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The Impact of European
Inflation-indexed Bonds on Dutch
Pension Fund Portfolio Management
Yun Li
11138297
MSc in Econometrics
Track: Financial Econometrics Data of final version: March 2017
Supervisor: prof. dr. Roel M.W.J. Beetsma Second reader: prof. dr. Kees Jan van Garderen
Abstract
Inflation-indexed bonds have never been off topic of long-term asset-liability portfolio management. However, most researches are limited to simulated data due to the avail-ability. Evidence has been found that there is a substantial difference between the actual and simulated series, caused by the liquidity premium and vigorous market conditions. In this study, we use actual pension fund data and Euro-zone indexed bond series, and find Dutch pension funds can gain significant efficiency with the inclusion of indexed bonds by applying a strategic asset-liability allocation method.
This document is written by Yun Li who declares to take full responsibility for the contents of this document. I declare that the text and work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
First of all, I would like to thank my supervisor, Prof. Roel Beetsma, who gave me this great opportunity to experience what empirical research in the domain of asset liability management is like. When I started, I had absolutely no idea who he is but thought the proposed topic is really interesting. After talking to some of his neighbours on his office floor, I realised that he is probably one of the busiest people in the faculty. However, a comprehensive and well-written guide was given by him when I started and he insisted weekly meetings to give me guidance and track my progress; not to mention, he actively helped me to find the right data and was always there when I needed help.
Secondly, I would like to thank Mr. Kin Lee from MN services, who unconditionally shared his knowledge about what data to use, helped me to find it, and organised it for me; who patiently lectured me on how inflation-indexed bonds work and what pension funds’ inflation hedging portfolio looks like. Even though we only met once, I already feel that he is like the big brother I never had.
Next, I would like to thank all the professors and lecturers from the Econometrics programme and my elected courses, Risk Management for Insurers and Pensions and Fixed Income Risk Management , who have organised a wonderful programme/courses. Here I would like to specially thank Prof. Kees Jan van Garderen, who was incredibly kind and patient when I just started to learn Matlab; who was always there for us asking if we did well or not; who helped chair my presentation when my supervisor was not available; and who is now my second reader when I was a bit worried if I would have one. One of my friends affectionately calls him the ’mother’ of the entire Econometrics programme.
Last but not least, I would like to thank my real mother, who was, is, and always will be there for me; who did not say a single no when I wanted to stop pursuing a career as a medical doctor, even though she always wanted me to be; who never complained even when I was never around for the past few years.
You have no idea of how grateful I am right now. And saying ’thank you’ once is definitely not enough.
Thank you all!
Statement of Originality i
Acknowledgements ii
1 Introduction 1
1.1 Dutch pension system in a nutshell . . . 1
1.2 Inflation-indexed bonds . . . 2 1.3 Outline . . . 3 2 Return Dynamics 5 2.1 Model . . . 6 2.2 Data . . . 9 2.2.1 PMT data. . . 9
2.2.2 Major investment categories. . . 10
2.2.3 Duration match-up . . . 11
2.2.4 State variables . . . 11
2.2.5 Data summary . . . 11
2.3 Estimation results . . . 13
3 Optimal portfolio choice 16 3.1 Literature review . . . 16
3.2 Outline . . . 17
3.3 The pensioner’s optimization problem . . . 18
3.4 Strategic asset-liability allocation . . . 18
3.5 Monte Carlo simulation of correlated asset returns . . . 22
3.6 Optimization results . . . 23
3.6.1 Unconstrained optimization I . . . 23
3.6.2 Unconstrained optimization II: equity adjusted . . . 25
3.6.3 Constrained optimization . . . 28 3.6.4 Simulation results . . . 31 4 Conclusion 34 A VAR(1) Estimation 36 B Simulation Output 39 iii
Introduction
1.1
Dutch pension system in a nutshell
There are in total three pillars in the Dutch pension system. The first one is the social security pension (AOW), which has a pay-as-you-go scheme; it is collective, compulsory and implemented by the Dutch government. The second pillar is the complementary pension, which has a investing-for-later scheme, it is also collective and compulsory, but implemented by pension funds and insurers. The third pillar is the extra additional pension, which is individual and voluntary.
The reason why the Dutch pension system is unique globally is that the second pillar arranged by pension funds and insurers is as big as the state pension that is arranged by the Dutch government. Most of the employees in the Netherlands build up their occupational pension through a company or a sectoral pension fund. For example, in this study, we use data from the Pensioenfonds Metaal en Techniek (PMT), which is a sectoral pension fund for employees who serve in the metal and engineering industry. The PMT is connected to 1.3 million participants, 379,005 of which are still active in this industry, 661,179 of which have worked in this industry before, and 211,070 of which already receive their pension payments; the fund itself manages 59 billion euro worth assets and carries 61 billion euro worth liabilities by the end of 2015. Overall, there are around 600 company pension funds and 100 industry-wide pension funds in the Netherlands. Similar to the PMT, other Dutch pension funds are often found operating with enormous amounts of assets and liabilities, since the second pillar is collective and compulsory. Moreover, the relative size of the second pillar is expected to grow, due to the fact that it is getting harder and harder to finance the first pillar as the average life expectancy keeps increasing.
Within the second pillar, employees build up their nominal pension rights (expressed in euros) through their contributions to the pension funds. These rights promise them a
nominally-fixed pension benefits after their retirement date. The nominally-fixed pen-sion benefit are partially or fully indexed with respect to price or wage inflation, de-pending on the financial health of the pension funds and indexation policies applying to the pension benefits.
However, the indexation policies are not forced by law. More than 85% of the pension funds in the Netherlands follow a conditional indexation policy based on the funding ratio of each individual fund. Funding ratio is regarded as one of the most important measurements of the fund health. Conditional indexation means that funds give nil-indexed to fully-nil-indexed pension payments when their funding ratios are above a target threshold; a long-term or even short-term restoration plan has to take place when the funding ratio is below some threshold and no indexation is applied at all. Additionally, the financial health of the pension funds is extremely sensitive to economic shocks. Take the PMT for example again; it has abolished its pension indexation since the 2008 global financial crisis, which substantially decreased the purchasing power of its pension payments in the past few years under a non-degenerate inflation.
1.2
Inflation-indexed bonds
Inflation-indexed bonds are bonds where the principal or/and coupons are indexed to a one-period lagged inflation index, such as the national CPI index excluding tobacco. They are designed to hedge inflation risk of the nominal bonds. The market primarily consists of sovereign bonds, with a small portion of privately issued ones.
It is often suggested to include inflation-indexed bonds (ILBs) in the pension funds’ portfolio, in order to decrease the vulnerability of the pension funds to some of those economic shocks; the inflation shock, for instance. However, the Dutch government does not issue such bonds and probably will not in the near future. It is because that inflation hedge for institutions, like pension funds, is not a legitimate reason to issue such bonds, whilst there are other inflation hedge instruments available, such as inflation linked swaps, which currently weight around 15%-20% on pension funds’ investment plans. But still, inflation-indexed bond has never been off the discussion board since the earliest asset-liability allocation study, where its nominal return moves up and down with inflation, which is usually regarded as a perfect inflation hedge for liabilities that contain real consumption flows.
The inflation-indexed bond has a relatively brief history and is only issued by a lim-ited number of foreign governments, and none of them are indexed to the Dutch price
level. Some of these foreign inflation-index bonds, for instance the U.S. Treasury
Inflation-Protected Securities (TIPS), have been criticized with liquidity problems and low-correlation with its domestic inflation due to its volatile yield return, especially in
its first few years of issuing. However, not all of them have such problems; for exam-ple, French and German sovereign inflation-index bond returns are perfectly correlated with their national CPI at least at a yearly frequency. Nearly all research regarding inflation-indexed bonds is based on simulated series, due to the limited data availabil-ity. However, there is a substantial difference between simulated series and actual data, due to the liquidity premium and vigorous market conditions. For example, Campbell
and Viceira (2009)[1] find that synthetic yields are lower on average and less volatile
compared to actual ones. Moreover, even though the inflation-indexed bond history is short, there is almost two decades of data available, which could be a good indication of a decent starting point to do active research with the actual data.
In this empirical study, the French sovereign inflation-indexed bonds, OATi, are chosen to demonstrate if Dutch pension funds can achieve efficiency gains by investing in it. The reason why we use the French ones is due to the fact that they have been available for the longest period in the Euro-zone. As we all know, the inflation indices of all industrialized countries after the Second World War are highly correlated. However, some of them, more than the others, are correlated with the Dutch CPI, especially after the establishment of Euro-system; for example, French and German CPI have correlations of over 0.80 with the Dutch CPI for the last two decades.
1.3
Outline
Our goal is to examine how impactful the Euro-zone (French) inflation-indexed bonds would have been if they have been included in the investment basket by Dutch pension funds since their first issuing, by finding and comparing the counter-factual asset-liability allocation path of the PMT data to its original path.
In order to make the comparison valid, the return dynamics among various asset classes and real pension liabilities of the PMT will be studied within the framework proposed
by Campbell and Viceira (2002, 2005)[2,3], which is a simple parsimonious VAR model
that explains the variance-covariance structure of all asset classes and pension liabilities in Chapter 2. This is followed by studying the strategic asset-liability optimal portfolio choice, which assumes constant unconditional variance-covariance structure and constant re-balancing to the initial state at each investment period, starting with unconstrained optimization, then constrained optimization and, last but not least, optimization based on simulated asset class return series, in Chapter 3.
We find that inflation-indexed bonds provide much better liability hedge properties than the nominal bonds, real estate, and high yield & emerging market bonds. However, these properties are not free, due to the 10% lower Sharpe ratio than nominal bonds in the past data from 1999 to 2014, when there is no inflation hike, which might have caused a substantial loss if pension funds have linearly substituted their nominal bond investment
to inflation-indexed bonds. However, the risk-return dynamics in a strategic allocation portfolio that includes more asset classes is more complicated than linearly substituting nominal bonds with inflation-index bonds. Our constrained optimal asset-liability alloca-tions suggest a strategic asset-liability allocation portfolio that includes inflation-indexed bonds using both actual and simulated data outperforms a strategic asset-liability al-location portfolio that does not include inflation-indexed bonds as an asset class, with significantly higher average funding ratios and funding ratio variances. However, while using the actual series, for some moderate risk aversion coefficient, it results with a reverse situation that a portfolio that excludes inflation-indexed bonds outperforms a portfolio that does not, simply because there is a significant, instead of moderate, shift from asset class such as high yield & emerging market bonds to inflation-indexed bonds that drives down the average volume of assets and funding ratio. Overall, we believe that the Euro-zone inflation-indexed bonds, such as the French inflation-indexed bonds have some excellent liability hedge property for Dutch pension funds. They might be resoundingly costly comparing to the nominal sovereign bonds, but pension funds might achieve a significant efficiency gain in their funds’ financial health even if there is flat or downward trending inflation.
Return Dynamics
To describe the dynamic behavior of asset returns, we use a simple parsimonious statis-tical model, the first-order VAR process, or VAR(1). This method was introduced by
Campbell and Viceira (2002, 2005)[2, 3]. It starts with a set of asset classes that can
join the basket of potential portfolio and whose returns we try to model. Moreover, they include a set of variables that are previously well documented by empirical researchers that are relevant for forecasting their set of asset classes returns, which are referred to as the ’state variables’. The one-period-ahead forecast of each asset return is obtained by regressing the return onto a constant, its own lagged value, the lagged value of the other asset returns and the lagged values of the state variables. Notice that an exten-sion of VAR(1) process to more lagged values is also viable in this setting. In addition, they mention that it is interesting to note that one can rewrite a VAR of any order in the form of a VAR(1) by adding more state variables which are simply lagged values of the original vector of variables. A more important consideration about additional lags would be choosing the most parsimonious model based on information criteria by using historical data. However, for technical simplicity reasons, the technical part of this paper is only written in terms of a VAR(1) model.
In our setting, asset classes would be stock, high-grade government bonds, high yield & emerging market bonds (both developed and emerging market low-grade sovereign and corporate bonds), real estate and Euro-zone inflation-indexed bonds. In addition to their setting, we introduce the actual PMT liabilities to the VAR model, in terms of its annual log (or continuously compounded) liability return, in order to capture the variance-covariance structure among our asset classes and the liabilities of an actual Dutch pension fund. For state variables, we choose variables that have strong predictive power to expected asset class returns which are already well-documented in previous empirical research. For this study we choose the short-term Dutch T-bill rate as the benchmark asset class. However, no leverage is allowed in our asset-liability management study, in accordance with the PMT data.
2.1
Model
The return dynamics of assets and liabilities are described by a first order VAR model
for yearly data as below: Let zt denote a column vector whose elements are the returns
on all asset classes under consideration, and values of the state variables at time (t), which can be written as
zt≡ rtb,t xt st ≡ rtb,t xA,t xL,t st , (2.1)
where xA,t is a vector of log (or continuously compounded) excess returns of our asset
classes, xL,tis log (or continuously compounded) excess return of pension liabilities and
stis a vector of state variables. The vector of state variables stcontains three predictive
variables: dividend yield (dy), term spread (ts) and sovereign spread (ss). Notice that
xt = rt− rtb,t, where rtb,t = rtb,tι, rtb,t represents the T-bill return and represents a
column vector of ones.
Thus, the VAR(1) model can be represented compactly as
zt+1= Φ0+ Φ1zt+ vt+1, (2.2)
where Φ0 is a vector of intercepts; Φ1 is a square matrix that stacks together with the
slope coefficients and vt+1 is a vector of zero-mean shocks to realization of returns and
return forecasting variables.
When dealing with time series, the stationarity condition has to be satisfied, which in
this case would be the determinant of Φ1 is bounded between -1 and 1. This condition
guarantees that in the absence of shocks, the variables of VAR(1) converge to their long-term means in a finite number of periods. Otherwise, co-integration has to be taken into consideration.
Last but not least, we have to more precise about the nature of the vector of shocks
to asset returns and state variables (vt+1). In particular, we assume that the vector of
shocks is multivariate normally distributed,
vt+1
i.i.d.
∼ N (0, Σv), (2.3)
where Σv denotes the contemporaneous variance-covariance matrix of shocks. In such a
Furthermore Σv can have following representation, in accordance with our predefined zt+1 Σv ≡ Var(vt+1) = σ02 σ00x σ00s σ0x Σxx Σ0xs σ0s Σxs Σs , (2.4) where σ2
0 is the variance of T-bill return, Σxx is the variance-covariance matrix of
un-expected excess returns, Σs is the variance-covariance matrix of state variables, Σ0x
is the covariances of T-bill return with excess returns, Σ0s is the covariances of T-bill
return with shocks to the state variables and Σxs is the covariances of excess returns
with shocks to the state variables. It is easy to find that
zt+1+ · · · + zt+k= "k−1 X i=o (k − i)Φi1 # Φ0+ k X j=1 Φj1 zt+ k X q=1 k−q X p=0 Φp1vt+q (2.5)
Now we can compute conditional k-period moments of the state vector. The conditional k-period mean is given by
Et(zt+1+ · · · + zt+k) = "k−1 X i=o (k − i)Φi1 # Φ0+ k X j=1 Φj1 zt, (2.6)
since the shocks vt+q have zero mean.
The conditional k-period variance is given by
Vart(zt+1+ · · · + zt+k) = Vart "k−1 X i=o (k − i)Φi1 # Φ0+ k X j=1 Φj1 zt+ k X q=1 k−q X p=0 Φp1vt+q = Vart k X q=1 k−q X p=0 Φp1vt+q , (2.7)
since all other terms are deterministic at time t. Expanding from Eq 2.7we have
Vart(zt+1+ · · · + zt+k) =Σv+ (I + Φ1)Σv(I + Φ1)0
+ (I + Φ1Φ1)Σv(I + Φ1Φ1)0
+ · · ·
+ (I + · · · + Φk−11 )Σv(I + · · · + Φk−11 )
0. (2.8)
which follows from reordering terms and VAR assumption of unchanged conditional
variance-covariance matrix of vt+1 at all lead i.
variables in VAR setting. So we have n − 1 asset classes and m − n − 1 state variables. Define matrix Hx Hx = h 0(n)×(1) I(n)×(n) 0(n)×(m−n−1) i
So the annualized conditional k-period moments of returns are
µ(k)t = 1 kEt(x (k) t+1) = 1 kHrEt(zt+1+ · · · + zt+k) (2.9) Σ(k)t = 1 kVart(x (k) t+1) = 1 kHrVar(zt+1+ · · · + zt+k)H 0 r = Σxx(k) = " ΣAA(k) σ0AL(k) σAL(k) σL2(k) # , (2.10)
where Hx helps pick the right elements that correspond to excess returns from the
ensuring vector/matrix, x(k)t+1 denotes the cumulative excess return over k periods, µ(k)t
denotes the annualized conditional k-period excess return mean and Σ(k)t denotes the
annualized conditional k-period excess return variance-covariance matrix. Estimating
µ(k)t and Σ(k)t is the main purpose of our time series analysis.
An additional remark would be about the Cholesky impulse response function.
Un-der stationarity condition, a VAR(1) model, see Eq 2.2, has a Moving Average (MA)
representation zt+1= ∞ X j=0 Φj1Φ0+ ∞ X j=0 Φj1vt−j = µ + ∞ X j=0 Φj1vt−j+1, (2.11) where µ =P∞ j=0Φ j
1Φ0 and vt is multivariate white noise follows from Eq2.3.
A Cholesky-type decomposition of the contemporaneous variance-covariance matrix
ma-trix Σv would be
Σv = LGL0, (2.12)
where L is some lower triangular matrix with 1 on its diagonal and G is the diagonal
matrix of Σv.
Therefore, transformed shocks would be
bt= L−1vt, (2.13)
which have diagonal variance matrix G and hence are orthogonal.
Then Eq 2.11 can be further written as
zt+1= µ +
∞ X
j=0
where Ψ∗j = Φj1L. And the sequence
Ψ∗ij = ∂zi,t+1 ∂bj,t−l+1
, ∀l = 0, 1 · · · (2.15)
is called the Cholesky impulse response function of zi,t+1 with respect to bj,t+1 and i
is the row index in zt+1. An impulse response function could be interpreted as effect
on zi,t+1 of shock in zj,t+1. Since the contemporaneous shocks in a VAR setting can be correlated with each other, and Cholesky decomposition can help isolate the effect of shocks.
2.2
Data
Our empirical study is based on yearly data, due to the fact that the PMT data is only available at a yearly frequency from their annual reports. However, we do obtain the rest of data, including T-bill returns, equity returns, nominal bond returns, real estate returns, high yield & emerging market bond returns, inflation-indexed bond returns, dividend yields, term spreads and sovereign spreads, at a monthly frequency and aggre-gate to a yearly frequency for the convenience of our simulation study. Not to mention that some of our data are in values of U.S. dollar. In that case, the corresponding log (or continuously compounded) return series is transformed to euro terms by adding the log (or continuously compounded) return of euro/dollar exchange rate from
Datas-tream1, which tracks the major currency returns, such as the Deutsche mark, before the
establishment of the Euro-system.
2.2.1 PMT data
The PMT data starts from 1997 and ends at 2014. The asset allocation of all asset classes is displayed and sums to 100%. For each year, the total volume of pension contributions, the total volume of pension payments, and the volume of assets and liabilities are presented in millions of euro. Notice that the surplus of the total volume of pension contributions and the total volume of pension payments goes to the total volume of assets and we have no way to know what it would have been under our
counter-factual path. So the actual series of their surplus is used to construct the
counter-factual asset series. Moreover, the funding ratio, discount rate and type of discount rate are also included in this data. An additional matter needs to be considered is that there are some major draw-downs in the funding ratio within our sample period, especially in 2008 and 2011, the funding ratio has been way below 100%, 84.7% and
88.5%, respectively. According to the Dutch pension system regulatory framework,
there should be no indexation when funding ratio is below105% and monetary restoration
needs to be taken place when the funding ratio is below a certain threshold, for example, 100%, under conditional/unconditional indexation policies. However, it is not clear to us that what exactly their restoration plans were, for instance, how much funds were buffered and how much pension payments were cut, since there are only two annual reports, 2013 and 2015, available on their website. Thus for simplicity, we assume there was no fund buffer at all. However, any restoration plan could lead to a different asset-liability portfolio optimization problem in reality, since we optimize the funding ratio. Overall, it should not be a matter in our study, due to the fact that we only use the exact volume of assets in 1999 and stick to it for our entire optimization problem.
2.2.2 Major investment categories
The investment categories are also narrowed down to four major asset classes, namely, equity, fixed income, real estate and high yield & emerging market bonds, since we have no way to know what exactly ’other investments’ are. To proxy these investment
categories, we use total return of the MSCI All Countries World index from Bloomberg2,
which includes both price and dividend returns, for the fully diversified global equity market; the BarCAp Gov’t France all maturities index from the Barclays Capital web
portal3, for the global high grade government fixed income market; the GPR Global
index also from Bloomberg, for the listed real estate market; and the BoA ML Global
HY & EM index from Bank of America Merrill Lynch web portal4 for the low-grade high
yield & emerging market bonds. For the French inflation-indexed bond market, we use the BarCap France Gov’t Inflation linked All Maturities index, also from the Barclays Capital web portal.
Since the French government inflation (CPI) linked bonds, OATi, were firstly issued in the end of September, 1998, the corresponding index is not available for longer. Similarly, the BoA ML Global HY & EM Index has a short period of availability, which starts from December, 1998. Other indices are available for longer periods. However, in order to avoid complication in constructing a restricted VAR model, we only use data from December, 1998, causing the first return series would be in January, 1999.
Moreover, the Dutch short-term T-bill rate is based on the 3-month T-bill rate, which
is from OECD5. However, the data is from the first day of every month and the rest
of our data is from the last work day of every month. In order to make them match, one-period lagged T-bill rate data is used.
2 www.bloomberg.com 3http://www.barcaplive.com 4 http://www.mlx.ml.com 5 https://data.oecd.org/interest/short-term-interest-rates.htm
2.2.3 Duration match-up
Notice that the average duration of the BarCap France Gov’t Inflation linked All Ma-turities index is 8.93 year and the average duration of the BarCap Gov’t France all maturities index is 6.20 year. To make the French inflation-indexed bonds more compa-rable to the fixed income category, we need to match up their durations based on the fixed-income immunization theory. Notice that the convexity is negligible comparing to duration in this case, so we only look at the duration match-up. Thus, we need to look at the break-downs of the BarCap Gov’t France all maturities index, which are also provided on the Barclays Capital web portal. Two sub-indices are used in the dura-tion match-up, namely, the 5 − 10 year sub-index and >10 year sub-index, which have average durations of 6.37 year and 12.84 year, respectively; the duration match-uped index is simply the adjusted index with respect to the linear combination that gives the same duration to the BarCap France Gov’t Inflation linked All Maturities index at each month.
2.2.4 State variables
Last but not least, dividend yield is based on the S&P Composite and from the ’Irrational
Exuberance’ book data of Shiller6. For the sovereign spread, we use the France-Germany
long term government bond spread from OECD7 as well; notice that the French credit
rating is only AA by S&P and our study covers the period of European debt crisis. The term spread is based on long term and short term French government bond spreads, also from OECD. These state variables are common in literature. Compbell and Shiller
(1988, 1991)[4, 5] are early references for the dividend yield. Campbell, Chan and
Viceira (2003)[6] and Brandt and Santa-Clara (2006)[7] use dividend yield and term
spread. Tang and Yan (2010)[8] explain the dynamics between market condition and
credit spread, which shows corporate credit spread has relatively good predictive power to credits and bonds. With loss of generality, sovereign spread should be predictive to nominal and inflation sovereign bonds as well.
2.2.5 Data summary
Table2.1reports the descriptive statistics of our data.
The real estate outperforms everyone else in terms of yearly returns, with an average return of 9.66%, followed by the high yield & emerging market bonds, with an average return of 8.19%. Nominal bonds outperform inflation-indexed bonds, which is logical to us, since French CPI has an overall downward trend during the entire sample period,
see Figure 2.1, and indexed bonds are relatively costly under such market conditions.
6http://aida.econ.yale.edu/ shiller/data.htm 7
Table 2.1: Descriptive Statistics
Sharpe ratio is the division of mean by standard deviation. All data except term spread and sovereign spread are log return series. All series are scaled by a multiplication
of 100, which represent log returns in percentage.
Mean Std Sharpe Min Max Kurt Skew Obs
Returns Equity 4.47 22.56 0.20 -54.81 29.73 1.78 -1.31 16 Bonds 6.44 5.34 1.21 -1.50 14.31 -1.20 -0.17 16 Real Estate 9.66 23.07 0.42 -56.95 28.93 3.71 -1.84 16 HY & EM 8.19 15.71 0.52 -25.20 42.30 1.20 -0.06 16 Inflation-indexed bonds 5.18 4.79 1.08 -3.95 13.47 -0.14 -0.15 16 Liability 8.22 8.10 -3.90 25.14 -0.04 0.43 16 State Variables Dividend Yield 1.84 0.47 1.17 3.23 4.52 1.50 16 Term Spread 1.39 0.80 -0.40 2.42 0.25 -0.93 16 Sovereign Spread 0.29 0.30 0.03 1.04 1.13 1.32 16
However, the inflation trend can look totally different if we look at a different time scale, just like any other time series; for instance, there is a clear upward trend in the French yearly inflation from 2008 to 2010. Surprisingly, equity has a much lower average yearly return than bonds, but the standard deviation is nearly quadrupled, which yields that the Sharpe ratio of equity is only one sixth of that of bonds. An explanation to this would be that anything can happen in a short sample period and our sample period covers the so called ’Lost decade’ 2000-2010, which started with the 2000s recession that badly affected the European Union and the U.S. successively from 2001 to 2003, followed with the global financial crisis from 2007-2008 and ended up with the European sovereign debt crisis. Nearly all equity markets failed for a persistently long period of time.
The positive dividend yield growth in the S&P 500 Composite reflects a lower growth or possibly a higher decline in equity prices than the corresponding dividends, especially in the tech sector where relatively lower dividends are paid out. The positive sovereign spread between French and German long term bonds is due to the compensation for expected default loss and increasing liquidity risk, explained by Elton, Gruber, Agrawal
and Mann (2001)[9]. An example would be the peak of the France-Germany bond return
spread in the end of 2011 and the beginning of 2012, before and during when the hot money in the Euro-zone bond market shifted to higher-grade German sovereign bonds.
Figure 2.1: French yearly inflation in %
2.3
Estimation results
The VAR(1) estimates are displayed in the Appendix; see Table A.1and Table A.2for
the cases with and without inflation-indexed bonds, respectively. Co-integration test and estimation of a VAR model of any higher order could not be carried on, due to the insufficient observations. However, data at a monthly level (all asset classes excluding state variables) passes the co-integration test and therefore their joint process is con-cluded as stationary. Moreover, both Akaike and Schwarz information criteria suggest to use a VAR(2) model based on the monthly frequency data. But the difference be-tween VAR(1) and VAR(2) model in terms of information criteria is negligible and most importantly no VAR estimation is conducted with data of a monthly frequency here. The normality test does not reject the assumption that residuals follow a multivariate normal distribution in both VAR(1) models, which means our normality assumption of log (or continuously compounded) returns is not violated.
The dividend yield shows some predictive power for the excess equity return. The
term spread is predictive for bond and inflation-indexed bond excess returns; however, the long-term sovereign spread between France and Germany has a relatively sizable coefficient over next period bond returns, and the corresponding t-statistic is small.
All the excess returns have quite large R2, which means that both our VAR(1) models
are statistically and economically meaningful, in terms of explaining the return dynamics and overall being helpful in improving portfolio performance. But one thing to know
here is that R2 based on yearly data usually tends to be higher.
Table 2.2: Error covariance and correlation structure: liability with asset classes
This table shows the covariance and correlation between excess return of each asset class and excess liability return
Equity Bonds Real estate HY & EM ILBs Without inflation-indexed bonds Cov -0.004813 0.000413 -0.003936 -0.000986
Corr -0.96 0.38 -0.67 -0.27
With inflation-indexed bonds Cov -0.004033 0.000169 -0.004831 -0.002036 -0.0000705 Corr -0.96 0.20 -0.97 -0.82 -0.06
AR coefficients of term spread and sovereign spread are relatively large, with values of 0.75 and 0.82, respectively, which indicates these series are very persistent. Notice that the bond excess returns are also persistent.
Moreover, as depicted in Table2.2, there are some significant shifts in the error
covari-ance and correlation structure of bond excess returns and liability excess returns, real estate excess returns and liability excess returns, and high yield & emerging market bond excess returns with liability excess returns in the residual. The shift in the resid-ual structure of high yield & emerging market bond excess return and liability excess return is the strongest. On the contrary, there is nearly no shift in the residual structure of equity excess return with liability excess return. This result will deeply impact the Liability Hedge portfolio allocation shift when inflation-indexed bonds are included in the investment basket, which will be explained in the later chapter.
Parts of the Cholesky impulse response functions of excess asset returns are depicted in
Figure2.2. All impulse response functions start with period one and end up with period
12. The vertical axis shows how large is effect of one standard deviation of shock on the first moment of asset excess returns. Both equity and bonds have similar responses to their own shocks, where they positively respond in the next period and bounce back at a lower magnitude. However, equity responds to its own shocks at a much larger scale than nominal bonds do. Furthermore, equity responds to its own shocks more persistently than nominal bonds do. Both nominal bonds and inflation-indexed bonds respond to equity shocks negatively in at least first three periods, which means positive equity shocks drive down and negative equity shocks drive up both nominal and indexed bonds excess returns. However, equity shocks to nominal bonds are less persistent than to indexed bonds. Similarly, as shown on the bottom of the figure, positive nominal bond shocks drive up inflation-indexed and high yield & emerging market bond excess
returns in the next period, and rebound in the 2nd or 3rd period for indexed bonds and
Figure 2.2: Impulse response function: shocks in unit of one std.
(a): Equity response to equity (b): Bond response to bonds
(c): Bond response to equity (d): ILB response to equity
Optimal portfolio choice
3.1
Literature review
It has not been a secret that long-term investors look at risk differently with short-term
investors in portfolio management, since Samuelson (1969)[10] and Merton(1969)[11]
es-tablished their first work. Samuelson looks at it from a mathematician point of view, who tries to solve it with dynamic programming. Till this day, the results of dynamic programming methods has been regarded as superior to those of mean-variance optimiza-tion methods. However, problems might get extremely complicated when the dimension
of asset classes rises up. Later on, Merton (1971, 1973) [12, 13] developed an
analyti-cal framework for understanding the risk dynamics from a long-term investor’s point of view, which he calls the Intertemporal Capital Pricing Model. The framework is found to be hard to implement due to some technical difficulties in solving it, by Campbell and
Viceira (2002)[2]. The importance of asset-liability management, instead of asset-only
management, has been long recognized by both practitioners and academia, however, for a long period of time an analytical solution was only under a static/one-period
mean-variance optimization framework. After a while, Campbell and Viceira (2005)[7] extend
from the approach that is suggested by Stambaugh (1997)[14], without liabilities in the
covariance structure, but it is a good demonstration of how asset price dynamics can implicate portfolio choices for asset only investors at different investment horizons. After
that, many researchers incorporated liabilities to their setting. Hoevenaars (2008)[15]
is a significant contributer in this literature. He assumes that long-term investors who have risky liabilities choose the optimal portfolio weights of the investment horizon and re-balance to the initial state prior to next investment horizon. He finds that there is a significant difference between asset-only and asset-liability investors in choosing the
port-folio weights. However, the method is criticised as myopic by Barberis (2000)[16], due
to the assumption, that investors re-balance to the static/one-period portfolio weight at/prior to the next investment horizon and ignore the new information available at
each re-balancing time, is too harsh. The static re-balancing to the initial state at ev-ery investment period is both psychologically impossible and physically expensive for practitioners to implement. Until now, the problem has not yet been fully solved in this framework. However, most of the pension funds still use this myopic approach to make decisions and review every 3 to 5 years. Recently, Giamouridis, Sakkas and Tessaromatis
(2017)[17] have proposed a backward recursion model that takes the new information
available at each re-balancing time into consideration. Before this, all kinds of extensions to the myopic method have been studied; for example, implementing shortselling and
borrowing constraints by Brandt, Santa-Clara and Valkanov (2009)[18] and minimum
and maximum funding ratio constraints under a regulatory framework. More advanced techniques in math and statistics have been incorporated by practitioners, for example, regime switching model, multi-stage stochastic dynamic programming with Monte Carlo sampling and other efficient heuristics.
3.2
Outline
The main goal in our study is to find the counter-factual path of the volume of the PMT assets when the French inflation-indexed bonds are put into the investment basket, based on the historical data. However, the obtained proxy indices are not precisely identical to the actual returns that PMT employed for each investment category. Therefore, the volume of assets calculated with the proxy index returns at each time period is significantly different from the actual PMT volume of assets. In order to make the counter-factual path with inflation-indexed bonds comparable to the original path, we also need to find the counter-factual path without inflation-indexed bonds. Therefore, two optimization problems are considered with an identical allocation method but with different sets of asset classes, with and without inflation-indexed bonds. Our goal would be to know if the alternative asset class inflation-indexed bonds add value for the long-term investor under this counter-factual path. We will reach our goal by answering two questions: (i) Is there a significant portfolio shift when the inflation-indexed bonds are available? (ii) How does the volume of assets/funding ratio develop under the counter-factual path with inflation-indexed bonds comparing to the other counter-counter-factual path without inflation-indexed bonds? In addition, we have only dealt with one set of series; if the difference in the result is not significant, one can easily argue that it might be an estimation error or caused by a special market circumstance to have this insignificant difference. The last part of the problem would be a simulation study, which is based on a large amount of simulated sets of series with respect to the mean, variance and correlation structure among our original set of asset classes.
3.3
The pensioner’s optimization problem
Long-term investors like, pension funds, want to determine their holdings such that, in the future, they will be able to finance their later liability streams without the need of further contributions; for instance, future pension payments. Therefore, they are more concerned with their asset value relative to their liability value, instead of being only concerned with their later wealth. Hence, within an ALM context, the pensioner’s utility
optimization problem is expressed with respect to their funding ratio Ftat time t. The
funding ratio is defined as
Ft= At/Lt. (3.1)
A pensioner is assumed to have Constant Relative Risk Aversion (CRRA) preferences on
their funding ratio Ft. Hence, a specification of his liability having a constant maturity
at each time horizon has to be assumed as well. Therefore, his utility function at time t would be
u(Ft) =
Ft1−γ
1 − γ, (3.2)
for γ 6= 1 and where γ is the relative risk aversion coefficient. Therefore, the pensioner’s problem would be max αt Et " FT1−γ 1 − γ # , (3.3)
where T = t + k is some future date and αt is the composition of portfolio weights at
time t.
3.4
Strategic asset-liability allocation
In order to solve the pensioner’s problem, we need to find a proper tool to optimize the expected utility function, which means it should not be extremely hard to implement or computationally demanding. Our ultimate goal is to generate a robust result, and this can not be achieved without Monte Carlo simulation. A possible simulation routine would be:
1. Simulate T-bill returns, all asset excess returns and all state variables as a joint process at a monthly frequency. Details will be provided later.
2. Aggregate monthly data to a yearly frequency
3. Estimate the VAR(1) model based on all simulated yearly series excluding inflation-indexed bond excess return, and including pension liability return
4. Optimize the pensioner’s objective function based on the VAR(1) estimation in Step 3 at each time period in the entire investment horizon and find the counter-factual allocation path without considering inflation-indexed bonds as an asset class
5. Calculate funding ratios based on the simulated yearly returns found in Step 2 and optimal portfolio choice found in Step 4 for all time periods
6. Estimate the VAR(1) model based on all simulated yearly series including inflation-indexed bond excess return, and including pension liability return
7. Optimize the pensioner’s objective function based on the VAR(1) estimation in Step 6 at each time period in the entire investment horizon and find the counter-factual allocation path with considering inflation-indexed bonds as an asset class 8. Calculate funding ratios based on the simulated yearly returns found in Step 2 and
optimal portfolio choice found in Step 7 for all time periods 9. Repeat Step 1-8 for a decent amount of times
The whole procedure definitely demands some computational power. The multi-period
backward recursion model proposed by Giamouridis, Sakkas and Tessaromatis (2017)[17]
is brilliant in the sense of taking the new information after each portfolio choice into con-sideration and dynamically re-balancing, which is more actionable for investors. More-over, their simulation result shows that the funding ratio does not diverge as fast when the sampling periods go up. However, it estimates the VAR model once for optimization at each optimization, and the optimization needs to be solved backwards recursively,
which means it at least needs 162 = 256 times of the myopic computational time;
no-tice that the myopic case needs to take quite an amount of time for a large simulation already. Besides, it is a little bit hard to implement since for each period, there is an an-alytical allocation function with three affine sub-functions of the risk aversion coefficient γ, coefficients of each VAR estimation and the remaining period, in it, each of sub-functions has about 15-25 terms and overall takes some time to derive and implement. In practice, pension funds’ strategic asset allocations tend to follow a myopic strategy that is re-visited every three to five years and usually ignores timevarying opportuni-ties. Therefore, the myopic method fits our spirit a bit better, since we do not need the best predictive power but make a decent comparison under a reasonable allocation
framework. The method that we implement here is proposed by Hoevenaars (2008)[15].
Following from Eq3.1, we have
where rF,tis the log (continuously compounded) funding ratio return at time t.
The log-linear approximation to the one-period portfolio return provided by Campbell and Viceira (2005)[] is rA,t+1= rtb,t+1+ α0t(xA,t+1+ 1 2σ 2 A) − 1 2α 0 tΣAAαt (3.5)
Substitute Eq3.5 into Eq3.4, we get one-period funding ratio return
rF,t+1= −xL,t+1+ α0t(xA,t+1+ 1 2σ 2 A) − 1 2α 0 tΣAAαt (3.6)
Then aggregate each one-period funding ratio return for k-period, based on the assump-tion that re-balancing to the initial weight at the end of each horizon, we have
rF,t+k(k) = k X j=1 rF,t+j = −x(k)L,t+1+ α(k) 0 t (x (k) A,t+1+ k 2σ 2 A) − k 2α (k)0 t ΣAAα(k)t , (3.7)
where r(k)F,t+k denotes the aggregate k-period funding ratio return at time t + k and α(k)t
denotes a set of constant portfolio weights over k-period starts from time t. Evaluating the mean and variance of the k-period funding ratio return we have
E(rF,t+k(k) ) = k α(k)t 0(µ(k)A,t+1 2σ 2 A) − 1 2α (k)0 t ΣAAα (k) t − µ (k) L,t (3.8) Var(rF,t+k(k) ) = khσL(k)2− 2αt(k)0σAL(k)+ α(k)t 0Σ(k)AAα(k)t i, (3.9)
where estimation of k-period moments follows from the earlier chapter. Recalling the
assumption of normally distributed excess returns, the pensioner’s problem Eq 3.3can
be reduced to max α(k)t Et h r(k)F,t+ki+1 2(1 − γ)Vart h rF,t+k(k) i, (3.10)
which can be easily solved by taking its first order derivative.
An additional point that needs to be noticed here is that short selling is not allowed, and neither is leverage, which means PMT does not borrow money at a T-bill rate and use that to invest. In reality, a pensioner might be more flexible about this under the regulatory framework and its own risk managing policy; however, we are looking for a counter-factual path based on the original data and there is neither short selling nor leverage in their annual reports. The short selling constraint would be
α(k)i,t ≥ 0, for all i, (3.11)
There are many approaches to implement these constraints. One of them would be the method of Lagrange Multipliers, which combines the constraints with the objective function and optimizes that. An easier approach would be the one introduced by Brandt,
Santa-Clara and Valkanov (2009)[18], which truncates the negative portfolio weights in
αt(k)at zero and re-normalizes the optimal weights in order to sum the portfolio weights
to one. A representation to this would be
α(k)+i,t = max [0, α (k) i,t ] Pm−1 j=1 α (k) j,t , (3.12)
which also takes care of the leverage constraint.
The optimal portfolio is obtained after taking the first order derivative of Eq 3.10
α(k)t = 1 k −(1 − 1 γ)Σ (k) AA+ 1 γΣAA −1 (µ(k)t +1 2σ 2 A) − (1 − γ)σ (k) AL , (3.13)
where ΣAA is Σ(1)AA and σA2 the diagonal elements of ΣAA. The portfolio choice has two
components: the speculative component, normally referred as the Performance Seeking portfolio (PSP), which exploits risk premium with usual risk-return trade-off
α(k)S,t = 1 k (1 − 1 γ)Σ (k) AA+ 1 γΣAA −1 (µ(k)t + 1 2σ 2 A) , (3.14)
and the hedge demand component, which is normally referred as the Liability Hedge portfolio (LHP) α(k)H,t= 1 −1 k (1 − 1 γ)Σ (k) AA+ 1 γΣAA −1 σ(k)AL. (3.15)
The LHP does not change over time if we assume that the strategic portfolio is held constant over time over the entire investment horizon. However, this is not our case, since there is cash inflow from the Pension benefits/payments surplus every year. Ideally,
the higher the covariance σ(k)ALis, the higher the fraction of wealth is allocated. Moreover,
in theory, if liability is based on real consumption flows, which the PMT liability is, the best hedge portfolio consists of inflation-indexed bonds.
If the pensioner is extremely risk averse, γ goes really large, the entire portfolio will shift to hedge demand only. For example, as γ → ∞, the hedge demand portfolio simplifies to
αH,t(k) = (Σ(k)AA)−1σAL(k), (3.16)
3.5
Monte Carlo simulation of correlated asset returns
After all, an empirical study is less convincing if it is just based on a set of series, especially when the sampling period is short and data frequency (yearly) is low, since anything can happen in a small sample. One can easily argue that a slight difference in the result between portfolios with and without inflation-indexed bonds is by cause of an estimation error or a huge return discrepancy in one of those asset classes. For instance, the equity market performed poorly in most of the sampling period, especially in 2008 when the global equity index dropped almost 60%. Simply reducing portfolio weight in equity investments to any other asset class would have increased the funding ratio and reduced the funding ratio variance, which means it is hard to conclude that the inflation-indexed bonds necessarily add value to the investors. In order to generate a more promising and robust result, the Monte Carlo simulation method is applied, such that the result would be based on a decent amount of sets of series, instead of one. Ideally, the PMT liability return should be simulated together with other asset class returns. However, the frequency of the PMT data including its liability is yearly, which makes it nearly impossible. Although, it is possible to interpolate the series at a monthly frequency by descaling its yearly variance to a monthly level, due to the fact that Defined-Benefit scheme liabilities are less dynamic than the other scheme liabilities. However, it is extremely hard to make it rigorous, since indexed pension liabilities fluctuates with the inflation, and we know in some of those years the PMT provides indexation and does not in the others. To keep everything simple and rigorous, the PMT liability return will not be part of simulation.One way to simulate is to apply time series analysis again, which in our case could be a VAR model. Then we simulate forward based on monthly data and aggregate to yearly. However, this method gives a relatively high signal to noise ratio in the covariance structure. Another way would be simulating based on the correlated asset returns. What we need to do is to find the variance-covariance matrix based on variances and correlation matrix to find variance-covariance matrix. According to our normality assumption of log (or continuously compounded) returns, asset returns over an interval of length dt can be randomly drawn by
dS
S = µdt + σdz = µdt + σ
√
dt, (3.17)
where S is the asset price, µ is the expected return, sigma is the volatility of S and represents a random drawing from a standardized normal distribution.
3.6
Optimization results
In order to find the counter-factual paths, the return dynamic among the asset classes is assumed to unconditionally follow the VAR(1) model without analysis about the market condition change, in the entire period from 1999-2014, as mentioned earlier. Then we apply myopic optimization based on prior k-period information to make an optimal portfolio choice, notice that k varies from 1 to 15 in our case. In such a way, we assume the asset return follows a k-period VAR(1) process starting from period 0. Counter-factual paths of the portfolios that exclude and include the French inflation-indexed bonds are found in the following sections. Results are shown for every five years and two other selected years that are the most representative. The result is based
on three different risk aversion coefficients, γ = 5, 10 and 20. Both unconstrained
and constrained optimizations are studied. The unconstrained one uses the analytical solution that we demonstrate earlier and use the truncating method to realise no-short-selling and no-leverage constraints. Notice that it is hard to give an analytical solution to the constrained pensioner’s problem. Therefore, the constrained problem will be solved by applying the non-linear optimization theory to a Lagrange problem. The results are as follows.
3.6.1 Unconstrained optimization I
As shown in Table 3.1, there is no investment in the equity market for both paths in
all years under three different risk aversion coefficients γ. An answer to this would be the extremely low Sharpe-ratio and overall return dynamic of the equity market. As we priorly know there is a ’Lost decade’ in our sample, the VAR model tells us to avoid its pitfall. One possible way to deal with this and make the optimized portfolio more realistic comparing to the historical PMT portfolio is to use commonly assumed parameters for the equity processes, which will be covered in the next section. Other than that, there is a significant portfolio shift from common asset classes to inflation-indexed bonds, when we include the information of pension fund liabilities in the ALM process. As we can see, the proportion of inflation-indexed bonds investment peaks
around the 9th year, which is very close to the average duration of the inflation-indexed
bonds, 8.93 years. It means that the French inflation-indexed bonds does not only add value to the PMT portfolio based on the return dynamic analysis, but also is an excellent hedge to the real consumption flow based PMT liabilities. As γ goes up, the proportion of of inflation-indexed bonds investment increases, which is in accordance with the theory, the portfolio demand shifts to the hedge demand when γ goes large and the best hedge portfolio consists of inflation-indexed bonds. Another thing needs to be noticed is that the optimal portfolio choice has a huge decrease in demand of inflation-indexed bonds, real estate and high yield & emerging market bonds; and a
Table 3.1: Counter-factual allocation paths (unconstrained I)
Without inflation-indexed bonds With inflation-indexed bonds 2000 2004 2008 2009 2010 2014 2000 2004 2008 2009 2010 2014 γ = 5 Equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Bonds 0.00 0.66 0.72 0.91 0.79 0.79 0.00 0.09 0.07 0.86 0.26 0.33 Real estate 0.00 0.19 0.26 0.09 0.03 0.01 0.00 0.17 0.24 0.14 0.06 0.03 HYEM 1.00 0.15 0.02 0.00 0.17 0.20 0.43 0.17 0.06 0.00 0.19 0.21 Inflation-indexed bonds 0.57 0.57 0.63 0.00 0.50 0.43 γ = 10 Equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Bonds 0.00 0.66 0.72 0.91 0.79 0.79 0.00 0.10 0.10 0.85 0.26 0.33 Real estate 0.00 0.19 0.26 0.09 0.03 0.01 0.00 0.16 0.21 0.13 0.05 0.02 HYEM 1.00 0.15 0.02 0.00 0.17 0.20 0.43 0.17 0.08 0.02 0.19 0.21 Inflation-indexed bonds 0.57 0.57 0.61 0.00 0.50 0.44 γ = 20 Equity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Bonds 0.00 0.68 0.74 0.90 0.80 0.80 0.00 0.13 0.14 0.78 0.27 0.32 Real estate 0.00 0.17 0.21 0.07 0.03 0.00 0.00 0.15 0.17 0.10 0.04 0.01 HYEM 1.00 0.15 0.05 0.03 0.18 0.20 0.43 0.17 0.09 0.04 0.19 0.21 Inflation-indexed bonds 0.57 0.55 0.59 0.07 0.50 0.46
gigantic increase in government bonds in 2009. This is due to the equity shock from global financial crisis and French inflation shock in the previous year 2008, where the equity world index drops almost 60% and French yearly inflation hikes 1.5%; see Figure
2.1. This is in accordance with the impulse response functions from the VAR(1) model
that we found earlier, see Figure2.2. Equity shocks to nominal bonds are less persistent
than to indexed bonds, which means there will be portfolio shift from inflation-indexed bonds to nominal bonds in the next period if it comes with negative equity shocks, and vice versa. However, the assumption that the average past observations has predictive power to the future is where the traditional Markowitz mean-variance optimization fails. It might have worked decades ago since the market was much more inefficient back then and this kind of inefficiency has been arbitraged away for a while. A better model, for instance a regime switching model that identifies the current market condition, will have much more predictive power. And once again, since we only look for the counter-factual paths and make a comparison, the framework that we apply is more than sufficient.
However, the liability hedge that inflation-indexed bonds provide is not free. As depicted
in Table3.2, the funding ratio of the portfolio with inflation-indexed bonds falls behind
nearly 13% compared to that of the one without inflation-indexed bonds, under all risk aversion coefficients. Notice the rule of thumb in the pension fund management: 1% more return means 30% lower pension cost. Although the average funding ratio is high, 13% difference still means an enormous amount to the fund. However, this is our hypothetical counter-factual path, which involves no constraint for each individual asset, and it is possible that the extremely high funding ratio comes from the overall
Figure 3.1: Funding ratio: γ = 10
risk that it over-takes. For example, a position of 100% in high yield & emerging market bonds is just way too risky. Moreover, in practice equity always shows up on the trading book of a pension fund. The minimum portfolio weight in equity from the PMT data is 25% in 1997; other than that, it is always higher than 30%. In order to make more realistic comparison, constrained optimization that takes restrictions in each individual asset class will be employed in the later sections.
The funding ratio in our unconstrained counter-factual paths develop quite well. Nega-tive shocks do not bring persistent effects and the funding ratio recovers or remains at
a certain level soon after the shocks through the toughest decade, see Figure 3.1. The
two paths look nearly parallel through our naked eyes. However, the portfolio includ-ing inflation-indexed bonds has almost 22% lower fundinclud-ing ratio variance than the one excluding inflation-indexed bonds. There are some major draw-downs, for instance, in 2002 and 2008.
Table 3.2: Funding ratio mean and variance: I
γ = 5 10 20
ILBs w/o w w/o w w/o w
Mean 1.527738 1.330644 1.527738 1.328107 1.520944 1.324652
Variance 0.028807 0.022519 0.028807 0.02266 0.028789 0.022796
3.6.2 Unconstrained optimization II: equity adjusted
Anything can happen in a short sampling period and it really did in the equity market within our sampling period, However, equity returns are extremely hard to predict
Table 3.3: Counter-factual allocation paths (unconstrained II: equity adjusted)
Without inflation-indexed bonds With inflation-indexed bonds 2000 2004 2008 2009 2010 2014 2000 2004 2008 2009 2010 2014 γ = 5 Equity 0.53 0.25 0.52 0.19 0.39 0.54 0.49 0.44 0.47 0.04 0.16 0.24 Bonds 0 0.6 0.48 0.81 0.44 0.44 0 0.09 0.07 0.84 0.15 0.23 Real estate 0 0 0 0 0 0 0 0 0.12 0 0 HYEM 0.47 0.15 0.02 0 0.17 0.2 0.36 0.1 0.06 0 0.19 0.21 Inflation-indexed bonds 0.15 0.37 0.4 0 0.5 0.32 γ = 10 Equity 0.33 0.19 0.26 0.11 0.22 0.29 0.27 0.22 0.28 0.13 0.18 0.25 Bonds 0 0.66 0.72 0.89 0.61 0.51 0 0.1 0.1 0.85 0.17 0.22 Real estate 0 0 0 0 0 0 0 0 0 0 0 0 HYEM 0.67 0.15 0.02 0 0.17 0.2 0.43 0.17 0.08 0.02 0.19 0.21 inflation-indexed bonds 0.3 0.51 0.54 0 0.46 0.32 γ = 20 Equity 0.25 0.1 0.22 0.09 0.19 0.24 0.25 0.16 0.17 0.09 0.18 0.21 Bonds 0 0.68 0.73 0.88 0.63 0.56 0 0.13 0.14 0.78 0.16 0.21 Real estate 0 0.07 0 0 0 0 0 0 0 0 0 0 HYEM 0.75 0.15 0.05 0.03 0.18 0.2 0.43 0.17 0.09 0.05 0.19 0.21 Inflation-indexed bonds 0.32 0.54 0.59 0.07 0.47 0.37
and most of the institutional investors still prefer to put a fairly large share of their assets in the equity market even during the hardest decade, in order to presume some stimulus effect. Therefore, it is also interesting to show some analysis that is based on commonly assumed parameters for the equity processes before we jump into the constrained problems. In order to do so, we add one quarter of the equity standard deviation during our sample period in our optimization process, which roughly gives equity market an average yearly return of 10.11% and a Sharpe ratio of 0.45, which are much closer to the commonly assumed equity parameters. An additional remark would be an average yearly return of 10.11% outperforms any other asset classes. Even though, a Sharpe ratio of 0.45 is still way below the Sharpe ratios of non-risky assets, like nominal bonds and inflation-indexed bonds; but close to the Sharpe ratios of the listed real estate market and high yield & emerging market bond market, 0.42 and 0.52, respectively.
As shown in Table 3.3, the investment in equity market no longer weights zero after
adjusting the equity parameters. But the weights on real estates seem disappeared. This is due to equity market is so much more competitive than the real estate market after the adjustment, with a higher average yearly return but a lower standard deviation, which yields a higher Sharpe ratio. However, real estate investments are not replaced in all years; for instance, in 2001, 2005 and 2005, which are not depicted in the table. The equity weights tend to grow in the long run, which makes sense because the
mean-reverting property of equity makes it an excellent inflation hedge[19]. There are a few
draw-downs in the equity weights, for example, in 2009, which is due to the negative equity shock in 2008. The portfolio shift to inflation-indexed bonds is quite similar to the unadjusted case, mainly from nominal bonds. However, it is hard to tell if there is a
Figure 3.2: Funding ratio II: γ = 10
shift from equity to indexed bonds. Because, in some years, there are shifts from equity to indexed bonds; in the other years, the inclusion of indexed bonds seems to exaggerate the equity weights. Moreover, the weight on inflation-indexed bonds peaks again around the 9th year, which is very close to the average duration of the French inflation-indexed
bonds. It demonstrates again the excellent hedge property of the French
inflation-indexed bonds to the real consumption flow based PMT liabilities. Furthermore, as investor becomes more risk averse, the investment weight on equity tends to decrease and the investment on indexed bonds tends to grow, which is in accordance with our theory. However, the result is still not satisfactory enough for us. The combined investment weight on risky assets happen to extend the risk limit of the pension fund and the replacement of investment in real estate market to equity market is not ideal for the purpose of diversification. In order to make more realistic comparison, we have to go to our next step, which is the constrained optimization that takes restrictions in each individual asset class.
The funding ratio in our equity adjusted unconstrained counter-factual paths develop not so well. Small negative shocks do not bring persistent effects, but huge ones do,
for example the one in 2008, see Figure 3.2. The two paths look quite different and
they cross each other for more than 5 times. In the end, the portfolio that includes the inflation-indexed bonds outperforms the one that excludes the inflation-indexed bonds with a slight advantage. However, both of them in the end are way below the 100% level. The portfolio including inflation-indexed bonds has almost 27% lower funding
ratio variance than the one excluding inflation-indexed bonds.
Table 3.4: Funding ratio mean and variance: II
γ = 5 10 20
ILBs w/o w w/o w w/o w
Mean 1.086123 1.095144 1.133445 1.144159 1.09523 1.108242
Variance 0.08712 0.07821 0.04795 0.03533 0.028789 0.022796
3.6.3 Constrained optimization
In general, asset classes, such as equity and high yield & emerging market bonds, have much better Earning-to-Price ratios than any other asset classes, which are genuinely considered as risky assets. Most pensioners and other institutional investors tend to favor a decent amount of portfolio weights on these risky assets, especially when the interest rate remains low, in order to achieve some stimulus effect on the fund performance. However, it is highly unacceptable that a total investment weight on these risky assets is too large, for instance 100% investment in the high yield & emerging market bonds in our previous hypothetical unconstrained optimization problem is just way too risky. We might not know how much precisely the PMT pensioner’s risk tolerance for each asset class is. But there is some information we can extract from the PMT data, which is the historical portfolio weight limit in each investment category, and that should be bounded by the pensioner’s risk limit. A reasonable set of constraints for each individual asset class would be:
• Equity: 20%-30% • Real estate: 0%-17% • HY & EM: 0%-15%
Notice that there is no constraint for non-risky assets, for instance, the nominal bonds and the inflation-indexed bonds, by assuming no credit risk in the high-grade sovereign bonds. The short-selling and leverage constraints are implemented as:
• Bonds: 0%-80%
• Inflation-indexed bonds: 0%-80% • Sum of all asset classes is 100%
The Lagrange has a slightly more complicated representation than this, due to the matrix notation. However, here above are all the constraints that we use.
The constrained pensioner’s problem, in Eq3.10, is solved as a non-linear optimization,
Table 3.5: Counter-factual allocation paths (constrained)
Without inflation-indexed bonds With inflation-indexed bonds 2000 2004 2008 2009 2010 2014 2000 2004 2008 2009 2010 2014 γ = 5 Equity 0.30 0.20 0.20 0.20 0.20 0.20 0.30 0.20 0.20 0.20 0.20 0.20 Bonds 0.38 0.48 0.49 0.80 0.65 0.65 0.00 0.00 0.00 0.80 0.11 0.64 Real estate 0.17 0.17 0.17 0.00 0.00 0.00 0.17 0.17 0.17 0.00 0.00 0.00 HYEM 0.15 0.15 0.14 0.00 0.15 0.15 0.15 0.15 0.02 0.00 0.15 0.15 Inflation-indexed bonds 0.38 0.48 0.61 0.00 0.54 0.01 γ = 10 Equity 0.30 0.20 0.20 0.20 0.20 0.20 0.30 0.20 0.20 0.20 0.20 0.20 Bonds 0.38 0.50 0.67 0.80 0.73 0.65 0.00 0.00 0.00 0.67 0.00 0.00 Real estate 0.17 0.15 0.13 0.00 0.00 0.00 0.17 0.17 0.17 0.00 0.00 0.00 HYEM 0.15 0.15 0.00 0.00 0.07 0.15 0.15 0.15 0.00 0.00 0.03 0.10 Inflation-indexed bonds 0.38 0.48 0.63 0.13 0.77 0.70 γ = 20 Equity 0.30 0.20 0.20 0.20 0.20 0.20 0.30 0.20 0.20 0.20 0.20 0.20 Bonds 0.38 0.68 0.80 0.80 0.80 0.75 0.00 0.00 0.00 0.11 0.00 0.00 Real estate 0.17 0.00 0.00 0.00 0.00 0.00 0.17 0.00 0.00 0.00 0.00 0.00 HYEM 0.15 0.12 0.00 0.00 0.00 0.05 0.15 0.12 0.00 0.00 0.00 0.00 Inflation-indexed bonds 0.38 0.68 0.80 0.69 0.80 0.80
term can be consistently estimated at time t for all k, as mentioned earlier. An finite-difference interior-point algorithm is applied, with respect to to the analytical gradient and analytical Hessian matrix of the constrained pensioner’s problem.
As shown in Table 3.5, there are two constrained counter-factual paths, with and
with-out inflation-indexed bonds in their baskets of asset classes, under each risk aversion coefficients, γ = 5, 10 and 20. For all equity investment choices, they are all precisely at the lower bound 20%, except in the first year, 2000, simply because equity market did extremely well in 2009 and that greatly impacts the optimal portfolio choice, regardless of the overall return dynamics for the entire horizon. There are some major shifts in asset-liability allocation from bonds to the inflation-indexed bonds category under all risk aversion coefficients for all most years, except in 2009, 2012 and 2014. The shift from bonds to inflation-indexed bonds in 2013 is also comparably much smaller than any other years which have major shifts. An explanation to this would be that the shocks in 2008 and 2011 from the equity and bond markets, respectively, to other markets are impactful, which is in accordance with our earlier analysis with the impulse response functions. In addition to this, the unconstrained counter-factual allocation paths are useful, in the sense of observing the effects from shocks to inflation-indexed bonds in-vestment, for instance, we would not be able observe impact of the shock in 2009 under γ = 20, if we set constraints for the real estate and high yield & emerging market bonds markets, that causes the upper bound of nominal bonds and inflation-indexed bonds below 69%. The portfolio weight in the inflation-indexed bonds peaks in 2008, which again is in accordance with the duration of inflation-indexed bonds. As the investment horizon goes beyond the duration of the inflation-indexed bonds, the less effective its hedging property is, to the real consumption flow based PMT liabilities, and thus the
