yield-x: the market works the workings of the yield-x yield curves,

Yield-X: The Market WorksThe workings of the Yield-X yield curves,

products and Calm margining system.

Rogan Etheredge

07 February 2005

Quant Financial Research 06 Feb 2005 2

Programme

• Introduction to Yield-X

• Yield curves

• Products

• Margining


Yield-X Products• jBonds: liquid RSA bonds (in Govi)• jCarries: 1 and 13 weeks on jBonds• jFutures: futures on jBonds• jTRIs and jGOVI: futures on Total Return Indices• jSwaps: bond lookalike swaps• jRods: swaps against overnight interest rate• jNotes: futures on notional swaps• jFRAs: futures on FRAs / Jibar• jOptions: options on all futures


Physical Futures

jBonds

jCarries

jSwaps

jRods

jFutures

jGOVI

jNotes

jFRAs

Implicit mark-to-market Explicit mark-to-market

jOptions

Underlying: R194, R153, R201, R157, R203, R186

Real and notional interest rate swaps on Jibar and RODI

Yield-X Products


Explicit mark-to-market

Loss Profit

Market and Position

Totalmargin


Implicit mark-to-market

Loss Profit

Market Position

Totalmargin

Unrealisedloss


Trading on Yield-X• Screen-based central order book

• Yield to three decimal places (1/10th basis point)

• Central counterparty:– settlement guaranteed– margining

• Report-only trades:– derivatives fall within risk positions and are guaranteed– jBonds only if both Clearing members agree


Mark-to-market

Closing yields

Yield curves

Market yields

Markto

market

Futures and

options


Margining

All positionsAll instruments: full offset.

Estimate maximum loss which position can suffer:next 24 hours, 99.95% confidence.

Loss is potential exposure of the Exchange:take the amount in margin now!

Margin earns interest at wholesale rates;

re-estimated daily; returned when position is closed.


Daily cashflows

• Explicit mark-to-markets

• Change in margin

• Settlements of spot bonds

• Net interest payments on jSwaps and jRods

• Conversion payments

• Interest on margin

• Booking fees


Yield curve methodology

• BEASSA

• Perfect fit

• Available on website daily to 30 years


Four curves

• Yield-X Curve– RODI, 3 mth Jibar, jFRAs, jNotes

• jBonds curve– RODI, 1 and 13 week jCarries, jBonds

• jSwaps curve– RODI, 3 mth Jibar, jSwaps

• jRods curve– RODI, jRods


Spreads

• Market price paramount• Instruments in generating set have zero spreads• Others have spreads calculated so that the Curve +

Spread prices them back to the market• Instruments without market prices priced off their

curve, using their latest recorded spread


Use of curves

• Market information

• Pricing illiquid instruments

• Resets

• Close-out prices of jFRAs and jNotes

• Margining


jBonds• R194, R153, R201, R157, R203, R186• Quoted on yield to maturity• Bond pricing formula ("BPF")• Spot: t+3• Forward: settlement date by agreement• On-screen through central order book:

– included in Risk Positions; – settlement guaranteed

• Report only:– both Clearing Members must explicitly agree

• Guaranteed forwards subject to yield resets• Settlement margins


jCarries• On all jBonds• 1st leg settled on t+3

– priced at current market price, using BPF

• 2nd leg settled 1week or 13 weeks after t+3• Quoted on carry rate as a simple yield between settlement

dates– quoted rate converted to ccr to find price of 2nd leg

• jCarries are on-screen through the central order book:– included in risk positions– settlement of both legs guaranteed

• 2nd leg subject to yield resets• Settlement margins


jFutures

• On all jBonds• Expire 1st business Thursday of Feb, May, Aug and Nov• Physically settled on t+3 of expiry date• Quoted on yield to maturity• Priced using BPF for settlement date• Settlement guaranteed• Settlement margins• Safex positions will be carried over

to Yield-X


jSwaps

• Bond lookalike swaps on all jBonds– j194, j153, j201, j157, j203, j186

• Expire on parent bonds' (mid) redemption dates• Semi-annual interest payments on same days as parent bonds'• Floating rate is 3-mth Jibar, compounded at mid interest period• Quoted on fixed rate: yield for the period• Standard fixed rate, reset on interest payment dates (or when market

has moved) to market rate, rounded to 25 bps• Dealt positions are converted to the standard rate• Conversion payments: PV off jSwaps Curve (+ spread) of difference

in fixed interest payments• Interest payments net through the Exchange• Guaranteed by the Exchange


jSwap Interest Calculations

Next interest date,di,ti

Last interest date,di-1,ti-1

Floating interest payments at f (known or implied)

Fixed interest payments at x

3-mth Jibar: fi1

Mid-periodreset date,

Di,Ti

3-mth Jibar: fi2

1 1 2100 1 1 1i i i i i iL f T t f t T

1100 i iX x t t


jSwap Equation

V N f T t f t T e x t t e ez ti i

z t

i

nz ti i n n

1 11 1 0 2 1 1 11

1 1


jRods

• Swaps of floating RODI against fixed 3-mth rate• Expire monthly on last business day of each month• Quarterly interest payments on last business day of month• Floating rate is jRODI (RODI converted from nacm to nacd),

compounded daily• Quoted on fixed rate: yield for the period• Standard fixed rate, reset on interest payment dates (or when market

has moved) to market rate, rounded to 25 bps• Dealt positions are converted to the standard rate• Conversion payments: PV off jRods Curve (+ spread) of difference in

fixed interest payments• Interest payments net through the Exchange• Guaranteed by the Exchange


jRod Interest Calculations

m-j: 0 1 2 3 … j 1 m





1100 i iX x t t

1

0

100 1 1365

mj

j

fL

jRODIs: fj


The jRod Equation

V N e x t t e ef

j

lZ T

i iz t

i

nz tj i i n n

1 3650

1

11

1 1


jNotes• Futures on notional swaps• Expire 1st business Thursday of Feb, May, Aug

and Nov• Swaps are for 2, 5 and 10 years from expiry date• Quarterly interest payments on 1bThu 2, 5, 8, 11

cycle• Floating rate is 3-mth Jibar• Quoted on fixed rate: yield for the period• Marked-to-market at closing fixed rate• Cash settled on expiry off the Yield-X Curve


jNote Interest Calculations





3-mth Jibar: fi

1100 i i iL f t t

1100 i iX x t t


jFRAs• Futures on FRAs / Jibar• Expire 1st business Thursday of Feb, May, Aug

and Nov• FRA interest payment date is 3 months from

expiry on 1bThu 2, 5, 8, 11 cycle• Floating rate is 3-mth Jibar• Quoted on fixed rate: yield for the period• Marked-to-market at closing fixed rate• Cash settled on expiry off the Yield-X

Curve / 3 mth Jibar


jFRA Interest Calculations

Interest date,d1,t1

Contact expiry date,d0,t0

Floating interest payment at f (known or implied)

Fixed interest payment at x

3-mth Jibar: f1

1 1 0100L f t t

1 0100X x t t


jOptions

• On all futures contracts• American calls and puts• Expire on underlying future's expiry date• Strike prices in same units as future's quote• Each option on one futures contract• Fully margined: premium not paid up-front, but realised over the life of the

option through the mark-to-market process• Naked options traded on price• Options with delta traded on volatility• Early exercise allowed but never optimal• Automatic exercise of options more than 2.5% in

the money on expiry• Long options may be abandoned• Pricing via Modified Black Option Formula


Volatilities and Skews

• Time structure: each expiry has its own atm vol• Skew: Moneyedness grid defines additive skew

for various degrees of moneyedness:Vol = Atm Vol + S[Moneyedness]

• Grid and atm vol found by weighted linear regressions of traded volatilities:

– weight by vega * number of options– atm vol also weighted by time since trade– historical trades also brought forward,

with decay factor

• Manual override possible


Yield-X Margining

• Comprehensive

• All-embracing

• Logical

• Margining

Calm


Two prerequisites

• Logical: all interest rate instruments may be valued by reference to a single construct:

The yield curve

• Technical: the existence of reliable yield curves


Margining Factors Program

Statistical analysis of historical data to create

margining factors

Margin Calculator

Calculation of value-at-risk of positions

Historicalmarket

data

Mark-to-market data for the day

Positionsof

participants

Marginsof

participants

Margining

factors

The two stages of margining


Data

• Zero coupon yield curves from 08 May 2002: 685 days– Data cleaned– Extend backwards– Yield-X Curve when available

• 1d, 1w, 1m, 3m-12m, 18m-60m, 6y-30y


Historical Yields

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04

Yie

ld (

cc

r)

1d

60m

30.00y

Historical yields


Yield changes

• Compare today’s yields with yesterday’s forwards one day

• Log of ratio

• Basic unit: Trading day


Yield changes – 1d ratesLPR1

-16.0%

-14.0%

-12.0%

-10.0%

-8.0%

-6.0%

-4.0%

-2.0%

0.0%

2.0%

4.0%

6.0%


1d


Yield changes – 60m ratesLPR15

-4.0%

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%


60m


Yield changes – 30y ratesLPR40

-12.0%

-10.0%

-8.0%

-6.0%

-4.0%

-2.0%

0.0%

2.0%

4.0%

6.0%

8.0%


30.00y


30 day exponential weighting30 day Exponential Weights

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

13-Jun-03 13-Jul-03 12-Aug-03 11-Sep-03 11-Oct-03 10-Nov-03 10-Dec-03 09-Jan-04 08-Feb-04 09-Mar-04


750 day exponential weighting750 day Exponential Weights

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0



Double exponential weightingDouble Exponential Weights

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


w1

750 day

Combined


1d Vols: alpha = 0.7

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

50.0%


xVol30

xVol750

xVol

Vols – 1d


60m Vols: alpha = 0.7

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%


xVol30

xVol750

xVol

Vols – 60m


30.00y Vols: wt = 0.7

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

50.0%


yVol30

yVol750

yVol

Vols – 30y


Corrs: 1d vs 30.00y; Wt = 0.7

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

08-05-02 08-11-02 08-05-03 08-11-03 08-05-04

Corr30

Corr750

Corr

Correlations: 1d vs 30y


Principal Components Analysis

• “The curse of dimensionality”

• PCA reduces it:– Find variance / covariance matrix for latest day– Find its eigenvectors: the components– First n eigenvectors: the principal components– Eigenvalues are variances of components– First n account for most of the variance


Volatilities: Bondsbd (Swaps): 04-Jun-04(67.3%, 84.1%, 91.8%, 97.2%, 98.9%)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

PCs

Ycs

Vols: Curves and Components


The First Component: 68%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Term

Yie

ld %

ag

e c

ha

ng

e

PC1

PC2

PC3

PC4

PC5

First component


The Second Component: 16%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Term

Yie

ld %

ag

e c

ha

ng

e

PC1

PC2

PC3

PC4

PC5

Second component


The Third Component: 8.5%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Term

Yie

ld %

ag

e c

ha

ng

e

PC1

PC2

PC3

PC4

PC5

Third component


The Fourth Component: 5%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Term

Yie

ld %

ag

e c

ha

ng

e

PC1

PC2

PC3

PC4

PC5

Fourth component


The Fifth Component: 1.5%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Term

Yie

ld %

ag

e c

ha

ng

e

PC1

PC2

PC3

PC4

PC5

Fifth component


Construct historical components1 n 1 n

1 1

1 n1 1 n

n

m n

* =Yieldchanges

Historicalcomponent

changes

Eigenmatrix


PC1 Vols: alpha = 0.7

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

140.0%


xVol30

xVol750

xVol

Vols - PC1


pc3

-25.0%

-20.0%

-15.0%

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

08

-Ma

y-0

2

08

-Ju

n-0

2

08

-Ju

l-0

2

08

-Au

g-0

2

08

-Se

p-0

2

08

-Oct

-02

08

-No

v-0

2

08

-De

c-0

2

08

-Ja

n-0

3

08

-Fe

b-0

3

08

-Ma

r-0

3

08

-Ap

r-0

3

08

-Ma

y-0

3

08

-Ju

n-0

3

08

-Ju

l-0

3

08

-Au

g-0

3

08

-Se

p-0

3

08

-Oct

-03

08

-No

v-0

3

08

-De

c-0

3

08

-Ja

n-0

4

08

-Fe

b-0

4

08

-Ma

r-0

4

08

-Ap

r-0

4

pc3

13 Sep 02: +100 pts

13 Jun 03:-150 pts

11 Sep 03:-100 pts

17 Oct 03:-150 pts

12 Dec 03:-50 pts15 Aug 03:

-100 pts

14 Jun 02: +100 pts

PC3: Yield Changes


PC3 Vols: wt = 0.7

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%


yVol30

yVol750

yVol

Vols – PC3


Corrs: PC1 vs PC3; Wt = 0.7

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

08-05-02 08-11-02 08-05-03 08-11-03 08-05-04

Corr30

Corr750

Corr

Correlations: PC1 vs PC3


PC1Vol = 77.6%; Skew = 0.07; Kurt = 11.2

0

2

4

6

8

10

12

14

16

-32.7% -28.8% -24.9% -21.0% -17.1% -13.1% -9.2% -5.3% -1.4% 2.5% 6.4% 10.4% 14.3% 18.2% 22.1% 26.0% 29.9% 33.9%

PC1 distribution


PC1: Smoothed distribution


ReproBars

• Choose 5 percentile ranges:– 5%, 25%, 40%, 25%, 5%– Find expected value in each range from

smoothed distribution

• Get a discrete distribution with just 5 points


PC5: ReprobarsPC5: 5-bar repro

-4.0% -3.3% -2.7% -2.0% -1.3% -0.6% 0.03% 0.7% 1.4% 2.1% 2.7% 3.4%


Construction of scenario curves

• All 5*5*5*5*5 = 3125 combinations of 5 abscissae/ordinates from 5 PC distributions

• Each combination is a set of 5 independent component changes, and has its own probability

• Apply the changes to the base curve to get 3125 scenario curves, each with its own probability.


Valuation of positions

• Value position 3125 times, to get 3125 values, each with its own probability

• Smooth the resulting distribution

• Find EWCV at .05% over 1 trading day


Expected worst case valueP&L Distribution

-4 -3 -2 -1 0 1 2 3 4

Cuttoff:.05%

ETL: expected loss, given that

loss is more than .05%


Margin

"On the one day in 2,000 when things are really bad, just how bad are they likely to be?"


Calm Results

• Assumptions:– Positions as at 28 Jan 2005– Yield curve and bond yields as at same date– Settlement day is 02 Feb 2005– Margining valuations for 31 Jan 2005– Margins calculated as the one-trading-day

Value-at-Risk (VAR) of the position at 99.95% confidence level


Physical Bonds• (NB: settlement margins are ignored. Once a deal has

been irrevocably committed to, it falls out of the risk position.)

Instrument Position MarginSpot R153 +R1m 15,806

-R1m 17,665 Spot R157 +R1m 28,470

-R1m 30,507 Spot R186 +R1m 49,029

-R1m 51,378


Forward Bonds

Instrument Settlement Position MarginR153 04-May-2005 +R1m 15,985

-R1m 17,692 02-Feb-2006 +R1m 14,344

-R1m 15,415


Carries

Instrument Settlement Position MarginR153 02-Feb-2005 -R1mR153 09-Feb-2005 +R1m 101

R153 02-Feb-2005 -R1mR153 02-Feb-2006 +R1m 4,318


Compound Spot Position

Instrument Settlement Position MarginR153 02-Feb-2005 -R1mR157 02-Feb-2005 +R2mR186 02-Feb-2005 -R1m 16,443


Positions with Futures

Instrument Settlement Position MarginMay 05 R153 -1 18,087

May 05 R153 -1 Spot R153 02-May-2005 +R1m 669


Positions with jSwaps

Instrument Settlement Position Marginj153 7.5 swap +R1m 15,743

j153 7.5 swap +R1mMay 05 R153 1 3,268


jNotes

Instrument Settlement Position MarginMay 05 jN02 +1 6,642

Aug 05 jN05 +1 13,777

Nov 05 jN10 +1 22,251


Distribution of Scenario ValuesScenario Values: May 05 jN02

(6,923 (6,091 (5,259 (4,427 (3,595 (2,762 (1,930 (1,098 (266) 566 1,399 2,231 3,063 3,895 4,727 5,560 6,392 7,224


Scenario Values: jN05Scenario Values: Aug 05 jN05

(14,

405)

(12,

757)

(11,

109)

(9,4

61)

(7,8

13)

(6,1

65)

(4,5

17)

(2,8

69)

(1,2

22)

426

2,0

74

3,7

22

5,3

70

7,0

18

8,6

66

10,

314

11,

962

13,

609


Scenario Values: jN10Scenario Values: Nov 05 jN10

(23,

417)

(20,

664)

(17,

910)

(15,

157)

(12,

403)

(9,6

50)

(6,8

96)

(4,1

43)

(1,3

89)

1,3

64

4,1

18

6,8

71

9,6

25

12,

378

15,

132

17,

885

20,

639

23,

392


jFRAs

Instrument Settlement Position MarginMay 05 jFRA +1 751

May 06 jFRA +1 1,052

Nov 06 jFRA +1 1,034


jRods

Instrument Settlement Position MarginJun 05 jRod +R1m 1,007

Jan 06 jRod +R1m 2,939


jGoviInstrument Settlement Position Margin

May 05 jGOVI +1 30,307

May 05 jGOVI +100R152 02-Feb-2005 -R14mR194 02-Feb-2005 -R39mR153 02-Feb-2005 -R55mR201 02-Feb-2005 -R14mR157 02-Feb-2005 -R30mR203 02-Feb-2005 -R3mR186 02-Feb-2005 -R17m 39,578

Questions

yield-x: the market works the workings of the yield-x yield curves,

Documents