Jun 16, 2003 - See application ?le for complete search history. (56). References Cited ... Analytical Approximation for
US007590581B1
(12) United States Patent
(10) Patent N0.: (45) Date of Patent:
Payne et al. (54)
COMPUTER BASED SYSTEM FOR PRICING
US 7,590,581 B1 Sep. 15,2009
Ferrenberg, Wong, Landau, and Wong, “Monte Carlo Simulations: Hidden Errors from “Good” Random Number Generators”, 1992, pp.
AN INDEX-OFFSET DEPOSIT PRODUCT
3382-3384.
(75) Inventors: Richard C. Payne, Mississauga (CA);
Gerber, Richard, “The Software Optimization Cookbook”, Intel
John A. Rose, Toronto (CA); Marc G.
Press, 2002. Golub, G.H. and C. Reinsch. Singular Value Decomposition and Least-Squares Solutions. in J .H. Wilkinson and C. Reinsch (editors), Handbook for automatic computation, vol. II: “Linear Algebra”,
Verrier, Caledon East (CA)
(73) Assignee: Genesis Financial Products, Inc., Mississauga, Ontario (CA) (*)
Notice:
Springer Verlag, 1974, pp. 134-151. Golub and Van Loan, “Matrix Computations” 1989, pp. 430-436. Hunter, C. J ., P. Jackel, and MS. Joshi, “Drift Approximations in a
Subject to any disclaimer, the term of this patent is extended or adjusted under 35
ForWard-Rate-Based LIBOR Market Model”, 2001, pp. 1-3.
“Intel Architecture Optimization Reference Manual”, Intel, 1998.
U.S.C. 154(b) by 1280 days.
Kloeden, Peter E. and Eckhard Platen, “Numerical Solution of Sto
chastic Differential Equations”, Springer-Verlag, 1995, pp. 221-226.
(21) Appl.No.: 10/463,180 (22) Filed: (51)
Primary ExamineriThu Thao Havan
(74) Attorney, Agent, or FirmiSimpson & Simpson, PLLC
Int. Cl.
G06Q 40/00 (52) (58)
(Continued)
Jun. 16, 2003
(2006.01)
(57)
U.S. Cl. ......................................... .. 705/36; 705/35 Field of Classi?cation Search ............ .. 705/35-36
See application ?le for complete search history. (56)
A computer-based method for determining a value of an
index-offset deposit product, having a principal amount P, a
References Cited
term T, a speci?ed guaranteed amount G, and an index credit
C, comprising:
U.S. PATENT DOCUMENTS 6,343,272 B1
ABSTRACT
d) setting trial values for ?xed-income-linked crediting parameters for said product implying an expected ?xed
1/2002 Payne et al. .................. .. 705/4
income-linked credit component F at the end of the term OTHER PUBLICATIONS
T;
Aho, Alfred V. and Jeffrey D. Ullman, “Principles of Compiler Design” 1977, pp. 491-497. Booth, R, “Inner Loops”, 1997, pp. 235-236. Duan, Jin-Chuan, Genevieve Gauthier and Jean-Guy Simonato, An
e) determining a cost for an equity option paying equity
Analytical Approximation for the GARCH option pricing model,
f) summing said equity option cost, present value of prin
linked credit component E such that the index credit
C:E+F, to be paid at T, together With the principal P, is at least equal to G; and
2001, pp. 75-116. Elton, Edwin J. and Martin J. Gruber, “The Management of Bond
cipal, and present value of ?xed-income-linked credit component to determine said value of said index-offset
Portfolios”, Chapter 19 of “Modern Portfolio Theory and Investment Analysis”, 4th Ed, 1991, pp. 542-572.
deposit product.
Espen Gaarder Haug, “The complete guide to option pricing formu
9 Claims, 7 Drawing Sheets
las” 1997, pp. 97-102.
Ourrent Yield Curve l
2
$215?“ new" |1.a4 [$00
5
1
10
20
1.12 |2.11
3
2.05
3.50
4.01
4.03
1,128
3.1115
3.657
‘.130
5.2“
2.100
Assumed Coupon
|a.00
Ship
NA-GARCH Equity Parameters e
|1.1s2s310
bQllZ 0. 0375553
lambda
0.050574
mm
25:
blll0
21550790
haka1 |0.055‘I01
Declared Rate Annuity 1
imam Flala 2.59
2
a
100
r
|1ss
5
5.50
|1s0
s
7
3.98
12.50
Get PV
2,9558
Capital Protection Annuity TlauAllaa 100
Equity Alla:
ll
Equilv Pullicipaliwl Rule 100
mm’, Plrlininlinn 0.». Rate
100
Unit]:
W
Dawn-id:
0.00
Avg Credit
Val vz. Hi?nrical [1970-2000]
3.500
3.00
3.500
3.600
Scanlrius |s0000
3.500
3.500
E=|Pv
W
3.5!]
10000]
Della DMODD
Intare51 Rats Btpaiuras - Change in MV(Liah)for1% increase in Forward Rate 1 -0.N43
2 0.9750
3 41.8571
I 41.9507
5 0.9499
B 41.51 S
7 -0. 5451
9 41.0031
10 0.0020
I1 41.0050
I2 0.01121
I3 0.0130
14 000107
15 0.0000
0 0.11121
3.500
US 7,590,581 B1 Page 2 OTHER PUBLICATIONS Liu, Jun, Francis A. Longstaff, and Ravit E. Mandell, “The Market Price of Credit Risk: An Empirical Analysis of Interest Rate Swap Spreads”, 2000, pp. 1-10. Marsaglia, George and Wai Wan Tsang, “The Ziggurat Method for Generating Random Variables”, 2000, pp. l-7. McKay, M.D., R.J. Beckman, and W.J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code”, Technometrics, 1979, 21(2), pp. 239 245.
Neftci, Salih N., “An Introduction to the Mathematics of Financial Derivatives” 2001.
Press, William H., William T. Vetterling, Saul A. Teukolsky, Brian P.
Flannery, “Numerical Recipes in C”, Cambridge University Press, 1992, pp. 309-315.
SalZberg, Betty Joan, “File Structures: An Analytic Approach”, 1988, pp. 20 -2 5.
Sedgewick, Robert, Algorithms, 1983, pp. 115 - l 24.
US. Patent
Sep. 15, 2009
Sheet 1 of7
US 7,590,581 B1
)1‘ Capital Protection Annuity - Interactive Pricing - Version 1.2
lCurrent Yield Curve 1
[Coupon] Treasury
Yield,
3
5
1.940
W“
1.720
2.100
Assumed
7
[1.34 [1.72 [2.17 [3.05
Resulting Zero
c
2
10
20
[3.50 [4.01
9.100
3.007
Coupon
[4.93
4.190
[3.00
_
Strip [
5.259
NA-GARCH Equity Parameters
|1.162B310
betaZ ‘0.0375993
lambda [0.050674
beta0 ‘2.155876'6
beta1 |0.099101
obs/yr
div
lnstVol I25
I253
1.25
Declared Rate Annuity 1
Declared Rate [3.00
2
3
[3.00
4
[3.00
5
[3.00
0
[3.00
7
[3.00
W
[3.00
Get Pv [ 0 88858
Capital Protection Annuity Treas Allor; I100
Equity Alloc
In
Treasury Participation
Equity Participation Hate [100 1
2
3
4
5
0
7
Base Rate [W FlnorHale [2.00 [2.00 [2.00 [2.00 [2.00 [2.00 [2.00 Upside 0.00 [Zap Rate [99.00 [99.00 [99.00 [99.00 [9*900 [99.00 [99.00 Downside [W
Avg Credit
Vol vs. Historical [1970-2000]
3.090
3.00
3.000
Scenarios
3.000
3.000
W
3.000
GelPV
3.000
W "00090 Della 0.00000
Interest Rate Exposures — Change in MV(Liob) for1% increase in Forward Rate 1
2
3
4
5
B
7
-0.9943
-0.9768
-0.9B72
0.9587
-0.9499
-0.9510
-0.9461
9 -0.0037
10 0.0020
11 -0.0050
12 0.0021
13 0.0130
14 [1020?
15 0.0000
Fig. 1
8
0.0021
3.000
US. Patent
Sep. 15, 2009
Sheet 2 017
US 7,590,581 B1
)3’ Capital Protection Annuity - Interactive Pricing - Version 1.2
Current Yield Curve 2
[Coupon] Treasury
Yield,
5
1.340
Yie'“
1.720
2.100
Assumed
7
[1.34 [1.72 [2.17 [3.05
Resulting Zero
c
3
[3.50
3.106
10
20
[4.01
[4.93
3.667
4.130
Coupon
[3.00
Strip [
5.259
NA-GARCH Equity Parameters
|1.1626318
beta2 |0.0375993
lambda |0.050674
beta0 |2.15587e-B
beta1 |l1.899101
obs/yr I253
div
lnstVol I25
I125
Declared Rate Annuity 1
Declared Rate [3.00
2
3
[3.88
4
[3.00
5
[3.00
1;
[3.00
7
[3.88
W
[3.60
Get w |
0.99050
Capital Protection Annuity Treas Alloc I100
Equity Alice
In
Treasury Partrcrpatron
Equity Participation Rate I100 1
2
3
4
5
B
7
Base Rate |1.72 Upside 1.00
Floor Rate [1.72 [1.72 [1.72 [1.72 [1.72 [1.72 [1.72 Cap Flate [99.00 [99.00 [99.00 [99.00 [99.00 [99.00 [99.00
Downside
Avg Credit
1.00
Vol vs. Historical [1970-2000]
1.720
3.00
2.424
Scenarios
2.759
2.990
I500!!!)
3.110
Get PV
3.215
W
[199908
Delta
0.00000
Interest Rate Exposures - Change in MV(Liab) for 1% increase in Forward Rate 1
2
3
4
5
B
7
-0.9843
-0.8741
-0.7550
-0.B342
-0.5174
-0.4017
-0.2889
9 0.7469
10 0.6339
11 0.4552
12 0.3319
13 0.2020
14 0.0387
15 0.0000
Fig. 2
8
0.8878
3.300
US. Patent
Sep. 15, 2009
Sheet 3 of7
US 7,590,581 B1
)1‘ Capital Protection Annuity - lnteiactive Pricing - Version 1.2
Current Yield Curve 1
[Coupon] Treasury
Yield,
3
5
Assumed
7
10
20
|1.34 |1.72 |2.17 |3.05 |3.5a [4.01
Resulting 2810
1.340
We'd‘
c
2
1.720
2.100
3.105
3,551
Coupon
[4.93
4.1 30
|3.00
_
snip |
5. 25s
NA-GARCH Equity Parameters
|1.1B2B310
betaZ |0.0375903
lambda |0.050B74
beta0 |2_15587e-6
bete? I009!“ 01
obs/pi
div
lnslVol I25
253
L25
Declared Rate Annuity 1
Declared Hate |3.ss
2
3
|3.s0
4
|3.ss
5
p.50
|3.sa
s
7
|3.s0
W
|a.ss
Get PV |
0.00050
Capital Protection Annuity T1868 A-"IJC Treasury Participation Equity AIIOC
1
2Equity Pa?icipalion 3 4Hale
Base Rate |4.00 Upside 0.00
Floor Rate |4.00 [4.00 Cap Rate |99.00 |09.00
Downside
Avg Credit
0.00
Vol vs. Historical [1070-2000]
4.000
3.00
4.000
Scenarios
5
B
|4_00 [4.00 |4.00 [4.00 |4.00 |99.00 |s9.00 |0a.00 |99.00 [99.00 4.000
4.000
|50000
4.000
Gel W
W
4.000
‘00025
080a 0.40009
Interest Rate Exposures — Change in MV(Liab) for1% increase in Forward Rate 1
2
3
4
5
7
6
T
-0.5B20
0.5716
0.5570
-0.5501
0.5418
-0.5438
0.5498
9 -0.0100
1U -0.00BT
11 -0.0155
12 -0.0050
13 0.0087
14 0.0189
15 0.0000
Fig. 3
8
-0. 0039
4.000
US. Patent
Sep. 15, 2009
Sheet 4 017
US 7,590,581 B1
['1' Capital Protection Annuity - Interactive Pricing - Version 1.2
Current Yield Curve 1
[Coupon] Treasury
Yields
3
5
Assumed
7
10
20
E34 [1.72 [2.17 [3.05 [3.50 [4.01
Resulting Zero
1.340
“8'”
c
2
1.726
2.199
3.101;
3.91;?
Coupon
[4.93
4.1 30
[3.00
_
Strip [
5.259
NA-GARCH Equity Parameters
|1.1B26318
beta2 ‘0.0375993
lambda |0.050674
beta0 |Z15507e~6
beta1 [0.893101
obs/yr I253
div
lnstVol I25
I125
Declared Rate Annuity 1
Declared Rate [3.68
2
3
[3.50
4
[3.09
5
[3.59
s
[3.59
1
[3.68
W
[3.09
Get Pv [
0.99959
Capital Protection Annuity Treas Alloc '50
Equity Alloc
'50
Treasury Partlcrpatron
Equity Participation Hate I100
1
2
3
4
5
6
7
Base Rate [1.63 Upside 1.00
FloorRate [1.153 [1.63 [1.63 [1.63 [1.63 [1.63 [1.63 Cap Hate [99.00 [99.00 [99.00 [99.00 [99.00 [09.00 [99.00
Downside
Avg Credit
1.00
Vol vs. Historical 11970-2000]
1.030
3.00
2.334
Scenarios
2.689
2.900
[50000
3.020
Get w
3.125
W
‘00034
Delta
0.30991
Interest Rate Exposures — Change in MV(Liab) for 1% increase in Forward Rate 1
2
3
4
5
6
7
05970
0.5452
0.4085
-0.4272
—0.3750
0.3203
-0.2097
0 0.2962
10 0.2552
11 0.1726
12 0.1304
13 0.0877
14 0.0313
15 0.0000
Fig. 4
8
0.3471
3.290
US. Patent
Sep. 15, 2009
Sheet 5 017
US 7,590,581 B1
111-‘ Capital Protection Annuity - Interactive Pricing - Version 1.2
Current Yield Curve 1
[Coupon] Treasury
Yield,
3
5
Assumed
7
10
20
[1.34 [1.72 [2.17 [3.95 [3.58 [4.91
Resulting Zero
1.949
We'd‘
0
2
1.726
2.199
3.196
3.667
Coupon
[4.93
4.1 99
[3.99
_
Strip [
5.259
NA-GARCH Equity Parameters
‘1.1525318
belaZ |0.0375993
lambda |0.050674
beta0 |2.155B7e-6
beta1 |0.85S101
obslyr
div
lnslVol I25
253
1.25
Declared Rate Annuity 1
Declared Rate [3.68
2
3
[3.69
4
[3.68
5
[3.69
6
[3.69
7
[3.69
[3.68
GetPVl
w
9.99959
Capital Protection Annuity Treas Alloc '50
Equity Alloc
'50
Treasury Parllcrpatron
Equity Participation Hate '75
1
2
3
4
5
B
7
Base Rate [2.96 upside 1.99
FloorRate [2.96 [2.96 [2.96 [2.96 [2.96 [2.96 [2.96 Cap Rate [99.99 [99.99 [99.99 [99.99 [99.09 [99.99 [99.99
Downside
Avg Credit
1.99
vdr vs. Historical [1979-2999]
2.969
3.99
3.664
Scenarios
3.999
4.239
|59999
4.359
6e1Pv
W
4.455
4.629
"00012
Delta 9.31699
Interest Rate Exposures — Change in MV(Liab) for 1 % increase in Forward Rate 1
2
3
4
5
6
7
43.6754 9
41.6158 19
{1.5539 11
9.4996 12
9.4399 13
41.3721 14
9.3229 15
0.3545
0.3029
0.2089
0.1575
0.1003
0.0257
0. 0000
Fig. 5
9 9.4162
______________________________________ .....ESE§¥R9§.!f!?i--. __
US. Patent
all
Sep. 15, 2009
Sheet 6 of7
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‘
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iniL finished
US 7,590,581 B1
2003 E B 16 J 52 80
2003 6 8 L6 3 54 70
0. 0337l3917582 0.7709757798
Fig. 6
US 7,590,581 B1 1
2 The last point may require explanation. Participation rates
COMPUTER BASED SYSTEM FOR PRICING AN INDEX-OFFSET DEPOSIT PRODUCT
are loW When interest rates are loW because most of the
amount deposited must be invested in ?xed income to guar
antee return of principal, leaving little left over to buy equity
FIELD OF THE INVENTION
indexed options. Similarly, higher equity index volatility leads to higher option prices for the most common types of
The present invention relates generally to ?nancial prod ucts, more speci?cally to computer-based systems for pricing
options, driving participation rates doWn.
?nancial products, and, even more particularly, to a com
The investor or consumer therefore must face the situation
puter-based system for pricing an index-offset deposit prod
that achievement of equity participation and a guarantee of
principal generally precludes earning an attractive interest
uct.
rate. A dif?cult choice must be made.
References useful in understanding the present invention
BACKGROUND OF THE INVENTION
include: An Introduction to the Mathematics ofFinancialDerivatives,
A call option is a ?nancial instrument that gives its holder
Salih N. Neftci (2001) Financial Calculus, Martin Baxter and AndreW Rennie
the right (but not the obligation) to purchase a given security at a pre-speci?ed price, called the strike price or exercise
price, from the option seller. This structure alloWs the option holder to pro?t if the price of the security exceeds the strike price at the time of expiry of the option. At the same time, the maximum possible loss to the holder is limited to the price paid for the option if the security is Worth less than the
(1996) Martingale Methods in Financial Modelling, Marek Musiela and Marek RutkoWski (1997) 20
Option Pricing, Geman, H., El Karoui, N. and Rochet, J. C.
(1995)
exercise price, since the holder is not forced to buy the secu rity at an above-market price.
Arbitrage Theory in Continuous Time, Tomas Bjork (1998) Beyond average intelligence, Michael Curran, Risk 5 (10),
Options usually have a limited lifespan (the term) and have tWo main styles of exercise, American and European. In an
Changes ofNumeraire, Changes ofProbability Measure and
25
(1992)
American-exercise call option, the security may be purchased
The complete guide to option pricing formulas, Espen
for its strike price at any time during the term. In a European
Gaarder Haug, 1997 Measuring and Testing the Impact of News on Volatility, Robert F. Engle & Victor K. Ng (1993) Option Pricing in ARCH-Type Models, Jan Kallsen & Murad
exercise call option, in contrast, the security may only be purchased at the end of the term. An equity-indexed call option is one in Which the role of “securities price” is played by an equity index such as the S&P 500 or the Nasdaq 100. Since delivering the basket of
30
S. Taqqu (1994) The GARCH Option Pricing Model, Jin-Chuan Duan (1995) Pricing Options Under Generalised GARCH and Stochastic Volatility Processes, Peter Ritchken & Rob Trevor (1997)
securities that comprise the equity index is usually impracti cal, equity-indexed call options are usually cash-settled. This means that if the equity index is greater than the strike price at
35
time of exercise, the option seller pays the option holder the difference in price in cash: if the equity index is less than or equal to the strike, no payment is made.
An Analytical Approximationfor the GARCH option pricing model by Jin-Chuan Duan, Genevieve Gauthier, and Jean Guy Simonato (2001) The Market Model of Interest Rate Dynamics, Alan Brace, DariusZ Gatarek, and Marek Musiela (1997)
Many investors currently purchase equity-indexed call options directly to help achieve a desired balance of risk and return in their investment portfolios. Many investors and con sumers also bene?t indirectly from investments in such
A Simulation Algorithm Based on Measure Relationships in
options When they buy equity-linked deposit products such as
LIBOR and swap market models and measures, Farshid Jam
the Lognormal Market Models, Alan Brace, Marek
Musiela, and Erik Schlogl (1998)
equity-indexed annuities or equity-indexed certi?cates of
shidian (1997)
deposit (CD’s). This is because equity-linked deposit prod
Rate Models Theory and Practice, Damiano Brigo & Fabio
ucts are usually constructed from a mixture of equity-indexed call options and ?xed-income instruments such as bonds or
Drift Approximations in a Forward-Rate-Based LIBOR Mar
Mercurio (2001)
mortgages.
ket Model, C. J. Hunter, P. Jackel, and M. S. Joshi (2001) The Market Price of Credit Risk: An Empirical Analysis of Interest Rate Swap Spreads by Jun Liu, Francis A. Long staff, and Ravit E. Mandell (2000) Modern Pricing ofInterest-Rate Derivatives, Riccardo Reb onato (2002)
Investors and consumers obtain valuable bene?ts through
the use of equity-linked deposit products currently available in the market, such as:
The ability to bene?t from increases in the equity index
While protecting principal; and Achievement of diversi?cation by linking investment
55
returns to an equity index aggregating the performance of multiple issuers, rather than just one.
els, K. C. Hsieh, Peter Ritchken (2000)
Modern Portfolio Theory and Investment Analysis (4th ed.), EdWin J. Elton and Martin J. Gruber (1991)
There are also some disadvantages associated With cur
rently available equity-linked products, including: The lack of ?xed-income linkage, i.e., the inability to take advantage of increases in interest rates after product
The Art of Computer Programming, Vol. 2, Donald E. Knuth, 60
Addison-Wesley (1973)
purchase, because returns are tied to one equity index for
Algorithms, Robert SedgeWick (1983)
LoWer-than-desired “participation rates” (the proportion
equity index volatility.
Addison-Wesley (1969) The Art of Computer Programming, Vol. 3, Donald E. Knuth,
the length of the term; and, of increases in the equity index credited to the product), especially during times of loW interest rates or high
An Empirical Comparison ofGARCH Option Pricing Mod
65
Handbook of Mathematical Functions (AMS55), Milton AbramoWitZ and Irene A. Stegun (1972) Matrix Computations, Gene H. Golub and Charles F. Van
Loan (1989)
US 7,590,581 B1 3
4
Numerical Methods, Germund Dahlquist and Ake Bjorck, Prentice-Hall (1974) Algorithmsfor Minimization without Derivatives, R. P. Brent, Prentice-Hall (1973)
value of ?xed-income-linked crediting component to deter mine said value of said index-offset deposit product.
Numerical Recipes in C, William H. Press, William T. Vetter
The nature and mode of operation of the present invention Will noW be more fully described in the folloWing detailed
BRIEF DESCRIPTION OF THE DRAWINGS
ling, Saul A. Teukolsky, Brian P. Flannery, Cambridge University Press, 1992 Numerical Solution ofStochastic Diferential Equations Peter
description of the invention taken With the accompanying
draWing ?gures, in Which:
E. Kloeden and Eckhard Platen, (1995)
FIG. 1 is a screen shot Which shoWs hoW the program can be used to calculate the rate sensitivities of a traditional rate
Stochastic Simulation, Brian D. Ripley, Wiley (1987) Intel Architecture Optimization Reference Manual, Intel
annuity; FIG. 2 is a screen shot Which shoWs hoW the program can
(1998) Inner Loops by Rick Booth (1997) The Software Optimization Cookbook, Richard Gerber, Intel Press (2002) Principles of Compiler Design by Alfred V. Aho and Jeffrey D. Ullman (1977) File Structures An Analytic Approach, Betty Joan SalZberg
be used to calculate the price and interest rate sensitivities of a product With ?xed income linked index credits With the equity index allocation still at Zero; FIG. 3 is a screen shot Which shoWs hoW the program can
be used to calculate the price and interest rate sensitivities of a product With a constant base rate and With an equity index 20
FIG. 4 is a screen shot Which shoWs hoW the program can
(1988) A Very Fast Shift-Register Sequence Random Number Gen erator, Scott Kirkpatrick and Erich P. Stoll, Journal of
Computational Physics 40, (1981) 517-526
25
Monte Carlo Simulations: Hidden Errorsfrom r‘Good” Ran dom Number Generators, A. M. Ferrenberg, Y. J. Wong, and D. P. Landau (1992)
The Ziggurat Method for Generating Random Variables, George Marsaglia and Wai Wan Tsang (2000) Remark on Algorithm 659: Implementing Sobol's quasiran
FIG. 7 is a screen print illustrating the method of operation 30
35
General Description A brief description of an index-offset deposit product is
variables in the analysis ofoutputfrom a computer code, M. D. McKay, R. J. Beckman, and W. J. Conover, Techno
one that provides the purchaser With a notional allocation of
principal to equity-linked and ?xed-income-linked alloca 40
Elements ofSampling Theory, Vic Barnett (1974) Singular Value Decomposition and Least-Squares Solutions, Reinsch (editors), Handbook for automatic computation, 45
Accordingly, there is a long-felt need for an indexed
deposit product structure permitting the purchaser to enjoy an attractive combination of equity-linkage and ?xed-income linkage While guaranteeing a speci?ed percentage of princi pal. There is correspondingly a long-felt need for a computer based system for pricing such an indexed deposit product
tions and tWo guarantees:
a guarantee that a speci?ed percentage (often 100%) of principal Will be paid to the holder at the end of a speci ?ed term, and a guarantee that the index credit computed from the equity
G. H. Golub and C. Reinsch, in J. H. Wilkinson and C.
vol. II: “LinearAlgebra”, Springer Verlag (1974)
of the cpp_patc operation. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
A comparison of three methods for selecting values of input
metrics, 21(2):239-245, (1979)
be used to calculate the price and interest rate sensitivities of a product With a treasury linked base rate and With an equity index allocation of 50%; FIG. 5 is a screen shot similar to that of FIG. 4 but shoWing that an equity participation rate of 75% has been introduced; FIG. 6 is a screen print illustrating the method of operation
of the ?nd_cpp operation; and,
dom sequence generator, Stephen Joe and FrancesY. Kuo, ACM Transactions on Mathematical Software, March 2003
allocation of 50%;
linked and ?xed-income-linked index credit compo nents at the end of the term Will be non-negative, i.e., that
positive and negative index credit components from the equity-linked and ?xed-income-linked notional alloca tions can offset each other so long as the index credit 50
structure.
itself is nonnegative. The index credit component for the equity-linked notional allocation Will normally be based on a published equity index such as the S&P 500 index or NASDAQ index. The index
credit component for the ?xed-income-linked notional allo SUMMARY OF THE INVENTION
The present invention comprises a computer-based method for determining a value of an index-offset deposit product, having a principal amount P, a term T, a speci?ed guaranteed amount G, and an index credit C, comprising a) setting trial values for ?xed-income-linked crediting parameters for said
55
declared by the issuer (internal index). We refer to a Constant
Maturity Treasury rate and Zero-coupon bond yields beloW 60
for the ?xed-income-linked notional allocation for the sake of concreteness, but the extension to different external and inter
65
nal interest indices is straightforward. Having de?ned a generic index-offset deposit product, We can de?ne speci?c index-offset deposit products, such as deferred annuities, life insurance, certi?cates of deposit, and bonds, as specializations of the generic product. Index credits
product implying an expected ?xed-income-linked crediting component F at the end of the term T, b) determining a cost for
an equity option paying equity-linked credit component E such that the index credit C:E+F, to be paid at T, together With the principal P, is at least equal to G; and c) summing said equity option cost, present value of principal, and present
cation may be based on a Treasury-based or Libor-based interest rate (external index) or may be based on rates
for such products are calculated from ?xed income and equity-indexed notional allocations and index credit compo
US 7,590,581 B1 5
6
nent parameters, With a guarantee that a speci?ed percentage of principal Will be paid at the end of the term. The index-offset deposit product has some features in com
has declined to 2.5% at the ?rst anniversary, then the base rate
mon With equity-indexed deposit structures that have previ ously been described in the literature, (see, e.g., U.S. Pat. No. 6,343,272, a system for managing equity-indexed life and
centage (e.g., 100%) of the (signed) increase in the equity end value, or to an average value such as the Weekly average
annuity policies). HoWever, there are important differences betWeen the present invention and prior products, Which lead
of the index over the last quarter (3 months) of the term. The interaction betWeen the tWo index credit components
for the second year Will be 3%+0.5*(2.5%—3.5%):2.5%. The equity-indexed credit component is based on a per
index over the term, measured from the starting point to the
to the present invention being a more e?icient product, i.e., providing a more attractive combination of equity-linkage and ?xed-income linkage under the constraint that a speci?ed
is that a decrease in the equity index can offset the interest
indexed credit component: hoWever, a speci?ed percentage of principal is guaranteed (e.g., the index credit itself cannot be negative). For example, assume that 50% of the deposit is notionally allocated to ?xed-income-linked and 50% is notionally allocated to equity-linked. If the interest-indexed
percentage of principal must be guaranteed. The key differ ences are:
The notional equity-linked and ?xed-income-linked allo cation of principal, and The guarantee that the index credit itself, although not each
credit component at the end of the term Were 20% of the
original deposit, and the equity index Were doWn 12%, then the index credit Would be 20%-0.5*12%:14% of premium: conversely, if the equity index Were up 12%, then the index credit Would be 20%+0.5*12%:26% of premium.
index credit component separately, Will be nonnegative. These differences make the product more dif?cult to price
than existing equity-indexed products, especially in the gen eral case in Which the ?xed-income-linked credit component is indexed to a Treasury or Libor-based rate, because the
20
Risk management for deposit products usually requires
interest-rate and equity-market risks interact. Key interac tions include:
the dependence of arbitrage-free pricing for equity options on realiZed short-term risk-free interest rates, so that
25
equity exposures vary depending on the shape, level, and volatility of the yield curve, and 30
companies have typically managed their assets and liabilities
35
The principal paid for the index-offset deposit product is 40
?xed-income-linked allocation. The notional allocation per
centages are determined by the issuer, not the purchaser, and
Duration and convexity measures, Which assume parallel shifts in the yield curve, are not very useful in managing a 45
lated at the end of the term. The length of the term might be 5 to 7 years for a typical product. The ?xed-income-linked credit component is determined
different than the exposure arising from guaranteeing a ?xed interest rate. 50
as large as a speci?ed ?oor rate (e.g., 2%, varying by year of the term), but no larger than a speci?ed cap rate (e.g., 8%,
Measuring the sensitivity of the market value of the liabil ity to changes in individual forWard rates is a more generally
useful methodology than measuring duration and convexity. The folloWing example (for a ?ve-year term) shoWs the dif
once again varying by year of the term). The base rate for the ?rst year of the term is declared by the
issuer (e.g., 2%). At successive intervals during the term (e.g., annually), the base rate changes by a percentage of the change
more general ?xed-income-linked product like the current
invention. The interest-rate exposure created by indexing to, for example, a Constant Maturity Treasury, is considerably
by compounding together the “base rates” for each year of the term. In this compounding, the base rate is taken to be at least
J. Elton and Martin J. Gruber). Use of duration and convexity for insurance carriers, for example, is so Widely accepted that it has been formalized in regulations such as NeW York Regulation 127, Which uses Macaulay duration as the criterion for determining hoW Well the assets and liabilities of a carrier are matched.
might be (for example) 50% each. Equity-linked and ?xed-income-linked credit components are determined over each term and the index credit is calcu
convexity and liability duration and convexity. A good dis cussion of duration and convexity for deposit products is found in The Management ofBond Portfolios (Chapter 19 of
Modern Portfolio Theory andlnvestmentAnalysis, by EdWin
Detailed Description of Product Mechanics notionally allocated to the equity-linked allocation and the
Sensitivity to Forward Interest Rates Deposit-taking institutions such as banks and insurance to try to minimiZe the difference betWeen asset duration and
components. Pricing a product With such interactions requires the devel opment of softWare speci?cally designed to take these inter acting risks into account.
attention to the equity-market exposures created by the prod uct (for equity-linked products) and to the interest-rate expo sures created by the product (for ?xed-income-linked and equity-linked products). Risk management considerations for the current invention are more complicated than for currently available products in at least three Ways:
the fact that the index credit at the end of the term, and hence interest rate exposures, depend on the expected
index credit component from the equity-linked notional allocation, because of the potential for offset betWeen the equity-linked and ?xed-income-linked index credit
Risk Management Considerations
ference in sensitivity to forWard interest rates for a ?xed 55
income-linked product With no equity-linkage:
in a benchmark yield, such as the 5-Year Constant Maturity Treasury rate, or the yield on a 5-year Zero-coupon bond.
Different percentages (participation rates) may apply to increases and decreases, and the percentages may be positive
60
or negative.
For example, suppose the upWard percentage is 100%, the doWnWard percentage is 50%, the base rate in the ?rst year is 3%, and the term is seven years. The 5-Year CMT rate at issue
is 3.5%. If, at the ?rst anniversary, the 5-Year CMT rate is 4.5%, then the base rate for the second year Will be 3%+1* (4.5%—3 .5%):4%. If, on the other hand, the 5-Year CMT rate
65
Forward Rate
GIC
Fixed-Income Linked
1 2 3 4 5 6 7 8
—0.98 —0.98 —0.97 —0.96 —0.95 0.00 0.00 0.00
—0.98 —0.77 —0.55 —0.37 —0.21 0.75 0.55 0.33
US 7,590,581 B1 7
8
-continued
The equity-linked notional allocation has a participation rate of 100%, and there is no averaging period at the end of the term;
Forward Rate
GIC
Fixed-Income Linked
9 10
0.00 0.00
0.16 0.00
—4.84
— l .09
Total
Options are priced using the NA-GARCH pricing model (a generalization of the well-known Black-Scholes formula) with an initial equity index volatility of 25%; The notional allocation of principal is 50% to each of the equity-linked allocation and the ?xed-income-linked alloca
tion; and The “Total” row shows duration, as traditionally measured. For a traditional deposit product, like a Guaranteed Invest
10
Given these assumptions, one simple strategy to hedge the return guaranteed to the purchaser is to buy an in-the-money equity-indexed call option with some of the amount deposited and a 7-year Zero-coupon bond with the rest. The problem is how much to invest in each of these investments to hedge the
ment Contract (GIC) the total is useful information, showing essentially the duration of a Zero-coupon bond broken out by the forward rates. The total in the Fixed-Income Linked expo sures (which assume 100% linkage to upward moves in the 5 -Year CMT rate and 50% linkage to downward moves) is not very useful, as it is a sum of positive and negative components that are not level by forward period. Backing a ?ve-year GIC with a ?ve-year Zero-coupon bond would achieve a good
asset/liability match, but backing the above ?xed-income
The expense factor (the assumed deduction from the earned rate to cover issuer expenses and pro?ts) is 1.37%.
return properly. We approach this by examining the index credit component for the ?xed-income-linked notional allo cation, ?rst under the assumption that it is offered in isolation 20
from the equity-linked notional allocation, and then allowing
linked product with a 1.1 year bond would be very risky if the
for the interaction between the index credit components.
yield curve were to steepen.
Stand-Alone Fixed-Income-Linked Crediting Rate
Forward Rate/Equity Index Interaction Interest-rate exposures for equity options are usually cap
The notional ?xed-income-linked allocation is 50% of 25
principal, the “annual excess guarantee cost” is 0%. If more than 100% of principal were guaranteed at the end of the term, this “annual excess guarantee cost” would be greater than
tured by a measure called rho, which assumes (like duration and convexity) that yield curve shifts are parallel. For this product, these interest rate exposures must be broken out by individual forward rates in the same way as in the previous
item, to allow them to be managed properly under the assumption that yield curve shifts need not be parallel.
Zero.
30
income-linked credits. Depending on the current and
The issuer could therefore afford to credit the following on the notional ?xed-income-linked allocation, if it were stand
alone rather than combined with the equity-linked notional
Equity-Index/Credited Interest Interaction The simple example above does not take into account the interaction between the equity-linked credits and the ?xed
principal. Since the overall product guarantee is 100% of
allocation: Earned Rate-expense factor-annual excess guarantee cost; 35 or
expected levels of the equity index, the amount to be credited
5.61%-1.37%-0%;
at the end of the term will vary, and so its present value (and hence the interest rate exposures of the product) will also vary.
or 4.24% at the end of each year.
If the product is extemally-indexed then the current yield
Discounted at 5.61% (the earned rate), the present value of 40
4.24% at the end of each year is 0.240009427.
curve and interest volatilities will also affect the expected amount of interest to be credited to the policy, which in turn
Combined Fixed-Income-Linked Crediting Rate We can solve (by bisection, regula falsi, Brent’s method, or
affects equity exposures because of the offset between the
other root-?nding method), that a rate of 4.828% can be credited on the ?xed-income-linked notional allocation, i.e.,
extemally-indexed index credit component and the equity indexed index credit component.
45
This is demonstrated by the following calculations:
Simple Pricing Example
At the end of the term, the value of the ?xed-income-linked notional allocation will grow to 50%>
