818 • Thus the value of the long duration portfolio will be Rs118.18.
• Now if interest rate declines to 9%, portfolio value will increase by Ru = DΔy. • Δy = (1.10 -1.09)/ 1.10 = 0.009090909 • DΔy = 10 (0.009090909) = 0.090909 • Thus the value of the long duration portfolio will be Rs109.09.
•
• Now if interest rate increases to 11%, portfolio value will decrease
by Ru = -DΔy. • Δy = (1.10 -1.11)/ 1.10 = -0.00909090 • -DΔy = 10 (-0.0181818) = -0.090909 • Thus the value of the long duration portfolio will be Rs90.91. • But to immunize the portfolio at 11% we need Rs 91.24 [i.e.,153.86 (1.11)-5} to get Rs 153.86. But the potfolio value reduced to Rs90.09 which is less than required amount of Rs 91.24. • Hence the portfolio manager is required to stop active portfolio management and use classical immunization with the remaining asset before the interest rate increases to 11%.
• Now if interest rate increases to 10.8%, portfolio value
will decrease by Ru = -DΔy. • Δy = (1.10 -1.108)/ 1.10 = -0.00727272 • -DΔy = -10 (0.00727272) = -0.07272 • Thus the value of the long duration portfolio will be Rs 92.73 (appx). • Now if the manager immunize the portfolio value of Rs 92.73 at 10.8% interest rate, he will get Rs 92.73 (1.108)5 = Rs 153.86 which is the exact amount required by the investor to get 9% potential return.
• Required Assets
• 136.36 • Safety Margin • • Trigger Point
• 114.93 • • 109.09 • 100 • 95.56 • 92.73
•
• 6 9 10 10.8 14 •
Potential Return
• If the portfolio were immediately immunized when market rates were 10%, it naturally earn the 10% market rate. • Alternatively if yields declined instantaneously to 6%, the portfolio’s asset value would increase to Rs136.36. • If this portfolio (Rs136.36) were immunized at the market rate of 6% over the remaining five year period, the portfolio would compound at 6% to a total value of Rs182.449 {i.e.,(136.36 x (1.06)5}. • This ending value of Rs182.45 represents an 12.75% realized (horizon) rate of return on the original Rs100 portfolio. • In contrast, if interest rate increases, the value of the portfolio will decline substantially and the potential return will decline. • For example if the market rates rise to 10.5% the asset value of bond will decline to • Rs100(1-DΔy) • Rs100(1-.0454545) since Δy = (1.1-1.105)/1.10 • Rs95.4545 = -0.004545 • If this portfolio of Rs95.45 were immunized for the remaining 5 Years at the prevailing market rate of 10.5%, the ending value would be Rs157.257. This ending value implies a potential return of 9.5%. • If interest rate goes to 10.8%, the portfolio value declines to Rs92.13 (the trigger point) and the portfolio would have to be immunized. • At this point, if the remaining assets of Rs92.13 were immunized at this current market rate of 10.8%, the value of portfolio would grow to Rs153.86 {i.e., 92.13x1.67}. • This ending value implies that the potential return for the portfolio would be exactly 9%. • Regardless of what happens to subsequent market rates, the portfolio has been immunized at the floor rate of 9%. The potential return concept
• Potential return • (percent)
• Contingent immunization
• 12.75
• Classical immunization • 10 • 9.5 Trigger • 9 • Floor return
• 7 • -4 -2 0 0.5 0.8 2 4 6 • Immediate yield change from 10% (percentage point) • INTEREST RATE SWAPS
• Consider a manager of a large portfolio that currently includes Rs100 crore par value of long term bonds paying an average coupon rate of 7%. The manager believes that interest rates are about to rise. As a result, he would like to sell the bonds and replace them with either short-term or floating rate issues. However, it would be expensive in terms of transaction costs to replace the portfolio every time the forecast for interest rates is updated. A cheaper and more flexible way to modify the portfolio is for the managers to “swap” the Rs7 crore a year in interest income the portfolio currently generates with a floating rate. • A swap dealer might advertise its willingness to exchange, or “swap” a cash flow based on LIBOR rate for one based on a fixed rate of 7%. • The portfolio manager would then enter into a “swap” agreement with the dealer to pay 7% on notional principal of Rs100 crore and receive payment of the LIBOR rate on that amount of notional principal. Now consider the net cash flow to the manager’s portfolio in three interest rate scenarios:
LIBOR rate LIBOR rate LIBOR rate of 6.5% of 7% of7.5%
Swap inflow 650,00,000 700,00,000 750,00,000 Swap outflow 700,00,000 700,00,000 700,00,000
Net gain -50,00,000 0 50,00,000 The Swap dealer
• What about the swap dealer? Why is the dealer, which is typically a financial intermediary such as a bank, willing to take on the opposite side of the swaps desired by this participant in the hypothetical swaps? • Consider a dealer who takes on one side of a swap, let’s say paying LIBOR and receiving a fixed rate. • The dealer will search for another trader in the swap market who wishes to receive a fixed rate and pay LIBOR. • For example, Company A may have issued a 7% coupon fixed-rate bond that it wishes to convert into a synthetic floating-rate debt, while Company B may have issued a floating rate bond tied to LIBOR that it wishes to convert into synthetic fixed-rate debt. Swap payment
• 6.95% 7.05%
• 7%
• Company A Swap Dealer Company B • • LIBOR
• LIBOR LIBOR
Swap inflow Swap outflow Interest outflow Net Flow To Debt holder Company A 6.95% -(LIBOR) -(7%) -(LIBOR+0.05%)
Company B LIBOR -(7.05%) -(LIBOR) -7.05%
Dealer (LIBOR+7.05%) -(LIBOR+6.95%) 0.10%
Riding the Yield Curve
• It is used by people whose primary objective is to have liquidity.
• 1. One way of investing is to simply purchase short term securities, hold them until they mature, and reinvest the proceeds.
• 2. An alternative way is to ride the yield curve, provided that certain conditions exist • (i) One condition is that the yield curve be upward-sloping, indicating that longer-term securities have higher yields. • (ii) Another condition is that the investor believe that the yield curve will remain upward-sloping.
• Given these two conditions, the investor who is riding the yield curve will purchase securities that have a somewhat longer term-to-maturity than desired and then will sell them before they mature. • Consider an investor who prefers investing in 90-days T/Bs. • Currently such bills are selling for Rs98.28 per Rs100 of face value. • If y is the quarterly yield rate, then • 98.28 = 100 /(1+y) • Or 1+y = 100 /98.28 = 1.0175, or y=.0175 • Yearly y (spot rate) = .0175 x4=0.07 or 7% • However, 180-day T/Bills are currently selling for Rs96.15 per Rs100 of face value. • If y is the half yearly yield rate, then • 96.15 = 100 /(1+y) • Or 1+y = 100 /96.15 = 1.04, or y=.04 • Yearly y (spot rate) = .04 x4=0.08 or 8%
Yield Curve
• 8% • 7%
• ¼ ½ 1 2 3 4 • Assumption: the yield curve remains upward-sloping over the next three month • If the investor buys the 180-day T/B for Rs 96.15 and subsequently sells them after 90 days, then the expected selling price would be • 100/(1.0175) = Rs98.28. • Thus the expected return = 4(98.28-96.15)/96.15 =8.86% • But had he bought 3 month T/Bill for Rs98.28 per Rs100 face value and hold it till maturity, his return would have been 7% which is less than 8.86 , the expected return from riding the yield curve. CAVEAT
• If the yield curve does change, then “riding it” might be detrimental to the investor’s return. • That is riding the yield curve has more risk than simply buying securities that mature at the appropriate time. • Similarly two transactions are necessary (buying and selling the security) when riding the yield curve, where as maturity strategy has only one transaction. Thus there will be larger transaction cost associated with riding the yield curve.