JPM Smooth Interpolation of Zero Curves

8/8/2019 JPM Smooth Interpolation of Zero Curves

1/48

J.P. Morgan/Reuters RiskMetrics

Monitor

Research, Development, and Applications

An investigation into term structure estimation methods for RiskMetrics 3

Prices of discount (zero coupon) bonds and their corresponding (spot) interest rates are animportant data source for the RiskMetrics product. RiskMetrics uses prices of discountbonds to calculate the volatility and correlation of future cashows as well as to mark-to-market those cashows. Since in many markets the prices of discount bonds are notobserved, RiskMetrics constructs such prices (and interest rates) by estimating a so-calledterm structure of interest rates from a set of coupon bonds issued by governments. This termstructure is estimated each day that RiskMetrics produces volatility and correlation data les.

The purpose of this article is three-fold. First, we explain an algorithm to generate spot ratesfrom coupon-bearing government bonds that RiskMetrics plans to implement in its produc-tion process. This algorithm, which is based on the bootstrap procedure, will replace the cur-

rent term structure estimation method. Second, as background material for the discussion onterm structure estimation methods, we present several denitions related to discount andcoupon bonds, and explain various bond market conventions (e.g., coupon payment fre-quency, day count basis, etc.). And third, we present alternative term structure estimationmethods that are popular among academics and practitioners.

When is a portfolio of options normally distributed? 33

When considering a portfolio of options, a risk manager is faced with two contrasting piecesof information. On the one hand, the manager is well aware that the distribution of the returnon any one of the options is asymmetric and certainly not normal. On the other hand, themanager knows that when a large number of independent options are considered, even if their individual distributions are not normal, the distribution of their sum will be close to nor-mal. In this article, we take up two practical questions: rst, how large must a portfolio of

independent options be for the portfolio distribution to appear normal; and second, is it pos-sible that reasonably uncorrelated portfolios of options will also appear to be normally dis-tributed if the portfolios are large enough?

Previous editions of the RiskMetrics Monitor 45

RiskMetrics News

SEC issues nal rule on the disclosure of market risks in nancial instruments and deriva-tives

The rst issue of the CreditMetrics Monitor is planned for release in November 1997

Reuters has taken over production of RiskMetrics datasets

RIMES Technologies offers its Windows application, HistDB, on the web

Third quarter 1997

New York September 15, 1997

Morgan Guaranty Trust CompanyRisk Management ResearchPeter Zangari

(1-212) 648-8641

[email protected]

Reuters LtdInternational MarketingMartin Spencer(44-171) 542-3260

[email protected]


2/48

RiskMetrics MonitorThird Quarter 1997page 2

Scott HowardMorgan Guaranty Trust CompanyRisk Management Advisory(1-212) [email protected]

SEC issues nal rule on the disclosure of market risks in nancial instruments andderivatives

Below is a synopsis of who, when and what has to be disclosed about the market risk in nancial instru-ment and derivatives, as dened, and other nancial instruments (i.e., those in scope of FAS 107). Mate-riality is measured by, a) fair value at the end of the period (netting is permitted to the extent of FIN 39),or b) potential loss of future earnings, fair values, or cash ows from reasonably possible near-term marketmovements.

Who

: SEC registrants with material market risk exposures (e.g., interest rate, foreign currency, commodityand equity price risk) arising from all

nancial instruments, including derivatives.

When

: Effective for periods ending after June 15, 1997 for banks, thrifts and registrants with market cap-italization of $2.5 billion as of January 28, 1997. For other registrants one year later.

What

: If a registrant has material market risk, they must provide outside of nancial statements quantita-tive and qualitative disclosures about such risks separately for trading and nontrading portfolios. Three al-ternative exist:

Tabular

- Fair value information and contract terms relevant to determining future cash ows, cat-egorized by maturity, grouped based on instrument characteristics.

Sensitivity Analysis

- Potential loss by risk type in future earning, fair values or cash ows fromselected hypothetical changes in market rates and prices.

Value at Risk

- Potential loss by risk type in future earning, fair values or cash ows from marketmovements over a selected period of time and with a selected likelihood of occurrence.

The rst issue of the CreditMetrics Monitor is planned for release in November 1997

The upcoming CreditMetrics Monitor will contain several articles. Likely topics will be: (a) additionalproduct coverage, e.g., credit derivatives, b) estimation of additional dimensions of risk,e.g.,credit spreadvolatility, c) recovery rate correlation modeling, d) a case study discussion of model outputs, sensitivitiesand applications, e)detailed illustrations of relevant calculations to include forward curve estimation

Reuters has taken over production of RiskMetrics

Since mid August the Reuters web and ftp sites (http:/ /www.riskmetrics.reuters.com/WDown4.htm or ftp:/ /ftp.riskmetrics.reuters.com/datasets/) are the only places to get the RiskMetrics datasets. They are postedon the same schedule as before.

RIMES Technologies offers its Windows application, HistDB, on the web

The RIMES HistDB offers access to historical data, e.g., economic, price, that can be combined and ana-lyzed together into a familiar Windows NT application. DEaR and VaR measurements usingthe RiskMet-rics methodology can be computed on-the-y using any of the assets (or custom porfolios) availablein the system. RIMES can be found at http://www.rimes.com.

RiskMetrics News

RiskMetrics and FourFifteen are registered trademark of J.P. Morgan in the United States and in other countries. They are written with the symbol on its rst occurrence and RiskMetrics and FourFifteen thereafter.


3/48

RiskMetrics Monitor Third Quarter 1997 page 3

Peter ZangariMorgan Guaranty Trust CompanyRisk Management Research(1-212) 648-8641

[email protected]

1. Introduction and Overview

Prices of discount (zero coupon) bonds and their corresponding (spot) interest rates are an important datasource for the RiskMetrics product. RiskMetrics uses prices of discount bonds to calculate the volatilityand correlation of future cashows as well as to mark-to-market those cashows. Since in many marketsthe prices of discount bonds are not observed, RiskMetrics constructs such prices (and interest rates) byestimating a so-called term structure of interest rates from a set of coupon bonds issued by governments.This term structure is estimated each day that RiskMetrics produces volatility and correlation data les.

In its most common representation, the term structure

1

of interest rates

is the relationship between theinterest rates on default-free xed income investments, which generate only one payment (at maturity),and the maturity dates of these cashows. In other words, the term structure is the relationship betweeninterest rates on discount bonds and the maturity dates of the cashows associated with those interest rates.For any bond market, the complete term structure of interest rates is unobservable, that is to say, prices of discount bonds (spot interest rates) for a continuum of maturity dates do not exist. Therefore, we must es-timate of the term structure by applying a

term structure estimation method

to a set of observed bonds.The observed bonds are usually taken to be coupon bonds issued by governments.

2

We classify

term structure estimation methods

into two groups: theoretical

and empirical

. Theoreticalterm structure methods posit an explicit structure for coupon bond prices, whose values depend on a setof parameters that govern the mean reversion and volatility of the so-called short interest rate. Variousforms of regression analysis can be used to estimate the value of these parameters. Examples of theoreticalmethods include Vasicek (1977), and Cox et.al (1985). In fact, RiskMetrics currently uses a theoretical

term structure estimation method

to compute spot interest rates from government bonds. Once estimat-ed, these spot rates are converted to prices of discount bonds using a simple formula that relates the priceand interest rate of a discount bond.

Alternatively,

empirical methods

are available to compute spot interest rates. Unlike the theoretical meth-ods, the empirical methods are

independent of any model or theory of the term structure

. Whereas thetheoretical methods attempt to explain typical features of the term structure, which may include how theterm structure evolves through time, the empirical methods merely try to nd a close representation of theterm structure at any point in time, given some observed interest rate data. Examples of empirical methodsinclude a procedure known as

bootstrapping (e.g., Fama and Bliss, 1987), applications of splines

(e.g.,Fischer et al 1994) and exponential polynomials

(e.g., Nelson and Siegel, 1992 and Buono et. al 1992),and the maximum smoothness approach (Adams and Van Deventer, 1994).

The purpose of this article is three-fold:

First, we explain an algorithm to generate spot rates from coupon-bearing government

bonds that RiskMetrics plans to implement in its production process

. This algorithm, whichis based on the bootstrap procedure, will replace the current (theoretical) term structure estima-tion method. The principal reason for changing the term structure estimation method is that thecurrent method relies on a set of parameters that are market specic and must be updated fre-quently. The proposed (bootstrap) method, on the other hand, is much simpler to implement andcan generate comparable results which are suitable to RiskMetrics.

1

In the academic and industry literature, the phrasesterm structure model and yield curve model are sometimes usedinterchangeably.

2

For a discussion about whether one should use government bonds or interbank rates when estimating the term structure, seeOda, 1996 and Malz, 1996.

An investigation into term structure estimation methods for RiskMetrics


4/48


Second, as background material for the discussion on term structure estimation methods, wepresent several denitions related to discount and coupon bonds, and explain various bondmarket conventions

(e.g., coupon payment frequency, day count basis, etc.). This discussionincludes a review of the relationship between spot interest rates, forward interest rates andprices of discount bonds

.

Third, we present alternative term structure estimation methods

that are popular among aca-demics and practitioners. The main goal of this discussion is to streamline and clarify a somewhatdisjointed literature on term structure estimation methods. By using the same notation and deni-tions to explain six term structure estimation methods, this presentation will facilitate compari-sons between the various term structure estimation methods

.

The rest of this article is organized as follows:

Section 2 provides an overview of bond pricing notation, formulae and terminology, and consistsof 4 sub-sections.

-- Section 2.1

presents some basic denitions related to the time value of money.-- Section 2.2

explains how the term structure of interest rates can be expressed in terms of (1)spot interest rates, (2) forward interest rates, and (3) prices of discount bonds. This sectionincludes a discussion on the relationship between spot and forward interest rates.

-- Section 2.3

relies on the denitions of the previous sections to dene a coupon bond. We present a numerical example to demonstrate how to nd a coupon bonds yield-to-maturity(YTM) and how to calculate accrued interest.

-- Section 2.4

shows how a price of a coupon bond can be written as a set of discount bonds.

Section 3

reviews the data on government bond spot rates that RiskMetrics currently provides.

Section 4 presents the bootstrap procedure to extract spot interest rates from coupon government bonds.

-- Section 4.1

presents the details of the bootstrap.-- Section 4.2

discusses the practical implementation of the bootstrap. Specically,we explain how to bootstrap when either no or multiple bonds mature at a given bootstrapmaturity date.

Section 5

, which consists of 5 sub-sections, presents alternative term structure estimation meth-ods. We discuss several popular models.

-- Section 5.1

explains how to bootstrap forward interest rates.

--

Section 5.2

reviews some research that estimates the term structure of interest rates withpolynomial and exponential splines.

--

Section 5.3

introduces the exponential polynomial method to estimate the term structure of interest rates.

--

Section 5.4

discusses modeling and smoothing forward interest rates as a way to estimatethe term structure.

-- Section 5.5

demonstrates how a particular class of splines--cubic B-splines--can be appliedto estimate the term structure of interest rates.

Section 6

concludes the article with a summary and discussion.



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6/48


The day count basis for coupon bonds is used to measure the time between coupon payments. Table 1 provides a listing of some day count bases currently employed in xed income markets.

Consider a simple example where coupon payments occur on August 1, 1997 and August 15, 1997 and weuse the Actual/Actual method to measure the amount of time between payments. In this case, the basisnumerator is 14 and the denominator is 364, which yields a basis of 0.0385 years.

The day count basis is also important for calculating accrued interest

. In the context of pricing coupon bonds, accrued interest is the amount of coupon payment that accrues between the last coupon payment

Table 1Day count conventions and bases

Counting the time between the j-1st and jth coupons

(1) Actual/Actual n = the actual number of days between and

d = the actual number of days between and multiplied by the coupon frequency.

(2) Actual/360 n = the actual number of days between and

d = 360

(3) Actual/365 n = the actual number of days between and

d = 365

(4) Actual/365L n = the actual number of days between and

d = 366 if the next coupon falls within a leap year, otherwise 365.

(5) 30/360(general)

Given that , , and and , , and denote the day, month and year for

and , respectively then:

n =

d = 360

(6) 30/360 n = same as (5) except if falls on the 31st of the month, then change it to the 30th; if falls

on the 31st of the month, then change it to the 30th if falls on either the 30th or the 31st

d = 360

(7) 30E/360 n = same as (5) except if either or falls on the 31st of the month, then change it to the 30th

d = 360

(8) 30E+/360 n = same as (5) except if falls on the 31st of the month, then change it to the 30th; if falls

on the 31st, then change it to the 1st and increase by 1

d = 360

t j 1 t j

t j 1 t j

t j 1 t j

t j 1 t j

t j 1 t j

d j 1 m j 1 y j 1 d j m j y j

t j 1 t j

d j d j 1 30 m j m j 1( ) 360 y j y j 1( )+ +

d j 1 d j

d j 1

d j 1 d j

d j 1 d j

m j



7/48


8/48


Unlike other bonds that can take on any type of compounding frequency, discount bonds have a com-pounding frequency equal to the reciprocal of their time to maturity . For example, if a discount bond

pays $1.00 in years, then the compounding frequency is and we have:

[2] or

The spot curve based on continuous compounding, referred to as the continuous spot curve , representsthe relationship between the spot rate, , and time to maturity . In contrast to the numerous discretely compounded spot curves, there is only one continuously compounded spot curve

With continuous compounding, the expression becomes where exp(x)= . In other words, converges to as approaches innity. There-fore, the price of a discount bond [1] can be written as

[3]

2.2.2 The relationship between spot and forward ratesIn this section we explain the relationship between spot and forward interest rates, assuming continuouscompounding. This relationship plays a fundamental role in various types of term structure estimationmethods.

A forward interest rate , assuming discrete compounding , is the annual interest rate contracted at timeto be paid from time and , compounded f times a year. We denote this forward rate by .

Note that the forward rate is simply an interest rate that takes effect at some future point in time. Similar to a spot rate, a forward rate is dened in terms of a compounding horizon and a time to maturity. Chart 3shows the relationship between current and future dates and the forward rate.

Chart 3Demonstration of forward interest rate time prole

Using continuous compounding , the forward interest rate in terms of the short and long in

terest rates can be written as:

[4]

where is the time between and , and is the continuously compounded forward rate between and .

Using [4], we can solve for the forward rate, , as

j 1 j

p j( ) 1 z j j1,( ) j+( ) 1= p j( ) 1 1 z j j

1,( ) j+( ) =

z j( ) j

1 z j f ,( ) f +( ) j f z j( ) j( )exp

e x 1 z j f ,( ) f +( ) j f z j( )

j( )exp f

p j( ) z j( ) j( )exp=

t 0 t 1 t 2 F t 1 t 2 f , ,( )

t0

t2t1

F(t1,t2,f)

z 1( ) z 2( )

z 1( ) 1( )exp F 2 2 1,( ) 2 1,( )exp z 2( ) 2( )exp=

2 1, t 1 t 2 F 2 2 1,( )t 1 t 2

F 2 2 1,( )



9/48


[5]

The second term in [5], , represents the slope of the continuous spot curve. In the limit, i.e., asapproaches , the forward rate for a very short period of time beginning at is

[6]

is the instantaneous forward rate for a maturity of . In general, the instantaneous forward ratedescribes the rate of return on a very short-term investment at time in the future. Using [6] and [3], wecan establish mathematical relationships between the instantaneous forward rate, the spot rate andprice of a discount bond . For example, take the natural logarithm of [3]

[7]

and then a total differential with respect to maturity

[8]

It follows from [6] that the right-hand side of [8] is the instantaneous forward rate, i.e.,

[9]

Therefore, we are left with

[10]

Next, write [10] as

[11]

integrate (sum) from time 0 to and then exponentiate. The result is,

[12]

Expression [12] is important because it allows us to write the price of a discount bond as a function of instantaneous forward interest rates . The forward rates exist at every instant in time from 0 to . Inaddition, we can derive the relationship between spot and instantaneous forward rates by taking the loga-rithm of [12], using the denition given in [3], and solving for the spot rate, .

F 2 2 1,( ) z 2( )= z 2( ) z 1( )

2 1------------------------------- 1+

z 2( )= 1 2,( ) 1+

1 2,( )2 z 2( )( ) 1 z 1( )( ) 1

F 1( ) z 2( )=dz 1( )

d 1--------------- 1+

F 1( ) 1 j

p j( )( ) z j j( ) j=log

jd d p j( )( ) z j jd

dz j j+=log

F j( ) z j jd dz j j+=

F j( ) jd d p j( )( )log=

d j F j( ) d p j( )( )log= j

F s( ) sd 0

j

p j( )=exp

j

z j( )



10/48


[13]

Expression [13] states that the spot interest rate is an average of instantaneous forward interest ratesbetween 0 and .

We now summarize the main ndings of this section . First, we derived the instantaneous forward ratefrom the simple relationship that links short, long and forward interest rates ([6]). Second, we showed howthe price of a discount bond can be written explicitly as a function of the forward rates ([12]). And third,we derived an expression that relates spot and forward interest rates ([13]). We will make use of such re-lationships when we discuss term structure estimation methods in sections 4 and 5.

2.3 Coupon bondsSuppose we have a set of N coupon bonds , i=1,2,...,N. The ith coupon bond makes regular payments,

(coupons), at times <


11/48


where denotes the analysis (current) date and and denote the previous and next coupon paymentdates, respectively.

2.2.1 A numerical exampleConsider the purchase of a coupon bond with the characteristics presented in table 2:

This bond was purchased on February 12, 1997 and it pays coupons of 4 US dollars every six-months upuntil August 15, 2001 when it matures and pays (in addition to the nal coupon) its par value of 100 USdollars. Table 3 shows the cash ow prole of this bond:

Given the price of the bond (98 US dollars) and the information presented in table 3, we use [14] to solvefor the bonds YTM which is 8.489%.

Calculating a bonds price between coupon payment days requires that we determine accrued interest.Consider the same bond discussed above, but now we would like to compute its price 21 days after (March8, 1997) the rst coupon payment (February 15, 1997). Using equation [16], the accrued interest, Ad , iscalculated as follows:

[17]

Recall that at any point in time over the life of a bond, its price can be decomposed into a clean price ,(the price that does not include accrued interest) and dirty price (the price that includes accrued inter-est).

Table 2Bond characteristicsPrice USD 98.00Coupon rate 8.000%Coupon f requency Semi-annualDay count basis Actual/Actual

Trade date 12 Feb 1997Effective date 15 Feb 1997Maturity date 15 Aug 2001

Table 3Bond characteristics

As of February 15, 1997 j (cashow) Date n d Coupon

-- 15-Feb-97 -- -- 0.0000 --

1 15-Aug-97 181 362 0.5000 42 15-Feb-98 365 368 0.9918 43 15-Aug-98 546 362 1.5083 44 15-Feb-99 730 368 1.9837 45 15-Aug-99 911 362 2.5166 46 15-Feb-00 1,095 368 2.9755 47 15-Aug-00 1,277 364 3.5082 48 15-Feb-01 1,461 368 3.9701 49 15-Aug-01 1,642 362 4.5359 104

t t t p t n

j

Ad 82--- 21

181--------- 0.4641= =

p cc

p c



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It follows that . We keep the dirty price at USD 98.000, therefore the clean price of this bond on March 8 is USD 97.536 and the YTM must satisfy:

[18]

Table 4 shows the bonds coupon dates, the bonds coupons and the number of days until the coupon pay-ments.

Table 4Bond characteristics

As of March 8, 1997, refer to table 1 for denitions

The YTM of the bond as of March 8, 1997 is 8.750%.

2.4 Expressing the price of a coupon bond as a set of discount bondsEach coupon and principal payment, taken individually, has the structure of a discount bond. Accordingly,a coupon bond can be viewed as a set of discount bonds. Such a structure allows us to relate couponbond prices to the spot curve . For example, for any coupon bond, the jth coupon payment is equivalentto a j times f period deposit at an annual rate . If this coupon payment were traded separately, its price(present value) at would be . Similarly, the price of the P principal payment at would be

.

We can formalize the relationship between coupon and discount bonds as follows. Starting with a set of Ncoupon bonds, the price of the ith bond (i=1,...,N) is given by the expression

[19]

where

is the number of cashows (including principal) generated by the ith coupon bond(j=1,..., ).

is the dirty price of the coupon bond (that is, the clean price plus accrued interest) that

j (cashow) Date n d Payment

-- 15-Feb-97 n/a n/a n/a n/a-- 8-Mar-97 21 181 0.1160 0.46411 15-Aug-97 160 362 0.4420 42 15-Feb-98 344 368 0.9348 43 15-Aug-98 525 362 1.4503 44 15-Feb-99 709 368 1.9266 45 15-Aug-99 890 362 2.4586 46 15-Feb-00 1,074 368 2.9185 47 15-Aug-00 1,256 364 3.5450 4

8 15-Feb-01 1,440 368 3.9130 49 15-Aug-01 1,621 362 4.4779 104

p c Ad p cc+=

0.4641 97.536+ 4 yc i( ) j( )exp

----------------------------- 104 yc i( ) M i( )exp

--------------------------------+ j 1=

9

=

j

c it 0 c i p j( ) t 0

P p M i( )

p c i M i,( ) w i j, p i j,( ) j 1=

M i

=

M i M i

p c i M i,( )



13/48


matures in years.

is the jth cashow generated from the ith coupon bond. Note that is equal to the lastcoupon payment plus principal.

is the time until the jth cashow of the ith coupon bond (in years)

is the price of a discount bond from time until time associated with the ith coupon bond.Recall from section 2.2.1, that the price of a discount bond, j=1,...,. , is given by

[20]

3. RiskMetrics discount bond prices and spot rates

RiskMetrics produces discount bond prices and spot interest rates ranging in maturity from 2 to 30 yearsfor the government bond markets included in the J.P. Morgan Government Bond Index as well as theIrish, ECU, and New Zealand markets. Table 5 shows the markets and the maturities of the discount

bonds for which RiskMetrics produces prices and interest rates.

Table 5RiskMetrics government bond spot interest rates

As was discussed in the Introduction and Overview to this article, RiskMetrics currently relies on a theo-retical term structure estimation method to construct spot interest rates from daily coupon bond prices.Often, this type of estimation method is used in the context of nding the theoretically correct value of

Term structureMarket 2y 3y 4y 5y 7y 9y 10y 15y 20y 30y

Australia Japan New Zealand

Belgium Denmark France Germany Ireland Italy Netherlands South Africa Spain Sweden U.K. ECU

Canada U.S.

M iwi j, w i M i,

i j,

p i j,( ) t 0 t j M i

p j( ) z j( ) j( ) (Continuous compounding)exp

1 1 z j f ,( ) f +( ) j f

(Discrete compounding)

=



14/48


a bond, or set of bonds. That is to say, when pricing bonds, the theoretical approach provides a way todescribe price of bonds as a function of certain parameters such as the mean-reversion of the short-terminterest rate.

RiskMetrics, however, provides rates and prices to be used for risk management--rather than pric-ing . Therefore, the methodology it uses to construct spot interest rates does not necessarily have to beconsistent with a bond pricing model that attempts to explain the variation of bond prices across variousmaturities. Moreover, as table 5 shows, RiskMetrics requires a set of spot rates (and prices of discount

bonds) on relatively few maturities. Such a requirement is unlike that found in a pricing framework where researchers seek to value bonds that may have slightly different maturities.

Considerations such as the number and type of spot interest rates required, as well as their use (risk man-

agement), motivate the new term structure estimation method that RiskMetrics plans to employ .

4. Bootstrapping spot interest rates and discount functions

Bootstrapping is probably the most widely used empirical method for estimating spot interest rates froma set of coupon bonds. According to the bootstrapping technique, spot (or forward) interest rates are iter-atively extracted using a standard formula for pricing a coupon bond. Once the (bootstrapped) spot interestrates corresponding to each coupon payment period (e.g., every six-months) have been obtained, we cangenerate a complete universe of spot rates by either smoothing or interpolating the bootstrapped rates.

The proposed RiskMetrics term structure estimation method is based on the bootstrap algorithm appliedto a set of coupon bonds. Table 6 presents a typical sample of coupon bond information, the universe of Finnish government bonds on March 10, 1997.

Table 6 Finnish government bonds

As of March 10, 1997, 30/360 basis, annual coupon payment ,

For a given set of coupon bonds, such as those shown in table 5, the bootstrap procedure solves for spotrates and prices of discount bonds by making use of the standard bond pricing formula [19].

4.1 The bootstrapThe starting point for the bootstrap procedure is equation [19] where we write the price of a coupon bondthat matures in years as follows:

[21]

For ease of exposition, we assume the following when describing the bootstrap:

Description Maturitydate

Price Accrual Coupon

FIN GOV 11.00% Jan 99 15-Jan-1999 112.782 1.772 11.00FIN GOV 10.00% S ep 01 15-Sep-2001 121.203 4.944 10.00FIN GOV 10.75% Mar 02 15-Mar-2002 125.877 10.69 10.75FIN GOV 9.50% Mar 04 15-Mar-2004 122.745 9.447 9.50FIN GOV 7.25% Apr 06 18-Apr-2006 108.916 6.545 7.25FIN GOV 8.25% Oct 10 15-Oct-2010 116.899 3.392 8.25

Mi

p c i M i,( ) p i 1,( )c i p i 2,( )c i p i M i,( ) P c i+( )+ + +=



15/48


16/48


[26] (price of discount bond)

[27] (spot interest rate)

We would then apply this price (or rate) when bootstrapping at eighteen months. In general, we can writeout the details of this averaging procedure as follows:

First, let represent the r th maturity date required in the bootstrap procedure (r=1,..,R) where R is the total number of maturity dates required to bootstrap. Each is referred to as a boot-strap maturity date . For example, if a bond has exactly 5 years to maturity with semi-annualcompounding then R = 5x2 = 10. Moreover, represents the six-month date, denotes theone-year date, and so on, up until which represents the ve-year date.

Second, calculate , the total number of bonds maturing at time . That is, is the totalnumber of bonds such that for r = 1,...,R.

Third, at every , calculate the prices of discount bonds (or spot rates). That is to say, cal-culate the prices of discount bonds for every bond that matures on a bootstrap maturity date.

Fourth, compute the average price of the discount bond (or spot rate) at each .

When no bonds exist at a required maturity date , the yield to maturity at that date is found by in-terpolating (e.g. linearly) the two nearest yields, i.e., interpolating between the yields that exist at times

and . Given this yield, the price of the bond and coupon are determined by assuming that the bond at that maturity is priced at par.

5. Alternative term structure estimation methods

In this section we explain alternative term structure estimation methods. The purpose of this section is to present an overview of term structure estimation methods so that the reader can obtain a better understand-ing of the various techniques used, by both academics and practitioners, to estimate the term structure of interest rates. We focus on empirical term structure estimation methods . Excluding the bootstrap, suchmethods consist of two basic steps:

Model : First, assume that the price of a discount bond (or some variation of it) can be modeledaccording to some mathematical function, such as a cubic spline or exponential polynomial,whose value is determined by a set of parameters.

Estimate : Second, use observed prices on coupon or discount bonds to estimate the parameters of the model. Exactly how the parameters are estimated is determined by some smoothness crite-rion. Smoothing refers to a statistical technique such as regression analysis that ts a line orcurve through a set of points. Smoothing in the context of term structure estimation assumes thatbond price data are measured with some error, i.e., they are noisy. Interpolation , on the otherhand, is predicated on nonnoisy data and simply links nearby data points by some mathematicalfunction.

p 2( )13--- p k 2,( )

k 1=

3

=

z 2( )13--- z k 2,( )

k 1=

3

=

'r 'r

'1 '2'10

N 'r 'r N 'r i M i, '= r

'r N 'r

'r 'r

'r 1 'r 1+



17/48


5.1 Bootstrapping forward ratesIn section 4 we explained a procedure to iteratively extract prices of discount bonds (spot rates) from a setof coupon bonds. We now show how to bootstrap forward interest rates from the same data. Recall that wecan write the price of the ith coupon bond as

[28]

Moreover, we can write the discount factor, assuming continuous compounding, in terms of forward ratesas follows;

[29]

where = and to simplify notation set . Note that with discrete compounding wewould have:

[30]

and = . Given this framework, it should seem obvious that we can apply the general principalsof section 4.1 to bootstrap forward rates. For example, from [22] we know that

[31]

We can solve for since . Given , [23] implies

[32]

We can solve for since . Proceeding in an iterative man-

ner we can extract all of the forward rates.In cases where there are either multiple or no bonds mature at a bootstrap maturity date, we apply the tech-niques described in section 4.2.

5.2 Polynomial and exponential splinesThere is an extensive research literature in the area of xed income on the best way to extract prices of discount bonds and spot rates from coupon bonds. A large portion of this research has been dedicated tonding mathematical functions that can approximate the true shape of the discount function or term struc-ture of interest rates. By discount function we mean simply the price of a discount bond evaluated overa set of maturities . Chart 4 shows a typical shape of the discount function .

p c i M i,( ) w i j, p j( ) j 1=

M i

=

p j( ) z j( ) j( )exp= F 1 1 0,( ) F 2 2 1,( )exp F j j j 1,( )exp exp=

F s s s 1,( )exps 1=

j

=

F 1 z 1( ) F s F s 1( )=

p j( ) 1 z j f ,( ) f +( )= j f 1 F 1 f +( )

1 0, f 1 F j f +( ) j j 1, f

=

1 F s f +( )s s 1, f

s 1=

j

=

F 1 z 1 f ,( )

p 1( ) p c 1 M 1,( ) P =

F 1 p 1( ) 1 F 1 f +( )1 0, f = p 1( )

p 2( ) p c 2 M 2,( ) p 1( )c2( ) P c 2+( ) =

F 2 p 2( ) 1 F 1 f +( )1 0, f

1 F 2 f +( )2 1, f

=

p j( )



18/48


Chart 4A plot of the discount function [from 0 to T]

A continuous and complete discount function is unobservable. That is, we do not observe prices of dis-count bonds at all possible maturities since coupon bonds will only yield a set of discrete discount bond

prices. One approach to estimating a complete discount function is to nd a mathematical function(polynomial) that has a similar shape to the true discount function . Such a function would be denedover all maturities from time 0 to time T. Chart 5 shows how the shape of an approximating polynomialmay appear in comparison to the discount function.

Chart 5Approximating the discount function with some polynomial

In many cases, rather than using one polynomial, dened over the entire set of maturities, we may moreaccurately model the shape of the discount function by applying a piecewise polynomial . That is, instead of approximating the function over the entire domain of maturities [0,T], we rst break the maturitiesup into segments, and nd functions that locally describe the discount function over each of these seg-ments. We then t a polynomial to each segment for j=1,...,n and =0 and =T. Finally, we

p( j)

0 T

p( j)

0 T

j 1 j,[ ] 0 n



19/48


attach each of these functions at their so-called join points. Such a piecewise polynomial is known as apolynomial spline and is represented in chart 6.

Chart 6 Applying a spline to the spot price curve

The original work on estimating the term structure of interest rates used splines to approximate the dis-

count function . Subsequent work on term structure estimation applied splines to some function of such as

, the negative logarithm of the discount function

, the spot interest rates, and

,the forward interest rates.

We will discuss such applications in more detail in section 5.5 when we explain a general class of splinefunctions known as B-splines.

McCulloch (1971,1975) and Shea (1984) where among the rst to use splines to approximate the discountfunction. McCulloch (1975) and Shea suggested the use of a cubic polynomial spline , which for the ith

bond and the jth maturity interval , models the discount function as

[33]

where and are parameters which are estimated from observed bond prices. Shea estimatesthese parameters by restricted least squares (RLS)8. In fact, he applied RLS to a polynomial spline andshows that this technique is identical to McCullochs cubic spline. The main practical problem with us-ing such splines , however, is that it is possible to generate unbounded positive and negative forward rates.

8This follows from the work of Buse & Lim (1977). Note that Shea did not use coupon bonds when he estimated the spline.Also, he applied his model to Nippon Telegraph and Telephone (NTT) zero-coupon issues--long-term discount bonds tradedin Japans bond market.

p( j)

0 1

2 3

4 T

p j( ) p j( )

p j( )( )log

z j( )

F j( )

j 1 j,[ ] p j( )

p j( ) 1 1 2 j 3 j2 4 j

3+ + + +=

1 2, 3, 4



20/48


Also, the term structure of interest rates tends to bend sharply toward the end of the maturity range ob-served in the sample of bonds.

Vasicek and Fong (1982) argue that while splines constitute a exible family of curves, there are severaldrawbacks to tting such functions. The problem with polynomial splines, they argue, is that they weavearound discount function resulting in highly unstable forward rates 9. They suggest approximating the discount function with the exponential spline such as

[34]

where and are parameters and is the instantaneous forward rate. Vasicek and Fong simply propose this model and suggest a methodology to estimate the parameters. The authors do not t the modelto any data.

Shea (1985) estimates the Vasicek and Fong model using the NTT zero-coupon bonds that were appliedin Shea (1984). The author concludes that the estimation of exponential splines is no more convenientthan estimation with polynomial splines , that is, both splines produce identical results.

5.3 Exponential polynomial methodChambers, Carleton, and Waldman (1984) assume that the spot interest rate may be expressed as a sim

ple polynomial

[35]

where L is the length of the polynomial. Under the assumption of continuous compounding (see [3]), thediscount function, becomes the exponential of a polynomial. That is,

[36]

Therefore, the price of a coupon bond can be written explicitly as a function of the parameters , l=1,...,L.

[37]

from which we get the regression equation

[38]

where is a random error term and we estimate . Chambers et. al. estimate the polynomials parame-ters using non-linear least squares.

9Refer to section 2.2 for a discussion on the relationship between zero and forward rates.

p j( ) 1 2 j( )exp 3 2 j( )exp 4 3 j( )exp+ + +=

1 2, 3, 4

z j( ) l jl 1

l 1=

L

=

p j( )

p j( ) l jl

l 1=

L

exp=

l

p c i M i,( ) w i j, l jl

l 1=

L

exp j 1=

M i

=

p c i M i,( ) w i j, l jl

l 1=

L

i+exp

j 1=

M i

=

i l



21/48


5.4 Modeling and smoothing forward ratesNelson and Siegel (1987) model the forward rate, , as an exponential polynomial

[39]

where and are parameters and and are constants. To obtain the spot rate as a function of maturity we integrate [39] from 0 to and then divide by . The resulting function is

[40]

Nelson and Siegel estimate the parameters of this model by least squares using US Treasury bills, thus

avoiding coupon bonds. For 37 samples covering January 22, 1981 through October 27, 1983, the matu-rities on these bills ranged from 3 days to 1 year.

Svensson (1994) proposed the following modied form of Nelson and Siegels forward model:

[41]

This form of the Nelson and Siegel model has been noted in Oda (1991) and Malz (1997). Recently, Bliss(1994) proposes estimating the Nelson and Siegel model using a nonlinear, constrained optimization pro-cedure that accounts the bid and ask prices of bonds as well bonds durations. See Bliss (1994) for moredetails.

Adams and Van Deventer (1994), while continuing to focus on the forward rate function, take a funda-

mentally different approach to estimating the term structure of interest rates. Their criterion for the besttted yield curve is in terms of the maximum smoothness for the forward rates . The starting point for their analysis is a measure of the smoothest possible forward rate curve on some interval [0,T] which theydene as

[42]

where is the second derivative of the forward rate curve at maturity s. To understand why this is anatural measure of smoothness, express [42] in discrete form:

[43]

Recall that the second derivative measures rate of change of a curve, i.e., how the slope of the curve chang-es as the independent variable (maturity, in this case) changes. Hence, the closer the second derivative isto zero, the more smooth the curve. Smoothness also requires that the second derivative is small at each

point in time from the beginning (time 0) to the end (time T). Therefore, we should require that the sum(or integral) of the squared deviations of the second derivatives from zero is as small as possible.

In practice, the minimization of [42] is meaningless unless it is combined with prices of observed discount bonds. In section 2.2, we showed how to write the prices of discount bonds in terms of forward rates:

F j( )

F j( ) 1 2 j c1 ( )exp 3 j c2 ( )exp+ +=

1 2, 3 c1 c2 j j

z j( ) 1 2 + 3( ) 1- j c1 ( )exp[ ] j c2 ( ) 2 j c1 ( )exp+ +=

F j( ) 1 2 j c1 ( )exp 3 j c2 ( ) j c2 ( ) 4 j c3 ( ) j c3 ( )exp+exp+ +=

Z F '' s( )[ ]2 sd 0

T

=

F '' s( )

Z d F '' s i( ) 0[ ]2 s i s i 1( )i 0=

T

=

F '' s( )



22/48


[44] j=1,....,M

Let represent the prices of observed discount bonds . Adams and Van Deventer dene the maximumsmoothness criterion as that which minimizes Z subject to the constraint:

[45]

Within this framework, we estimate a term structure of interest rates by modeling the forward rates as a polynomial and estimate the parameters of the polynomial using the maximum smoothness criterion, i.e.,

minimize Z subject to [45]. Two suggested parameterizations of the forward rate include:

[46] (forward rate in polynomial form)

[47] (forward rate in exponential form)

The authors compare this maximum smoothness approach to alternative term structure estimation methodsusing yen and US interest rate swap and money market data.

5.5 A detailed look at a general class of splines: B-splinesIn this section we explain how a particular class of spline functions, known as cubic B (for basis)splines, can be applied to estimate the term structure of interest rates . This class of splines has beenapplied recently in Steely (1991) and Fisher et. al (1995). B-splines are appealing because they avoidsome of the difculties associated with estimating the parameters of polynomial splines discussed in sec-tion 5.2. The purpose of this section is to show exactly how to estimate the parameters of (B-) splinesfrom a set of coupon bonds by applying simple regression analysis. In addition, we provide a detailedexposition of the mechanics of B-splines--which are often not published--to show the mathematical un-derstanding necessary to implement a robust spline methodology.

To explain how B-splines work, we begin (again) with a general denition of the price of a coupon bond,now written in vector form.

[48]

where

and we assume that there are N bonds (i=1,...,N)

p j( ) F s( ) sd 0

j

exp= p j

p j F s( ) sd 0

j

exp=

F j( ) s js 1

l 1=

L

=

F j( ) s js 1( )exp

l 1=

L

=

p c i M i,( ) wiT p i( )=

wi is the vector wi 1, w i 2, w i M i,, , ,[ ](M i 1 )

i is the vector i 1, i 2, i M i,, , ,[ ](M i 1 )

p i( ) is the vector p i 1,( ) p i 2,( ) p i M i,( ), , ,[ ](M i 1 )



23/48


Splines10 require that we divide the maturity space of all N bonds into K-1 intervals by using K knots(break points). denotes the K break points (k=1,...,K) and and and M is thelongest maturity bond.

In practice, when using a special class of B-splines known as cubic B-splines , it is suggested that research-ers work with augmented knot points which are denoted by where ,

and for .

Let (kappa) represent the total dimension of the spline space, i.e., is the number of independent B-splines that can be used to approximate any function.

[49]

A cubic B-spline is given by the following recursive expression where r=4 and .

[50]

[51]

For each maturity, , we need to calculate for .

A cubic B-spline basis which is a set of B-splines that are used to approximate a function is represented by the vector

[52]

From the B-spline basis we construct a cubic spline which is a weighted summation of B-splines. For ex-ample, if we dene a vector of coefcients (weights), , and let , denotethe function we want to spline, then the cubic spline is given by expression

[53]

where s denotes spline. Recall, that for each bond we have future payments that occur at particular maturities, denoted by the vector , then for any of the k bases , we have

[54]

Using the k bases we can construct an (collocation) matrix

10This section is based on Fisher et. al (1995)

sk { }k 1=K s1 0= sK M =

d k { }k 1=K 6+ d 1 d 2 d 3 0= = =

d k 4+ d k 5+ d k 6+ sK = = = d k 3+ sk = 1 k K

K 2+= # of intervals= K 1

degree of polynomial3

+

1 k K

k r j( )

k r 1 j( ) j d k ( )

d k r 1+ d k ( )------------------------------------------

k 1+r 1 j( ) d k r + j( )

d k r + d k 1+( )-----------------------------------------------------+=

k 1 j( )

1 dk j d k 1+


24/48


[55]

This matrix will prove useful when we try to estimate the coefcients.

B-splines are used to approximate the discount function ( ), the negative logarithm of the dis-count function ( ) and the forward rate ( ). We denote the function that we approximate by

. It follows that can take on any of three different forms-- , and --such thatthere exists some function g() where .

We can write the splined function for the ith bond as which is an vector.It follows that each bond of the N bonds has an vector of splined functions . We now ex-

plain three parameterizations of .

1. We approximate (spline) the discount function directly , i.e.,

[56]

By denition we know that . Solving for we get

[57]

The splined function is simply

[58]

where for each bond (i=1,...,N) we have discount functions

[59]

Chart 7 shows how the splines relate to the discount function.

i( )

14 i 1,( ) 2

4 i 1,( ) 4 i 1,( )

14 i 2,( ) 2

4 i 2,( ) 4 i 2,( )

1

4 i M i,( ) 24 i M i,( )

4 i M i,( )

M i

=

p j( )l j( ) F j( )

h j( ) h j( ) p j( ) l j( ) F j( )g h j( ) .,( ) p j( )=

h s i ,( ) i( )= M i 1 M i 1 h s i ,( )

h s i ,( )

h j( ) p j( )=

h j( ) z j( ) j( )exp= z j( )

z j( )p j( )( )log j

--------------------------=

p s j( ) j( )T =

M i

p s i 1,( ) k i 1,( )k k 1=

=

p s i M i,( ) k i M i,( )k k 1=

=



25/48


Chart 7 Using B-splines to approximate discount func-

tion

Lets write the present value of the ith coupon bond as

[60]

we can re-write this as

[61]

If we add the error term to the denition , then we can get the regression equation

[62] (i=1,...,N)

where

(1 X )

Let Y represent the vector of observed prices of coupon bonds. That is,. And let X denote the N x matrix of regressors, i.e., .

Given the structure in [62], we can estimate the spline parameters by ordinary least squares (OLS). Thesimple OLS estimator is

[63]

p( i1 )

0 i1i2

p(i2 )

d1 d2 d3T

k 4 i1( ) k

k 1=

K

k 4

i2( ) k k 1=

K

p c i M i,( ) i ( ) c i 1, p s i 1,( ) c i 2, p s i 2,( ) c i M i, p s i M i,( )+ + += =

c= i 1, k i 1,( )k k 1=

c i 2, k i 2,( )k k 1=

c i M i, k i M i,( )k

k 1=

+ + +

i ( )c i j, 1 i j,( )

j 1=

M i

1

X i 1,

c i j, 2 i j,( ) j 1=

M i

2

X i 2,

c i j, i j,( )

j 1=

M i

X i ,

+ +=

i pc i M i,( ) i ( )=

p c i M i,( ) X i i+=

X i X i 1, X i 2, X i ,, , ,( )T =

N 1Y pc 1 M 1,( ) p c 1 M N ,( ), ,( )= X X 1 X 2 X , , ,( )=

OL S X T X ( ) 1 X T Y =



26/48


Next, when we estimate , we need to impose the restriction that the discount function at the current timeis one, i.e., . This is equivalent to

[64]

In order to estimate the spline parameters while imposing this restriction, we dene the 1 x vector and set q = 1 such that .The restricted OLS estimator (ROLS)

is given by the expression.

[65]

Finally, we obtain estimates of the prices of the discount bonds at each maturity using the relationship

[66]

Similarly, estimates of the spot rates are given by:

[67]

2. We approximate (spline) the negative logarithm of the discount function , i.e.,

[68]

In this case the splined function is given by the expression

[69]or

We can write the spot rate in terms of as

[70]

Solving for the discount function yields

[71]

p 0( ) 1=

k 0( )k k 1=

1=

R 14 0( ) 2

4 0( ) 4 0( ), ,,( )= R q=

ROLS

OL S

X T

X ( )1

RT

R X T

X ( )1

RT

[ ]1

ROL S

q( )=

p s i 1,( ) k i 1,( )k ROLS

k 1=

=

p s i M i,( ) k i M i,( )k ROLS

k 1=

=

z j( )p s j( )( )log j

----------------------------=

h j( ) l j( )=

l j( ) j( )( )T =

l j( ) p j( )( )log z j( ) j= =

l j( )

z j( )l j( ) j

-----------=

p j( ) l j( )( )exp j( )( )exp= =



27/48


In this case, the discount function associated with the ith bond is written as

[72]

We can write the price of a bond as

[73]

Or, more succinctly, as

[74]

Now, dene the regressor variable

[75]

So the regression model can be written as:

[76] (i=1,...,N)

In this situation there is a nonlinear relationship between the price of the bonds and the coefcient vector , so we must solve for using nonlinear least squares algorithm .

3. Lastly, we approximate (spline) the forward rate , i.e.,

[77]

Recall from section 2.2.2 that instantaneous forward rate is given by

[78]

p s i( ) k 1=

i( )k

exp=

p c i M i,( ) i ( ) c i 1, c i 2, c i M i,, , ,[ ]== k 1=

i 1,( )k

exp

k 1=

i 2,( )k

exp

k 1=

i M i,( )k

exp

pc i M i,( )

ci

T k

k 1=

i( )

k

exp=

X i ( ) c iT k

k 1=

i( )k

exp=

p c i M i,( ) X

i ( ) i+=

h ( ) F ( ) ( )T ==

F j( )d p j( )( )log

d j-----------------------------=



28/48


29/48


[85]

so that the regression model is

[86]

As in the previous case, there is a nonlinear relationship between the price of the coupon bonds and thecoefcient vector so we must solve for using nonlinear least squares algorithm .

6. Discussion and Summary

RiskMetrics currently uses a theoretical term structure estimation method to generate spot interest ratesand prices on discount bonds from coupon bonds traded in 17 government bond markets. In this article,we presented an alternative term structure estimation method, which is based on the bootstrap procedure,that we plan to employ in the RiskMetrics production process. This algorithm will replace the (theoretical)term structure estimation method that RiskMetrics currently uses.

The search for the best term structure estimation method requires that we analyze and assess the perfor-mance of several competing estimation methods.The nal decision on which term structure estimationmethod to use depends on three key factors:

1. Use : The way in which the spot interest rates and their corresponding prices will be used. In RiskMet-rics, for example, prices of discount bonds are used to calculate the volatility and correlations of futurecashows as well as to determine the present value of those cashows.

2. Coverage : The number and characteristics of the government bond markets used to estimate the termstructure of interest rates. RiskMetrics requires estimates of the term structure of interest rates for 13bond markets. On any given day, the number of bonds traded in these markets ranges from approxi-mately 160 (U.S.) to 8 (Finland).

3. Practical : We must consider the practical implementation of term structure estimation methodology.That is, how many parameters are required to estimate the term structure and how do these parametersvary across markets?

A quick perusal of the literature on term structure estimation methods will reveal that there is no shortage

of suggestions for ways to estimate the term structure of interest rates. In order to better understand thealternative estimation methods, we presented six term structure estimation methods and explained, in de-tail, how to estimate B-splines using standard regression analysis. The purpose of this discussion is to al-low for a comparison between the bootstrap and alternative methods in terms of what it takes to implementsuch algorithms in practice.

References

Adams, K.J., and D.R. Van Deventer. Fitting Yield Curves and Forward Rate Curves With MaximumSmoothness, Journal of Fixed Income, 2 (June 1994) pp. 52-62.

X i ( ) c iT k

k 1=

i( )k

exp=

p c i M i,( ) X i ( ) i+=



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Bliss, R.R. Testing Term Structure Estimation Methods, Federal Reserve Bank of Atlanta Working Pa- per 96-12a, (November 1996)

Buono, M., R.B Gregory-Allen, and Uzi Yaari. The Efcacy of Term Structure Estimation Techniques:A Monte Carlo Study, Journal of Fixed Income, 1 (March 1992) pp. 52-59.

Buse, A., and L. Lim. Cubic Splines as a Special Case of Restricted Least Squares, Journal of the Amer-ican Statistical Association , 72, (1977) pp. 64-68.

Chambers, D., W. Carleton, and D.W. Waldman. A New Approach to Estimation of the Term Structureof Interest Rates, Journal of Financial and Quantitative Analysis , 19 (September 1984), pp. 233-252.

Cox, J.C., J.E. Ingersoll, Jr., and S. Ross. A Theory of the Term Structure of Interest Rates, Economet-rica , 53 (March 1985) pp. 385-407.

Fama, E.F. and R.R. Bliss. The Information in Long-Maturity Forward Rates, American Economic Re-view, 77 (September 1988) pp. 893-911.

Fisher, M., D. Nychka, and D. Zervos. Fitting the Term Structure of Interest Rates with SmoothingSplines, Working Paper 95-1, Finance and Economics Discussion Series, Federal Reserve Board, (Janu-ary 1995).

de Munnik, J.F.J, and P.C. Schotman. Cross-sectional versus Time Series Estimation of Term StructureModels: Empirical Results for the Dutch Bond Market, Journal of Banking and Finance 18, (1994) pp.997-1025.

Malz, A. M. Interbank Interest Rates as Term Structure Indicators manuscript, (June 1997)

Malz, A. M. Interest-rate Mathematics manuscript, (June 1997)

McCulloch, J.H. Measuring the Term Structure of Interest Rates. Journal of Finance , 34 (January 1971) pp.19-31.

McCulloch, J.H. Tax-Adjusted Yield Curve, Journal of Finance , 30 (June 1975), pp. 811-829.

Nelson, C.R., and A.F. Siegel. Parsimonious Modeling of Yield Curves, Journal of Business , 60 (Octo ber 1987), pp. 473-489.

Oda, N. A Note on the Estimation of Japanese Government Bond Yield Curves, IMES Discussion Paper

96-E-27, (August 1996)

Shea, G.S. Pitfalls in Smoothing Interest Rate Term Structure Data: Equilibrium Models and Spline Ap- proximations. Journal of Financial and Quantitative Analysis , 19 (September 1984), pp. 253-269.

Shea, G.S. Interest Rate Term Structure Estimation with Exponential Splines: A Note. Journal of Fi-nance , 40 (March 1985), pp. 319-325.

Steely, J.M. Estimating the Gilt-Edged Term Structure: Basis Splines and Condence Intervals, Jour-nal of Business Finance and Accounting , 18, (June 1991) pp. 513-529.

Svensson, L.E.0. Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, IMF WorkingPaper, WP/94/114, (1994).



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Vasicek, O.A, An Equilibrium Characterization of the Term Structure. Journal of Financial Economics ,37 (June 1977), pp. 177-188.

Vasicek, O.A., andH.G. Fong. Term Structure Modeling Using Exponential Splines. Journal of Finance ,37 (May 1982), pp. 339-348.

Wegman, E.J., and I.W. Wright. Splines in Statistics, Journal of the American Statistical Association ,78, (June 1983) pp. 351-363.



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Christopher C. FingerMorgan Guaranty Trust CompanyRisk Management Services(1-212) 648-4657

[email protected]

Introduction

When considering a portfolio of options, a risk manager is faced with two contrasting pieces of infor-mation. On the one hand, the manager is well aware that the distribution of the return on any one of theoptions is asymmetric and certainly not normal. On the other hand, the manager knows that when alarge number of random variables are considered, even if their individual distributions are not normal,the distribution of their sum will be close to normal. Furthermore, the normal assumption is attractivefrom a practical point of view, as it allows the manager to compute Value-at-Risk statistics through onlyan analysis of the volatilities and correlations of the portfolio, rather than through full-blown simulationtechniques.

Thus is the dilemma: one option displays non-normal returns, while a portfolio of a large number of reasonably uncorrelated options displays returns which are normally distributed. In this article, we willattempt to shed some light on the questions of how many options constitute a large number andwhat is the meaning of reasonably uncorrelated

.

As the goal of this article is to give intuitive results rather than describe option portfolio distributionswith pinpoint accuracy, we will make a number of simplifying assumptions. We consider only Europe-an options, and assume the following:

1. Every option in each portfolio is struck at the money forward

1

.

2. Every option has the same expiration date.

3. Returns on the underlying assets are normally distributed, each with the same volatility.

4. Each pair of underlying assets has the same correlation.

Additionally, we will only consider the portfolio distribution at the options expiration date. Certainly,departures from these assumptions will inuence the portfolio distribution, but making these assump-tions allows us to isolate the effects of the two parameters we wish to consider -- correlations andportfolio size

.

We begin by examining various option portfolios through simulations, and show that in almost all cas-es, there is signicant non-normality in the portfolio distributions. We then present a simple analyticalmodel which gives some intuition to these somewhat surprising ndings. Finally, we summarize andconclude.

Simulation results

In this section, we investigate portfolios of options through simulations. Our procedure will be to x alevel of correlation, generate a large number (5000 in most cases) of scenarios for returns on the assetsunderlying our options

2

, and then compute the value of the option portfolio in each scenario. We willbe concerned with the shape of the portfolio distribution, and will examine two types of output. One issimply a histogram of the portfolio scenarios

, which we may compare visually with the normal dis-tribution. The second set of output consists of percentile levels of the portfolio distribution

. Recallthat one of the most useful properties of the normal distribution is that its percentile levels may be ob-

1

That is, the strike price is equal to the expected value of the price of the underlying at the expiration date.

2

For more information on the generation of return scenarios, refer to the RiskMetrics Technical Document, 4th edition

,Chapter 7 and Appendix E.

When is a portfolio of options normally distributed?


34/48


tained by multiplying the standard deviation by an appropriate scaling factor. To compare the distribu-tions of our option portfolios, then, to the normal, we will calculate how many standard deviations eachportfolios 90th, 95th, and 99th percentiles lie above the portfolio mean. If the portfolio distributionsare close to normal, we will expect to see these values close to 1.28, 1.65, and 2.33, respectively.

To begin, consider the distribution of value for one call option, as presented in Chart 1. Note the sig-nicant skew due to the asymmetric payoff prole for the option -- in roughly half of the cases the op-tion expires worthless, while in the other half, the option takes on one of many possible positive values.This skewness manifests itself in the distributions percentile levels, as the 90th, 95th, and 99th percen-tiles are 1.50, 2.12, and 3.34 standard deviations, respectively, above the mean.

Chart 1

Distribution of value for a single call option.

Our intuition is that for large enough portfolios with weak enough correlations, this asymmetrywill disappear, and the portfolio distribution will become normal

. Indeed, if we examine a portfolioof just twenty options with independent underlyings, we obtain a distribution that appears close to nor-mally distributed, as in Chart 2. If we examine the percentiles of portfolios of independent options, wesee a nice agreement with the normal distribution as well. In Chart 3, we present these percentiles as a

function of portfolio size. The at lines represent the 90th, 95th, and 99th percentiles (in standard de-viations above the mean) for the normal distribution, while the curves represent the same percentilesfor option portfolios. We see that the 90th and 95th percentiles are well predicted by the normal distri-bution for portfolios of as few as twenty options, while beyond a portfolio size of about fty, even the99th percentile is well predicted.

0.09 0.41 0.72 1.03 1.34 1.65 1.96 2.28 2.59 2.900

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Chart 2

Distribution of value for a portfolio of twenty independent options.

Chart 3Percentiles of portfolios of independent options as a function of portfolio size.

We may pursue the same investigations for portfolios of weakly correlated (here, we assume that re-turns on the underlyings have a 10% correlation) options. In Chart 4, we present a distribution for aportfolio of twenty weakly correlated options, and see that a slight skew still persists in the distribution;in fact, the 90th, 95th, and 99th percentiles are 1.36, 1.81, and 2.87, respectively, far from those whichwould be predicted by the normal distribution. (For contrast, recall the distribution for a portfolio of twenty independent options in Chart 2.) We hope for such small correlations that there will still be asignicant benet due to diversication, and that the portfolio distribution will appear normal once theportfolio is large enough.

But we do not see this. Referring to Chart 5, we see that for portfolios of weakly correlated options, the90th percentile is predicted well by the normal approximation, but the more extreme percentiles arepre-

1.3 3.2 5.1 7.0 9.0 10.9 12.8 14.7 16.7 18.60

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dicted poorly. What is of more concern, however, is that the portfolio distributions do not appear to be-come more normal as portfolio size increases; in other words, the error from using the normalapproximation for a portfolio of two hundred options is as severe as the error in the case of fty options.

Chart 4

Distribution of value for a portfolio of twenty weakly correlated options.

Chart 5Percentiles of portfolios of weakly correlated options as a function of portfolio size.

At higher levels of correlation, these problems are even greater. Chart 6 and Chart 7 present results forportfolios of options whose underlyings have a 40% correlation. Here, the twenty option portfolio ex-hibits an even stronger skew, and we see again the phenomenon that the portfolio distributions do notbecome more normal as portfolio size increases.

Thus, it appears that even at low levels of correlation, option portfolios are not normally distributed,regardless of the portfolio size. In fact, portfolio size has very little inuence at all on the shape of

0.32 3.55 6.78 10 13.23 16.46 19.68 22.91 26.14 29.370

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the portfolio distribution

; that is, the percentile levels we have examined do not depend on the sizeof the portfolio. In the next section, we present a simple analytical model which provides some insightinto these curious observations.

Chart 6

Distribution of value for a portfolio of twenty strongly correlated options.

Chart 7 Percentiles of portfolios of strongly correlated options as a function of portfolio size.

Analytical results

In order to explain the results of the previous section, we present here a simple analytical model. Letdenote the change in value in each of the underlying assets for our options. (This change

0.5 5.3 10.1 15.0 19.8 24.6 29.5 34.3 39.1 44.00

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in value is from the present to the options expiration

3

.) If all of our options are calls, then we may writethe value of each by[1] ,

and then the value of our portfolio is given by

[2] .

Our assumption that each option is at the money implies that each has mean 0. Further, we assumethat each of the is normally distributed, each with the same variance . Finally, we assume that thecorrelation between distinct and is .

A set of returns with the covariance structure we have assumed may be expressed as follows:

[3]

where are independent standard normal random variables. We see that the correlationbetween and , for example, comes only from these variables dependence on . For this reason,we refer to as the market return

, and as the idiosyncratic returns

for each underlyingasset. The interpretation is that each asset moves somewhat due to the broad market (factors that inu-ence all assets) and somewhat due to factors which only inuence the particular asset

4

. Observe thatthe correlation between any underlying asset and the broad market is .

Note that using the decomposition in Eq. [3], we may rewrite the value of each option in Eq. [1] as

[4] ,

where

[5] .

Eq. [4] allows us to interpret the option returns in terms of market and idiosyncratic returns as well.Note that there are two parts to the option value

: the rst, , is simply a constant times the marketreturn; the second may be thought of as an option payoff where the underlying is and the strike priceis . Thus, if the market falls (that is, is negative), then the market piece of the option falls aswell, but the idiosyncratic piece of the value goes well into the money, and at worst compensates forthe market loss.

An important observation on Eq. [4] is that the market piece is common to the value of all options,while the idiosyncratic pieces, conditional on the value of , are independent. Thus, it will be usefulto consider the distribution of each option value conditional on the market move. Note that for a stan-dard normal random variable and any constant , the expectation of the maximum of and is[6] ,

3

Technically, this should be the difference in value between the assets forward value at present and the realized value atoption expiry. However, this does not inuence our analysis.

4

This is very similar to the Capital Asset Pricing Model (CAPM).

V i max 0 Y i,( )=

V P V ii 1=

n=Y i

Y i 2

Y i Y j

Y 1 Y M 1 Y 1Y 2

+ Y M 1 Y 2

Y n

+

Y M 1 Y n+

==

=

Y M Y 1 Y 2 Y n, , , ,Y 1 Y 2 Y M

Y M Y 1 Y 2 Y n, , ,

V i 1 Y M max Y M Y i,( )+{ }=

1 ------------=

Y M Y i

Y M Y M

Y M

Z s s Z

E max s Z ,( ) s s( ) s( )+=



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appear normal. What remains then is to specify which correlation levels lead to situations in which themajority of the portfolio variance is of this second type.

In Chart 10, we show the percentage of portfolio variance which comes from the market component,plotted against the pairwise correlation for the portfolio. We see that the percentage increases veryquickly, and only for correlations less than about 3% does the market component contribute less thanhalf of the total portfolio variance. Thus, the underlying assets must be virtually independent in or-der for the portfolio distribution to be normal .

Chart 10Market contribution to portfolio variance for a portfolio of 50 options.

Conclusions

We have shown that for all but the most minute levels of average portfolio correlation, increasing thesize of a portfolio of options does not lead to a more normal portfolio distribution . The most basicintuition behind this conclusion is that a portfolio of options on (albeit weakly) correlated underlyingassets is implicitly an option on the market. In our example, where all of our options were calls withthe same strike, this option on the market had a distribution which looks like the distribution for a call.In general, one can expect the portfolio distribution to be greatly inuenced by the portfolios positionon the market. Thus, a portfolio with half calls and half puts at the same strike, while not expressing adirectional view on the broad market, will still behave like a straddle on the broad market, and will notdisplay a normal distribution.

In the end, there are only two ways to guarantee that the portfolio distribution is normal . The rstis to construct the portfolio of options on independent underlyings , which is clearly not practical.The second is to maintain the portfolio such that the implicit position on the market is neutral .Note that this goes beyond just delta hedging the portfolio -- the example above with equal numbersof calls and puts would have zero delta -- but means that all positions on the market must be offset with-in the portfolio. The job of maintaining a normally distributed portfolio thus falls on the shoulders of the investor; the mathematician has little recourse.

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0%

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RiskMetrics Monitor Third Quarter 1997page 45

2nd Quarter 1997: June 17, 1997

A detailed analysis of a simple credit exposure calculator

A general approach to calculating VaR without volatilities and correlations

1st Quarter 1997: March 14, 1997

On Measuring credit exposure

The effect of EMU on risk management

Streamlining the market risk measurement process

4th Quarter 1996: December 19, 1996

Testing RiskMetrics volatility forecasts on emerging markets data.

When is non-normality a problem? The case of 15 time series from emerging markets.

3rd Quarter 1996: September 16, 1996

Accounting for pull to par and roll down for RiskMetrics cash ows.

How accurate is the delta-gamma methodology.

VaR for basket currencies.

2nd Quarter 1996: June 11, 1996

An improved RiskMetrics methodology to help risk managers avoid underestimating VaR.

A Value-at-Risk analysis of foreign exchange ows exposed to OECD and emerging marketcurrencies, most of which are not yet covered by the RiskMetrics data sets.

Estimating index tracking error for equity portfolios in the context of principal variables thatinuence the process of portfolio diversication.

1st Quarter 1996: January 23, 1996

Basel Committee revises market risk supplement to 1988 Capital Accord.

A look at two methodologies that use a basic delta-gamma parametric VaR precept but achieveresults similar to simulation.

Previous editions of the RiskMetrics Monitor


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Third Quarter 1997page 47


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RiskMetrics Monitor

Third quarter 1997page 48

RiskMetrics products

Introduction to RiskMetrics:

An eight-page document thatbroadly describes the RiskMetrics methodology for measuringmarket risks.

RiskMetrics Directory:

Available exclusively on-line, a listof consulting practices and software products that incorporatethe RiskMetrics methodology and/or data sets.

RiskMetricsTechnical Document: A manual describingthe RiskMetrics methodology for estimating market risks. It

species how nancial instruments should be mapped anddescribes how volatilities and correlations are estimated inorder to compute market risks for trading and investmenthorizons. The manual also describes the format of the volatilityand correlation data and the sources from which daily updatescan be downloaded. Available in printed form as well as Adobepdf format.

RiskMetrics Monitor:

A quarterly publication that discussesbroad market risk management issues and statistical questionsas well as new software products built by third-party vendors tosupport RiskMetrics.

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A RiskMetrics Regulatory data set, which incorporates thelatest recommendations from the Basel Committee on the useof internal models to measure market risk, is also available.

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Documents

JPM Smooth Interpolation of Zero Curves