### Modeling and Hedging Non-Parallel Term Structure Shifts

After completing this reading you should be able to:

• Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques.
• Describe the principal components analysis and explain its use in understanding term structure movements.
• 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.
• Apply key rate and multi-factor analysis to estimating portfolio volatility.

## Major Weakness Attributable to Single-factor Approaches When Hedging Portfolios or Implementing Asset Liability Techniques

The main weakness attributable to single-factor approaches to portfolio hedging has much to do with the assumption that movements in the entire term structure can be exhaustively described by one interest rate factor. In other words, the single-factor approach erroneously assumes that all rate changes within the term structure of interest rates are driven by a single factor.

From a practical point of view, rates in different regions of the term structure are not always correlated. As an example, the single-factor approach tells us that the 6-month rate can perfectly predict the change in the 30-year rate. This in turns informs the hedging of the 30-year bond with a 6-month bill. Such a move is unlikely to hedge the total risk inherent in the 30-year bond.

Predicted changes in the 30-year rate based purely on changes in the 6-month rate can be quite misleading. That’s because rates in different regions of the term structure (yield curve) are not always correlated. The risk of such non-parallel shifts along the yield curve is known as yield curve risk.

## Key Rate Exposures

Key rate exposures help to describe the risk distribution along the term structure given a bond portfolio. They help describe how to execute the perfect hedge using highly liquid benchmark bonds. Bonds used for this purpose are normally government bonds issued in the recent past, which means they are likely to be trading at or near par.

Partial ‘01s are used to measure and hedge the risk of portfolios of swaps or portfolios that combine both bonds and swaps in terms of the most liquid money market and swap instruments.

Forward bucket ‘01s are also used to measure and the hedge risk of portfolios of swaps/bond combinations, but the difference here is that instead of measuring risk based on other comparable securities on the market, they measure risk based on changes in the shape of the yield curve. Forward bucket ‘01s present an intuitive way to understand yield curve risk of a portfolio, but they are otherwise not efficient at recommending the perfect hedges to neutralize such risks. To compute forward ‘01s, the yield curve is divided into several defined regions.

## A Description of the Key-rate Shift Analysis

The assumption behind key rate shift analysis is that the entire spectrum of rates can be considered a function of a few select rates at specified points along the yield curve. Thus, to measure risk and predict interest rate movements, a small number of key rates are used, usually those of highly liquid government bonds.

The rates most commonly used are the U.S. Treasury 2-, 5-, 10-, and 30-year par yields. As the words suggest, a “key rate shift” occurs when any of these rates shifts by one basis point. The key rate technique indicates that changes in each key rate will affect rates from the term of the previous key rate to the term of the subsequent key rate.

The key rate shift approach enables analysts to estimate changes in all rates based on a few select rates.

## Definition, Calculation, and Interpretation of Key Rate ’01 and Key Rate Duration

By definition, a $$Key\quad rate\quad ’01$$ (key rate DV01) is the effect of a dollar change of a one basis point shift around each key rate on the value of the security.

The $$Key\quad rate\quad ’01$$ is computed using the same logic as the DV01 formula used in the single-factor approach

$$Key\quad rate\quad ’01=-\frac { \Delta BV }{ 10,000\times \Delta y }$$

Where:

$$\Delta BV$$=change in bond value

$$\Delta y$$=change in yield (0.01%)

Note that yield here implies the yield to maturity.

The change in bond value here is measured in reference to the initial bond value.

Example: Key Rate DV01s and Durations of the May 15, 2045, C-STRIP as of May 28, 2015

In the table below, column (1) gives the initial price of a C-STRIP and its present value after application of key rate one basis point shifts.

$$\begin{array}{|l|l|l|c|} \hline {} & \left( 1 \right) \quad Value & \left( 2 \right) \quad Key\quad Rate\quad ‘01 & \left( 3 \right) Calculation \\ {} & {} & {} & \\ \hline Initial\quad value & 26.11485 & {} & {} \\ \hline 2-year\quad shift & 26.11582 & { -0.001} & {-\frac {26.11582-26.11485}{10,000∗0.01\%}} \\ \hline 5-year\quad shift & 26.11885 & { -0.040} & {-\frac {26.13885-26.11885}{10,000∗0.01\%}} \\ \hline 10-year\quad shift & 26.13885 & -0.024 & { -\frac {26.13885-26.11485}{10,000∗0.01\%} } \\ \hline 30-year\quad shift & 26.01192 & { 0.103} & {-\frac {26.01192-26.11485}{10,000∗0.01\%}} \\ \hline \end{array}$$

The key rate ’01 with respect to the 10-year shift is calculated as:

$$Key\quad rate\quad ’01=-\frac { \Delta BV }{ 10,000\times \Delta y }$$

$$=-\frac { 26.13885-26.11485 }{ 10,000\times 0.01\% } =-0.024$$

#### Important note:

Before looking at the second example, it is important to understand exactly what key rates stand for. When we say that, for example, the 5-year key rate changes, what we mean is that the 5-year par rate changes; all other par rates are unchanged.  It is easy to think of the 5-year key rate as the 5-year spot rate, but it is not; it’s the par rate. Key rates are not spot rates. (Par rate denotes the coupon rate for which the price of a bond is equal to its nominal value (or par value).

This leads us to a very important observation: a bond priced at par (i.e., purchase price = par value = $100) only has price sensitivity to key rates at the same tenor as its maturity. For instance, a 5-year coupon-paying par bond has zero sensitivity to a change in the 2-year key rate. However, a 5-year premium/discount bond will have some sensitivity to the 2-year key rate. The reason, as we have seen above is that the 5-year par rate doesn’t change. We compute the price of a bond by discounting all its cash flows by its YTM. If the 5-year par rate doesn’t change, then the YTM on a 5-year par bond doesn’t change, and therefore the price of a 5-year par bond doesn’t change. #### Example 2 on Multi-factor Hedging Let’s use an example to illustrate how to pull off the perfect hedge under multi-factor hedging: Suppose we have a 30-year option-free bond paying semiannual coupons of$5,000 in a flat rate environment of 5% across all maturities. Using the concepts learnt in the preceding learning outcome statements, we can compute the following key rate ‘01s and key rate durations, assuming a one-basis point shift in the key rates used:

$$\begin{array}{|l|l|l||} \hline {} & Value & Key\quad Rate\quad ‘01 & Key\quad Rate\quad Duration \\ \hline Initial\quad value & 145,066.45 & {} & {} \\ \hline 2-year\quad shift & 145,061.23 & 5.22 & 0.36 \\ \hline 5-year\quad shift & 145,050.68 & 15.77 & 1.09 \\ \hline 10-year\quad shift & 144,989.02 & 77.43 & 5.34 \\ \hline 30-year\quad shift & 145,000.95 & 65.50 & 4.52 \\ \hline Total & {} & 163.92 & 11.31 \\ \hline \end{array}$$

For example,

The $$key\quad rate\quad ’01$$ with respect to the 5-year shift is calculated as:

$$Key\quad rate\quad ’01=-\frac { \Delta BV }{ 10,000\times \Delta y }$$

$$=-\frac { 145,050.68-145066.45 }{ 10,000\times 0.01\% } =15.77$$

And the corresponding key rate duration is:

$$duration=\frac { DV01 }{ 0.0001\times bond\quad value }$$

$$=\frac { 15.77 }{ 0.0001\times 145066.45 }$$

$$=1.09$$

A key rate ’01 of 15.77 implies that the bond decreases in value by $15.77 for a one basis point shock to the 5-year key rate. We can easily come up with the other key rate 01’s and key rate durations by performing similar calculations. Now to illustrate how hedging is carried out in this scenario, assume we have four other different securities, each with the following key rate exposures: To illustrate how hedging is carried out based on key rates, assume we have four other different securities, each with the following key rate exposures: $$\begin{array}{|l|lll|} \hline Security & Exposure & ( per\quad 100\quad & face\quad value ) & {} \\ \hline {} & 2-year\quad key & 5-year\quad key & 10-year & 30-year \\ {} & rate & rate & key\quad rate & key\quad rate \\ \hline 2-year\quad security & 0.001 & {} & {} & {} \\ \hline 5-year\quad security & 0.0015 & 0.045 & {} & {} \\ \hline 10-year\quad security & 0.002 & 0.001 & 0.1 & {} \\ \hline 30-year\quad security & {} & {} & {} & 0.20 \\ \hline \end{array}$$ Note: In the table above, we assume that the 2-year bond and the 30-year bond are trading at par, in which case they are only exposed to the key rate corresponding to their maturity dates (2 years and 30 years, respectively). On the other hand, the 5-year and 10-year securities are trading at a premium. For the hedge to work, we must neutralize the key rate exposure at each key rate. Let $${ F }_{ 2 }$$, $${ F }_{ 5 }$$, $${ F }_{ 10 }$$, and $${ F }_{ 30 }$$ be the face amounts of the bonds in the hedging portfolio to be sold. 2-year key rate exposure: Three bonds, namely the two-year, five-year and 10-year, have an exposure to the two-year key rate. Therefore, for the two-year key rate exposure of the hedging portfolio to equal that of the underlying position, it must be the case that $$2-year\quad key\quad rate\quad exposure:\frac { 0.001 }{ 100 } \times { F }_{ 2 }+\frac { 0.0015 }{ 100 } \times { F }_{ 5 }+\frac { 0.002 }{ 100 } \times { F }_{ 10 }=5.22$$ 5-year key rate exposure: Only two bonds, namely the five-year and 10-year, have an exposure to the five-year key rate. Therefore, for the five-year key rate exposure of the hedging portfolio to equal that of the underlying position, it must be the case that $$5-year\quad key\quad rate\quad exposure:\frac { 0.045 }{ 100 } \times { F }_{ 5 }+\frac { 0.001 }{ 100 } \times { F }_{ 10 }=15.77$$ 10-year key rate exposure: Only the ten-year bond has an exposure to the ten-year key rate. Therefore, for the ten-year key rate exposure of the hedging portfolio to equal that of the underlying position, it must be the case that $$10-year\quad key\quad rate\quad exposure:\frac { 0.1 }{ 100 } \times { F }_{ 10 }=77.43$$ 30-year key rate exposure: Only the ten-year bond has an exposure to the ten-year key rate. Therefore, for the ten-year key rate exposure of the hedging portfolio to equal that of the underlying position, it must be the case that $$30-year\quad key\quad rate\quad exposure:\frac { 0.2 }{ 100 } \times { F }_{ 30 }=65.50$$ Solving equations (1) through (4) simultaneously gives the following solution for the face value of the hedging bonds in the hedging portfolio: $${ F }_{ 30 }=32,750$$ $${ F }_{ 10 }=77,430$$ $${ F }_{ 5 }=33,324$$ $${ F }_{ 2 }=317,150$$ What then, do these figures imply? The investor needs to short$317,150 face amount of the 2-year security, short $33,324 face amount of the 5-year security, short$77,430 face amount of the 10-year security, and finally short $32,750 face amount of the 30-year security. Only then would the initial bond position be insured from changes in rates close to the key rates used. However, such a hedge portfolio is not perfect, and the hedged position is actually only approximately immune due to two main reasons As is the case whenever derivatives are used for hedging purposes, the quality of hedge deteriorates as the size of the interest rate change increases. The hedge will work only if the par yields between key rates move as assumed (linearly). Other reasons why the hedge may not work include: • Hedging implies more instruments and more transaction costs which may eat up the scooped gains; • Under the key rate model, the number of key rate durations to be used and the corresponding choice of key rates remain quite arbitrary ## Partial ‘01s and Forward-bucket ‘01s ### Partial ‘01s Key rate shifts make use of a few key rates to determine risk exposures and execute hedging strategies. For, example, in this reading, we’ve used the 2-year, 5-year, 10-year, and 30-year par yields. The hedging strategy must involve all four. However, when the securities involved contain swaps, we need more to assess the effect of interest rates at more points along the yield curve. There’s a need to measure more frequently. This leads us to partial ‘01s and forward-bucket ‘01s. 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 swap rate at least once every day from a set of observable par swap rates or futures rates. Using the fitted swap rate curve, the sensitivity of a portfolio can be measured in terms of changes in the rates of the fitting securities. It follows that by definition, partial ’01 (PV01) is the change in value of a portfolio after a one-basis-point decline in that fitted rate and a refitting of the curve. All other fitted rates are unchanged. With partial ‘01s, yield curve shifts are able to be fitted more precisely because we are constantly fitting securities. #### Example of Partial ‘01s 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. ### Forward-bucket ’01s While key rates and partial ’01s do a fantastic job expressing the exposures of a position in terms of hedging securities, forward-bucket ’01s present a far more direct and intuitive way to convey the exposures of a position to different parts of the curve. A bucket is jargon for a region of the term structure of interest rates. Forward-bucket ’01s are computed by shifting the forward rate over each of several defined regions of the term structure on region at a time. They, however, aren’t the quickest way to determine the hedges required to pull off that perfect “immunization”. The first step under this methodology is to subdivide the term structure into buckets. The 5 most common buckets are 0-2 years, 2-5 years, 5- 10 years, 10-15 years, and 20-30 years. After that, each forward-bucket ’01 is computed by shifting the forward rates in that bucket by one basis point. In so doing, the analyst may have to shift all of a bucket’s semiannual forward rates, quarterly forward rates, or even shorter-term rates. #### Example: Computation of forward-bucket ‘01s of a 5-year swap given a 0-2 year bucket and a 2-5 year bucket. The table below lists the cash flows of the fixed side of 100 notional amount of a swap, the current forward rates (marked “current”) as of the pricing date, and the three shifted forward curves. Credit: Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today’s Markets, 3rd Edition • For the “0-2 Shift,” forward rates of term 0.5 to 2.0 years are shifted up by one basis point while holding all other forward rates constant. • For the “2-5 Shift,” forward rates of term 2.0 to 5.0 years are shifted up by one basis point while, again, holding all other forward rates constant. • Lastly, for “Shift All,” the forward rates in the curve are shifted The row labeled “Present Value” gives the present value of the cash flows first under the initial forward rate curve and then under each of the shifted curves. The forward-bucket ’01 for each shift can then be computed as the negative of the difference between the shifted and initial present values, i.e., Forward bucket ’01 = –(shifted present value – initial present value) For the 0-2-year shift, for example, the ’01 is −(99.976 − 99.9955), or .0.0196 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. ## Hedging across Forward-Bucket Exposures Referring to GARP-assigned Tuckman reading, let us say we have a counterparty enters into a euro 5×10 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. The figure below 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. The figure below shows forward-bucket exposures of three different ways to hedge this payer swaption (as of May 28, 2010) using securities presented in the previous figure: As is apparent, 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). Credit: Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today’s Markets, 3rd Edition ## Applying Key Rate and Multi-factor Analysis to Estimate Portfolio Volatility Although we have studied at length the term-structure of interest rates, we are yet to look at volatility. Just like there is a term-structure for interest rates, there is also a term-structure for volatility. In fact, the volatility term structure typically slopes downwards when plotted against maturity. This implies that the shorter the maturity of the par-rate, the more volatile it tends to be. 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 but changes in these rates are not perfectly correlated. How can we go about estimating volatilities for the key-rates? Step 1: Estimate the volatility for each key rate as well as the correlation for each pair of key rates. Step 2: Compute the key-rate 01s of the portfolio Step 3: Compute the variance and volatility of the portfolio For example, let’s make some assumptions: • There are two key-rates C1 and C2 • The key rates of the portfolio are KR011 and KR012. • P gives the value of our portfolio Then, by the definition of key rates, $$\Delta P = KR01_1 \times \Delta C_1 + KR01_2 \times \Delta C_2$$ Furthermore, let $$\sigma_P^2$$, $$\sigma_1^2$$ and $$\sigma_2^2$$ denote the variances of the portfolio and of the key rates and let ρ denote the correlation of the key rates. By applying the usual formula for finding the variance of a portfolio, we can estimate portfolio variance: $$\sigma_P^2 = \sigma_1^2 KR01_1^2 + \sigma_2^2 KR01_2^2 + 2 \rho_{1,2}\sigma_1\sigma_2KR01_1KR01_2$$ Note that this methodology can be applied equally well to partial ’01s or forward-bucket ’01s. ## Questions ### Question 1 A risk manager at an Indian bank helps manage a portfolio of investment-grade option-free bonds. After lengthy market analysis, the manager strongly recommends portfolio hedging using the key rates of 2-year, 5-year, 7-year, and 20-year exposures. According to the manager, the 2-year rate has increased by 10 basis points in the recent past. How will the increase affect the 20-year rate? 1. It will increase by 10 basis points 2. It will decrease by 10 basis points 3. It will increase by 20 basis points 4. It will increase by zero basis points The correct answer is D. The key rate technique indicates that changes in each key rate will affect rates from the term of the previous key rate to the term of the subsequent key rate. In this case, the 2-year key rate will affect all rates from 0 to 5 years; the 5-year key rate affects all rates from 2 to 7 years; the 7-year key rate affects all rates from 5 to 20 years; and the 20-year key rate affects all rates from 7 years to the end of the curve. ### Question 2 Suppose we have a 30-year option-free bond paying semiannual coupons of$4,000 in a flat rate environment of 5% across all maturities. The following table provides the initial price of the bond and its present value after application of a one basis point shift in four key rates:

$$\begin{array}{|l|l|} \hline {} & Value & Key\quad Rate\quad ‘01 \\ \hline Initial\quad value & 138,200.55 & {} \\ \hline 2-year\quad shift & 138,195.23 & 5.32 \\ \hline 5-year\quad shift & 138,187.33 & 13.22 \\ \hline 7-year\quad shift & 138,172.91 & 27.64 \\ \hline 30-year\quad shift & 138,180.25 & 20.30 \\ \hline Total & {} & 66.48 \\ \hline \end{array}$$

Suppose further that there are four other different bonds with the following key rate exposures:

$$\begin{array}{|l|lll|} \hline Security & Exposure & ( per\quad 100\quad & face\quad value ) & {} \\ \hline {} & 2-year\quad key & 5-year\quad key & 7-year & 30-year \\ {} & rate & rate & key\quad rate & key\quad rate \\ \hline 2-year\quad security & 0.001 & {} & {} & {} \\ \hline 5-year\quad security & 0.0015 & 0.045 & {} & {} \\ \hline 7-year\quad security & 0.002 & 0.001 & 0.1 & {} \\ \hline 30-year\quad security & {} & {} & {} & 0.20 \\ \hline \end{array}$$

If we wish to fully hedge our initial position using these four securities, determine the face amount of the 5-year security we need to short (assume that the 2-year bond and the 30-year bond are trading at par):

1. 27,640
2. 10,150
3. 28,764
4. 30,000

If we assume that the 2-year bond and the 30-year bond are trading at par, they are only exposed to the key rate corresponding to their maturity dates (2 years and 30 years, respectively). The face amount we need for each security is given by $${ F }_{ i }$$.

$$2-year\quad key\quad rate\quad exposure:\frac { 0.001 }{ 100 } \times { F }_{ 2 }+\frac { 0.0015 }{ 100 } \times { F }_{ 5 }+\frac { 0.002 }{ 100 } \times { F }_{ 7 }=5.32$$

$$5-year\quad key\quad rate\quad exposure:\frac { 0.045 }{ 100 } \times { F }_{ 5 }+\frac { 0.001 }{ 100 } \times { F }_{ 7 }=13.22$$

$$7-year\quad key\quad rate\quad exposure:\frac { 0.1 }{ 100 } \times { F }_{ 7 }=27.64$$

$$30-year\quad key\quad rate\quad exposure:\frac { 0.2 }{ 100 } \times { F }_{ 30 }=20.30$$

$$\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots$$

$${ F }_{ 30 }=10,150$$

$${ F }_{ 7 }=27,640$$

$${ F }_{ 5 }=\frac { 13.22-0.2764 }{ 0.00045 } =28,764$$