Fixed Income Markets and Products

Transcription

1 PART I ANALYSIS AND VALUATION OF BONDS Fixed Income Markets and Products Raquel M. Gaspar Sérgio F. Silva 1. Bonds and Money-Market Instruments 2. Bond Prices and Yields 3. Term Structure of Interest Rates 4. Hedging Interest Rate Risk 5. Investment Strategies Passive Strategies 5. BondInvestmentStrategies Look to have a performance equal to the market Passive investors act like that if the market were efficient If the market is fully effective, no active strategy is able to beat the market Straightforward(direct) Replication Replication by Stratified Sampling Tracking-Error Minimization Factor Based Replication 3 4

2 Direct Replication Stratified Sampling Exact index replication -Holdingallassetsintheexactproportionstheshowintheindex Once the index is replicated, trades are made only to keep track of the index changes Although widely used in the case of stock indices, it is not so common in the case of bonds indices - High number of emissions - The composition of the index changes regularly Tries to replicate the different components/features of the index with a smaller number of assets Different components/features: Sectors (treasury, corporate, ) Rating Classes (AAA, AA, A, BBB, ) Duration Maturities 5 6 Minimize the Tracking Error Risk models allow you to replicate an index creating portfolios with a minimal tracking error The problem of optimization is: -create a portfolio of N bonds -choosing weights(w i, i =1, 2 N) in order to replicate as closely as possible the return of the bond index(r B ) These models are based on the historical volatilities and correlations between the returns of different asset classes or different risk factors It is expected that such portfolios have at least a correlation of 0.95 with the index they are trying to track The technique involves two steps: - Estimation of variances and covariances of returns obligations -Use this matrix to minimize the tracking error - portfolio return - variance of portfolio returns -covariance between the return of bond iand the return of bond j (element ij of the variance-covariance matrix) -covariance between the return of bond iand the return of the index 7 8 -variance of the index return

3 Sample variance-covariance matrix The key aspect of the problem is the variance covariance matrixof the bonds returns Different approaches: Historical estimation of variance-covariance matrix Exponentially weighted variance-covariance matrix Factor models based variance-covariance matrix T sample size (number of periods of time); N number of bonds; R t vector(nx1) of bond returns in period t vector (Nx1) of average returns of bonds Exponentially weighted variance-covariance matrix (gives more importance to the most recent observations) λ-decreasing factor (is common to use λ= 0,94) 9 10 Factor Model based variance-covariance matrix 11 Reduce the number of estimates through the use of a model of factors We know the term structure dynamics is driven by a limited set of factors (2 or 3) -F jt is the factor j at date t(j = 1, 2,..,k) -e it is specific return of asset i - β ij is the measure of sensitivity of R i to the factor j Using estimates of β ij and to get estimates of Cov(R i,r j ) considering 2 factors : -variance - covariance -These expressions simplify it even more if we assume thatcov(f 1,F 2 ) = 0 12 Example-replicate the index JP Morgan Treasury-Bond using the following treasuries 6.25% 31-Jan % 15-Feb % 15-Nov % 15-Aug % 15-Feb % 15-Aug % 15-May % 15-Feb Sample: daily observations for the period 2/08/ /01/ Obtaining the tracking error for a portfolio: - arbitrary with equal weights - optimal with no restrictions on short selling - optimal when short selling is not allowed

4 Example the case of sample variance-covariance matrix Example the case of sample variance-covariance matrix Portfolio with equal weights (Traking error = 0,14%) Optimal portfolio without short sales (Traking error = 0,07%) Example the case of sample variance-covariance matrix Optimal portfolio with short sales (Traking error = 0,04%) Example: Results (see excel file) Sample Covariance Matrix Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8 TE with short-sales contraints 12,93% 14,19% 0,00% 0,00% 0,00% 62,41% 8,33% 2,13% 0,07% without short-sales contraints 1,99% 39,92% -1,43% 20,93% -62,38% 83,59% 3,44% 13,93% 0,04% Equal weight 0,14% Exponentially-Weighted Estimator Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8 TE with short-sales contraints 0,37% 34,44% 0,27% 4,02% 0,00% 44,79% 0,00% 16,10% 0,02% without short-sales contraints 0,20% 29,00% 5,21% 18,80% -23,80% 54,74% -4,97% 20,82% 0,01% Equal weight 0,12% 15 Single-Index Covariance Matrix* Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8 TE beta 0,021 0,361 0,695 0,821 1,071 1,216 1,930 1,963 with short-sales contraints 8,20% 10,84% 8,29% 7,81% 7,85% 48,73% 4,74% 3,53% 0,08% without short-sales contraints 8,20% 10,84% 8,29% 7,81% 7,85% 48,73% 4,74% 3,53% 0,08% *Considers as a factor the index itself 16 TE tracking error = Equal weight 0,10%

5 Factor-Based Replication Consists of choosing weights so that the exposure of the portfolio and the index to the different factors are the same In case of 3 factors: (i = 1,2, N) 5.2 Active Strategies Investors who don't accept the hypothesis of market efficiency would rather active investment strategies 2 types of strategies Market Timing (to trade based upon expectations on the future evolution of interest rates) Minimizing tracking error considering additional restrictions: Bond Picking (to trade to explore possible market inefficiencies) Market Timming The portfolio managers do "bets" on the future changes of the yield curve -"Bets" based on the assumption the yield curve will not change(riding the yield curve) -"Bets" based on their opinions on what will be the changes in the levelof interest rates (naive strategies or roll-over) -"Bets" based on their opinions on what will be the level, slopeand/or curvatureof the yield curve (butterflies) Managers need scenario analysis tools both to estimate returns and to assess the risk of the strategies implemented - Evaluation of the break-even point - Risk assessment in case their expectation is not verified Riding the yield curve Technique traditionally used by managers to increase the returns when: the yield curve is increasing; anditisexpecteditwillremainunchanged Allowstheinvestortoobtainahigherrateofreturnby: buying bonds with maturities exceeding the intended investment time horizon selling them when the yields are reduced as the maturity decreases bythesimplepassageoftime It can be implemented considering different classes of bonds or assets (treasury vs corporate bonds vs. equities;) 19 20

6 Example Consider the following zero-coupon curve (column 2) and 5 bonds with a coupon rate of 6% In column 3 contains the current (t=0) prices of the bonds (for a nominal value of 100 $) and in column 4 the prices in 1 year (t=1) assuming the zerocoupon curve remained unchanged Maturity Zero-coupon Price at t=0 Price at t=1 rate 1 y 3.90% $ $ y 4.50% $ $ y 4.90% $ $ y 5.25% $ $ y 5.60% $ Assume the portfolio manager has funds available to invest with a 1-year time horizon Example (cont.) -Option 1: investing in the bond with a maturity of 1 year -Option 2: riding the yield curve -option 2.1: buy a 2-year bond and sell it in 1 year -option 2.2: buy a 3-year bond and sell it in 1 year -option2.3: buy a 4-year bond and sell it in 1 year -option2.4: buy a 5-year bond and sell it in 1 year Returns: Option 1 - Option Option Option Option Example (cont.) Expectations about the level of interest rate - Best option: 2.4 (return of 6.633%) -The longer the maturity of the bond, the greater the return Strategies based on the changes of the level of interest rates are simple Assumes that the yield curve depends on just one factor the focus is in the YTM But, only considers parallel movements - What if zero-coupon rates would have increased Two possible moves: an increase in interest rates (upward movement) or a reduction in interest rates (downward movement) -the return would have been less than 6.633%; -and it could even be less than 3.90% If there is an expectation of a reduction of the interest rate level, then one should buy bonds with high duration on the other hand, if there is an expectation of an increase of the level of interest rates, then we should reduce the $ duration or modified duration of the portfolio of bonds or sell uncovered bonds (or futures contracts) or alternatively hold short-term instruments and at roll over them at maturity

7 Expectations of a decrease in interest rates Recall that: -The higher the maturity T and the higher coupon rate C, the higher the $ duration of a bond -The higher the maturity T and the lower coupon rate C, the greater the modified duration Strategies: -Investing in bonds with high T and high C, to optimize the absolute gain -Investing in bonds with high T and low C, to optimize the relative gain Example expectation of a decrease in the interest rate level Consider on date t, a flat curve at the 5% level and 5 bonds with the following characteristics Bond Maturity Coupon rate YTM Price 1 2 anos 5% 5% $ anos 5% 5% $ anos 5% 5% $ anos 7,5% 5% $138, anos 10% 5% $176,86 -A portfolio manager has the expectation that the YTM curve will decrease to the 4.5% level -What bonds must he negotiate if you want to: - maximize the absolute gain maximize the relative gain Example (cont.) expectation of a decrease in the interest rate level Bond Modified duration $ Duration Relative gain Absolute gain 1 1, ,9 0,936% $0, , ,2 3,956% $3, , ,2 8,144% $8, , ,3 7,538% $10, , ,4 7,196% $12,727 -If the focus is the relative gain: the 30 years bond with a coupon rate of 5% (bond number 3) -If the focus is the absolute gain: the 30 years bond with a coupon rate of 10% (bond number 5) Example expectation of an increasse in the level Consider a flat curve at the 5%level -the investment time horizon is 5 years -you expect an increase in interest rates by 1% in 1 year Option 1 -Buy a 5 years maturity bond -Hold the bond until maturity Option 2 - Buy a 1-year obligation (T-bill) - Hold the 1-year T-bill until maturity -A year from now buy a 4-year maturity bond - Hold the 4-year bond until maturity 27 28

8 Example (cont.) expectation of an increase in level Assume your expectations were realized Option 1 Date t=0 t=1 t=2 t=3 t=4 t=5 Cash Flow Annual rate of return: Option 2 Date t=0 t=1 t=2 t=3 t=4 t=5 Cash Flow Expectations on the level, slope and curvature of the yield curve The yield curve is potentially affected by other movements beyond parallel movements(level) there are also movements in the slope and curvature In general it is quite complex to know under what market conditions a concrete strategy will generate a positive or negative results when all possible movesoftheyieldcurvearetakenintoaccount. Wewillsee: Simple strategies bullet, barbell e Ladders More complex butterfly strategies Other strategies that can be understood as contingent hedging strategies Annual rate of return: Bullets -Definition: a bullet portfolio is built concentrating all the investment on a particular maturity we put a bullet on a particular point of the yield curve - Example: portfolio 100% invested in 5 years bonds Barbells -Definition: a barbell portfolio is built concentrating investments in short and long segments of the yield curve -Example: portfolio in which 50% is invested in 6-month Treasury bills and the other 50% in 30 years bonds Ladders -Definition: a ladder portfolio is built by investing equal proportions in various bonds of different maturities -Example: portfolio in which 20% is invested in 1 year OTs, 20% in 2 year OTs, 20% in 3 years OTs, 20% in 4 years OTs and the last 20% in 5 years OTs Butterfly -Itisoneofthemostcommonlyusedstrategies -Resultsfromthecombinationofabarbellwithabullet - The weights of the components are adjusted so that the combination presents a null $duration and is a self-financed transaction(cash-neutral) - To presenting a null $duration, it guarantees an almost perfect immunity to minor parallel changes of the yield curve - Usually the butterfly strategy is structured to present a positive convexity which generates a positive gain for parallel yield curve shifts - There are different types of butterflies that are structured to generate a positive pay-off in the event a particular yield curve movement. 31 OBS:Bullets, Barbells e Ladders constitute the basis for building more complex strategies 32

9 Butterfly Cash and $Duration Neutral -Ensures a positive pay-off if the yield curve movement shifts in a parallel way ( no matter if a positive or negative shoft) Example(excel): Maturity Coupon rate YTM Price $ $Duration Quantity 2 5% 5% ,9 q s 5 5% 5% ,9 α Example I (cont.) -For a given value of α, for example1000, find out q s and q l that guarantee: $Duration = 0: cash neutral: 10 5% 5% ,2 q l Designing the butterfly: -Sell 5 year bonds (in quantity α) -buy 2 years and 10 years bonds (in quantity q s e q l respectively) q s = 578,65 q l = 421, Example I (cont.) Example (see Buterfly excel file): Consider the following bonds: Maturity Coupon rate YTM Price $ $duration 2 5% 4,5% 100, , % 5,5% 97, , % 6% 92,64-701,139 Structure a butterfly Cash and $Duration Neutral, considering you sell year bonds: 35 The butterfly has a positive convexity; whatever the changes that occur in the YTM, this strategy generates always gain but the yield curve can suffer non-parallel movements in which case looking to the YTM is not enough. 36

10 Butterfly weighted 50/50 -Adjust the weights so that the $duration of the portfolio is null and that the $duration of the wing" (short and long segments) are the same -The aim is to make the strategy neutral to small movements, increases or decreases, off the slope(steepening and flattening respectively), such as: - Steepening scenario: -30/0/30 - Flattening Scenario: 30/0/-30 Using the data in the previous example: Butterfly regression weighted - Since short-term rates are more volatile than long term rates one expects larger variations in the yield curve short segment than long segment - Do the regression that explains the variation of the spread between the longsegment and intermediatesegment ( S 1 )based uponthe variationof the spread between the intermediate segment and the short segment ( S 2 ): S 1 = a + b S 2 + e - Adjust the positions so that the portfolio presents a null $duration and that the short segment position $duration (weighted by the factor b) equals the $duration of the position in the long segment Example: if b =0.5, the strategy is almost neutral to a steepening scenario of the type "-30/0/15" or a flattening scenario of the type "30/0/-3". Using the previous example data we would have: Butterfly maturity weighted -Similar to the previous case, but instead of considering the weight as a regression coefficient, it is defined using the maturities of the bonds: Results (one day) for different scenarios Net profit (in $) assuming a 4% financing rate Type of Butterfly equal +20 pb -20 pb -30/0/30 30/0/-30-30/0/15 30/0/-15 In the Example, we would have: Cash-neutral / Regresão Maturidade

11 5.2.2 Bond Picking - Comparison of bonds relative value technique of detecting undervalued and overvalued bonds - There are two different types of investment opportunities: Pure arbitrage opportunities -It compares the price of two products with identical cash flows -Typically an bond and a sum of STRIPs (zero coupon bonds) that replicate that bond -If there is any price difference, then there exists an arbitrage opportunity without risk Speculative arbitrage opportunities - It detects expensive or cheap assets that historically have abnormal YTM -Taking as reference a theoretical zero coupon yield curve that should apply to that concrete product. Pure arbitrage opportunities Exemplo: - In 14/09/ Prices of Strips with maturities 15/02/2005, 15/02/2006 and 15/02/2007 were 99,07%, 96,06% and 92,54%, respectively - The (dirty) price of a bond maturing in 5/02/2007, paying an annual couponrateof10%,wasatthattime121,5% - Is there any arbitrage opportunity? We have to compute the price of the portfolio of strips that replicate the bond: For a face value of the bond equal to 100, we have the replica: 10 face value on the Strip 15/02/ face value on the Strip 15/02/ face value on the Strip 15/02/2007 Example(cont.): Replica price: Replica price(121,307%) < Bond price(121,5%) There is a pure arbitrage strategy: -buythe replica -(short) sellthe bond Result for traded nominal value: (121,307%) Speculative arbitrage opportunities(rich-cheap analysis) Steps 1. Building a zero coupon yield curve using data on bonds that are homogeneous and have similar characteristics to the bonds under analysis, atleast,intermsofliquidityandrisk 2. Using that theoretical zero coupon yield curve, compute the theoretical prices of the bonds under analysis 3. Using the theoretical prices derive the implied YTM for each bond 4. The spread (yield de mercado yield teórica) allows you to identify the bonds that are expensive (spread < 0) and the obligations that are cheap (spread>0) 5. Using statistical analysis (Z-score analysis) on the historical spreads of each bond in order to distinguish abnormal or current inefficiencies of yields 43 44

12 Steps(cont.) 6. Combine long and short positions to create a portfolio almost insensitive to interest rate risk 7. Reverse the positions according to a criterion defined a priori -for instance, when the spread is back to "normal level Z-score Analysis (assuming normality) Calculate the mean m and standard deviation σ the spread based upon, for example, the last 60 days Assume that the spread, S, is normally distributed, defining the standardized spread as U =(S-m)/kσ with k =2 We the get Prob(-1<U<1) = 0,9544 Thus, if the standardized spread value is higher (lower) than 1 (-1), the bond can be seen as relatively cheap(expensive) We can change the confidence level using different k values Example: -The average value and standard deviation of the last 60 days spreads are: m = 0,03% e σ= 0,04% -The observed spread is S = -0,11%, taking k = 3 (confidence level equal to 99,73%), we know that U = -1,166, thius we consider the bond is expensive Performance Analysis Example: Measures of return - value-weighted -equivalent to an internal rate of return -initialinvestment50 -after6 months, the investmentvalueis25 -newentraceof 25 (CF = -25) -aftermore 6 month, the final investmentvalueis100 (CF=100) - considers the various cash flows (inputs and outputs) -value-weighted: r = 40,69% - time-weighted - calculation of returns between exits/entrances - calculation of the return of the period (geometric mean) - time-weighted: -first6 months: -second6 months: -total rate of return of the period: 47 48

13 Return Measures- comments Measures of risk-adjusted return - value-weighted measures are better: -when the manager has control over the entrances and exits of funds - the manager should be rewarded when you take good decisions - Sharpe ratio: - Sortino ratio: - time-weighted: - should be used in other situations(e.g. for Pension Fund Managers) - MAR Minimum Acceptable Return -The calculation of risk only contemplates the observations to which R t < MAR (T m ) - Penalizes managers by taking negative risk 49 50