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VALUATION AND RISK MODELS—PART I EXAM WEIGHT 30%

Level 1

Multi-Factor Risk Metrics and Hedges

1RIFT Pvt. Ltd. /FRM L1/Valuation & Risk Models


• Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques. • Define key rate exposures and know the characteristics of key rate exposure factors including partial ‘01s and forward-bucket ‘01s. • Describe key-rate shift analysis. • Define, calculate, and interpret key rate ‘01 and key rate duration. • Describe the key rate exposure technique in multi-factor hedging applications; summarize its advantages and disadvantages. • Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile. • Relate key rates, partial ‘01s and forward-bucket ‘01s, and calculate the forward-bucket ‘01 for a shift in rates in one or more buckets. • Construct an appropriate hedge for a position across its entire range of forward-bucket exposures. • Apply key rate and multi-factor analysis to estimating portfolio volatility.

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➢ A major weakness of the single-factor approach, is the assumption that movements in the entire term structure can be described by one interest rate factor.

➢ It is widely recognized that rates in different regions of the term structure are far from perfectly correlated.

➢ The risk that rates along the term structure move by different amounts is known as curve risk.➢ This topic will discusses how to measure and hedge the risks of a security or portfolio in terms of several

other securities, where each hedging security is most sensitive to a different part of the term structure.➢ Hedging with one security requires extremely strong assumptions about how rates move together.➢ Key-rate exposures are used for measuring and hedging the risk of bond portfolios in terms of a

relatively small number of the most liquid bonds available, usually the most recently issued, near-par, government bonds.

RIFT Pvt. Ltd. /FRM L1/Valuation & Risk Models

WEAKNESS OF SINGLE-FACTOR APPROACHES

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➢ Key-rate exposures are designed to describe how the risk of a bond portfolio is distributed along the term structure and how to implement any desired hedge, all in terms of some set of benchmark bonds, usually the more liquid government securities.

➢ Table below, as an example, shows a key-rate exposure report for the U.S. Lehman Aggregate Bond Index, a benchmark portfolio of U.S. governments, agencies, mortgages, and corporates.

➢ The duration of the portfolio with respect to U.S. government rates is 4.339, as reported in the last row of the table.

➢ For example, more than half of the portfolio’s duration risk is closely related to—and could be hedged with—5- and 10-year bonds.


KEY RATE EXPOSURES, PARTIAL’01 & FORWARD BUCKET’01

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➢ Partial ’01s are used for measuring and hedging the risk of portfolios of swaps or portfolios that contain both bonds and swaps in terms of the most liquid money market and swap instruments.

➢ As these instruments are almost always those whose prices are used to build a swap curve, the number of securities used in this methodology is usually greater than the number used in a key-rate framework.

➢ Forward-bucket ’01s, mostly used in the swap or combined bond and swap contexts as well, measure the risk of a portfolio not in terms of other securities but in terms of direct changes in the shape of the term structure.

➢ As a result, forward-bucket ’01s are often the most intuitive way to understand the curve risks of a portfolio, but not the quickest way to see which hedges are required to neutralize such risks.


KEY RATE EXPOSURES, PARTIAL’01 & FORWARD BUCKET’01

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➢ The crucial assumption of the key-rate approach is that all rates can be determined as a function of a relatively small number of key rates.

➢ Therefore, the following decisions have to be made in order to implement the methodology:✓ the number of key rates, ✓ the type of the key rates (e.g., spot rates, par yields), ✓ the terms of the key rates, and ✓ the rule for computing all other rates given the key rates.

➢ In order to cover risk across the term structure, to keep the number of key rates as few as reasonable, and to rely only on the most liquid government securities, one popular choice of key rates for the U.S. Treasury and related markets are the 2-, 5-, 10-, and 30-year par yields.

➢ Then, the change in the term structure of par yields given a one-basis point change in each of the key rates is assumed to be as in Figure.


KEY RATE SHIFT TECHNIQUE

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KEY RATE SHIFT TECHNIQUEA Specification of Key-Rate Shifts

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➢ Each of the four shapes is called a key-rate shift. ➢ Each key rate affects par yields from the term of the previous key rate (or zero) to the term of the next

key rate (or the last term).➢ By construction, the four key-rate shifts sum to a constant shift of one basis point. ➢ This allows for the interpretation of key-rate exposures as a decomposition of the total DV01 or duration

of a security or a portfolio into exposures to four different regions of the term structure.


KEY RATE SHIFT TECHNIQUE

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The following example demonstrates the calculation of key rate ’01 and key rate duration, using a 30-year zero-coupon bond. Zero-coupon securities are also referred to as STRIPS (separate trading of registered interest and principal securities). Investors of zero-coupon bonds receive payment from STRIPS at maturity.Figure : Key Rate ‘01s and Duration of a C-STRIP

(1)Value (2) Key Rate ’01 (3) Key Rate DurationInitial value 25.115842-years shift 25.116815- years shift 25.11984 -0.00040 -1.5910-years shift 25.1398430-years shift 25.01257


CALCULATING KEY RATE ’01 AND KEY RATE DURATION

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➢ For every basis point shift in a key rate, the corresponding key rate ’01 provides the dollar change in the value of the bond.

➢ Similarly, key rate duration provides the approximate percentage change in the value of the bond. ➢ Key rate duration works off 100 basis point changes, so it is the percentage of price movement for every

100 basis point change in rates.

➢ Key rate shifts allow for better hedging of a bond position, and when summed across all key rates, assume a parallel shift across all maturities in the maturity spectrum.

➢ Key rate exposure analysis is a useful tool for measuring bond price sensitivity; it makes very strong assumptions about how the term structure behaves.

➢ It assumes that the rate of given term is affected only by the key rates that surround it. ➢ In reality, shifts are not always perfect linear.


HEDGING APPLICATIONS

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➢ Example:Suppose a 30-year semiannual-paying non callable bond pays a $4,500 semiannual coupon in a flat rate environment of 5% across all maturities. If we assume a one basis point shift in the key rate soused (2-, 5-, 10- and 30-year key rates), the subsequent key rate effects on the security are as shown in Figure.Figure : Key Rate Exposure o f 30-Year Semiannual Pay Non-Callable Bond

Value (2) Key Rate ’01 (3) Key Rate DurationInitial value 139,088.952-year shift 139,083.96 4.99 0.365-year shift 139,074.21 14.74 1.0610-year shift 139,015.04 73.91 5.3130-year shift 139,024.25 64.70 4.65Total 158.34 11.38



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➢ Example:To illustrate hedging based on key rates, suppose that four other securities (shown below) exist in addition to the non callable bond just discussed and that each of these new hedging securities have the following key rate exposures:A 2-year security only has a 2-year key rate exposure of 0.015 per $100 face value.A 5-year security has exposures over the 2-year and 5-year key rate of 0.0025 and 0.035, respectively, per $100 face value.A 10-year security has exposures over the 2-years, 5-years and 10-year key rates of 0.003, 0.015, and 0.1, respectively, per$100 face value.A 30-year security only security only has exposure to the 30-year key rate of 0.15 per $100 face value.



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➢ Example:It is assumed in this example that the 2-year bond and the 30-year bond rate are trading at par, so their only exposure is to the key rate corresponding to the maturity date. Using the key rate exposures from Figure 3 generates the following set of equations the hedge:

2-year key-rate exposure: 0.015

100× 𝐹2 +

0.0025

100× 𝐹5 +

0.003

100× 𝐹10 = 4.99

5-years key-rate exposure:0.035

100× 𝐹5 +

0.0015

100× 𝐹10 = 14.74

10-year key-rate exposure:0.1

100× 𝐹10 = 73.91

30-year key-rate exposure:0.15

100× 𝐹30 = 64.70

By simultaneously solving for 𝐹2, 𝐹5, 𝐹10 𝑎𝑛𝑑 𝐹30, these equations indicate that the investor needs to short the 2-year security in the face amount of $16,745, short the 5-year in the face amount of $10,439, short the 10-year in the face amount of $73,910, and short the 30-years in the face amount of $43,133. Combination from changes in rates close to the rates selected.



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➢ The assumption of interest rate movements drives the effective of immunized strategies.➢ Using the key rates in an immunized setting will only be an approximation of the effectiveness of

immunization.➢ The only way immunization will work perfect in a real-world setting is if all sources of interest rate

changes are perfectly matched.



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➢ Key rate shifts utilize just a few key rates, and express position exposures in terms of hedging securities. ➢ For example, if we assume the key rates are 2-year, 5-year, 10-year and 30-year par yield, each exposure

is measured and hedged separately, and all four securities are needed to hedge the fixed income position.

➢ When swaps are taken as the benchmark for interest rates in complex portfolios, 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.


PARTIAL ’01S AND PV ‘01

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➢ 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.

➢ 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.


PARTIAL ’01S AND PV ‘01

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➢ While key rates and partial ’01s conveniently express the exposures of a position in terms of hedging securities, forward-bucket ’01s convey the exposures of a position to different parts of the curve in a much more direct and intuitive way.

➢ Forward-bucket ’01s are computed by shifting the forward rate over each of several defined regions of the term structure, one region at a time.

➢ The starting point of the methodology is the division of the term structure into buckets. ➢ For the illustration of this section, the term structure is divided into five buckets: 0-2 years, 2-5 years, 5-

10 years, 10-15 years, and 20-30 years. ➢ The best choice of buckets depends, of course, on the application at hand. ➢ A financing desk that does most of its trading in very short-term securities would define many, narrow

buckets in the short end and relatively few, wide buckets in the long end. ➢ A swaps market-making desk, with business across the curve, might use the buckets defined for this

section, although it would likely prefer a greater number of narrower buckets and, particularly in Europe, might need buckets to cover maturities beyond 30 years.


FORWARD-BUCKET ’01s

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➢ Each forward-bucket ’01 is computed by shifting the forward rates in that bucket by one basis point. Depending on how rate curves are stored, this may mean shifting all of a bucket’s semi annual forward rates, quarterly forward rates, or rates of even shorter term.

➢ Consider a 2.12% five-year swap as of May 28, 2010.➢ Table lists the cash flows of the fixed side of 100 notional amount of the swap, the “Current” forward

rates as of the pricing date, and the three shifted forward curves. ➢ For the “0-2 Shift,” forward rates of term .5 to 2.0 years are shifted up by one basis point while all other

forward rates stay the same. ➢ For the “2-5 Shift,” forward rates in that bucket, and that bucket only, are shifted. ➢ Lastly, for “Shift All,” the entire forward curve is shifted.➢ The row of Table labeled “Present Value” gives the present value of the cash flows under the initial

forward rate curve and under each of the shifted curves. ➢ The forward-bucket ’01 for each shift is then the negative of the difference between the shifted and

initial present values. ➢ For the 2-5-year shift, for example, the ’01 is −(99.9679 − 99.9955), or .0276.


Forward-Bucket Shifts and ’01 Calculations

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Computation of the Forward-Bucket ’01s of a Five-Year 2.12 Percent EUR Swap as of May 28, 2010

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An Example of Spot-Starting and Forward-Starting Swaps

The ’01 of the “Shift All” scenario is analogous to a DV01. The forward bucket analysis decomposes this total ’01 into .0196 due to the 0-2-year part of the curve and .0276 due to the 2-5-year part of the curve.

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Hedging across Forward-Bucket Exposures

Suppose a counterparty enters into a euro 5x10 payer swaption with a strike of 4.044% on May 28, 2010. This payer swaption gives the buyer the right to pay a fixed rate of 4.044% on a 10-year euro swap in five years. The underlying is a 10-years swap for settlement on May 31, 2015. Figure gives the forward-bucket ‘01s of this swaption for four different buckets, along with other swaps for hedging purposes.Since the overall forward-bucket ’01 of the year swapation is negative (-0.0380), as rates rise, the value of the option to pay a fixed rate of 4.044% in exchange for a floating rate worth par also rises.

Figure: Forward-Bucket ExposuresSecurity Rate 0-2 2-5 5-10 10-15 All5x10 payer 4.044% 0.0010 0.0016 -0.0218 -0.0188 -0.380Swaption5-year swap 2.120% 0.0196 0.0276 0.0000 0.0000 0.047210-yer swap 2.943% 0.0194 0.0269 0.0394 0.0000 0.085715-year swap 3.290% 0.0194 0.0265 0.0383 0.0323 0.11645x10 swap 4.044% 0.0000 0.0000 0.0449 0.0366 0.815

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Hedging across Forward-Bucket Exposures

Figure shows forward-bucket exposures of three different ways to hedge this payer swaption (as of May 28, 2010) using securities presented in Figure 5. As you can see, the third hedge is the best option since this hedge best neutralizes risk in each of the buckets (the lowest net position indicates when risk is best neutralized).

Figure : Hedging with Forward-Bucket ExposuresSecurity / Portfolio 0-2 2-5 5-10 10-15 All5x10 payer swaption 0.001 0.0016 -0.0218 -0.0188 -0.0380Hedge # 1:

Long 44.34% of 10-year swaps 0.0086 0.0119 0.0175 0.038Net position 0.0096 0.0135 -0.0043 -0.0188 0.000Hedge # 2:

Long 46.66% of 5x10 swaps 0.0209 0.0171 0.038Net position 0.001 0.0016 -0.0009 0.0017 0.000Hedge # 3:

Long 57.55% of 15-year swaps 0.0112 0.0153 0.022 0.0186 0.067Short 61.55% of 15-year swaps -0.012 -0.017 -0.029

Net position 0.0002 -0.0001 0.0002 -0.0002 0.000

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ESTIMATING PORTFOLIO VOLATILITY

➢ As there is a term structure of interest rates, there is a term structure of volatility. ➢ The 10-year par rate, for example, is usually more volatile than the 30-year par rate.➢ In general, portfolios are exposed to interest rates all along the curve and changes in these rates are

not perfectly correlated.➢ First, estimate a volatility for each of the key rates and estimate a correlation for each pair of key

rates. Second, compute the key-rate 01s of the portfolio. ➢ Third, compute the variance and volatility of the portfolio.➢ Say that there are only two key rates, C1 and C2, that the key rates of the portfolio are KR011 and

KR012, that the value of the portfolio is P, and that changes are denoted by . ➢ Then, by the definition of key rates,

∆𝑃 = 𝐾𝑅011 × 𝐶1 + 𝐾𝑅012 × ∆𝐶2

➢ Furthermore, letting 𝜎𝑃2, 𝜎1

2 , and 𝜎22 denote the variances of the portfolio and of the key rates and

letting ρ denote the correlation of the key rates, equation implies that𝜎𝑃2 = 𝐾𝑅011

2𝜎12 + 𝐾𝑅012

2𝜎22 + 2𝐾𝑅011𝐾𝑅012𝜌𝜎1𝜎2

➢ This reasoning can be applied equally well to partial ’01s or forward-bucket ’01s.

Thank You

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