Investments Analysis Last 2 Lectures: Fixed Income Securities Bond Prices and Yields Term Structure of Interest Rates This Lecture (#7): Fixed Income Securities Term Structure of Interest Rates Interest Rate Risk Bond Portfolio Management
Part II: Fixed Income Markets
Interpreting the Yield Curve TSOIR and interest rate uncertainty link between Various cases» interest rates» liquidity premia possible cases interpretation
TSOIR and Interest Rate Uncertainty Interpreting the term structure Short perspective liquidity preference theory (investors) liquidity premium theory (issuer) Long perspective Expectations hypothesis Market Segmentation vs. Preferred Habitat Examples
TSOIR and Interest Rate Uncertainty 2 Short perspective liquidity preference theory ( short investors)» investors need to be induced to buy LT securities» example: 1-year zero at 8% vs. 2-year zero at 8.995% liquidity premium theory (issuer)» issuers prefer to lock in interest rates f 2 E[r 2 ] f 2 = E[r 2 ] + risk premium
TSOIR and Interest Rate Uncertainty 3 Long perspective long investors wish to lock in rates» roll over a 1-year zero at 8%»or lock in via a 2-year zero at 8.995% E[r 2 ] f 2 f 2 = E[r 2 ] - risk premium Expectation Hypothesis E[r 2 ] = f 2 (risk premium = 0) idea: arbitrage
TSOIR and Interest Rate Uncertainty 4 Market segmentation theory idea: clienteles (ST and LT bonds are not substitutes) (un)reasonable? Preferred Habitat Theory investors prefer some maturities but can be tempted In practice liquidity preference + preferred habitat» these hypotheses are thought more reasonable
TSOIR and Interest Rate Uncertainty 5 Example 2 y 1 = r1 = short term rates: r 1 = r 2 = r 3 = 10% liquidity premium = constant 1% per year YTM 10% y2 = ( 1+ r1 )(1 + f2) 1= (1 + 10%)(1 + 10% + 1%) 1= 10.5% y 3 ( 1 )(1 )(1 ) 1 3 3 = + r1 + f2 + f3 = (1 + 10%)( 1+ 11%)( 1+ 11%) 1= 10.67%
TSOIR and Interest Rate Uncertainty 6 Example 3 Zero-Coupon Rates Bond Maturity 1yr Fwd Rate 12.00% 1 spot = 12% 11.75% 2 11.5% 11.25% 3 10.26% 10.00% 4 6.33% 9.25% 5 6.30%
TSOIR and Interest Rate Uncertainty 7 Example 3 (continued) 1yr Forward Rates 1yr from now [(1.1175) 2 / 1.12] - 1 = 0.115006 2yrs from now [(1.1125) 3 / (1.1175) 2 ] - 1 = 0.102567 3yrs from now [(1.1) 4 / (1.1125) 3 ] - 1 = 0.063336 4yrs from now [(1.0925) 5 / (1.1) 4 ] - 1 = 0.063008
Interpreting the TSOIR Types of yield curve cases Fig 15.5» upward sloping (most common)» downward sloping» hump-shaped Interpretative assumptions either short rates are the culprit or the liquidity premium is positive
Interpretation 2: Rising yield curves Causes either short rates are expected to climb: E[r n ] E[r n-1 ] or the liquidity premium is positive Fig. 15.5a Interpretative assumptions estimate the liquidity premium assume the liquidity premium is constant empirical evidence» liquidity premium is not constant; past > future?!
Interpretation 3: Inverted yield curve Easy interpretation if there is a liquidity premium then inversion expectations of falling short rates why would interest rates fall? Example» inflation vs. real rates» inverted curve recession? 2000 yield curve
Interpretation 4: Hump-Shape curve Interpretation liquidity? Most recent example Spring 00 yield curve
Arbitrage Strategies Question: The YTM on 1-year-maturity zero coupon bonds is 5% The YTM on 2-year-maturity zero coupon bonds is 6%. The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is 5.8%. What arbitrage opportunity exists for an investment banking firm? What is the arbitrage profit?
Answer: Arbitrage Strategies The price of the coupon bond, based on its YTM, is: 120 PA(5.8%, 2) + 1000 PF(5.8%, 2) = $1,113.99. If the coupons were stripped and sold separately as zeros, then based on the YTM of zeros with maturities of one and two years, the coupon payments could be sold separately for The arbitrage strategy is to: [120/1.05] + [1,120/1.06 2 ] = $1,111.08. buy zeros with face values of $120 and $1,120 and respective maturities of 1 and 2 years simultaneously sell the coupon bond. The profit equals $2.91 on each bond.
Fixed Income Portfolio Management In general bonds are securities just like other > use the CAPM Bond Index Funds -- NOT Exam Material Immunization net worth immunization target date immunization contingent immunization -- NOT exam material
Bond Index Funds -- NOT Exam Mat l Idea US indices composition» Solomon Bros. Broad Investment Grade (BIG)» Lehman Bros. Aggregate» Merrill Lynch Domestic Master» government, corporate, mortgage, Yankee» bond maturities: more than 1 year Canada: Scotia-McLeod (esp. Universe Index)
Bond Index Funds 2 -- NOT Exam Mat l Problems lots of securities in each index portfolio rebalancing» market liquidity» bonds are dropped (maturities, calls, defaults, )
Bond Index Funds 3 -- NOT Exam Mat l Solution: cellular approach idea classify by maturity/risk/category/ compute percentages in each cell match portfolio weights effectiveness average absolute tracking error = 2 to 16 b.p. / month
Special risks for bond portfolios cash-flow risk call, default, sinking funds, early repayments, solution: select high quality bonds interest rate risk bond prices are sensitive to YTM solution» measure interest rate risk»immunize
Equation: Interest Rate Risk P = PV(annuity) + PV(final payment) = T t= coupon Par + t 1 (1 + r) (1 + r) T Yield sensitivity of bond Prices: P yield Measure?
Interest Rate Risk 2 Determinants of a bond s yield sensitivity time to maturity coupon rate»maturity sensitivity (concave function)» coupon sensitivity discount bond vs. premium bond zeroes have the highest sensitivity» intuition: coupon bonds = average of zeroes YTM» initial YTM sensitivity
Duration Idea maturity sensitivity to measure a bond s yield sensitivity, measure its effective maturity Measure Macaulay duration: w t = C P.( 1+ YTM ) t t T T 1 Ct P wt = = t = 1 P t= 1 (1 + YTM ) P t T D = t. w t= 1 t = 1
Duration 2 Duration = effective measure of elasticity P P (1 + YTM ) = D. 1+ YTM Proof Modified duration P P = D * [ YTM ]. with D * D = 1+ y
Duration 3 Interpretation 1 T t= 1 D = t. wt = average time until bond payment Interpretation 2 % price change of coupon bond of a given duration = % price change of zero with maturity = to duration
Example (Table 16.3) 8% Bond Duration 5 first bond is zero coupon with 1.8853 years to maturity second bond is 2-year, 8% coupon bond Time years Payment PV of CF (10%) Weight Col.1 times Col.4 0.5 40 38.095 = 40/1.05 0.0395 0.0197 1 40 36.281 0.0376 =36.281/964.54 1.5 40 34.553 0.0358 0.0376 0.0537 2.0 1040 855.611 0.8871 1.7742 sum 964.540 1.000 1.8852
Duration 5 Example (Continued, Table 16.3) P P suppose YTM changes by 1 basis point (0.01%) zero coupon bond with 1.8853 years to maturity old price new price = = 1000 ( 1.05) 3.7706 1000 ( 1.0501) 831.6636 831.9623 = 831.9623 3.7706 = 0.0359% = 831.9623 = 831.6636 (1 + YTM ) = D. 1+ YTM
Duration 6 Example: Table 15.3 suppose YTM changes by 1 basis point (0.01%) coupon bond» either compare the bond s price with YTM = 5.01% relative to the bond s price with YTM = 5%» or simply compute the price change from the duration P P (1 + YTM ) 5.01% 5% = D. = 1.8853x2x = 0.0359% 1+ YTM 1.05
Duration 7 Properties of duration (other things constant) zero coupon bond: duration = maturity time to maturity»maturity duration» exception: deep discount bonds coupon rate» coupon duration YTM»YTM duration» exception: zeroes (unchanged)
Duration 8 Properties of duration duration of perpetuity = D y = 1+ y less than infinity! coupon bonds ( annuities + zero ) see book simplifies if par bond
Duration 9 Importance simple measure essential to implement portfolio immunization measures interest rate sensitivity effectively
Possible Caveats to Duration 1. Assumptions on term structure Macaulay duration uses YTM D = T T t t. wt = t. t t= 1 P t= 1 (1 + YTM ) 1 C» only valid for level changes in flat term structure Fisher-Weil duration measure (NOT exam material) D = T T t t. wt = t. t t= 1 P t= 1 (1 + rs ) s= 1 1 C
Possible Caveats to Duration 2 problems with the Fisher-Weil duration» assumes a parallel shift in term structure» need forecast of future interest rates» bottom line: same problem as realized compound yield Cox-Ingersoll-Ross duration (NOT exam material) bottom line: let s keep Macaulay
Possible Caveats to Duration 3 2. Convexity Macaulay duration first-order approximation: P P = D *. (1 + YTM ) small changes vs. large changes» duration = point estimate» for larger changes, an arc estimate is needed solution: add convexity
Possible Caveats to Duration 4 Convexity (continued) second-order approximation: convexity = 1 P(1 + YTM ) 2 T 2 ( t + t). t= 1 (1 + Ct YTM ) t P P = D * 1. YTM +. convexity. YTM 2 ( ) 2
Possible Caveats to Duration 5 Convexity: numerical example P = Par = 1,000; T = 30 years; 8% annual coupon computations give D*=11.26 years; convexity = 212.4 years suppose YTM = 8% > YTM = 10% P P P P P P = D = D * *. YTM. YTM = 11.26x0.02 = 22.52% + 1 2 811.46 1,000 = 1,000. convexity. = 18.85% 2 ( YTM ) = 18.27%
Bottom Line on Duration Very useful But take it with a grain of salt for large changes
Immunization Why? How? obligation to meet promises (pension funds)» protect future value of portfolio ratios, regulation, solvency (banks)» protect current net worth of institution measure interest rate risk: duration match duration of elements to be immunized
Immunization What? net worth immunization» match duration of assets and liabilities target date immunization» match inflows and outflows» immunize the net flows Who? banks» net worth immunization insurance companies, pension funds» target date immunization
Net Worth Immunization Gap management assets vs. liabilities long term (mortgages, loans, ) vs. short term (deposits, ) match duration of assets and liabilities decrease duration of assets (ex.: ARM) increase duration of liabilities (ex.: term deposits) condition for success portfolio duration = 0 (assets = liabilities)
Target Date Immunization Idea Example: suppose interest rates fall good for the pension fund price risk» existing (fixed rate) assets increase in value bad for the pension fund reinvestment risk» PV of future liabilities increases» so more must be invested now
Target Date Immunization 2 Solution match duration of portfolio and fund s horizon single bond bond portfolio» duration of portfolio» = weighted average of components duration» condition: assets have equal yields
Target Date Immunization 3 Question: Pension funds pay lifetime annuities to recipients. Firm expects to be in business indefinitely, its pension obligation perpetuity. Suppose, your pension fund must make perpetual payments of $2 million/year. The yield to maturity on all bonds is 16%. (a) duration of 5-year bonds with coupon rates of 12% (paid annually) is 4 years duration of 20-year bonds with coupon rates of 6% (paid annually) is 11 years how much of each of these coupon bonds (in market value) should you hold to both fully fund and immunize your obligation? (b) What will be the par value of your holdings in the 20-year coupon bond?
Target Date Immunization 4 Answer: (a) PV of the firm s perpetual obligation = ($2 million/0.16) = $12.5 million. duration of this obligation = duration of a perpetuity = (1.16/0.16) = 7.25 years. Denote by w the weight on the 5-year maturity bond, which has duration of 4 years. Then, w x 4 + (1 w) x 11 = 7.25, which implies that w = 0.5357. Therefore, 0.5357 x $12.5 = $6.7 million in the 5-year bond and 0.4643 x $12.5 = $5.8 million in the 20-year bond. The total invested = $(6.7+5.8) million = $12.5 million, fully matching the funding needs.
Target Date Immunization 5 Answer: ( b ) Price of the 20-year bond = 60 x PA(16%, 20) + 1000 x PF(16%, 20) = $407.11. Therefore, the bond sells for 0.4071 times Par, and Market value = Par value x 0.4071 => $5.8 million = Par value x 0.4071 => Par value = $14.25 million. Another way to see this is to note that each bond with a par value of $1,000 sells for $407.11. If the total market value is $5.8 million, then you need to buy 14,250 bonds, which results in total par value of $14,250,000.
Dangers with Immunization 1. Portfolio rebalancing is needed Time passes duration changes bonds mature, sinking funds, YTM changes duration changes example: Table 16.6 duration YTM 5 8% 4.97 7% 5.02 9%
Dangers with Immunization 2 2. Duration = nominal concept immunization only for nominal liabilities counter example solution» children s tuition» why?» do not immunize» buy assets
An Alternative? Cash-Flow Dedication Buy zeroes to match all liabilities Problems difficult to get underpriced zeroes zeroes not available for all maturities ex.: perpetuity
Contingent Immunization Idea try to beat the market while limiting the downside risk Procedure (Fig. 16.12) -- NOT Exam Material compute the PV of the obligation at current rates assess available funds play the difference immunize if trigger point is hit