###### Options Markets

After completing this reading, you should be able to: Describe the various types,... **Read More**

**After completing this reading you should be able to:**

- 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.
- Compute the positions in hedging instruments necessary to hedge the key rate risks of a portfolio.
- 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.

This is a statistical technique that can be used to explain movements in term structure in historical data. Daily movements in the rates of various maturities are observed and certain factors (term structure movements) are identified. Term structure movements have the following properties:

- The daily term structure movements observed are a linear combination of the factors (e.g., a combination of 2 units of the first factor, one unit of the second factor, and 4 units of the third factor);
- The factors are uncorrelated; and
- The first two or three factors account for most of the observed daily movements.

The importance of a factor can be measured by the standard deviation of its factor scores.

The following are the three most important factors driving Treasury rates:

- Factor 1 is a shift in the term structure where all rates move in the same direction by roughly (but not exactly) the same amount.
- Factor 2 is a shift where short-term rates move in one direction and long-term rates move in the other direction. It corresponds to steepening or flattening of the term

structure. - Factor 3 is a bowing of the term structure where relatively short-term and relatively long-term rates move in one direction while intermediate rates move in the other direction.

In chapter 12, we looked at just one factor (a parallel shift in the term structure).

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 turn 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.

Using the **principal components analysis**, multiple factors are identified and assessed in relation to their relative importance in describing movements in the term structure.

This chapter discusses how the metrics introduced in Chapter 12 can be extended to multi-factor models.

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 to hedge the 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 the 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.

The assumption behind the key rate shift analysis is that the entire spectrum of rates can be considered as 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.

By definition, a \(\text{Key rate ’01}\) (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 \(\text{Key rate ’01}\) is computed using the same logic as the DV01 formula used in the single-factor approach.

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

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.

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

$$

\begin{array}{l|c|c|c}

{} & \textbf{Value} & \textbf{Key Rate ‘01} & \textbf{Calculation} \\\hline

\text{Initial value} & 26.11485 & {} & {} \\ \hline

\text{2-year shift}& 26.11582 & { -0.001} & {-\frac {26.11582-26.11485}{10,000∗0.01\%}} \\ \hline

\text{5-year shift} & 26.11885 & { -0.004} & {-\frac {26.11885-26.11485}{10,000∗0.01\%}} \\ \hline

\text{10-year shift} & 26.13885 & -0.024 & { -\frac {26.13885-26.11485}{10,000∗0.01\%} } \\ \hline

\text{30-year shift} & 26.01192 & { 0.103} & {-\frac {26.01192-26.11485}{10,000∗0.01\%}} \\

\end{array} $$

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

$$\begin{align*} \text{Key rate ’01}&=-\frac { \Delta BV }{ 10,000\times \Delta y }&=-\frac { 26.13885-26.11485 }{ 10,000\times 0.01\% } =-0.024 \end{align*} $$

A key rate of -0.024 implies that the C-STRIP increases in price by 0.024 per $100 face value for a one basis point 10-year shift.

**Exam tip:** Just like the DV01, a negative key rate ’01 implies an increase in value after a given shift, relative to the initial value. A positive key rate ’01 implies a decrease in value after a given shift, relative to the initial value.

The effective duration calculates expected changes in price for a bond or portfolio of bonds given a basis point change in yield, but it is only valid for parallel shifts in the yield curve. The key rate duration presents an improvement to the effective duration because it gives the expected changes in price when the yield curve shifts in a manner that is not perfectly parallel.

The key rate duration is actually analogous to duration so that:

$$ \text{Key rate duration}=-\frac { 1 }{ P } \left( \frac { \partial P }{ \partial y } \right) $$

Thus, the key rate duration with respect to the 10-year shift is calculated as:

$$\begin{align*}\text{Key rate duration}&=-\left( \frac { 1 }{ 26.11485 } \right) \times \left( \frac { 26.13885-26.11485 }{ 0.01\% } \right)\\ & =-9.19\end{align*} $$

Alternatively, recall that:

$$ \text{DV01}=\text{Duration}\times 0.0001\times \text{Bond value }$$

Thus,

$$ \begin{align*} \text{Duration}&=\frac { \text{DV01} }{ 0.0001\times\text{ Bond value} }\\ &=\frac { -0.024 }{ 0.0001\times 26.11485 }\\ &=-9.19 \end{align*} $$

How do we interpret the key rate duration?

Interpreting each key rate duration in isolation can be quite difficult. That’s because, in practice, it’s highly unlikely that a single point on the yield curve will exhibit an upwards or downwards shift while all other points remain constant. For this reason, analysts tend to compare key rate durations across the curve.

**Exam tip:** The sum of all the key rate durations along a portfolio yield curve is equal to the effective duration of the portfolio.

Before looking at some examples, let’s try to map out a typical problem investors often find themselves in.

Let’s say an investor holds a (long) bond position and is afraid that the bond will lose some value in the future. We call such a position the “underlying exposure.” How exactly can the position lose value? Suppose the position has a 5-year key rate exposure of $0.05. This implies that the position will drop in value by $0.05 if there’s a one basis point shock to the 5-year key rate. If the bond is trading at, say, $94 per $100 face value, then the new price will be $93.95 per $100 face value following a one basis point increase in the 5-year key rate (Remember that bond prices fall when interest rates rise). To avoid the loss, the investor must identify and sell short another bond whose key rate exposure matches that of the underlying exposure.

Selling short is all about selling high and buying low. The investor will borrow the bond and sell it, anticipating an increase in the 5-year key rate which will result in a decrease in the bond’s price. The investor will still be obligated to “return” the bond to its owner but he will buy it at a lower price and get to keep the difference, which will offset the loss incurred on the long position (underlying exposure). This is the argument behind key rate exposure hedging.

Note that the hedging security need not have a 5-year key rate exposure of exactly $0.05. It could have an exposure of, say, 0.045, implying that the investor will not sell short exactly $100 face value of the bond; the amount will be slightly more.

In key rate exposure hedging, therefore, the secret lies in determining the face amount “F” that’s needed to **neutralize** the key rate exposure of the underlying position.

An underlying exposure (bond position) has a ten-year key-rate ’01 of +$880. If this key rate exposure can be hedged by trading a ten-year bond that itself has a 10-year KR01 of $0.0520 per 100 face amount, what is the hedge trade?

A positive key rate ’01 implies a decrease in value after a given shift, relative to the initial value. Thus, a ten-year key rate ’01 of +$880 implies that the bond position stands to lose $880 if there happens to be a one basis point shock to the ten-year key rate. To avoid this scenario, we must determine the face amount F(10) of the ten-year bond that must be sold short to neutralize the key rate exposure. We proceed as follows:

$$ \begin{align*} \frac{0.0520}{100} × F(10)& = 880\\ F(10)& = \frac{880}{0.00052} = $1,692,308 \end{align*} $$

Note that since the key rate 01s are reported per 100 face value, they need to be divided by 100 in the hedging equation. However, the key rate 01 of the initial bond position (underlying exposure) is reported for the face amount to be hedged, so it stands as it is.

The hedge trade requires us to short the $1.692 million face amount of the ten-year bond so as to neutralize the exposure to the ten-year key rate. If there’s a one basis point shock to the ten-year key rate, the long position will lose $880, while the short position will gain approximately $880 (= 0.052/100 × 1,692,000).

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 if 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.

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 semi-annual coupons of $5,000 in a flat rate environment of 5% across all maturities. Using the concepts learned 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|c|c|c}

{} & \textbf{Value} & \textbf{Key Rate ‘01} & \textbf{Key Rate Duration} \\ \hline

\text{Initial value} & 145,066.45 & {} & {} \\ \hline

\text{2-year shift} & 145,061.23 & 5.22 & 0.36 \\ \hline

\text{5-year shift }& 145,050.68 & 15.77 & 1.09 \\ \hline

\text{10-year shift} & 144,989.02 & 77.43 & 5.34 \\ \hline

\text{30-year shift }& 145,000.95 & 65.50 & 4.52 \\ \hline

\text{Total} & {} & 163.92 & 11.31 \\

\end{array}

$$

For example,

The \(\text{key rate ’01}\) with respect to the 5-year shift is calculated as:

$$ \begin{align*} \text{Key rate ’01}&=-\frac { \Delta BV }{ 10,000\times \Delta y }\\ &=-\frac { 145,050.68-145066.45 }{ 10,000\times 0.01\% } =15.77 \end{align*} $$

And the corresponding key rate duration is:

$$\begin{align*} \text{duration}&=\frac { \text{DV01} }{ 0.0001\times \text{bond value }}\\ &=\frac { 15.77 }{ 0.0001\times 145066.45 }\\ & =1.09 \end{align*}$$

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|c|c|c}

\textbf{Security} & \textbf{Exposure} & (\textbf{ per 100 }& \textbf{face value} ) & {} \\ \hline

{} & \text{2-year key} & \text{5-year key }& \text{7-year} & \text{30-year} \\

{} & \text{rate} & \text{rate} &\text{ key rate} & \text{key rate} \\ \hline

\text{2-year security} & 0.001 & {} & {} & {} \\ \hline

\text{5-year security} & 0.0015 & 0.045 & {} & {} \\ \hline

\text{10-year security} & 0.002 & 0.001 & 0.1 & {} \\ \hline

\text{30-year security} & {} & {} & {} & 0.20 \\

\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

$$ \text{2-year key rate exposure}:\frac { 0.001 }{ 100 } \times { F }_{ 2 }+\frac { 0.0015 }{ 100 } \times { F }_{ 5 }+\frac { 0.002 }{ 100 } \times { F }_{ 10 }=5.22 $$

**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

$$ \text{5-year key rate exposure}:\frac { 0.045 }{ 100 } \times { F }_{ 5 }+\frac { 0.001 }{ 100 } \times { F }_{ 10 }=15.77 $$

**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

$$ \text{10-year key rate exposure}:\frac { 0.1 }{ 100 } \times { F }_{ 10 }=77.43 $$

**30-year key rate exposure**: Only the 30-year bond has an exposure to the 30-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

$$ \text{30-year key rate exposure}:\frac { 0.2 }{ 100 } \times { F }_{ 30 }=65.50 $$

Solving equations (1) through (4) simultaneously gives the following solution for the face value of the hedging bonds in the hedging portfolio:

$$ \begin{align*}{ F }_{ 30 }&=32,750\\ { F }_{ 10 }&=77,430\\ { F }_{ 5 }&=33,324\\ { F }_{ 2 }&=317,150 \end{align*}$$

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

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 the value of a portfolio after** a one-basis-point decline in that fitted rate and 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.

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 refitting of the curve, where the three-month LIBOR and the par 5-, 10-, and 30-year rates are kept the same.

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 a 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 the 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.

The table below lists the cash flows of the fixed side of the 100 notional amount of a swap, the current forward rates (marked “current”) as of the pricing date, and the three shifted forward curves.

$$\begin{array}{lc|cccc}

& & & \textbf{Forward Rates} & & \\\hline

\textbf{Term} & \textbf{Cash flow} & \textbf{Current} & \textbf{0-2 shift} & \textbf{2-5 shift} & \textbf{shift-all} \\

0.5 & 1.06 & 1.012 & 1.022 & 1.012 & 1.022 \\

1 & 1.06 & 1.248 & 1.258 & 1.248 & 1.258 \\

1.5 & 1.06 & 1.412 & 1.422 & 1.412 & 1.422 \\

2 & 1.06 & 1.652 & 1.662 & 1.652 & 1.662 \\

2.5 & 1.06 & 1.945 & 1.945 & 1.955 & 1.955 \\

3 & 1.06 & 2.288 & 2.288 & 2.298 & 2.298 \\

3.5 & 1.06 & 2.614 & 2.614 & 2.624 & 2.624 \\

4 & 1.06 & 2.846 & 2.846 & 2.856 & 2.856 \\

4.5 & 1.06 & 3.121 & 3.121 & 3.131 & 3.131 \\

5 & 101.06 & 3.321 & 3.321 & 3.331 & 3.331 \\

& & & & & \\

\text{Present value} & & 99.9955 & 99.976 & 99.9679 & 99.9483 \\

01 & & & 0.0195 & 0.0276 & 0.0472

\end{array}$$

*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.,

The ’01 of the “Shift All” scenario is analogous to a DV01. The forward bucket analysis decomposes this total ’01 into 0 .0195 due to the 0-2-year part of the curve and 0.0276 due to the 2-5-year part of the curve.

Referring to the GARP-assigned Tuckman reading, let us say 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.

$$\begin{array}{l|c|clc|c|c|c}

\textbf{Security} & \textbf{Rate} & \textbf{0-2} & \textbf{2-5} & \textbf{5-10} & \textbf{10-15} & \textbf{All} \\ \hline

\text{5×10 payer swaption} & 4.04\% & 0.0010 & 0.0016 & -0.0218 & -0.0188 & \textbf{-0.0380} \\

\text{5-year swap} & 2.120\% & 0.0196 & 0.0276 & 0.0000 & 0.0000 & 0.0472 \\

\text{10-year swap} & 2.943\% & 0.0194 & 0.0269 & 0.0394 & 0.0000 & 0.0857 \\

\text{15-year swap} & 3.290\% & 0.0194 & 0.0265 & 0.0383 & 0.0323 & 0.1165 \\

\text{5×10 swap} & 4.044\% & 0.0000 & 0.0000 & 0.0449 & 0.0366 & 0.0815

\end{array}$$

Since the overall forward-bucket ’01 of the year swaption 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:

$$\begin{array}{l|cccc|c}

\textbf{Security/Portfolio} & \textbf{0-2} & \textbf{2-5} & \textbf{5-10} & \textbf{10-15} & \textbf{All} \\ \hline

\text{5*10 payer swaption} & 0.001 & 0.0016 & -0.0218 & -0.0188 & -0.0380 \\

\text{Hedge #1:Long 44.34% of 10-year swaps} & 0.0086 & 0.0119 & 0.0175 & & 0.038 \\

\text{Net position} & 0.0096 & 0.0135 & -0.0043 & -0.0188 & 0.000 \\ \hline

\text{Hedge #2:Long 46.66% of 5*10 swaps} & & & 0.0209 & 0.0171 & 0.038 \\

\text{Net position} & 0.001 & 0.0016 & -0.0009 & 0.0017 & 0.000 \\ \hline

\text{Hedge #3:} & & & & & \\

\text{Long 57.55% of 15-year swaps} & 0.0112 & 0.0153 & 0.022 & 0.0186 & 0.067 \\

\text{Short 61.55% of 15-year swaps} & -0.012 & -0.017 & & & -0.029 \\

\text{Net position} & 0.0002 & -0.0001 & 0.0002 & -0.0002 & 0.000

\end{array}$$

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 *

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 C
_{1}and C_{2} - The key rates of the portfolio are KR01
_{1}and KR01_{2}. - 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 a 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?

A. It will increase by 10 basis points

B. It will decrease by 10 basis points

C. It will increase by 20 basis points

D. 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|c}

{} & \textbf{Value} & \textbf{Key Rate ‘01} \\ \hline

\text{Initial value} & 138,200.55 & {} \\ \hline

\text{2-year shift}& 138,195.23 & 5.32 \\ \hline

\text{5-year shift} & 138,187.33 & 13.22 \\ \hline

\text{7-year shift} & 138,172.91 & 27.64 \\ \hline

\text{30-year shift} & 138,180.25 & 20.30 \\ \hline

\text{Total} & {} & 66.48 \\

\end{array}

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

$$\begin{array}{l|ccc}

\textbf{Security} & \textbf{Exposure} & (\textbf{ per 100 }& \textbf{face value} ) & {} \\ \hline

{} & \text{2-year key} & \text{5-year key }& \text{7-year} & \text{30-year} \\

{} & \text{rate} & \text{rate} &\text{ key rate} & \text{key rate} \\ \hline

\text{2-year security} & 0.001 & {} & {} & {} \\ \hline

\text{5-year security} & 0.0015 & 0.045 & {} & {} \\ \hline

\text{7-year security} & 0.002 & 0.001 & 0.1 & {} \\ \hline

\text{30-year security} & {} & {} & {} & 0.20 \\

\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):

A. 27,640

B. 10,150

C. 28,764

D. 30,000

The correct answer is

C.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 \({ F }_{ i }\).

$$\begin{align*} \text{2-year key rate exposure}:&\frac { 0.001 }{ 100 } \times { F }_{ 2 }+\frac { 0.0015 }{ 100 } \times { F }_{ 5 }+\frac { 0.002 }{ 100 } \times { F }_{ 7 }=5.32\\ \text{5-year key rate exposure}:&\frac { 0.045 }{ 100 } \times { F }_{ 5 }+\frac { 0.001 }{ 100 } \times { F }_{ 7 }=13.22\\ 7-\text{year key rate exposure}:&\frac { 0.1 }{ 100 } \times { F }_{ 7 }=27.64\\ \text{30-year key rate exposure}:&\frac { 0.2 }{ 100 } \times { F }_{ 30 }=20.30 \end{align*}$$

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

$$\begin{align*} { F }_{ 30 }&=10,150\\ { F }_{ 7 }&=27,640\\{ F }_{ 5 }&=\frac { 13.22-0.2764 }{ 0.00045 } =28,764 \end{align*}$$