mfe8812 bond portfolio management · 1 hedging interest-rate risk with duration overview basics of...

Hedging Interest-Rate Risk with DurationBeyond Duration

MFE8812 Bond Portfolio Management

William C. H. Leon

Nanyang Business School

January 8, 2018

1 / 86 William C. H. Leon MFE8812 Bond Portfolio Management


1 Hedging Interest-Rate Risk with DurationOverview

Basics of Interest-Rate Risk

Hedging with Duration

2 Beyond DurationOverview

Relaxing the Assumption of a Small ShiftUsing a Second-Order Taylor ExpansionProperties of Convexity

Relaxing the Assumption of a Parallel ShiftA Common PrincipleRegrouping Risk Factors Using Principal Components AnalysisHedging Using a Three-Factor Model of the Yield Curve

2/ 86 William C. H. Leon MFE8812 Bond Portfolio Management


Basics of Interest-Rate RiskHedging with Duration

Overview

Bond price risk is often measured in terms of the bond interest-ratesensitivity, or duration. This is a one-dimensional measure of the bond’ssensitivity to interest-rate movements.

The traditional hedging method that is intensively used by practitioners isthe duration hedging method. This approach is based on a series of veryrestrictive and simplistic assumptions, the assumptions of a small andparallel shift in the yield-to-maturity (YTM) curve.

There are three related notions of duration: Macaulay duration, $durationand modified duration.

Macaulay duration is defined as a weighted average maturity for theportfolio. The Macaulay duration of a bond or bond portfolio is theinvestment horizon such that investors with that horizon will not careif interest rates drop or rise as long as changes are small, as capitalgain risk is offset by reinvestment risk on the period.




Overview(Continue)

Three related notions of duration (continue):

$duration is used to compute the profit and loss of a bond portfoliofor a small change in the YTM. $duration also provides a convenienthedging strategy: to offset the risks related to a small change in thelevel of the yield curve, one should optimally invest in a hedgingasset a proportion equal to the opposite of the ratio of the $durationof the bond portfolio to be hedged by the $duration of the hedginginstrument.Modified duration is used to compute the relative profit and loss of abond portfolio for a small change in the YTM.




Five Theorems of Bond Pricing

The basics of bond price movements as a result of interest-rate changes arebest summarized by the five theorems on the relationship between bond pricesand yields.

As an illustration, consider the percentage price change for 5 bonds withdifferent annual coupon rates (5 and 8%) and different maturities (5, 15 and25 years), starting with a common 8% YTM:

New 5% Coupon 8% Coupon

Yield (%) 5-Year 15-Year 25-Year 5-Year 25-Year

5.00% 13.608% 34.550% 47.111% 12.988% 42.282%

7.00% 4.292% 10.041% 12.824% 4.100% 11.654%

7.99% 0.042% 0.094% 0.117% 0.040% 0.107%

8.01% −0.042% −0.094% −0.117% −0.040% −0.107%

9.00% −4.068% −8.832% −10.689% −3.890% −9.823%

11.00% −11.585% −23.502% −27.225% −11.088% −25.265%




Five Theorems of Bond Pricing(Continue)

(1) Bond prices move inversely to interest rates.

Investors must always keep in mind a fundamental fact about therelationship between bond prices and bond yields: bond prices moveinversely to market yields.

When the level of required yields demanded by investors on new issueschanges, the required yields on all bonds already outstanding will alsochange. For these yields to change, the prices of these bonds must change.

This inverse relationship is the basis for understanding, valuing andmanaging bonds.





(2) Holding maturity constant, a decrease in rates will raise bond prices on apercentage basis more than a corresponding increase in rates will lower bondprices.

Bond price volatility can work for, as well as against, investors. Moneycan be made, and lost, in risk-free Treasury securities as well as in riskiercorporate bonds.





(3) All things being equal, bond price volatility is an increasing function ofmaturity.

Long-term bond prices fluctuate more than short-term bond prices.

Although the inverse relationship between bond prices and interest rates isthe basis of all bond analysis, a complete understanding of bond pricechanges as a result of interest-rate changes requires additionalinformation. An increase in interest rates will cause bond prices to decline,but the exact amount of decline will depend on important variables uniqueto each bond such as time to maturity and coupon.

An important principle is that for a given change in market yields, changesin bond prices are directly related to time to maturity. Therefore, asinterest rates change, the prices of longer-term bonds will change morethan the prices of shorter-term bonds, everything else being equal.





(4) The percentage price change that occurs as a result of the directrelationship between a bond’s maturity and its price volatility increases at adecreasing rate as time to maturity increases.

In other words, the percentage of price change resulting from an increasein time to maturity increases, but at a decreasing rate.

Put simply, a doubling of the time to maturity will not result in a doublingof the percentage price change resulting from a change in market yields.





(5) The change in the price of a bond as a result of a change in interest ratesdepends on the coupon rate of the bond.

Holding other things constant, bond price fluctuations (volatility) andbond coupon rates are inversely related.

Note that the above principle holds for percentage price fluctuations; thisrelationship does not necessarily hold if we measure volatility in terms ofdollar price changes rather than percentage price changes.





These 5 relationships provide useful information for bond investors bydemonstrating how the price of a bond changes as interest rates change.Although investors have no control over the change and direction in marketrates, they can exercise control over the coupon and maturity, both of whichhave significant effects on bond price changes.

An important distinction needs to be made between two kinds of risk:

Reinvestment risk.

Capital gain risk.




Reinvestment Risk

It is important to note that the YTM is a promised yield, because investorsearn the indicated yield only if the bond is held to maturity and the couponsare reinvested at the calculated YTM.

Obviously, no trading can be done for a particular bond if the YTM is tobe earned. The investor simply buys and holds. What is not so obvious tomany investors, however, is the reinvestment implications of the YTMmeasure.

The YTM calculation assumes that the investor reinvests all couponsreceived from a bond at a rate equal to the computed YTM on that bond,thereby earning interest on interest over the life of the bond at thecomputed YTM rate.

If the investor spends the coupons, or reinvests them at a rate differentfrom the YTM, the realized yield that will actually be earned at thetermination of the investment in the bond will differ from the promisedYTM.




Reinvestment Risk(Continue)

In fact, coupons almost always will be reinvested at rates higher or lowerthan the computed YTM, resulting in a realized yield that differs from thepromised yield. This gives rise to reinvestment rate risk.

This interest-on-interest concept significantly affects the potential totaldollar return. The exact impact is a function of coupon and time tomaturity, with reinvestment becoming more important as either coupon ortime to maturity, or both, rise. Specifically,

Holding everything else constant, the longer the maturity of a bond,the greater the reinvestment risk;Holding everything else constant, the higher the coupon rate, thegreater the dependence of the total dollar return from the bond onthe reinvestment of the coupon payments.




Reinvestment Risk(Continue)

The reinvestment portion of the YTM concept is critical. In fact, forlong-term bonds the interest-on-interest component of the total realizedyield may account for more than 75% of the bond’s total dollar return.

Consider a 20-year standard bond purchased at a $100 face value,which delivers an annual 10% coupon rate. Lets look at realizedyields under different assumed reinvestment rates when the bond isheld to maturity:If the reinvestment rate is exactly 10%, the investor would realize a10.000% compound return, with 55.407% (= $372.75/$672.75) ofthe total dollar return from the bond attributable to thereinvestment of the coupon payments.At a 12% reinvestment rate, the investor would realize an 11.098%compound return, with 63.438% (= $520.52/$820.52) of the totalreturn coming from interest on interest.With no reinvestment of coupons (spending them as received), theinvestor would achieve only a 5.647% return.




Reinvestment Risk vs. Capital Gain Risk

Consider an investor with a horizon of 1 year who buys a 2-year, 10%semiannual coupon bond at a yield of 8% and a price of 103.630.

Scenario 1: The interest rate drops to 7.8%, soon after the bond ispurchased, and stays there.

The bond price rises immediately to 104.002.After 6 months, the bond price rises to 108.058 (= 104.002 ×

1.039). At this time, a coupon of 5% is paid, whereupon the price ofthe bond drops by an equal amount to 103.058.The bond increases in value at the end of the year to 107.078 (=103.058 × 1.039), while the 5% coupon has been reinvested at anannual yield of 7.8% and has grown to 5.195 (= 50 × 1.039), for atotal of 112.273.

Scenario 2: The interest rate rises to 8.2%, and stays there.

The price immediately drops to 103.259.However, the bond and coupons thereafter rise at the rate of 4.1%every 6 months, culminating in a value of 111.900 at the end of theinvestment horizon.




Reinvestment Risk vs. Capital Gain Risk(Continue)

As bond yield drops, bond price rises, and vice versa. These create capitalgains and losses for bond investors.

There is an interplay between capital gain risk and reinvestment risk: if interestrates drop and stay there, there is an immediate appreciation in the value of abond portfolio, but the portfolio then grows at a slower rate; on the otherhand, if interest rates rise and stay there, there is a capital loss, but theportfolio then appreciates more rapidly.

Hence, there is some investment horizon D , such that investors with thathorizon will not care if interest rates drop or rise (as long as the changes aresmall). The value of this horizon depends on the characteristics of the bondportfolio; specifically, D is the Macaulay duration of the bond portfolio.




Qualifying Interest-Rate Risk

Consider at date t a portfolio of fixed-income securities that delivers m certaincash flows Fi at future dates ti , for i = 1, 2, . . . ,m.

The price P of the portfolio (in $ value) can be written as the sum of thefuture cash flows discounted with the appropriate zero-coupon rate withmaturity corresponding to the maturity of each cash flow:

Pt =m∑i=1

Fi B(t, ti) =m∑i=1

Fi(1 + R(t, ti − t)

)ti−t,

where B(t, ti) is the price at date t of a zero-coupon bond paying $1 at date t

(also called the discount factor) and R(t, ti − t) is the associated zero-couponrate, starting at date t for a residual maturity of ti − t years.




Qualifying Interest-Rate Risk(Continue)

Note that price Pt is a function of m interest-rate variables R(t, ti − t)and of the time variable t. This suggests that the value of the portfolio issubject to a potentially large number m of risk factors. To hedge theportfolio, we need to be hedged against a change in all of these factorrisks.

In practice, it is not easy to hedge the risk of so many variables. We mustcreate a global portfolio containing the portfolio to be hedged in such away that the portfolio is insensitive to all sources of risk (the m

interest-rate variables and the time variable t).

One suitable way to simplify the hedging problem is to reduce the numberof risk variables. Duration hedging of a portfolio is based on a single riskvariable, the YTM of this portfolio.




A First Order Taylor Expansion

The idea behind duration hedging is to bypass the complication of amultidimensional interest-rate risk by identifying a single risk factor, the YTMof the portfolio, which will serve as a “proxy” for the whole term structure.

To examine the sensitivity of the price of bond to changes in YTM, write theprice of the portfolio Pt in $ value as a function of a single source ofinterest-rate risk, its YTM yt as follow:

Pt = P(yt) =m∑i=1

Fi(1 + yt

)ti−t.

In this case, the interest-rate risk is (imperfectly) summarized by changesin the YTM yt . The YTM is a complex average of the whole termstructure, and it can be regarded as the term structure if and only if theterm structure is flat.




A First-Order Taylor Expansion(Continue)

Derive a Taylor expansion of the value of the portfolio P to quantify themagnitude of value changes dP that are triggered by small changes dy in yield.We get an approximation of the absolute change in the value of the portfolio as

dP(y) = P(y + dy)− P(y) ≈ P′(y) dy ,

where

P′(y) = −

m∑i=1

Fi (ti − t)(1 + y

)ti−t+1.




$Duration

The derivative of the bond value function with respect to the YTM is known asthe $duration (or sensitivity) of portfolio P,

$Dur(P(y)

)= P

′(y) = −

m∑i=1

Fi (ti − t)(1 + y

)ti−t+1.

Note that $Dur < 0. This means that the relationship between price andyield is negative; higher yields imply lower prices.




Modified Duration

Dividing dP(y) by P(y), we obtain an approximation of the relative change invalue of the portfolio as

dP(y)

P(y)≈

P ′(y)

P(y)dy = −MD

(P(y)

)dy ,

where

MD(P(y)

)= −

P ′(y)

P(y)= −

$Dur(P(y)

)P(y)

=m∑i=1

Fi (ti − t)

P(y)(1 + y

)ti−t+1

is known as the modified duration (MD) of portfolio P.




$Duration & Modified Duration with Semiannual Payments

Considering a bond with semiannual coupon payments. From the price of thebond

P(y) =

m∑i=1

Fi(1 + y

2

)2(ti−t),

we obtain the derivative of the bond price with respect to the YTM y as

P′(y) = −

m∑i=1

Fi (ti − t)(1 + y

2

)2(ti−t)+1.

$Dur(P(y)

)= P

′(y) and MD(P(y)

)= −

P ′(y)

P(y).




Application of $Duration & Modified Duration

The $duration and the modified duration enable us to compute the absoluteprofit and loss (P&L) and the relative P&L of portfolio P for a small changeΔy of the YTM (e.g., 10 bps or 0.1% as expressed in percentage) using:

Absolute P&L ≈ $Dur ×Δy ,

Relative P&L ≈ −MD ×Δy .

The $duration and the modified duration are also measures of thevolatility of a bond portfolio.




Basis Point Value

Another standard measure is the basis point value (BPV), also known as pricevalue of a basis point (PVBP) and dollar value of a one basis point (DV01),which is the change in the bond price given a basis point change in the bond’syield.

BPV is given by the following equation:

BPV =MD × P

10, 000= −

$Dur

10, 000.

BPV is typically used for hedging bond positions.




Exercise

Consider below a bond with the following features:

Maturity: 10 years.

Coupon rate: 6%.

YTM: 5%.

Price: 107.72.

Assume that the coupon frequency and compounding frequency are annual.

1 What is $duration, modified duration and BPV of the bond?

2 If the YTM increases by 10 bps, what is the absolute and relative P&L?




Answer




Macaulay Duration, $Duration & Modified Duration

There are three different notions of duration. We have defined the $durationand the modified duration, which are used to compute the absolute P&L andthe relative P&L of the bond portfolio for a small change in the YTM.

The third one is the Macaulay duration, and it is defined as

D = D(P(y)

)= −(1 + y)

P ′(y)

P(y)=

m∑i=1

Fi (ti − t)

P(y)(1 + y

)ti−t.

The Macaulay duration may be interpreted as a weighted averagematurity for the portfolio. The weighted coefficient for each maturityti − t is equal to

ωi =Fi

P(y)(1 + y

)ti−t.




Macaulay Duration, $Duration & Modified Duration(Continue)

Note that the Macaulay duration is always less than or equal to thematurity, and is equal to it if and only if the portfolio has only one cashflow (e.g., zero-coupon bond).

The Macaulay duration of a bond or bond portfolio is the investmenthorizon such that investors with that horizon will not care if interest ratesdrop or rise as long as changes are small. In other words, capital gain riskis offset by reinvestment risk.




Exercise

Consider a 3-year standard bond with a 5% YTM and a $100 face value, whichdelivers a 5% coupon rate. Coupon frequency and compounding frequency areassumed to be annual. Its price is $100.

1 What is the Macaulay duration D of the bond?

2 Suppose that the YTM changes instantaneously to the following level, andstays at this new level during the life of the bond.

1 4%.2 4.5%.3 4.8%.4 5.2%.5 5.5%.6 6%.

After D years, what is the bond price at the new YTM level? And what isthe amount of reinvested coupons?




Answer




Answer




Relationships between the Different Duration Measures

Note that there is a set of simple relationships between the three differentdurations:

MD =D

1 + y,

$Dur = −MD × P = −D

1 + y× P.

When the coupon frequency and the compounding frequency of a bond areassumed to be semiannual, the two relationships are affected in the followingmanner:

MD =D

1 + y

2

,

$Dur = −MD × P = −D

1 + y

2

× P.




Properties of the Different Duration Measures

The main properties of Macaulay duration, modified duration and $durationmeasures are as follows:

The Macaulay duration of a zero-coupon bond equals its time to maturity.

Holding the maturity and the YTM of a bond constant, the bond’sMacaulay duration (or modified duration or $duration) is higher when thecoupon rate is lower.

Holding the coupon rate and the YTM of a bond constant, its Macaulayduration (or modified duration) increases with its time to maturity as$duration decreases.

Holding other factors constant, the Macaulay duration (or modifiedduration) of a coupon bond is higher as $duration is lower when thebond’s YTM is lower.




Properties of the Different Duration Measures(Continue)

The Macaulay duration of a perpetual bond that delivers an annualcoupon over an unlimited horizon and with a YTM equal to y is (1+ y)/y .

Macaulay duration is a linear operator. In other words, the Macaulayduration of a portfolio P invested in n bonds denominated in the samecurrency with weights wi is the weighted average of each bond’s Macaulayduration Di :

DP =

n∑i=1

wi Di .

Note that this linearity property (also holds for modified duration), is onlytrue in the context of a flat curve. When the YTM curve is no longer flat,this property becomes false and may only be used as an approximation ofthe true Macaulay duration (or modified duration).




Hedging in Practice

Suppose we want to hedge a bond portfolio with YTM y and price P(y) (in $value).

The idea is to consider one hedging asset with YTM y1 (a priori different fromy), whose price is denoted by H(y1), and to build a global portfolio with valueP∗ invested in the initial portfolio and some quantity δ of the hedginginstrument, i.e.,

P∗ = P(y) + δH(y1).

The goal is to make the global portfolio insensitive to small interest-ratevariations. Assuming that the YTM curve is only affected by parallel shifts sothat dy = dy1, we require

dP∗ = dP(y) + δ dH(y1) ≈

(P

′(y) + δH ′(y1))dy = 0.




Hedging in Practice(Continue)

Hence,

δ = −P ′(y)

H ′(y1)= −

$Dur(P(y)

)$Dur

(H(y1)

) = −

P(y)MD(P(y)

)H(y1)MD

(H(y1)

) .The optimal amount invested in the hedging asset is simply equal to theopposite of the ratio of the $duration of the bond portfolio to be hedgedby the $duration of the hedging instrument. The hedge requires taking anopposite position in the hedging instrument. The idea is that any loss(gain) with the bond has to be offset by a gain (loss) with the hedginginstrument..

In practice, it is preferable to use futures contracts or swaps instead ofbonds to hedge a bond portfolio because of significantly lower costs andhigher liquidity.




Hedging Ratio

When standard swaps are used as hedging instruments, the hedge ratio (HR)δS is

δS = −$DurP$DurB

,

where $DurP and $DurB are, respectively, the $duration of the portfolio to behedged and that of the fixed-coupon bond contained in the swap.

When futures are used as hedging instruments, the HR δf is

δf = −$DurP$DurCTD

× CF ,

where $DurCTD is the $duration of the cheapest-to-deliver bond, and CF theconversion factor.




Basis Point Value

Another measure commonly used by market participants to hedge their bondpositions is the BPV measure. In this case, the HR is

HR =BPVB

BPVH

×Change in bond yield

Change in yield of the hedging instrument,

where BPVB and BPVH are, respectively, the basis point value of the bond tobe hedged and that of the hedging instrument. The second ratio in theequation is sometimes called the yield ratio.



Relaxing the Assumption of a Small ShiftRelaxing the Assumption of a Parallel Shift

Overview

Duration hedging is very simple, but it is based upon the following, veryrestrictive, assumptions: (1) it is explicitly assumed that changes in theyield curve will be small, and (2) it is also assumed that the yield curve isonly affected by parallel shifts.

As large variations can affect the yield-to-maturity (YTM) curve and thatthree main factors (level, slope and curvature) have been found to drivethe dynamics of the yield curve, these strongly suggests that durationhedging, based on a one-dimensional measure of the bonds sensitivity tointerest-rate movements, is inefficient in many circumstances.

Relaxing the assumption of a small change in the yield curve can beperformed though the introduction of a convexity adjustment in thehedging procedure. Convexity is a measure of the sensitivity of $durationwith respect to yield changes.




Overview(Continue)

Accounting for general, nonparallel deformations of the term structure isnot easy because it increases the dimensionality of the problem. Becauseit is never easy to hedge the risk associated with too many sources ofuncertainty, it is always desirable to try and reduce the number of riskfactors and identify a limited number of common factors. The ways thiscan be done are to some extent arbitrary. In that context, it is importantto know the exact assumptions one has to make in the process, and try toevaluate the robustness of these assumptions with respect to the specificscenario in mind.




Duration Hedging

Duration hedging is very simple. However, one should be aware that themethod is based upon the following, very restrictive, assumptions:

It is explicitly assumed that the value of the portfolio could beapproximated by its first-order Taylor expansion. This assumption is allthe more critical, as the changes in interest rates are larger. In otherwords, the method relies on the assumption of small YTM changes. Thisis why the hedge portfolio should be readjusted reasonably often.

It is also assumed that the yield curve is only affected by parallel shifts. Inother words, the interest-rate risk is simply considered as a risk on thegeneral level of interest rates.




A Second-Order Taylor Expansion

Duration hedging only works for small yield changes, because the price of abond as a function of yield is nonlinear. In other words, the $duration of abond changes as its yield changes. When a portfolio manager expects apotentially large shift in the term structure, a convexity term should beintroduced. The price approximation can be improved if one can account forsuch nonlinearity by explicitly introducing a convexity term.

To consider the impact of a larger change dy in the yield on a bond value, oneneeds to write at least a second-order version of the Taylor expansion:

dP(y) ≈ P′(y) dy +

1

2P

′′(y) (dy)2

= $Dur(P(y)

)dy +

1

2$Conv

(P(y)

)(dy)2,

where




$Convexity & Convexity

P′′(y) =

m∑i=1

Fi (ti − t)(ti − t + 1)(1 + y

)ti−t+2≡ $Conv

(P(y)

)is the second derivative of the bond value function with respect to YTM y , alsodenoted by $Conv

(P(y)

), and is known as the $convexity of the bond P.

The relative change in the value of the bond is

dP(y)

P(y)≈ −MD

(P(y)

)dy +

1

2RC

(P(y)

)(dy)2,

where RC(P(y)

)=

P ′′(y)

P(y)is called the (relative) convexity of bond P.




Exercise

Consider a bond with the following features:

Maturity: 10 years.

Coupon rate: 6%.

YTM: 5%.

Price: 107.72173.

Coupon frequency and compounding frequency are assumed to be annual.

1 What is the $convexity of bond?

2 If the YTM changes by 50 bps, what is the absolute and relative gain orloss due to convexity?




Answer




$Convexity & Convexity with Semiannual Payments

The price of a bond with semiannual payments given by

P(y) =m∑i=1

Fi(1 + y

2

)2(ti−t).

From this we obtain the following expression for P ′′(y), which is the secondderivative of the bond price P(y) with respect to the YTM y ,

P′(y) = −

m∑i=1

Fi (ti − t)(1 + y

2

)2(ti−t)+1

P′′(y) =

m∑i=1

Fi (ti − t)(ti − t + 1

2

)(1 + y

2

)2(ti−t+1).




Properties of Convexity

The properties of the convexity and $convexity measures are as follows:

For a given bond, the change in value due to the convexity term is alwayspositive.

Holding the maturity and the YTM of a bond constant, the lower thecoupon rate, the higher its convexity and the lower its $convexity.

Holding the coupon rate and the YTM of a bond constant, its convexityand $convexity increase with its time to maturity.

Holding other factors constant, the convexity and $convexity of a couponbond are higher when the bonds YTM is lower.

Another convenient property of convexity is that it is a linear operator. Inother words, the convexity of a portfolio P invested in n bondsdenominated in the same currency with weights wi is the weightedaverage of each bonds convexity:

RCP =

n∑i=1

wi RCi .




Convexity When the YTM Curve is Not Flat

This property of linearity is only true in the context of a flat yield curve (as forMacaulay duration and modified duration). When the YTM curve is no longerflat, this property becomes false and may only be used as an approximation ofthe true convexity.

Consider three bonds with the following features:

Bond Maturity Coupon YTM Price Macaulay Modified Convexity(Years) Duration Duration

1 2 5% 5.00% 100.00 1.95238 1.85941 5.26941

2 7 5% 6.50% 91.77 6.02855 5.66061 40.35356

3 15 5% 7.50% 77.93 10.28578 9.56816 123.80798

Coupon frequency and compounding frequency are assumed to be annual.




Convexity When the YTM Curve is Not Flat(Continue)

Create a bond portfolio with a unit quantity of each of these three bonds.

To determine the modified duration and the convexity of this portfolio, we firstestablish the cash flows of the portfolio and search for the YTM of theportfolio. Then, we use the standard formula of Macaulay duration, modifiedduration and convexity.

The details about the bond portfolio is as follow:

Maturity YTM Price Macaulay Modified Convexity(Years) Duration Duration

15 6.84397% 269.70542 5.92492 5.54540 54.59394




Convexity When the YTM Curve is Not Flat(Continue)

If we use the weighted average of the bond’s Macaulay duration, modifiedduration and convexity, which are approximation of the real values, the errorsare given in the following table:

Macaulay Modified ConvexityDuration Duration

Approximation 5.74735 5.38032 51.45964

True value 5.92492 5.54540 54.59394

Difference -0.17758 -0.16508 -3.13429

% difference -2.99709% -2.97683% -5.74110%

Using approximation in this example, we can see that we underestimate theMacaulay duration by 3.00%, the modified duration by 2.98% and theconvexity by 5.74%.




Hedging Method

One needs to introduce two hedging assets with prices in $ denoted by H1 andH2, and YTM by y1 and y2, respectively, in order to hedge at the first andsecond order, the interest-rate risk of a portfolio with price in $ denoted by P,and YTM by y .

The goal is to obtain a portfolio that is both $duration-neutral and$convexity-neutral. The optimal quantity (δ1, δ2) of these two assets to hold isthen given by the solution to the following system of equations, at each date,assuming that dy = dy1 = dy2:

H′1(y1) δ1 + H

′2(y2) δ2 = −P

′(y),

H′′1 (y1) δ1 + H

′′2 (y2) δ2 = −P

′′(y).




Hedging Method(Continue)

These translates into

$Dur(H1(y1)

)δ1 + $Dur

(H2(y2)

)δ2 = −$Dur

(P(y)

),

$Conv(H1(y1)

)δ1 + $Conv

(H2(y2)

)δ2 = −$Conv

(P(y)

).

Therefore,(δ1

δ2

)= −

($Dur

(H1(y1)

)$Dur

(H2(y2)

)$Conv

(H1(y1)

)$Conv

(H2(y2)

))−1 (

$Dur(P(y)

)$Conv

(P(y)

)).




Exercise

Suppose the portfolio to be hedged is a Treasury-bond portfolio (with bonds ofvarious maturities) that has the following characteristics:

YTM Price Modified Duration Convexity

5.143% $32,863,500 6.760 85.329

The hedging instruments have the following features:

Asset Maturity Coupon YTM Price Modified Convexity(Years) Duration

1 3 7% 4.098% 108.03836 2.70488 10.16837

2 7 8% 4.779% 118.78811 5.48609 38.96242

3 12 5% 5.233% 97.96173 8.81336 99.08101

What hedging strategy would you recommend?




Answer




Multi-Factor Models

A major shortcoming of single-factor models is that they imply that allpossible zero-coupon rates are perfectly correlated, making bondsredundant assets.

We know, however, that rates with different maturities do not alwayschange in the same way. In particular, long-term rates tend to be lessvolatile than short-term rates.

An empirical analysis of the dynamics of the interest rate term structureshows that two or three factors account for most of the yield curvechanges. They can be interpreted, respectively, as level, slope andcurvature factors.

This strongly suggests that a multi-factor approach should be used forpricing and hedging fixed-income securities.




A Common Principle

There are different ways to account for nonparallel deformations of the termstructure. The common principle behind all techniques as follows:

Let us express the price in $ of the portfolio using the whole curve ofzero-coupon rates, where we now make explicit the time-dependency ofthe variables:

Pt =m∑i=1

Fi(1 + R(t, ti − t)

)ti−t.

i.e., we consider Pt to be a function of the zero-coupon rates R(t, ti − t),which will be denoted by R i

t for simplicity of exposition.

The risk factor is the yield curve as a whole, represented a priori by m

components, as opposed to a single variable, the YTM y . The wholepoint is to narrow down this number of factors in the least arbitrary way.




A Common Principle(Continue)

Starting from a first-order Taylor expansion of the portfolio value, wherewe treat Pt as a function of the rates R1

t ,R2t , . . . ,R

mt :

dPt ≈

m∑i=1

∂Pt

∂R it

dRit .

Assume that the investor is willing to use as many hedging assets, withprice in $ denoted by H(j), as there are different risk factors, i.e., m in thiscase. This assumption is restrictive as it is not convenient, and possiblyexpensive, to use many hedging assets.

The price of each of these hedging assets is also a function of the differentthe rates R1

t ,R2t , . . . ,R

mt . Thus, for j = 1, 2, . . . ,m,

dH(j)t ≈

m∑i=1

∂H(j)t

∂R it

dRit .





Construct a global hedge portfolio

P∗t = Pt +

m∑j=1

δ(j)t H

(j)t ,

such that, up to the first order, dP∗t = 0. Since

dP∗t ≈

m∑i=1

(∂Pt

∂R it

+m∑j=1

δ(j)t

∂H(j)t

∂R it

)dR

it ,

a necessary and sufficient condition to have dP∗t = 0, up to a first-order

approximation for any set of small variations dR it , is

∂Pt

∂R it

+m∑j=1

δ(j)t

∂H(j)t

∂R it

= 0, for i = 1, 2, . . . ,m.





Solving this linear system for δ(j)t , for j = 1, 2, . . . ,m, at each trading date

t gives the optimal hedging strategy.

Let

H′t =

(∂H

(j)t

∂Rit

)m×m

δt =(δ(i)t

)m×1

and P′t =

(∂Pt

∂Rit

)m×1

The linear system of equations is

P′t + H

′t δt = 0.

If the matrix H ′t is invertible, which means that no hedging asset price

may be a linear combination of the other m − 1 prices,

δt = −

(H

′t

)−1P

′t .




Cross-Hedge Risk

It sometimes happens that the value of the hedging assets may depend on riskfactors slightly different from those affecting the hedged portfolio. This iscalled correlation risk or cross-hedge risk.

Assume, for the sake of a simple exposition, that there is only one risk factor,which we denote by Rt . We write

Pt = P(Rt) and Ht = H(R ′t ),

where R ′t is a priori slightly different from Rt .

A priori one should always try to minimize that difference by selecting anappropriate hedging asset.




Cross-Hedge Risk(Continue)

The question is: once the hedging asset has been selected, what can be done a

posteriori to improve the hedge efficiency?

We have

dP∗t ≈

∂Pt

∂Rt

dRt + δt∂Ht

∂R ′t

dR′t .

The usual prescription

δt = −∂Pt

∂Rt

/∂Ht

∂R ′t

will fail to apply successfully because dR ′t may be different from dRt , which is

precisely what correlation risk is all about.





One may handle the situation in the following way:

Let us first consider the convenient situation in which one could express R ′t as

some function of Rt :R

′t = f

(Rt

).

In that case, we have

dP∗t ≈

∂Pt

∂Rt

dRt + δt∂Ht

∂R ′t

f′(Rt

)dRt .

Then

dP∗t ≈ 0 ⇐⇒ δt = −

∂Pt

∂Rt

/(∂Ht

∂R ′t

f′(Rt

)).

Therefore, we may keep the usual prescription provided we amend it in order toaccount for the sensitivity of one factor with respect to the other.





Unfortunately, it is not generally possible to express R ′t as a function f

(Rt

).

However, a satisfying solution may be found using a statistical estimation of thefunction f

(Rt

). We may, for example, assume a simple linear relationship, i.e.,

R′t = f

(Rt

)= a+ b Rt + εt ,

where εt is the usual error term, and the parameters are estimated usingstandard statistical tools. Because of cointegration and non-stationarity of theseries, it is better to consider a linear relationship in variations rather than inlevel. In this case, we have

dR′t = a+ b dRt + εt .





In either case, we getdR

′t ≈ b dRt .

Hence, we should amend the hedge ratio in the following way:

δt = −∂Pt

∂Rt

/(b∂Ht

∂R ′t

).

This method is as accurate as the quality of the linear approximation(measured through the squared correlation factor). This will change thehedging strategy and improve the efficiency of the method in the case when b

is significantly different from 1.




Exercise

Suppose that a portfolio manager invests a nominal amount of $50,000,000 ina bond A whose gross price and modified duration are, respectively, 93.274 and8.319. He fears a rate increase and wants to protect his investment.

Suppose that the hedging instrument is a bond B whose gross price andmodified duration are, respectively, 105.264 and 7.04. Nominal amount of bondB is $1,000.

Assume that changes in yields are not equal and that the relationship betweenthe YTM of bond A denoted by yA and the YTM of bond B denoted by yB isequal to

ΔyA ≈ 1.18 ×ΔyB .

What is the optimal hedging strategy for the portfolio manager?




Answer




Remark

In practice, it is more realistic that a hedger does not want to use as manyhedging assets as there are different risk factors. The principle isinvariably to aggregate the risks in the most sensible way to reduce thenumber of risk factors.

There is a systematic method to do so using results from a principalcomponents analysis (PCA) of the interest-rate variations. This is thestate-of-the-art technique for dynamic interest-rate hedging.




Regrouping Risk Factors Using PCA

The purpose of PCA is to explain the behavior of observed variables usinga smaller set of unobserved implied variables.

From a mathematical standpoint, PCA consists of transforming a set of mcorrelated variables into a reduced set of orthogonal variables thatreproduces the original information present in the correlation structure.

PCA can yield interesting results, especially for the pricing and riskmanagement of correlated positions. Using this tool with historicalzero-coupon rate curves, one can observe that the first three principalcomponents of spot curve changes explain the main part of the returnvariations on fixed-income securities over time.

These three factors, namely, level, slope and curvature, are believed todrive interest-rate dynamics and can be formulated in terms ofinterest-rate shocks, which can be used to compute principal componentdurations.




Regrouping Risk Factors Using PCA(Continue)

Express the change ΔR(t, si ) = R(t +Δt, si )− R(t, si ) of the zero-couponrate R(t, si ) with maturity si at date t such that

ΔR(t, si ) =

p<m∑k=1

ckiCkt + εti ,

where εti is the residual that is not explained by the factor model; C kt is the

value of the k-th factor at date t; and cki is the sensitivity of the i-th variableto the k-th factor defined as

Δ(ΔR(t, si )

)ΔC k

t

= cki ,

which amounts to individually applying a 1% variation to each factor, andcomputing the absolute sensitivity of each zero-coupon yield curve with respectto that unit variation. These sensitivities are commonly called the principalcomponent $durations.





Suppose p = 3. In this case, we need 3 hedging assets. The change in thevalue of a fixed-income portfolio is

ΔP∗t ≈

m∑i=1

[(∂Pt

∂R it

+

3∑j=1

δ(j)t

∂H(j)t

∂R it

)ΔR

it

].

Since ΔR(t, si ) ≈∑3

k=1 ckiCkt , we have

ΔP∗t ≈

m∑i=1

[(∂Pt

∂R it

+3∑

j=1

δ(j)t

∂H(j)t

∂R it

)×

3∑k=1

ckiCkt

]

=

3∑

k=1

[m∑i=1

cki

(∂Pt

∂R it

+3∑

j=1

δ(j)t

∂H(j)t

∂R it

)]C

kt .





The quantity∑m

i=1 cki

(∂Pt

∂Rit

+∑3

j=1 δ(j)t

∂H(j)t

∂Rit

)is commonly called the principal

component $duration of portfolio P∗ with respect to factor k .

To set the first-order variations in the hedged portfolio P∗ to zero, i.e.,ΔP∗

t = 0, for any possible evolution of the interest rates R(t, si ), orequivalently for any possible evolution of the C k

t , a sufficient condition for thisis

m∑i=1

cki

(∂Pt

∂R it

+3∑

j=1

δ(j)t

∂H(j)t

∂R it

)= 0, for k = 1, 2, 3,

i.e., neutral principal component $durations.

On each possible date, there are three unknowns δ(j)t in three linear equations.





Let us introduce

H′t =

(∑m

�=1 ci�∂H

(j)t

∂R�t

)3×3

δt =(δ(i)t

)3×1

and P′t =

(∑m

�=1 ci�∂Pt

∂R�t

)3×1

The linear system of equations may be represented in the following way:

P′t + H

′t δt = 0.

The solution is given by

δt = −

(H

′t

)−1P

′t .

In practice, we need to estimate the principal component $durations usedat date t. They are derived from a PCA performed on a period prior to t,e.g., the prior 3 months. Hence, the result of the method is stronglysample-dependent.




Hedging Using PCA

How does one use a PCA of the yield curve changes to hedge a portfolio?

Consider a portfolio to be hedged P that contains a set of Treasury bonds withvarious maturities.

The price Pt of that portfolio at date t is given by

Pt =m∑i=1

Ft+si(1 + R(t, si )

)si ,where m is the number of cash flow of portfolio P, Ft+si is the cash flow ofportfolio P to be received at date t + si , and R(t, si ) is the zero- coupon rateat date t for maturity si , for i = 1, . . . ,m.




Hedging Using PCA(Continue)

The change ΔPt in the portfolio value between dates t and t +Δt is

ΔPt = Pt+Δt − Pt =

m∑i=1

∂Pt

∂R(t, si )ΔR(t, si )

=

m∑i=1

(−si Ft+si(

1 + R(t, si ))si+1

)ΔR(t, si ).

Using the factor representation of ΔR(t, si ), we have

ΔPt ≈

m∑i=1

(−si Ft+si(

1 + R(t, si ))si+1

) (p∑

k=1

ckiCkt

).





Suppose we only hedge the first three factors so that

ΔPt ≈ −

3∑k=1

(m∑i=1

cki

(si Ft+si(

1 + R(t, si ))si+1

))︸︷︷︸

βPk

Ckt = −

3∑k=1

βPk C

kt .

Suppose, at date t, the hedging portfolio contains assets with prices H(j)t and

quantities δ(j)t . Given the additional self-financing constraint, the hedging

portfolio contains four assets instead of three. Using a similar reasoning asabove, the variation in the hedging portfolio value ΔHt δt is given by

ΔHt = −

(3∑

k=1

βH(j)t

k Ckt

)1≤j≤4

,

ΔHt δt = −

4∑j=1

3∑k=1

βH(j)t

k Ckt δ

(j)t .





The idea is to set the sensitivity of the global portfolio (i.e., the portfolio to behedged plus the hedging portfolio) to zero with respect to the factors C k

t . Asufficient condition for this is where the global portfolio value is zero at date t,i.e.,

ΔPt +ΔHt δt = 0 and Pt + Ht δt = 0,

or

βPk +

4∑j=1

βH(j)t

k δ(j)t = 0, for k = 1, 2, 3, and Pt +

4∑j=1

H(j)t δ

(j)t = 0.

The quantities δ(j)t to hold in the hedging assets are the solutions to that linear

system.




Hedging Using a Three-Factor Model of the Yield Curve

The estimated principal component $durations is sample-dependent. Forestimation purposes, it is more convenient in practise to use somefunctional specification for the zero-coupon yield curve that is consistentwith results from a PCA.

The idea here consists of using a model for the zero-coupon rate functionsuch as the Nelson and Siegel model and the Svensson model.




Nelson-Siegel and Svensson Models

Nelson and Siegel (1987) modeled the continuously compounded zero-couponrate at time zero with maturity s, denoted by Rc(0, s), as

Rc(0, s) = β0 + β1

1− e−s/τ1

s/τ1+ β2

[1− e

−s/τ1

s/τ1− e

−s/τ1

],

a form that was later extended by Svensson (1994) as

Rc(0, s) = β0 + β1

1− e−s/τ1

s/τ1+β2

[1− e

−s/τ1

s/τ1− e

−s/τ1

]

+ β3

[1− e

−s/τ2

s/τ2− e

−s/τ2

].




Nelson-Siegel and Svensson Models(Continue)

Interpretation of the parameters:

β0 is the limit of Rc(0, s) as s goes to infinity. In practice, β0 should beregarded as a long-term interest rate.

β1 is the limit of Rc(0, s)− β0 as s goes to 0. In practice, β1 should beregarded as the long-term to short-term spread.

β2 and β3 are a curvature parameters.

τ1 and τ2 is a scale parameter that measure the rate at which theshort-term and medium-term components decay to zero.

The parameters β0, β1, β2 and β3 are estimated daily by using an ordinary leastsquares (OLS) optimization program, which consists, for a basket of bonds, inminimizing the sum of the squared spreads between the market price and thetheoretical price of the bonds as obtained with the model.





The evolution of the zero-coupon rate Rc(0, s) is entirely driven by theevolution of the beta parameters, the scale parameters being fixed.

In an attempt to hedge a bond, one should build a global portfolio with thebond and a hedging instrument, so that the global portfolio achieves a neutralsensitivity to each of the beta parameters. To implement the method, onetherefore needs to compute the sensitivities of any arbitrary portfolio of bondsto each of the beta parameters.

Consider a bond that delivers principal or coupon and principal denoted by Fi

at dates si . Its price in $ at date t = 0, denoted by P0, is given by the followingformula:

P0 =∑i

Fi e−si R

c (0,si ).





In the Nelson and Siegel, and Svensson models, at date 0, the $durationDi = ∂P0/∂βi of the bond P to the parameter βi , for i = 0, 1, 2, 3, are given bythe following formulas:

D0 = −

∑i

si Fi e−si R

c (0,si ),

D1 = −

∑i

1− e−si/τ1

si/τ1si Fi e

−si Rc (0,si ),

D2 = −

∑i

(1− e

−si/τ1

si/τ1− e

−si/τ1

)si Fi e

−si Rc (0,si ),

D3 = −

∑i

(1− e

−si/τ2

si/τ2− e

−si/τ2

)si Fi e

−si Rc (0,si ).




Exercise

Suppose the values of the parameters in the Nelson-Siegel model are:

β0 β1 β2 τ1

8% −3% −1% 3

Consider three bonds with the following features:

Bond Maturity Coupon Price

1 2 5% 98.62726

2 7 5% 90.78630

3 15 5% 79.60619

Coupon frequency is annual and compounding frequency is continuous.

Calculate the $durations of the bonds.

Calculate the $durations for a portfolio composed of one unit of bond 1,bond 2 and bond 3.




Answer




Hedging Method

Hedging in the Svensson model requires one to create a global portfolio withthe bond portfolio to be hedged, whose price in $ is denoted by P, and fourhedging instruments, whose prices in $ are denoted by Gi , for i = 1, . . . , 4, andto make it neutral to changes in parameters βi ’s.

We look for the quantities δi to invest in the hedging instruments Gi , fori = 1, . . . , 4, which satisfy the following linear system:

δ1∂G1

∂β0+ δ2

∂G2

∂β0+ δ3

∂G3

∂β0+ δ4

∂G4

∂β0= −D0

δ1∂G1

∂β1+ δ2

∂G2

∂β1+ δ3

∂G3

∂β1+ δ4

∂G4

∂β1= −D1

δ1∂G1

∂β2+ δ2

∂G2

∂β2+ δ3

∂G3

∂β2+ δ4

∂G4

∂β2= −D2

δ1∂G1

∂β3+ δ2

∂G2

∂β3+ δ3

∂G3

∂β3+ δ4

∂G4

∂β3= −D3.




Hedging Method(Continue)

In the Nelson and Siegel model, we need only three hedging instrumentsbecause there are only three parameters. Set δ4 = 0, and the last equationof linear system disappears.

There is ample empirical evidence that changes in the yield curve can belarge and multidimensional, thus duration hedging techniques achievelimited efficiency in most market conditions.

By implementing semi-hedged strategies based on three-factor models, aportfolio manager can take specific bets on particular changes in the yieldcurve while being hedged against the others.


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