P1.T4.Valuation Tuckman, Chapter 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also violates GARP s ethical standards.
P1.T4. Tuckman, Chapter 5: Multi-Factor Risk Metrics and Hedges Tuckman, Fixed Income Securities Chapter 5: Multi-Factor Risk Metrics and Hedges (Including Key Rate and Bucket Exposures) Note: If you are unable to view the content within this document we recommend the following: MAC Users: The built-in pdf reader will not display our non-standard fonts. Please use adobe s pdf reader (http://get.adobe.com/reader/otherversions/) PC Users: We recommend you use the foxit pdf reader (http://www.foxitsoftware.com/secure_pdf_reader/) or adobe s pdf reader (http://get.adobe.com/reader/otherversions/). Mobile and Tablet users: We recommend you use the foxit pdf reader app or the adobe pdf reader app. All of these products are free. We apologize for any inconvenience. If you have any additional problems, please email Suzanne at suzanne@bionicturtle.com. 2
Related Learning Spreadsheets Spreadsheet T4.Tuckman.5 Multi-factor Exam Relevance (XLS not topic) Low 3
Chapter 5: Multi-Factor Risk Metrics and Hedges
Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques. One-factor approaches (duration) use a single interest rate factor to describe the entire term structure This tends to assume a parallel shift in the yield curve, or at least a strong relationship between yields of different maturities. Hedging (with a model that assumes parallel shifts) fails to protect against shape changes and/or twists in the yield curve. 5.012% 5.010% 5.008% 5.006% 5.004% 5.002% 5.000% 4.998% Parallel Shift + 1 basis point (+0.01%) 0 5 10 15 20 25 30 35 40 45 50 55 60 Maturity (Semi-Annual, 30 Years = 60 Periods) 5
Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques. Weaknesses attributable to single-factor approaches Up until now, we have worked with fixed income portfolios using duration and convexity to assess the impact on the portfolio given a change in yields. o But this presupposes the term-structure is governed by a single factor. The problem is that different maturities do not shift by the same relative or absolute magnitude o o o For example, we can observe 10-year rates go down while 3-year rates go up This called curve risk. Most term-structure rate models nowadays have at least three (3) factors: 1. an intercept, 2. a slope factor and 3. a concavity factor. These 3 factors combines explain more than 95% of movements in the term-structure of interest rates; however, a single factor alone, such as DV01 performs much less favorably. 6
Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques. Twist in slope (flattening) Short Intermediate Long 7
Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques. Curvature (- butterfly) Curvature (+ butterfly) Short Intermediate Long 8
Define key rate exposures and know the characteristics of key rate exposure factors including partial 01s and forward-bucket 01s. Three approaches: key-rate shifts, partial 01s and forward-bucket 01s. The common theme of key rate exposures is that different maturity rates do not change in tandem. So rather than matching the total duration, we disaggregate the information about duration. We look at the distribution of risk across maturities, and see what combination of different maturities will contribute (constitute) a good hedge for our portfolio. 9
Describe key-rate shift analysis. Key-rate shifts make two simplifying assumptions: 1) Shifts in the key-rates are linear 2) The rate of a given maturity is affected solely by its closest key-rate. For example, a shift in the 10-year rate is determined by the changes in the 5-year and 30-year rates. That is, if the 5-year and 1-year rates were to stay constant, a change in the 1-year rate would not change the 10-year rate. Both of these assumptions are violated by theory and empirical evidence. However, this is not reason to abandon the key-rate shift approach. The key-rate shift approach is not a theory: it is a model, a simplification of the world, and it performs well in practice. 10
Describe key-rate shift analysis. What is the key-rate shift? In simplest form, the key-rate shift technique consists of selecting a few rates along the term-structure that are representative of the curve. Typically selected are maturities that match highly-liquid instruments: maturities of 2-year, 5-year, 10-year and 30-year rates, as these are among the most heavily traded instruments. The rates chosen are normalized such that they are one basis point at their maturity and decline linearly to 0 at the maturity of the closest key-rate. For example, the 10-year rate would decline linearly from 1 basis point at the 10- year mark to zero (0) basis points at the 5-year, and 30-year mark (this forming a tent shape), as the 5-year and 30-year rates are the neighbor key rates. The sum of a shift in the key-rates is then defined to sum to ~ 1.0 basis point, such that we can disaggregate the total DV01 or duration change into our rate components. 11
Describe key-rate shift analysis. 5.01% 5.01% 5.01% 5.01% 5.00% 5.00% 5.00% 5.00% 5.00% 4.99% Key Rate Shifts Initial Yield Curve Flat at 5% 2 Yr 5 Yr 10 Yr 30 Yr 0.5 2.0 3.5 5.0 6.5 8.0 9.5 11.0 12.5 14.0 15.5 17.0 18.5 20.0 21.5 23.0 24.5 26.0 27.5 29.0 12
Describe key-rate shift analysis. 13
Describe the key rate exposure technique in multi-factor hedging applications and summarize its advantages and disadvantages. The key rate shift technique assumes that a set of key rates describes the movements of the entire term structure. The following choices must be made: The number of key rates, The type of rate to be used (usually spot rates or par yields), The terms of the key rates (e.g., 2 years, 5 years), and The rule for computing all other rates (e.g., interpolation) give the key rates 14
Describe the key rate exposure technique in multi-factor hedging applications and summarize its advantages and disadvantages. Appealing characteristics of key rate shift technique: Each region of the yield curve is affected by a 1 combination of its neighbors (i.e., the nearest key rates). Each rate is most impacted by its closest 2 neighbors. 3 The impacts of the key rates change are smooth. The sum of the key rate shifts equals a parallel 4 shift in the par yield curve. 15
Define, calculate, and interpret key rate 01 and key rate duration. Value Key-rate '01 Key-rate Duration Initial curve 26.22311 2-year shift 26.22411-0.0010-0.38 5-year shift 26.22664-0.0035-1.35 10 year shift 26.25763-0.0345 46.49 = -13.16 30-year shift 26.10121 0.1219 0.1219 / 46.49 Total 0.0829 26.22311 10,000 31.60 The table above (Tuckman s Table 5.2), shows $100 face C-STRIPs due 2040 along with key-rate duration and key-rate 01 calculations. The initial curve is the basis for our calculations, and the 2, 5, 10 and 30-year is the present value after applying a one basis point shift. DV01 = P D/10,000 D = DV01 10,000/P 16
Define, calculate, and interpret key rate 01 and key rate duration. We can write the key-rate 01 w.r.t. to the key-rate a one-basis point shift up is given as so, 01 = 01 =, such that the price change of Let us use this formula and give an example of the change in price of the $100 face C-STRIPs. Applying the formula to the 5-year shift we get that 1 26.22664 26.22311 10,000 0.01% = 0.035, which we can see from the table does indeed correspond to the key-rate 01 for the 5-year shift. 17
Define, calculate, and interpret key rate 01 and key rate duration. Similarly, we can write the key-rate duration as so, = 1. Applying this formula to the same example as for 01, we get 1 26.22664 26.22311 26.22311 0.01% = 1.35, which is equal to the key-rate duration calculated in the table. 18
Describe the key rate exposure technique in multi-factor hedging applications and summarize its advantages and disadvantages. Advantages of key rate exposure techniques Key rate durations can be used in decomposing portfolio returns, identifying interest rate risk exposure, designing active trading strategies, or implementing passive portfolio strategies such as portfolio immunization and index replication. The primary advantage is we can hedge a non-parallel term structure shift Another advantage is, despite using multiple rates to hedge, there is no need to calculate the covariance matrix nor make assumption about correlations between rates. In general, the greater the number of key rates we use, the better the quality of the resulting hedge. For example, using five key rates and thus five securities to hedge a portfolio can protect against a wider variety of yield curve changes than DV01-hedging alone. This effectively reduces our basis risk. 19
Describe the key rate exposure technique in multi-factor hedging applications and summarize its advantages and disadvantages. Disadvantages of key rate exposure techniques However, taking our argument to its logical conclusion, does that imply that we should use as many key-rate buckets as possible? As you might have guessed the answer to this is no. If rates and our portfolio did not change over time, it could theoretically be true. However, in a real-world setting the rates fluctuate, as does the value and composition of a typical portfolio. Adjusting our hedge with a large amount of securities every time this happened would be prohibitively expensive as well as impractical. Disadvantages thus include the added complexity of managing multiple instruments, And higher transaction costs incurred. 20
Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. Scenario 1. The trader shorted $100 million face amount of a 30-year STRIPS to a customer, buying about $47 million face of the 30-year bond to hedge the resulting interest rate risk. 2. The trader facilitated a customer 5s-10s curve trade by shorting $40 million face of the 10-year note and buying about $72 million of the 5-year note. Face Amount ($ millions) Bond Position Hedge Alternate Hedge.75s of 5/31/15-5.19 2.125s of 5/31/18 72.446-80.006-80.008 3.5s of 5/15/23-40.000-0.487 0s of 5/15/43-100.000 4.375s of 5/15/43 47.077 22.633 21.806 21
Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. 1. The trader shorted $100 million face amount of a 30-year STRIPS to a customer, buying about $47 million face of the 30-year bond to hedge the resulting interest rate risk. 2. The trader facilitated a customer 5s-10s curve trade by shorting $40 million face of the 10-year note and buying about $72 million of the 5-year note. Key-rate '01 per 100 face amount Bond 2-year 5-year 10-year 30 year Sum.75s of 5/31/15 0.0199.0000 0 0.0199 2.125s of 5/31/18 0.0480 0 0.0480 3.5s of 5/15/23 0 -.0001 0.0870 0.0869 0s if 5/15/43 -.0010 -.0035-0.0345 0.1219.0829 4.375s of 5/15/43.0000.0001 0.0010 0.1749.1760 Total position $1,000 $38,377 $198 $(39,578) $(3) Hedge $(1,000) $(38,377) $(198) $39,578 $3 Alternate Hedge $31 $(38,379) $217 $38,131 $- Total + Alt. Hedge $1,031 $(2) $415 $(1,447) $(3) 22
Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. The hedger is long $72.4m of the five year and short $40m of the 10-year. Using the key-rate 01s, we see that. x $40m = $72.4m. The trader is also long $47.077m.. of the 30-year bonds, with a short of $100m of the 30-year STRIPS: x $100m =. $47.077m. The 5-year key-rate 01 is given by: 72.446 x. 40 x. 100 x. + 47.077 x. = 0.038361, which translates to ($38,361 due to rounding) = $38,377 as can be seen in the table 23
Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. The trader is DV01 neutral (by construction). However, the trader has a 5s-30s steepener, and needs to hedge this to get a flatter key-rate profile. The face amount of each of the key-rate 01s hedging securities must be set to zero such that we have the following system of equations: 0.0199 100 0.048 100 + 0 + 0 + 0 + $1,000 = 0, 0.0001 100 + 0.0001 + $38,377 = 0, 100 0.087 100 + 0.001 + $198 = 0, 100 0.1749 100 $39,578 = 0, 24
Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. Solving for this yields $22.633m that indeed corresponds to the hedge amount for the 30-year in the first of the tables above. This seemed like a lot of work, so let s look at a short-cut method, which gives a fairly good approximation. Looking at the second table, we see that the 01 of the 30-year is 0.1749 and the amount to be hedge is -$39,578. The approximate face amount of the 30-year bond we need to go long can be found by $. % = $22,63m. 25
Describe the relationship between key rates, partial '01s and forward-bucket 01s, and calculate the forward bucket 01 for a shift in rates in one or more buckets. Swaps have become the most popular interest rate benchmark. Interest rate risk is measured in terms of swap curves by many market participants. When swaps are taken as the benchmark for interest rates, risk along the curve is usually measured with Partial 01s or Partial PV01s rather than with key-rate 01s. Swap market participants fit a par swap rate curve every day, if not more frequently, from a set of traded or observable par swap rates and shorter-term money market and futures rates. Leveraging this curve-fitting machinery, sensitivities of a portfolio or trading book are measured in terms of changes in the rates of the fitting securities. More specifically, the partial 01 with respect to a particular fitted rate is defined as the change in the value of the portfolio after a one-basis-point decline in that fitted rate and a refitting of the curve. All other fitted rates are unchanged. For example, if a curve fitting algorithm fits the three-month London Interbank Offered Rate (LIBOR) rate and par rates at 2-, 5-, 10-, and 30-year maturities, then the two-year partial 01 would be the change in the value of a portfolio for a one-basis point decline in the two-year par rate and a refitting of the curve, where the three-month LIBOR and the par 5-, 10-, and 30-year rates are kept the same. 26
Describe the relationship between key rates, partial '01s and forward-bucket 01s, and calculate the forward bucket 01 for a shift in rates in one or more buckets. With key-rate shifts defined in terms of par yields, the key-rate profile of the 10-year bond, for example, would be its DV01 for the 10-year shift and zero for all other shifts only if the 10-year bond matured in exactly 10 years and were priced at exactly par. By contrast, in the case of partial 01s, the shifts are defined precisely in terms of the fitting securities. Therefore, by construction, all of the 01 of a fitting security is concentrated in the partial 01 calculated by shifting its rate, making calculating hedges particularly easy. Nevertheless, since there are typically many fitting securities, market practice is to trade enough of the fitting securities so as to achieve an acceptable profile of partial 01s rather than trading every single fitting security so as to zero-out all partial 01s. The PV01 of a security is defined as the change in the value of the security if the rates of all fitting securities decline by one basis point. Hence is conceptually equivalent to DV01, where the underlying curve-fitting methodology defines rates at all terms given the changes in the rates of the fitting securities. Furthermore, since the sum of all the partial 01 shifts is the shift with one caveat (*) the partial 01s may be thought of as a decomposition of the PV01 into risks along the curve. 27
Describe the relationship between key rates, partial '01s and forward-bucket 01s, and calculate the forward bucket 01 for a shift in rates in one or more buckets. (*) The technical caveat is that money market rates and swap rates are quoted under different day-count conventions, namely, actual/360 for LIBOR-related rates and 30/360 for the fixed side of swaps. So, if money market rates and swap rates are mixed when fitting swap curves, as they usually are, changing each market rate by a basis point is not the same as changing all actual/360 rates by a basis point or all 30/360 rates by a basis point. To ensure that the sum of the partial 01s does equal the PV01, all rates could be converted into a single day-count convention. But this normalization sacrifices the desirable property that the 01 of each fitting security equals its 01 with respect to its own quoted rate. 28
Construct an appropriate hedge for a position across its entire range of forward bucket exposures. AIM will be updated in next version 29
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. Following a stylized example from the Tuckman reading, let us say we have a portfolio that has a DV01 of $10,000 and we observe that interest rates have a volatility of 100bp per annum. This implies that our portfolio has an volatility of $10,000x100 = $1.0 million per annum. However, this assumes that the volatility term-structure is governed by only 1 factor. We have seen the term-structure of interest rates, however, we have not yet raised any questions regarding volatility. However, just like there is a term-structure for interest rates, there is also a term-structure for volatility. The volatility termstructure is typically downward sloping when plotted against maturity. That is, the shorter the maturity of the par-rate, the more volatile it tends to be. We typically look at the volatility term structure by promptness. That means, that, e.g., every month when a rate expires, the next rate moves ahead in line and becomes the spot rate, and the following rate becomes the prompt rate. An alternative to looking at the volatility term-structure by promptness is to look at the volatility year-overyear. 30
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. Why do we call it, volatility term-structure by promptness when clearly the spot rate is ahead of the prompt rate (the prompt rate is spot + t1)? This is partially due to the fact that during the last month of trading, there are factors such as delivery, closing out of positions and so forth that makes the spot month volatility look very different than the rest of the volatility term-structure. Moreover, for some instruments typically consumption commodities there is very little actual trading in the spot month. Now let s look at how one might go about estimating volatilities for the key-rates 31
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. Now let s look at how one might go about estimating volatilities for the key-rates: Start off by estimating the volatility for each keyrate, as well as the correlation for each pair of key- 1 rates. Proceed to compute the key-rate 01s of your 2 portfolio. Then, compute the variance and volatility of your 3 portfolio. 32
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. For example, we make the following assumptions: There are two key-rates and. The key rates of the portfolio are 01 and 01. P gives the value of our portfolio. By using the definition of the key rates, we have that the change in our portfolio value is given by, = + Then applying the usual formula for finding the variance of the portfolio, we get, = + +,. 33
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. This approach can be applied in just the same manner with Partial PV01s and forward-bucket 01s. Do note however, that this example was not chose by accident: while the methodology is the same, Partial PV01s and forward-bucket 01s generally have more reference rates than the key-rate approach. As a corollary, it would require the estimation of a greater number of volatilities and a greater number of correlation pairs. Those approaches are therefore highly unlikely to be tested on the exam than is this relatively simple case. 34
Explain how key rate and multi-factor analysis may be applied in estimating portfolio volatility. Key rate and bucket analysis may be used to generalize a one-factor estimation of portfolio volatility. In the case of key rates, the steps are as follows: Estimate volatility for each of the key rates and 1 estimate a correlation for each pair of key rates. 2 Compute the key rate 01s of the portfolio. 3 Compute the variance and volatility of the portfolio. 35
End of P1.T4. Tuckman, Chapter 5: Multi-Factor Risk Metrics and Hedges Visit us on the forum @ www.bionicturtle.com/forum