UNIVERSITE DE SHERBROOKE
Fa c u l t e d’a d m i n i s t r a t i o n
ST R A T E G IE D ’IN V E ST ISSE M E N T G U ID E PAR LES PASSIFS ET IM M UNISATION D E PO R TE FE U IL L E : Un e a p p r o c h e D YN A M IQ U E
P ar
Mi g u e l M o i s a n- Po i s s o n
Memoire presente a la Faculte d ’adm inistration en vue de l’obtention du grade de
M A ITRE ES SC IEN C ES (M .S c.)
Ao u t 2 0 1 3
1+1
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UNIV ERSITE DE SHERBROOKE
Fa c u l t e d'a d m i n i s t r a t i o n
St r a t e g i e d'i n v e s t i s s e m e n t g u i d e p a r l e s p a s s i f s e t i m m u n i s a t i o n DE PO R TEFEU IL LE :
U n e a p p r o c h e d y n a m i q u e
Mi g u e l Mo i s a n- Po i s s o n
a ete evalue par un ju ry compose des personnes suivantes :
___________________________D irecteur de recherche Alain Belanger ___________________________Lecteur Guy Bellemare __________ Lecteur Anastassios Gentzoglanis Memoire accepte le
A b stract
In a previous MITACS project in collaboration with Addenda C apital, two basic liability m atching strategies have been investigated: cash flow m atching and mom ent matching. These strategies per formed well under a wide variety of tests including historical backtesting. A potential shortcom ing for both of these m ethods is th a t the optim ization process is done only once at the beginning of the investm ent horizon and uses determ inistic moment m atching constraints to immunize th e portfolio against interest rate movements. Though the portfolio subsequently need to be frequently rebal anced, this static optim ization does not take into account the relatively high rebalancing costs it involves.
The main objective of this present project is to further enhance th e mom ent m atching m ethod by implementing and testing a stochastic dynam ic optim ization and by com paring its efficiency w ith th e static one. O ur dynam ic optim ization problem is to minimize the portfolio cost and its expected rebalancing costs one m onth ahead over a set of interest rate scenarios by the use of stochastic moment m atching constraints.
O ur backtesting results show some improvements with th e 6 moments m atching strategy as the dynam ic optim ization slightly shrinks the difference in asset-liability gap between scenarios com pared with the static optim ization. However, after analyzing the realized periodic rebalancing costs each m onth (a constant bid-ask spread has been assigned to each asset’s position change in th e optim al portfolio), the im munization improvements are m itigated by substantialy higher costs. We also noticed, in th e case of the duration/convexity m atching strategy, th a t th e dynam ic optim ization is not th a t much more efficient th an the static m ethod.
Thus, these results confirm th a t the 6 moments m atching technique is still more efficient with both the static and stochastic dynam ic optim ization. O ur extensive dynam ic analysis of transaction costs through backtesting showed th a t from an efficiency to cost ratio and an efficiency to simplicity ratio, the static 6 moments m atching m ethod seems so far to be a more practical solution for liability matching.
R esu m e
Dans le contexte des marches financiers turbulents, les strategies de gestion de portefeuille dont les actifs doivent etre apparies a des passifs (ex : caisses de retraite, compagnies d ’assurance, etc.) sont devenues un enjeu im portant. P ar exemple, les caisses de retra ite dont les actifs ont litteralem ent fondus lors de la crise financiere et dont les passifs eventuels augm entent de plus en plus a cause de la retraite des baby-boomers presentent actuellement des deficits actuariels et doivent reflechir a de nouvelles strategies pour pallier a ce probleme. Une partie de la solution est dans la gestion accrue des risques de variations de valeur dans les portefeuilles. C ette gestion du risque provient en partie de la recherche de strategies optim ales d ’im munisation de portefeuilles, nouveau domaine appele ’investissement guide par le passif’ (Liability Driven Investm ent) . Ceci a pour objectif d ’optim iser et surpasser l’appariem ent des flux monetaires des actifs et des passifs d ’un portefeuille en utilisant de nouvelles techniques d ’optim isation dynam ique basees sur la duree, la convexite e t d ’autres mom ents d ’im munisation d ’un portefeuille.
Dans la litterature, on retrouve plusieurs etudes sur l’im munisation de portefeuille. On peut classer ces techniques d ’im m unisation en deux grandes categories : le m oment matching et le cash
de Redington (1952). Fong and Vasicek (1984) et Nawalkha and Cham bers (1997). La seconde implique differentes m ethodes de program m ation lineaire que Ton peut retrouver. par exemple. dans Kocherlakota et al. (1990).
Dans le cadre d ’un precedent projet MITACS en collaboration avec Addenda C apital. Augustin et al. (2010) etudient les deux strategies d ’appariem ent precedentes dans un contexte de passifs multiples. Leur strategie de m oment matching est inspiree des resultats de Theobald and Yallup (2010) qui m ontrent que l’utilisation de 6 mom ents offre une efficacite d ’im m unisation optim ale. Augustin et al. (2010) m ontrent que ces deux strategies performent bien sous une large variete de tests, y compris en backtesting. Un inconvenient potentiel de ces deux m ethodes est que le processus d ’optim isation est seulement effectue une fois au debut de l’horizon de placement et utilise des contraintes de m oment matching determ inistes pour immuniser le portefeuille contre les fluctuations des tau x d ’interet. Alors que le portefeuille necessite ensuite d ’etre frequemment reequilibre. cette optim isation statique ne tient pas en com pte les couts relativem ent eleves de ce reequilibrage.
L’objectif de ce projet est d ’ameliorer la m ethode de m oment matching par l’im plantation et la validation d ’un modele d ’optim isation dynam ique stochastique et en com parant son efficacite avec l’optim isation statique. Le probleme d ’optim isation dynam ique est de minimiser le cout du portefeuille ainsi que les couts de reequilibrage esperes sur un horizon d ’un mois pour un ensemble de scenarios de tau x d ’interet. Cela est possible via l’utilisation de contraintes stochastiques de
m oment matching. D ’autres modeles interessants d ’optm isation stochastique tels que ceux etudies
pas Schwaiger et al. (2010) ont ete envisage, mais n ’ont pu etre utilises faute de performance com putationnelle.
Nos resultats de backtesting m ontrent quelques am eliorations avec la m ethode des 6 moments car Ton observe que l’optim isation dynam ique perm et de reduire la difference de l’ecart actif-passif entre les differents scenarios com parativem ent a l’optim isation statique. Cependant, apres analyse des couts de reequilibrage periodiques realises chaque mois, il s ’avere que les am eliorations en term es d ’efficacite d ’im m unisation de portefeuille soient attenuees p ar une hausse substantielles des couts. Ces couts de transaction ex-post ont ete approxim e par un ecart bid-ask constant attrib u e au changement de positions de chaque actif du portefeuille optim al. Dans le cas de la strategie duree/convexite, on rem arque egalement que l’optim isation dynam ique n ’apporte pas d ’efficacite supplem entaire par rap p o rt a la m ethode statique.
Ces resultats confirment done que la technique des 6 mom ents est. encore une fois, la plus efficace, a la fois avec l’optim isation stochastique et l’optim isation statique. N otre analyse etendue des couts de transaction via le backtesting m ontre toutefois que le rapport couts-benefices ainsi que le rapport parcimonie-couts rend mitige l’efficacite de la m ethode des 6 moments dans le cadre de l’optim isation stochastique. Ainsi, dans le cadre de cette etude, il semble que l’optim isation statique soit une solution plus praticable pour l’appariem ent du passif en comparaison avec l’optim isation dynamique.
A ck n ow led gem ents
I would like to express my g ratitud e to my supervisor Alain Belanger who has willingly shared his precious tim e through the learning process of this m aster thesis. I appreciate the useful comments and rem arks which always helped me to develop my thoughts. Furtherm ore. I would like to thank MITACS Accelerate C a n a d a s research internship program (12-13-5629). FQ R N T Acceleration Quebec (171527) and A ddenda C apital for giving me th e o pp ortunity to im plem ent and te st this LDI strategy. In particular. I would like to th an k Bernard Augustin and the quantitative research team for their technical support and helpful comments. Finally, I would like to th an k th e Faculte d ’adm inistration for their financial support.
C on ten ts
L ist o f F ig u r es vii
L ist o f T ab les viii
I n tr o d u c tio n 1
1 B a ck g ro u n d in fo rm a tio n 1
2 T h e o r e tic a l fram ew ork 3
2.1 Yield curve modeling and shock s c e n a r i o s ... ... 3
2.2 Bonds and liabilities v a lu a tio n ... 4
2.3 Moments c a lc u la tio n ... 4
2.4 O ptim ization model ... 5
3 B a c k te s tin g m e th o d o lo g y 7 3.1 D ata and lim ita tio n s ... 7
3.1.1 Bond universe and liabilities ... 7
3.1.2 Transaction c o s t s ... 8
3.1.3 Yield curves and shock s c e n a r i o s ... 9
3.1.4 O ptim ization settings ... 9
3.2 Backtesting a l g o r i t h m ... 10
3.2.1 Shortfall liquidation algorithm ... 11
3.2.2 Rebalancing adjustm ents a lg o rith m ... 12
3.2.3 Asset-Liability gap measures ... 14
4 R e s u lts o u tc o m e s and fu tu r e resea rch 14 5 C o n c lu sio n 19 6 R e fe r e n c e s 20 A P o r tfo lio p o sitio n s tra ck in g 21 A .l D u ratio n /co n v ex ity -m atch in g ... 21
A .2 6 M oments m a tc h in g ... 24
B G ra p h s o f p o rtfo lio im m u n iza tio n re su lts 27 B .l W ithout additional liquidity injection ... 27
B.1.1 D u ratio n /co n v ex ity -m atch in g ... 28
B .l.2 6 Moments m a tc h in g ... 33
B.2 W ith additional liquidity injection ... 38
B.2.1 D u ratio n /co n v ex ity -m atch in g ... 39
B.2.2 6 M oments m a tc h in g ... 40
C D a ta an d s e ttle m e n t d a te s 41
C .l Bond universe d e ta ils ... 41
C.2 L ia b ilitie s ... 43
C.3 Settlem ent d a t e s ... 44
D O th er sto c h a stic p ro g ra m m in g m o d els 45 D .l Moment m atching m e th o d ... 45
D .1.1 Stochastic program m ing (SP) m o d e l ... 45
D .l .2 Chance-constrained programm ing (CCP) m o d e l... 46
D.2 Cash flow m atching m e t h o d ... 46
D.2.1 SP m o d e l ... 47
D.2.2 C CP m o d e l ... 47
D.2.3 Integrated chance-constrained program m ing (ICCP) m o d e l ... 48
List o f Figures
1 Tim e scale setting for b a c k te s tin g ... 102 D uration/convexity-m atching: asset and liability P V tracking (w ithout additional liquidity injection a t rebalancing dates) ... 15
3 6 mom ents matching: asset and liability PV tracking (w ithout additional liquidity injection at rebalancing dates) ... 16
4 DC matching: portfolio tracking with static optim ization ... 22
5 DC matching: portfolio tracking with stochastic o p tim iz a tio n ... 23
6 6M matching: portfolio tracking with static o p tim iz a tio n ... 25
7 6M matching: portfolio tracking with stochastic o p tim iz a tio n ... 26
8 DC matching: asset-liability gap (w ithout additional liquidity injection a t rebalanc ing d a t e s ) ... 28
9 DC matching: asset cash-flows and liability stream (w ithout additional liquidity injection a t rebalancing dates) ... 29
10 DC matching: net investments costs and bid-ask costs (w ithout additional liquidity injection a t rebalancing dates) ... 30
11 DC matching: additional liquidity injection needs at rebalancing d a t e s ... 31
12 DC matching: difference between A-L gap under each scenarios (w ithout additional liquidity injection a t rebalancing dates) ... 32
13 6M matching: asset-liability gap (w ithout additional liquidity injection at rebalanc ing d a t e s ) ... 33
14 6M matching: asset cash-flows and liability stream (w ithout additional liquidity injection a t rebalancing dates) ... 34
15 6M matching: net investments costs and bid-ask costs (w ithout additional liquidity injection a t rebalancing dates) ... 35
16 6M matching: additional liquidity injection needs at rebalancing d a t e s ... 36
17 6M matching: difference between A-L gap under each scenarios (w ithout additional liquidity injection a t rebalancing dates) ... 37
18 DC matching: asset-liability gap with additional liquidity injection a t rebalancing d a t e s ... 39
19 6M matching: asset-liability gap with additional liquidity injection at rebalancing
d a t e s ... 40
20 Bond u n iv e rse ... 42
21 L ia b ilitie s ... 43
22 O ptim ization and backtesting settlem ent d a t e s ... 44
List o f Tables
1 Yield curve PCA scen ario s... 9In trod u ction
Since the recent financial meltdown and because of th e current economic instability, many fund m anagers tend to shift their total-return-oriented investment approach toward a liability-driven investm ent (’LD I’) strategy. Its main objective is to find investment strategies th a t will m atch or outperform a liability stream (pensions, insurance claims, etc.). The popularity of this strategy among pension fund managers, in particular, is not surprising because of m ajor changes in the demographics of developed and emerging countries; among others, life expectancies are increasing. In N orth America, baby-boomers have also started to retire massively. These effects heighten the need for pension funds to properly fund their rising liabilities, especially because of th e recent fall in th e equity m arkets and bond yields reaching their lowest historical levels. Moreover, recent accounting standards and regulatory changes force pension fund managers to adopt a new view on their asset allocation to reduce the volatility of their funding statu s and financial results.
This docum ent presents the results of a LDI strategy for portfolio moment m atching immuniza tion techniques. This study follows results upon a previous MITACS project. T he objective is to minimize the portfolio cost and its expected rebalancing costs by allowing it to be regularly rebal anced over time. In order to take into account th e uncertainty of possible movements of the term stru ctu re of interest rates, th a t is the interest rates risk, we use dynamic stochastic optim ization. It is the second MITACS project on this topic in collaboration with A ddenda C ap ital1.
T he Section 1 gives an overview of the cash flow m atching and moment m atching literature and explains how the present study could improve the previous project results. In Section 2 we explain, in a generic m anner, the theoretical framework of our optim ization model. In Section 3 we give details on d ata used in our analysis and explains the assum ptions we had to make. We also describe how we performed th e backtesting of our model and how we analysed its efficiency. We explain in Section 4 our main results and gives some ideas of possible future research.
1
B ackground inform ation
There are two large classes of LDI methods: portfolio im munization and cash flow m atching. The purpose of immunization is to construct a portfolio for which the change in value will m atch the change in liability value over a given horizon. There exist many im munization techniques th a t are in th e lineage of classical papers like Redington (1952) and Fong and Vasicek (1984). T he classical m ethods like duration/convexity m atching are used to immunize the portfolio change in value against parallel movements in the term stru cture of interest rates (’T S IR ’). More recently, Nawalkha and Cham bers (1997) and Theobald and Yallup (2010) improved im munization techniques by using non-centered moment m atching for m ultiple liabilities. These more advanced techniques give the possibility to immunize the portfolio against different movements of th e TSIR. Theobald and Yallup (2010) conducted a very comprehensive empirical analysis in the UK bond m arket and showed th a t using the first 6 moments m atching is optim al for an immunized portfolio. T he cash flow m atching is a technique which consists in selecting securities w ith a m aturity th a t m atch the tim ing and am ount of the liabilities. Linear program m ing can be employed to construct a least-cost cash flow m atching portfolio (see K ocherlakota et al., 1990).
In a previous MITACS project in collaboration w ith Addenda Capital. B. A ugustin. A. Belanger,
’w w w .addenda-capital.com.
K. Haniidya and Y. Wagner (hereafter referred to as A ugustin et al.. 2010) investigated th e previous two basic liability m atching strategies: cash flow m atching and mom ent m atching. Their results first show th a t the cash flow m atching algorithm works well to reduce deviations from liability cash flow needs. They also show th a t this strategy is relatively expensive in term s of portfolio cost. For the second matching strategy, the results of Augustin et al. (2010) show that the moment m atching technique substantially reduces the deviation risk between th e portfolio value and the liabilities, w hat we refer to as the asset-liability gap.
The two basic static liability m atching m ethods tested in A ugustin et al. (2010) performed well under a wide variety of tests including historical backtesting. However, in order to m aintain an optim al portfolio over time, th e portfolio has to be frequently rebalanced to meet the m atching constraints. Thus, it involves relatively high periodic rebalancing costs, which are not included in the optim ization problems used in Augustin et al. (2010). As their optim ization can be viewed as ’sta tic ’, then one might w ant to dynam ically include theses additional rebalancing costs over tim e into th e optim ization process. This dynamic p art of th e cash flow m atching and mom ent m atching optim ization problems becomes an optim al control problem (dynamic optim ization) where the controls are the positions in the bonds which are now allowed to discretely change w ith time. For both m atching techniques, the sta te variable is still the portfolio cost which now also includes a penalty function for the expected rebalancing costs over different yield curve scenarios for a given horizon. Moreover, w ith moment matching, the constraints in th e optim ization problem are translated into a dynam ical way such th a t the first k mom ents of th e portfolio and the liabilities are now allowed to change with tim e and depend on the TSIR scenarios.
T h e main objective of this present project is to further enhance th e static mom ents m atching m ethod described previously. As mentioned before, a potential shortcoming w ith basic techniques used by Augustin et al. (2010) is th a t the optim ization process is only done one tim e a t the beginning of the horizon and uses determ inistic constraints to immunize the portfolio. O ur goal is to model the dynam ics of the optim ization process by allowing the portfolio to be rebalanced a t a minimum cost. In order to take into account the uncertainty of possible movements of the TSIR, th a t is th e interest rate risk, we initially wanted to use th e three stochastic program m ing approaches studied in Schwaiger et al. (2010) for both cash flow m atching and mom ent m atching techniques: a stochastic linear program m ing (’SLP’) model, a chance-constrained program m ing (’C C P ’) model and an integrated chance-constrained program m ing (T C C P ’) model.
We have to mention th a t for com putational issues, a slightly modified version of only th e first stochastic programm ing m ethod (the SLP) has been studied here. However, because they could be useful to use for these types of LDI strategies, these stochastic programm ing models are explained w ith more details in A ppendix D. Along the lines of w hat was suggested by a MITACS referee of this project, we used a stochastic dynam ic approach through ’backward in tim e’ to handle the problem. The stochastic program m ing m ethods mentioned above will be analyzed in a future research project.
In this study, we investigate th e effects of th e dynam ic stochastic optim ization on th e portfolio immunization efficiency and the resulting strategy costs. T he cash flow m atching m ethod being close to the duration/convexity m atching m ethod (2 mom ents), we tested only the two following liability moments matching strategies: duration/convexity (2 moments) and 6 m om ents matching. We then analysed for both of these m atching techniques the efficiency of the dynam ic optim ization over the static one. Because of com puter performance issues, we also reduced th e num ber of TSIR scenarios. We should however try it w ith more scenarios in a future research project. The
stage dynam ic stochastic optim ization m ethod developed here as well as the static m ethod were backtested over a twelve m onths window with m onthly portfolio rebalancing and weekly valuation.
As explained in the ’Results outcom es’ section, we obtained some improvements of th e 6 mo ments matching strategy by th e use of stochastic dynam ic optim ization but it is m itigated by substantial higher rebalancing costs. Our results thus highlight the 6 moments m atching efficiency in both static and dynam ic optim ization.
The next section explains with more details the technical aspects of the model used in our analysis.
2
T heoretical fram ework
Let B be a set of bonds which constitutes the bond universe and £ be a set of actuarial liability stream th a t has to be m et over multiple periods prior to an investm ent horizon H . Note th a t these liabilities are assumed to be determ inistic and have been given by A ddenda C apital. Each bond
n G B. n = 1 , 2 can be described with several characteristics, (fci, &2, &3, As m arket
conditions change over tim e, the bond universe B{t) at tim e t > 0 is described by th e following set:
B(t) = {n = n ( k i , k2, k 3,...) : ki € {c/assi}, &2 € {class2}, k3 € {cZasss},...} .
We will explain later in the ’Backtesting methodology’ section which d a ta and class filter we applied to our bond universe.
2 .1 Y ie ld c u r v e m o d e lin g a n d s h o c k s c e n a r io s
We define the current (spot) yield curve at tim e t w ith a tim e-to-m aturity T as a function r(t, T). The splines technique has been used to interpolate th e spot yield curve function for any m aturity. We also define the discount function2 D (t, T ) which gives at tim e t the discounted value of 1$ paid a t tim e T.
We define a finite set of yield curve scenarios w ith an horizon of h months. To generate TSIR shock scenarios, we used th e historical principal com ponent analysis (PCA). We did so by giving a shock on different com binations of the first three PCA com ponents of the actual spot curve (see L itterm an and Scheinkman, 1991). T h at is, each shock scenario ui € 0 is of three types of TSIR change: level (PCA1), steepness (PCA2) or curvature (PCA3). We can define the scenario universe as:
ft = {to = (L, S, C) : L = {shock in P C A l} , S = {shock in PCA2}, C = {shock in P C A 3 }}
The m agnitudes M - ^ P C A x ). x = 1,2,3. of these shocks is given by a m ultiple of the historical standard deviation of each PCA factor. As the set R is finite, the assigned scenario probability for each u j is P({w}) = p u and it is calculated by its historical frequency.
T he yield curve function a t tim e t for each scenario and for any m aturity is com puted using the spot curve at tim e t — h and each of the P C A ’s set of shocks ui. Thus, the yield curve function is defined as:
r ( t , T ; u ) = f ( r ( t - h,T);uj), Vw € fb (1)
2In our an alysis, we have used a d iscrete tim e d iscou n t function.
We also define the stochastic discount function D {t,T \uj).
The ’Backtesting m ethodology’ section explains in more details which yield curve d a ta we used and the characteristics of the generated PCA shock scenarios.
2 .2 B o n d s a n d lia b ilit ie s v a lu a t io n
Recall the bond index n G B. If tln is the i-th cash flow date after tim e t of th e n -th bond. We write B P V n (t\uj). the present value of th a t bond evaluated at tim e t. depending on scenario w. as:
B P V n (t; u) = ' ^ c n {ti„ ) D ( t ,tin\u>) (2)
in
where cn {tin) is th e n -th bond ’s cash flow at period tin . We let a generic sum m ation over all cash flow i up to the bo n d ’s m aturity. We define
B P V (t;uj) 4 ( B P V 1(t-,u;),BPV2(t;Lo),...,BPVK (t-,u;)) (3)
as a (1 x /Q -vector which contains the present value of each bond in the portfolio. We also define
P ( t ) ^ ( P 1( t) ,P 2( t) ,...,P K (t)) (4)
as a (1 x K )-vector which represents the m arket price for each bond.
Furtherm ore, if tj is the j- t h liability stream d ate after tim e t. we write L P V (t;uj). th e present value of the liabilities, as:
L P V f r u , ) = ' £ i l(tj ) D ( t , t j ;U) (5)
j
where l(tj) is th e liability stream a t period tj and the sum is over all liabilities prior or a t the investment horizon H .
Finally, let the cheapest (optimal) bond portfolio 11(f) C B(t) which covers th e liabilities over time. Note th a t in our optim ization model, we have a portfolio whose composition depends on tim e
t. As explained later in this section, our optim ization model is used to find the optim al position for
each bond n th a t will minimize the initial portfolio cost and its expected rebalancing cost. These positions are denoted by the vector u (t) 4 (un (t)), w ith un (t) £ R+. Vn £ 11(f). T he portfolio value at tim e t, under scenario ui. denoted by A P V ( t; tu) for ’Asset P V ’, is expressed as:
A P V { t - u ) = un (t)B P V n (t;u ) = BPV(f ;w)-u'(t) (6)
nen(t) where ' is the transpose operator.
2 .3 M o m e n t s c a lc u la t io n
Following Theobald and Yallup (2010), the k-th m om ent of the portfolio at tim e t. noted Ik{t;u>), is com puted as:
I k ( t ' , 0 j ) = A p y U - i J } z L / ^ „ ltn ( 0 cn ( ^ n ) ^ ( ^ t i n tu;)- (7)
^ ’ ' nen(t) in
Note th a t this portfolio’s moment expression is equivalent to the weighted sum of each b o n d ’s moment:
where u ) is the A~-t,h moment of the n-th bond and the weight wn = Un^ p y n ■ w ith wn = 1.
Respectively, the Ar-th mom ent of the liabilities, noted Jk(t; oj) is computed as:
For both of these mom ent measures, we can have A; = 1,2, where p is the desired num ber of moments to be considered. Theobald and Yallup (2010) show th a t p — 6 is optim al. This conclusion is also supported by A ugustin et al. (2010).
2 .4 O p t im iz a t io n m o d e l
O ur optim ization model consists of two general steps. First, we need to generate yield curve scenarios. Since we assume th a t the liability stream is known, the m ajor source of random ness (risk) in our strategy is th e interest rate (we use high credit quality bonds in our analysis). Thus, as explained in the previous section, we have to sim ulate m ultiple T S IR shocks to generate different yield curve scenarios which are used in our optim ization model. Second, we perform a two-stage optim ization process, which depends on T S IR scenarios.
To incorporate th e rebalancing dynamics and the interest rate risk into the optim ization process, we use a two-stage stochastic dynam ic optim ization w ith stochastic-dependent constraints and an objective function th a t minimizes the portfolio cost and its expected rebalancing cost one m onth ahead. As m entioned previously, using the baseline yield curve r(to ,T ) at tim e to, we first generate a finite set fi of T SIR scenarios of h = 1 m onth horizon to have different yield curves scenarios
r(ti,T;co) at tim e t\ > to- The num ber of scenarios generated is explained in th e ’Bakctesting
methodology’ section. After, for each scenario, we find an optim al portfolio I I ( t i ; w) C B{t\) with optim al positions {itn (fi,u;)}. This step is done by minimizing th e portfolio cost subject to the moment m atching constraints. In fact, since we generated multiple TSIR scenarios, th e moment m atching constraints and the portfolio cost a t tim e t\ are acting as random variables which depend on these scenarios. These constraints are referred to as stochastic constraints. T he ’first stage’ optim ization problem is formulated as follows, for each scenario uj € fi:
n € F I( £ )
L P V (t;ui) (8)
minimize cost(ti;u>) = B P V (fi;w ) • u'(fi;u;)
u(q ,u>)
subject to A P V {t\\tjj) > L P V {t\\u j)
Ik(ti',u) = Jfc(fi;u;), \/k = 2 m — 1, m = 1 ,2 ,3 ,... (odd mom ents) (®)
> Jis(ti;u), Vfc = 2m, m — 1 ,2 ,3 ,... (even moments)
A • u(ti,u>) € a
F irst note th a t at this stage, we find a set {II(fi; cj)}weQ containing optim al portfolio for each scenario. In this optim ization problem, the first constraint shows th a t we want th e portfolio value to outperform the liability value. Moreover, we want the portfolio to have even m om ents greater
th a t those of the liability. This is because of th e positive convexity phenomenon. For example, if k = {1, 2 } (and thus, m — 1 only), the two m om ents m atching constraints are respectively the duration (odd mom ent k = 1) and the convexity (even moment k = 2). W ith 6 mom ents matching, we have m, = 1,2 such th a t k = {1, 2 ,3 ,4 , 5 ,6 }. The m atrix A in the last constraint can include different m andate constraints such as rating limit, individual weight limit, industry limit, etc. We will discuss about this below. Finally, note th a t the moment m atching are non-linear constraints because of th e denom inator A P V in the 4 expression (see equation (7)).
At this first stage, when we have the optim al portfolio for each scenario, we go backward through tim e to perform a second optim ization a t tim e to to find the needed optim al portfolio Il(to) whose positions are denoted by {un(4)}- As we have an assigned empirical probability m easure pw for each scenario u € fL we perform th e optim ization by minimizing th e portfolio cost a t tim e to and its expected rebalancing cost from to to t\. This is done by using the determ inistic portfolio cost (m arket price) and the moment m atching constraints (with the use of th e baseline yield curve
r(to ,T )). To model the rebalancing costs, we assign a constant bid-ask spread a n as a function
of th e positions traded. The bid-ask we used is defined in the ’Backtesting m ethodology’ section. Thus, if we note 4 ( 4 ) and 4 ( 4 ) respectively th e determ inistic fc-th moment of the portfolio and the liabilities a t tim e to, we can form ulate th e ’second stage’ problem as follows:
cost(to) = P ( 4 ) • u '( t0) +
X
PtJ \X
Q"A P V ( t 0) > L P V (to ) (10)
4 4 o) = 4 ( 4 ) , Vfc = 2m - 1, m = 1, 2 ,3 ,... 4 (4 ) > 4 (4 ), Vfc = 2m, m = 1, 2, 3, .. .
A • u (4 ) € a
Note th a t th e first term in the objective function is the m arket cost of th e portfolio at time to-The second term is th e expectation of th e rebalancing cost function over each yield curve scenario at tim e t\. This term can be viewed as a penalty function for the portfolio rebalancing costs one m onth ahead. Since th e bid-ask cost is calculated w ith the position changes between the portfolio II(fo) and Il(<i;u;), A un is defined as follows:
( un (ti\oj) - u n (to) if n € n ( t 0) D II(fi; w)
= < un (t\]ui) if n £ Il(to) bu t n € II(<i;u;) (11) { — u n (to) if n € II(to) b ut n £ II(fi;u/) and m at{n) > t\
where m a t(n ) is the m aturity d ate of th e n -th bond. Note th a t if m a t( n ) < t\. it means th a t this obligation have m atured between to and t\.
As m andate constraints in A , we included a maximum individual asset weight. T his maximum weight is to force the optim ization to chose a larger num ber of assets in the portfolio and thus to limit concentration. We discuss further on this constraint in th e ’Backtesting m ethodology’ section. Note th a t if this weight constraint is removed, then when one performs a k moment m atching optim ization, there will only be k assets in the optim al portfolio. For example, with duration/convexity-m atching w ithout a weight constraint, the optim al portfolio contains only two bonds. In th e ’Research outcom es’ section, we discuss the effects of imposing this constraint on our results.
min u(«o) s.t.
Finally, as mentioned before, th e three stochastic optim ization models described in Schwaiger et al. (2010) should be tested in a future research project. These models are: the Stochastic Linear Program m ing model (SLP). the Chance Constrained Program m ing model (CCP) and the Integrated Chance Constrained Program m ing model (ICCP). They are interesting because one could formulate our optim ization problem using a two-stage stochastic linear programm ing with recourse decision (control) variables. Instead of finding independently the optim al portfolios under each scenario at tim e t\ and going backward through tim e to the optim ization at tim e to as it is the case here, the two-stage SLP formulation would find simultaneously th e optim al portfolio at to by taking into account the expected optim al portfolio and rebalancing costs of the second stage optim ization (the recourse action) at t\ over all TSIR scenarios. At this stage, the scenarios’ dependent control variables would be used to meet th e stochastic moment m atching constraints (adjustm ent variables). Then, for each scenario at ti, th e objective would still be a function of the rebalancing costs, which depends on optim al control variables choice. W ith th e CCP model, the objective function and the constraints would still be the same, but we would relax the stochastic constraints a t tim e 11 so th a t there is a non-zero probability of not meeting constraints for a ’sm all’ set of scenarios. In other words, we include a user-specified reliability level of reaching the stochastic constraints for the T SIR scenarios at t\. Finally, th e IC C P model would not only limit th e probability of constraints mism atching, b u t would also constraints the am ount of th e portfolio underfunding. T h a t is, we would include an expected shortfall constraint at t\. which is calculated over all T SIR scenarios. This can be viewed as a portfolio conditional value-at-risk (’CVaR’) type of constraint. We give more technical details of these models in A ppendix D.
3
B a ck testin g m eth o d ology
In our analysis, we com pared th e backtesting results of the stochastic dynam ic optim ization with th e static optim ization used in A ugustin et al. (2010). We performed this by com paring the im m unization efficiency (asset-liability gap) and rebalancing costs a t each m onth w ith the two following liability m atching techniques: duration/convexity and 6 mom ents m atching. For each im m unization strategy, we first optim ized the portfolio a t the beginning of the first m onth of our backtesting window. We then evaluated the portfolio each week (and under different TSIR scenarios) until th e next m onth. At this time, we performed a new portfolio optim ization and com pared the change of each asset’s position to calculate the realized bid-ask costs. After, we re-evaluated the new portfolio each week up to the next rebalancing m onth and so on for a total of twelve months.
3 .1 D a t a a n d lim it a t io n s
There are several assum ptions/lim itations th a t must be m ade on inputs d a ta of our optim ization model and backtesting algorithm .
3 .1 .1 B o n d u n iv erse an d lia b ilitie s
A ddenda has a large universe of over 300 liquid bonds th a t can be used to construct an optim al portfolio th a t would best m atch the liabilities (by cash flow m atching an d /o r by mom ent matching). However, we needed to apply different filters for credit quality, m andate policy and other technical
reasons. First, for credit quality, we limited the universe Group to ’G overnm ent’. For m andate policy, we added an additional filter on Sector to include only ’Provincial’ issuers. We limited
Industry to issuers with highest credit quality. We included ’Agency’ and ’Non-agency’ issuers.
For technical reasons, we also excluded bonds with m aturity greater than 2020-01-01. The set of determ inistic liabilities C is detailed in Appendix C.2. One can see th a t the last liability is on 2015-12-31. which can be seen as the liabilities horizon H (or the investment horizon). However, our 12 m onths backtesting window range only in [2010-07-01. 2011-07-01]. which is smaller than th e investment horizon H (see Appendix C.3). Note also th a t the m aturities of the initial large universe spread up to year 2050. which is far beyond the last liability date. We thus applied the m aturity filter for m atu rity dates beyond 2020-01-01 to avoid some difficulties w ithin the M atlab optim ization algorithm .
We furtherm ore included additional money m arket assets, i.e. bonds with m atu rity less th a n 1 year and 1 m onth C anadian Government Index as ’T bills’. These adjustm ents were made since we have a large liability stream very close to some settlem ent dates of optim ization.
In fact, the m aturity filter and the inclusion of money m arket assets are m ade so to have a m aturity distribution in the bond universe th a t spreads over the 22 liability dates. It thus allows the moment m atching to be more efficient. W ith those filters, the bond universe B(t) for each date
t becomes:
ki = Government
&2 = Provincial
k 3 e {BC,AB,QC,ON,MA,NB} k4 < 2020-01-01
and it includes a new 1 m onth TBill a t each d ate t. The details of the tim e zero filtered universe are in Appendix C .l.
3 .1 .2 T ra n sa ctio n c o s ts
For simplicity, we assumed a constant bid-ask spread measure a n = a for all bonds to calculate the transaction costs. However, for Tbills, we assigned a zero bid-ask spread because these are very liquid assets. O ur bid-ask spread is defined as ’basis points’ per bond unit, or ’dollars’ per bond per 100$ notional. Thus, we have the following definition for a:
0.05$ per 100$ notional if the asset is a bond
0 if the asset is a TBill
Note th a t, instead of a constant bid-ask, we could also use a bid-ask spread measure which would be defined as a fraction of th e m arket mid-price. In such case however, the transaction costs would have been overstated if bonds were priced at prem ium and understated if bonds were priced at discount. Overall, the results would have been relatively similar.
Finally, a m ajor assum ption is th a t our bid-ask spread m easure does not depend on bonds char acteristics. We should u se a bid-ask spread a function of several param eters such as bond-specific characteristics or m arket liquidity/credit conditions as explained later in th e ’Results outcom es’ section.
3 .1 .3 Y ie ld cu rv es an d sh o ck scen a rio s
Our baseline yield curve r(t, T ) is the C anadian Government yield curve at each d ate t. We used it for all bonds and we did not add any bond-specific spread.
According to A ddenda C ap ital’s PCA of the C anadian Government yield curve historical move ments. the different shocks are classified by three types: level, steepness and curvature. As explained in the previous section, these shocks have different m agnitude and frequency. For com putational purpose, we limited th e num ber of scenarios by generating shocks only 011 PCA1 (level) and PCA2 (steepness), but not on PC A3 (curvature) because of its small contribution to the T S IR movements. We generated a to tal of 9 yield curve scenarios given by the function r(t, T ; uj) defined in equation
(1). As the portfolio is rebalanced each month, we used a T S IR shock scenario horizon of h = 1 m onth for each optim ization date, for a to tal of 12 settlem ent dates (see the tim e scale setting on Fig. 1). For the weekly backtesting evaluation process, we took a scenario horizon of h = 0.25 month, or one week.
The m agnitude M ^ { P C A X) of each PCA shock is defined as a multiple of their respective historical stan d ard deviation. Each scenario has its assigned probability (historical frequency). The following Tab. 1 contains a sum m ary of the scenarios used in our analysis. Note th a t th e 5th scenario has no meaning because it is equivalent to th e baseline curve. In our analysis (e.g. on graphs in Appendix B). we voluntarily om itted this scenario and labeled the scenarios as 1 to 8 .
T a b . 1: P C A ’s scenario sets
Scenario ui Pu> (%) M U( P C A X) Shock type
PCA1 PCA2 PCA3
1 6.25 -1.4 -1.4 0 2 12.50 0 -1.4 0 negative steepness 3 6.25 1.4 -1.4 0 4 12.50 -1.4 0 0 negative level 5 25.00 0 0 0 6 12.50 1.4 0 0 positive level 7 6.25 -1.4 1.4 0 8 12.50 0 1.4 0 positive steepness 9 6.35 1.4 1.4 0 100.00
3 .1 .4 O p tim iz a tio n s e ttin g s
Since our optim ization model involves a lot of d a ta (large scale optim ization), we used th e ’Global Search’ option in M atlab to avoid sub-optim al solutions. We also let a relatively high tim e limit to give enough tim e for the algorithm to come up w ith a solution. Note th a t w ith the dura tion/convexity case, we alm ost reached this maximum tim e limit, bu t th a t w ithout any unfeasible solution message.
To limit large variations in portfolio’s positions between scenarios, we forced th e first stage optim ization process a t tim e t.\ to begin with a pre-com puted optim al portfolio. This portfolio was calculated using a determ inistic mom ent m atching optim ization w ith the baseline TSIR, th a t is,
by the simple static optim ization described previously in the ’Background inform ation’ section. As m andate constraint in A. as explained in the previous section, we imposed a maximum individual weight of 8% for each asset. This was to increase the num ber of assets in th e optim al portfolio and limit concentration. Note th a t w ithout these constraints, we found respectively an optim al portfolio composed of only two assets with duration/convexity-m atching and only G assets with G moments matching.
Finally, to avoid nonlinear constraints in optim ization problems (9) and (10). we forced the portfolio value to be equal to liability value, th a t is, we assumed th a t A P V = L P V . We then optimized to find optim al weights {u>n } instead of unit positions {un }. Thus, we com puted the portfolio fc-th mom ent constraint as the weighted sum of each bo nd’s fc-th mom ent. However, because the cost in the objective function depends of each bond ’s un it position {un }, we used the assum ption above to calculate these positions as: un = wn g p y . We thus added the additional constraint wn = 1 in the m atrix A.
3 .2 B a c k t e s t in g a lg o r it h m
Let a set of m onth indices with M being the number of rebalancing m onths w ithin the back testing window. For this analysis, we took a one year backtesting- window and initially optimized the portfolio at the beginning of the first m onth and then re-optimized to rebalance adequately the portfolio a t th e beginning of the remaining eleven m onths, for a to tal of M = 12 optim izations (m onths). Define also { A j } ^ the number of weeks in each m onth i (note th a t they can be different because of working holidays, etc.) As we evaluate the portfolio each week w ithin each m onth, we have a set of evaluation dates index {?C( } ^ 0 for each m onth i.
We thus have the following tim e scaling: are the optim ization dates, where i = 1 is th e initial portfolio creation index and i > 1 are the rebalancing period indexes. For each index
i. the set {tWltirit contains th e weekly evaluation dates within each m onth. In our settings,
the settle d ate of th e first optim ization is for exam ple t w0,mi = 2010-07-01 (the complete lists of optim ization and evaluation settlem ent dates are in A ppendix C.3). Note th a t tWo<mi are the dates of the beginning of the first week of each m onth m* (where the optim ization is done) and tWhTni are the dates of the end of each week in each m onth m i. Note also th a t the last week d ate of a given m onth is approxim ately th e same date as th e next m onth’s first date, th a t is tWN,_ mi_j « tWQjnt. T he Fig. 1 illustrates this tim e scale setting.
M o n th # 1 M o n th # 2 M o n th # A I
i 1 11 * i i * i
^U'v, jn2 ^wy
C0,mi ^h-0,wv/
o p tim . 1 o p tim . 2 o p tim . M
F i g . 1 : T im e scale settin g . For th is an alysis, we have M = 12.
T he value of the portfolio over time is equal to the asset value plus the available cash, net of any shortfall. Here we will describe how we have tracked these measures over th e backtesting window.
The end of week’s cash equation is defined as:
.ha \ - I max { cash(tu!,-.},mi )(l + r 0) + A C F ( tll.(,mi) ,0 } if 1 < 1 < Ni, Vi
CaS 1 Wl'mi) ~ { - netRebalAdj(t.Wo,mi) if I = 0 ' '
where3 A C F ( t Whl1li) = C ( t Wh71lj) — L (tWh7Tli) and with cash(tWOtlJll) A 0. In fact, at tim e zero, we inject enough cash to construct the startin g portfolio. Thus, immediately after, there is no available cash in the portfolio. So The netRebalAdj term comes from the net rebalancing costs (investment cost and bid-ask cost) and is com puted a t th e beginning of each m onth (at th e rebalancing date). Note th a t this net rebalancing cost can be either positive or negative, the negative case occurs when we are net seller (th a t is, we sell more assets th a n we buy to rebalance the portfolio). In our analysis, we assumed a money m arket ra te of ro = 0 .
Because there are some weeks in which a liability stream can be greater th an the available cash in the portfolio, we define the following shortfall equation:
shfl{tWumi) = min {cash{tWl_umi) (l + r 0) + A C F ( t Whirii),0 } - shflAdj(tWhini) (14)
for 1 < I < Ni, Vi. In presence of a shortfall (shfl > 0). after using all available cash, we need to liquidate the corresponding am ount of the portfolio. This is w hat is called here shflAdj. Note th a t the value of (14) should equal zero since shflAdj is a cash inflow which comes from the portfolio liquidation to meet the shortfall.
In the following subsections, we explain how we com puted the shflAdj and netRebalAdj.
3 .2 .1 S h o rtfa ll liq u id a tio n a lg o rith m
Let
n(tW
Jim4)
be th e optim al portfolio at m onth rril which contains optim al positions { u n } ^ ^ . Noteth a t K* can be different each m onth (the optim al portfolio has not necessarily th e same number of positions each m onth). The net shortfall value (if different from zero), is assumed to be equally distributed for each asset for liquidation purpose. Thus, the shortfall value for each bond n is:
n \ _ shfl{twi,mi)
s h fln itw i.m i) — ■ (15)
However, to take into account the bid-ask spread, we need to liquidate a few more positions to fund these transaction costs. The quantity of th e n -th asset to be liquidated must be4:
A un = shfln (tWhTn() ———— - — (16)
vn\J'Wi,rrn) 01
3S ince we can have m any cou p on s w ithin a w eek, n o te th a t C ( t w,,m i ) — X)t€u'| S)n6n(t„.| _ 1, m J Cr,(t)un , for 1 < I < N i , w here n ( t UJi_ ] ,m i) is th e op tim al portfolio a t th e end o f th e previous week (or a t th e begin n in g o f the actu al w eek) and Cn and u n are th e n -th bond cou p on and position ( c( t ) = 0 if there is no cou p on a t t for a given bon d ). T h e to ta l week liability L ( t wliT,l t ) is calcu lated in a sim ilar m anner.
4W e w ant to liq u id ate som e part o f th e portfolio to fund th e bid-ask, B A n and have a net value o f shfln from the liquidation. T h en , w e m ust have
sh fln = liq u id a tio n n — B A n <=> sh,flrl = A u n B P V n — A u na
shfln
Aun —
B P V n - a
where B P V n is the present value of th e bond. In our analysis, as explained previously, we used the constant bid-ask spreads defined in equation (12). Hence, for each asset, the liquidation value is A u nB P V n (tWhmi). with A u n defined in equation (16). and the transaction cost is B A n = A u na. Thus, the available cash from selling (net liquidation value) is
UquidatioTin(tWl,ni) — A unB P V n (tu./ nii) B A n (17)
and it m ust be equal to shfln (tWhmi).
Note th a t in fact, we must liquidate th e quantity m in{A un , u n }. since short sells are not allowed (we cannot liquidate more units th an the actual position in the portfolio). So we have to calculate th e liquidated quantity in a iterative m anner, startin g with th e smallest position in the portfolio. For example, let the smallest position u\ < A iti. In this case, we can only sell th e entire position
u \ and will receive a net liquidation value of liquidation = u \ B P V \ — B A \ . Then, the initial
total shortfall is reduced to shfl’ = s h fl— liquidation. To com pute again if a new position has to be liquidated, we iteratively use equation (15) and (16) using shfl’ and the new num ber of assets becomes K* —¥ K* — 1 for th e rem aining n > 1. Hereafter, we com pare one more tim e m in{A un , u n } and so on.
3 .2 .2 R eb a la n cin g a d ju stm e n ts a lg o r ith m
At rebalancing dates, we optimize to find an optim al portfolio, called th e target portfolio B i9i(tWo<mi) w ith a value approxim ately equal to the present value of the liabilities. Since the real portfolio a t this date, n ^ ^ o ^ ) is different from the targ et portfolio (because m arket conditions have changed and some bonds may have m atured), we need to rebalance it. If we define the positions in B t9t(twomi) as we have, similarly to (11) in the ’Theoretical framework’ section, the following definition for the change in each position:
( u T - u n if n € (two,mi) n n(tu,0,TOi)
A u n = < if n € but n £
n(ttlJo,mi)
(18)[ - u n if n ^ n i9t{tW0^mi) but n £ n ( t jj;0imi) and m a t( n ) > t WOtTni where m a t( n ) is th e m aturity date of the n -th asset.
Thus, for each asset, the net rebalancing cost (the net investment cost) is calculated by
netRebalCostnftwQ^) — Au^iB PVji(tmQ^rn^.
Note th a t if A u n < 0. we have a negative rebalancing cost. This could be possible if we are net seller when rebalancing. T he corresponding bid-ask cost is calculated by
B An — [Au/ilcr,
where th e bid-ask spread a is defined in equation (12). Finally, we have the following definition of the to tal rebalancing costs adjustm ent a t rebalancing dates:
netRebalAdj(tWOtmi) = [netRebalCostn(tWOtmi) + B A n}. (19)
nen<9<((u,0,mi)un(tu,0,mj)
Now. let
£ u ^ B P V n ( t u,0,m i ),
nen '9f(tu.0,mi)
= y ' i(tiB P v n (fifo,mj),
n ( / u 'Q . rt l j )
be respectively the present value of the target portfolio and th e present value of th e actual (real) portfolio before rebalancing. At rebalancing dates, there are two possible cases:
1. A P V t9t(tWo<mi) < A P V : The actual portfolio value is greater th an the target portfolio value. In this case, we can easily rebalance the portfolio by using cash an d /o r liquidating part of the actual portfolio to meet the net rebalancing costs (net investment plus bid-ask). Note th a t this situation did not appear in our backtesting analysis;
2. A P V t9t(tW0tini) > A P V (fmo.mj : The value of the target portfolio is greater th an the actual portfolio value. In this case, the rebalancing algorithm will depend on th e am ount of cash available in the portfolio:
(a) cash(tWOjmi) > netR ebalA dj(tWo^mi) : There is enough cash to meet the target portfolio, i.e. paying the net investm ent costs and bid-ask costs. In this case, the new positions in the portfolio becomes un (tWOiTni) -> u n (tw0,mi) + A un . We also have A P V n (tW()jrii) -»
A P V n 9t(tWo<mi) and th e available cash is given by the second case of (13). Note th a t in
this case, this term (cash in th e portfolio) is greater or equal to zero after rebalancing. (b) cash(tWo<mi) < n etR ebalA dj(tWo<rni) : In this case, we can only partially rebalance the
portfolio because we do not allow for additional liquidity injection. In fact, we can only invest an am ount corresponding to cash{tWQ^mi) minus bid-ask costs from this investment. As we want to reach th e target portfolio, we need to find what fraction of each optim al new positions we are able to invest. Define this fraction /? such th a t we have our limited am ount of net investm ent netRebalAdj*(tWo<TTli) equal to cash(tWOtTni):
= netRebalAdj* (two<mi)
= y^{Pui9t -Un)BPVn + Y , W 9t - U n\a
n n
= J 2 ( P u t 9 t - u n ) B P V n + / 3 ( < s t - « n ) a - £ P ( u t 9 t - u n ) a
n n(buy) n(sell)
_ c a s h ( t u i o , mi ) + Y i n u n B P V n + Y n ( b u y ) u n a — Y n ( s e l l ) u n a Y n Un B P V n + Y n ( b u y ) u n a ~~ Y n ( s e l l ) Un a
where n(buy) are the n -th asset such th a t u 9t — un > 0 (we need to buy additional positions) and n(sell) are such th a t u 91 — u n < 0 (we need to sell positions). Note th a t we always have 0 < (3 < 1, th e case of (} = 1 retu rn s to (a).
In the ’Results outcom es’ section, we will see th a t (3 is frequently slightly less th an one, so it creates system atic p atterns of negative gaps at rebalancing dates. Thus, we also analyzed w hat happens if we allow additional liquidity injections to meet th e entire target portfolio.
cash(tWo<mi)
=► cash(tWo,mi)
A P V t9t(tW0,mi)
A P V (iu.-o.mJ
3 .2 .3 A sse t-L ia b ility g ap m ea su res
Our goal is to create and m aintain (rebalancing) an optim al portfolio at a minimum cost th a t will best match the liability value when there are variations in interest rates (th a t is. portfolio im munization). Thus, the first measure of im m unization efficiency we analyzed is th e asset-liabilitv absolute gap (both under the baseline yield curve scenario and the T SIR shocks scenarios):
gap(t;u>) = portfolio value — liability value
= AP V(t]ui) — L P V {t\uj), w ithout adding available cash (21)
or
= [APV(t-,u) + cash(t-,uj)] — LPV(t-,Lj), including available cash (22) Note th a t because of the large value of the portfolio ( ~ 65 M$), the difference in the gaps between each scenario is very small. In fact, we were not able to see the difference on graphs. We thus only analyzed the gap efficiency using the baseline yield curve r ( t , T ) . However, the difference becomes more evident when we look at the relative gaps, i.e. the difference between the gap under each scenario and the baseline gap:
Agap(t-, u ) = gap(t; u ) - gap(t), (23)
where gap(t) is th e asset-liability gap com puted with the baseline yield curve r ( t , T ) .
4
R esu lts ou tcom es and future research
In this section, we give a big picture of th e results we obtained by analysing graphs and other inform ation given in the Appendices. All these figures contain more detailed explanation in them selves. A ppendix A illustrates the optim al portfolio positions tracking over time for both strategies (static optim ization and stochastic dynam ic optim ization) and for both m atching techniques (du ration/convexity and 6 moments). Moreover, graphs in A ppendix B illustrate the portfolio im mu nization efficiency and costs for both strategies and m atching techniques. Note th a t th e first graph section B .l shows results for which we do not perm it additional liquidity injection at rebalancing dates. A ppendix B.1.1 presents the duration/convexity-m atching and A ppendix B .l.2 presents the 6 moments matching. The second graph section B.2 illustrates what happens if we allow for additional liquidity injection.
At first glance, we observed th a t our results are different of what we initially expected. T h a t is, the stochastic dynam ic optim ization technique is not th a t much more efficient in comparison with the static version. However, in accordance with results of Augustin et al. (2010), th e im m unization is b etter as we increase th e number of mom ents to be matched but with an associated higher rebalancing cost (with both optim ization techniques). T h a t is, by comparing the PV tracking results of the duration/convexity-m atching on Fig. 2 w ith the 6 moments m atching on Fig. 3 on the next pages, one can rem ark th a t the im munization is b etter as we increase th e num ber of moments to be m atched (for both static and stochastic optim ization). We can see th a t th e time-zero optim al portfolio has a value ~ 65M $. In ’absolute’ dollars term s, there is no big difference between graphs. Thus, in Appendices, we also illustrate other ’relative’ measures as gap (equation(22)) and A gap (equation (23)).
\ f ' VA.
DurCvx - P r e se n t v a lu e s T im e s e r ie s A T A A - A P V I LPV I R e b a la n c in g d a t e s j -\J \ J V \ 2 0 1 0 -0 7 -0 1 2 0 1 0 -0 8 -0 3 2 0 1 0 -0 9 -0 2 2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2 2 0 1 0 -1 2 -0 2 2 0 1 1 -0 1 -0 4 2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2 2 0 1 1 -0 4 -0 2 2 0 1 1 -0 5 -0 3 2 0 1 1 -0 6 -0 2 2011-0 7 -0 1 x t g ’ DurCvx • A vailable c a s h T im e s e r ie s _ y \ / “ V i 2 0 1 0 -0 7 -0 1 2 0 1 0 -0 0 -0 3 2 0 1 0 -0 9 -0 2 2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2 2 0 1 0 -1 2 -0 2 2 0 1 1 -0 1 -0 4 2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2 2 0 1 1 -0 4 -0 2 2 0 1 1 -0 5 -0 3 2 0 1 1 -0 6 -0 2 2011-0 7 -0 1DurCvx - P r e se n t v a lu e s T im e s e r ie s (Including availab le c a s h in th e portfolio)
- A PV * C a sh LPV
R e b a la n c in g d a te s j
2 0 1 ^-0 7 -0 1 2 0 1 0 -0 8 -0 3 2 0 1 0 -0 9 -0 2 2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2 2 0 1 0 -1 2 -0 2 2 0 1 1 -0 1 -0 4 2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2 2 0 1 1 -0 4 -0 2 2 0 1 1 -0 5 -0 3 2 0 1 1 -0 6 -0 2 2011-0 7 -0 1
( a ) D u r a tio n /C o n v ex ity S tatic case.
" — \T " DurCvx (ST O C H A ST IC ) - P r e se n t v a lu e s T im e s e r ie s A / - A P V i LPV R e b a la n c in g d a t e s | NT... - I I - I _ l I _ i 2 0 1 6 -0 7 -0 1 2 0 1 0 -0 8 -0 3 2 0 1 0 -0 9 -0 2 2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2 2 0 1 0 -1 2 -0 2 2 0 1 1 -0 1 -0 4 2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2 2 0 1 1 -0 4 -0 2 2 0 1 1 -0 S -0 3 2 0 1 1 -0 6 -0 2 2011-0 7 -0 1
* , o’ DurCvx (ST O C H A ST IC ) - A vailable c a s h T im e s e r ie s
Z 7 Vr ~ l j 1/ 1____ / i _____ A_
2010 07-01 2010 08 03 2010-09 02 2010-10 05 2010- 11-02 2010 12 02 2011-01-04 2011-02 02 2011-03-02 2011 04-02 2011-05-03 2011-06-02 2011-07-01
. « io ’ DurCvx (ST O C H A ST IC ) - P r e se n t v a lu e s T im e s e r ie s (including av a ila b le c a s h in th e portfolio)
-t*—*----^ r- |— —- ' i — ( •— — [— - i r A PV ♦ C a s h LPV j • R e b a la n cin g d a te s 4 k . I . L I . . . . I . . . . i... 1 _ I _ ... i L . I I 2 0 1 0 -0 7 -0 1 2 0 1 0 -0 8 -0 3 2 0 1 0 -0 9 -0 2 2 0 1 0 -1 0 -0 5 2 0 1 0 -1 1 -0 2 2 0 1 0 -1 2 -0 2 2 0 1 1 -0 1 -0 4 2 0 1 1 -0 2 -0 2 2 0 1 1 -0 3 -0 2 2 0 1 1 -0 4 -0 2 2 0 1 1 -0 5 -0 3 2 0 1 1 -0 6 -0 2 2011 -0 7 -0 1
( b ) D u ra tio n /C o n v ex ity S to ch a stic case.
F i g . 2: In b oth graphs, there is a decrease o f th e liability P V at regular d a te s caused by a liab ility stream (cash o u tflow ). W e have a to ta l o f four liab ility stream s in our b acktesting w indow . T h e n egative gaps betw een th e asset P V and liability P V are because som e bon d s are m atured and becom e cash (see th e p ortfolio cash-flow s on Fig. 9 in A p p en d ix B .1 .1 ). B u t th e available cash from a ssets (given by equation (1 3 )) alm ost fills th ese n egative gaps. T h e large cash peak before th e 6-th rebalancing d a te is due to three large m atured p osition s and drop to zero after b ecau se it en tirely funds th e rebalancing co sts. A s we can see on (b ), th e use o f sto ch a stic o p tim iza tio n d o es not im prove th e d u ra tio n /co n v ex ity m atching efficiency. It is also confirm ed by observing th e relative m easures graphs in A p p en d ix B .1 .1 . See th e a sset-lia b ility gap graph on F ig. 8 and th e A in A -L gap on F ig. 12.
As we can see on the portfolio’s positions tracking graphs in Appendix A, the higher rebalancing costs associated with higher mom ents (6 mom ents in th a t case) are because it needs a larger number of assets to reach an optim al portfolio and it consequently increases the frequency of rebalancing. By com paring portfolio positions for duration/convexity (static case) on Fig. 4 w ith portfolio positions for 6 moments (static case) on Fig. 6 , one can see th a t the num ber of assets and the
