investor under cointegration approach
Eleni Orfanou
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: Eleni Orfanou
Student nr: 10229566
Email: orfanouel@live.com
Date: November 7, 2015
Supervisor: Dr L.J. van Gastel Second reader: Prof. dr H.P. Boswijk
Abstract
This paper aims to explore the co-movements between different as-set classes that compose an investment portfolio for an investor with liabilities mainly in the long run from an asset-liability management perspective. With the intention of constructing an optimal asset allo-cation, we analyze a portfolio consisting of stocks, government bonds, T-bills, listed real estate, credits and commodities. Using a vector er-ror correction model (VECM) that identifies the presence of long-run dynamics that drive the movements of assets and liabilities prices, we investigate the covariance structure of assets and liabilities tak-ing into account the co-movements between them and among different asset classes, and we examine the inflation and real interest rate hedg-ing properties of each asset class. The results suggest diversification effects, real interest rate hedging qualities and inflation hedging prop-erties as well. Differencies comparing to Hoevenaars outcome are iden-tified and explained further. Finally, we construct our optimal mean variance portfolio which indicates that the majority of the investment made is on the 20-year government bond with a declining percentage as investment horizon increases.
Keywords Asset allocation; Cointegration; Efficient frontier; Diversification effects; Optimal portfolio
Contents
Preface v
1 The concept of Cointegration 4
2 Data Modeling 8
3 Term structure of assets and liabilities 16
3.1 Term structure of risk . . . 16
3.2 Risk diversification properties . . . 18
3.3 Inflation hedging potential . . . 20
3.4 Real interest rate hedging properties . . . 21
4 Mean Variance Portfolio Efficient Frontier 23 4.1 Optimal Portfolio . . . 25
4.2 Efficient Frontier . . . 26
5 Conclusion 28
References 30
At this moment I would like to thank my parents, Panagiota Georgios Orfanos, for their huge support during my studies and for the values they have taught me. They have encouraged me not to quit, even if it was very hard to write the thesis while I was working in very tight schedule. I would also like to thank my supervisor, Leendert Van Gastel, for his patience during the very long duration of this thesis. His comments and suggestions helped me improve excessively my thesis. In addition, I would like to express my gratefulness for Prof. Rob Kaas, who was very supportive and replied to any e-mail at anytime. Thank you so much.
This study intends to find an appropriate approach for managing assets when liabilities are considered. The most common investors with risky liabilities in the long run are mostly pension funds and insurance companies. Hence, this thesis may be a valuable analysis for the purposes of these kinds of institutions. This analysis is concentrated on interest rate risk and inflation risk that these institutions have to deal with.
In this context, their management aims to decide about their strategic asset allocation, their contributions policy and their indexing policy. They have to meet the benefit pay-ments while remaining solvent according to the Regulator. Funding levels below 100% are not desired because the liabilities cannot be covered in total by the assets. In ad-dition, indexing policy is of great importance because markets are incomplete, in the sense that available asset classes cannot fully hedge the liabilities, since the latter are inflation-dependent. Pension funds intend to index their benefit payments on inflation according to their funding level. Indexing can be achieved by inflation linked securities such as Index-linked Bonds or inflation swaps, but the market is limited compared to the size of the liabilities that should be covered.
Asset-liability modeling, therefore, is a tool that assists managers to decide about their strategic policy by developing appropriate scenarios of the assets’ and liabilities’ evolu-tion in the future, especially at long horizons. The quesevolu-tion is: ”what is an appropriate approach for asset-liability modeling?”. The answer to our problem could be cointe-gration analysis, which is a powerful statistical technique for finding asset allocations that may be able to act as a hedge against a variety of liabilities. This technique is advantageous when it comes to liability-driven investing (LDI) where asset owners seek to minimize the volatility of their funding ratios. Cointegration is a very important concept for determining asset allocation at intermediate and long investment horizons. Asset allocation based on the cointegration approach can be much different to a tradi-tional Vector Autoregression (VAR) approach which ignores the cointegrating relations between asset classes as well as between assets and liabilities. This has been proven by R. Bansal and D.Kiku (2011). According to their study, VAR approach, used for modeling expected returns, captures the short-term dynamics, but in the presence of cointegration, it considerably misses the intermediate and long horizon dynamics of as-set returns. On the other hand, cointegration technique captures much larger benefits of time-diversification. Hence, the main research topic of this thesis is to support the evidence that cointegration approach is a good asset-liability modeling technique by practical data analysis.
Apart from our concern of whether cointegration could be a good approach for asset-liability modeling, our attempt is to also investigate the following questions: (i) What is the time and risk diversification properties of a liability-driven investor’s portfolio un-der cointegration approach? (ii) What is the interest rate risk and inflation risk hedging properties of the aforementioned portfolio? (iii) What are the optimal weights of the different asset classes in the portfolio and what is the efficient frontier under cointegra-tion approach? The steps followed to answer these quescointegra-tions will be further analyzed extensively.
In this thesis we will use Hoevenaars et al. (2008) as the main reference for our analysis investigating the risk characteristics and the hedging potential of the following asset classes, which have been introduced by Hoevenaars: stocks, government bonds,
2 Eleni Orfanou — Asset allocation under cointegration approach
rate bonds, T-bills, listed real estate and commodities. In Hoevennars’ paper (2008) a vector autoregression for returns, liabilities and macro-economic state variables has been used for the analysis of the intertemporal covariance structure of assets and lia-bilities. Time diversification, risk diversification, inflation and real interest rate hedging qualities have been studied for different investment horizons and for different types of investors (asset-only and asset-liability investors).
Instead of using the vector autoregression method that has been used by Hoevennars, we will examine the risk term structure, the risk diversification effects, the inflation hedging and real interest rate hedging qualities of the different asset classes by taking into account cointegration relations between them or between assets and liabilities. In order to accomplish it, as we will explain extensively later on, we obtain forecasts for the model-implied volatilities and correlations and we examine whether the outcome deviates from Hoevenaars’s results. Before starting building our model, we are going to identify the presence of cointegration between the return on typical pension fund liabil-ities and the return of the traditional and alternative asset classes that we mentioned previously.
To answer questions (i) and (ii), we investigate the risk characteristics and the hedg-ing properties of the asset classes portfolio for an investor with liabilities in the long run (pension funds liabilities) under the cointegration approach. Specifically, we use a Vector Error Correction Model (VECM) in order to obtain the covariance and the correlations between these assets. The asset classes we study are stocks, government bonds, corporate bonds, T-bills, listed real estate, credits and commodities, which are the same as Hoevenaars et al. (2008) data set for reasons to comparison. After having found the proper structure of the Vector Error Correction Model, then we construct the evolution of the annualized variance-covariance matrix and the evolution of the correla-tions matrix of our assets using Monte-Carlo simulacorrela-tions, which enables us to examine our asset classes time and risk diversification properties. Our asset classes time and risk diversification properties are compared to those of Hoevenaars’. As a next step, we use the outcome of the Vector Error Correction Model so as to determine how the infla-tion and real interest rate are correlated to the different asset classes. In particular, we study our asset classes potential as a hedge against real interest rate risk and inflation risk at different investment horizons. The Vector Error Correction set up gives us the appropriate correlations matrices which will be used for the construction of correlations evolution by Monte Carlo simulations.
Using model-implied forward-looking variances and expected returns we answer to ques-tion (iii) by considering Hoevenaars et al. (2008) approach where optimal mean-variance portfolios are obtained with respect to liabilities driven by inflation and real interest rate. Hoevenaars et al. (2008) model returns with a vector auto-regressive (VAR) model. This approach misses out any information on price dependencies and long-term equilib-ria and focuses purely on short-term effects in return series. For this reason, we include cointegration relationships in the model and assess sensitivities of model-implied dynam-ics with respect to the factors that capture price dependencies. Finally, we construct a mean variance portfolio for an asset-liability investor by obtaining optimal weights for each asset class and then we proceed to the efficient frontier. As a result we illustrate an as good as possible ALM policy.
The reason we choose not to base our analysis on the VAR approach is the fact that it uses the correlation between asset returns as a reflection of co-movements in returns. But correlation is a short-term measure since it is based on returns and thus it is not sufficient to ensure the long-term performance of assets. Moreover, some cointegrating
variables may deviate from their relationship in the short run, but their association would return in the long run. On the other hand, cointegration measures long-run co-movements in prices which may occur even through periods when static correlations appear low, as it has been mentioned by Carol Alexander (1999). This could result to a suboptimal asset allocation. Therefore, we need a methodology that takes into account common long-run trends in prices, a feature offered by cointegration.
Structure of the Thesis
In the first chapter we introduce the concept of cointegration and the model structure that takes into account the presence of cointegration. The second chapter presents the data which is used in modeling process in order to find evidence for our thesis hypothe-sis. Our cointegration model parameters are estimated and they are used so as to report predictive power and the different risk characteristics of our asset classes. Chapter 3 discusses the term structure of risk implied by cointegration model. First we investi-gate the time diversification properties within an asset class by exploring the implied volatilities. Then we examine the diversification possibilities between asset classes at different horizons by looking into the correlations of stocks and bonds with the other asset classes. Furthermore, the potential of our asset classes as a hedge against inflation and real interest rate risk is analyzed in the same chapter. Chapter 4 demonstrates the impact of cointegration on a mean-variance portfolio and illustrates the structure of the efficient frontier. The last chapter deals with the results.
Chapter 1
The concept of Cointegration
Now let us describe the concept behind Cointegration. Our view is that optimization portfolio is based upon the cointegration of prices rather than the correlation of returns. Cointegration refers to co-movements in asset prices, exchange rates or yields instead of co-movements in returns. This technique provides a rigorous scheme that may be implemented with basic statistical techniques.
The concept of cointegration was first introduced by Clive Granger (1981). Granger and his collaborator Robert Engle were jointly awarded the 2003 Nobel Economics Prize for their work in analyzing economic time series. According to Granger (1981), there are economic forces that make some economic variables which move apart from each other in the short run to be brought together in the long run. Thus, a cointe-grating relationship can be described as an equilibrium phenomenon. Cointegration is expressed into co-movements in asset prices, exchange rates or yields. As far as spreads are mean-reverting, asset prices are highly correlated in the long run and there is a com-mon stochastic trend which drives the process. Thus, cointegration is such a useful tool which helps us to find common trends between variables and it assists us in dynamic modeling for both long and short horizons. The cointegration process is conducted in two stages. First, any long-run cointegration relationships between asset prices are iden-tified and then a dynamic correlation model of returns is estimated.
Let us describe the mathematical concept of cointegration:
A process ytis integrated of order d (denoted by I(d)) if ∆dytis stationary while ∆d−1yt
is not stationary (L¨utkepohl 1993). In addition, usually if two variables, that are I(d), are linearly combined, then the combination will be I(d) as well. The same holds for more than two variables. But if this linear combination is I(0), meaning stationary, the variables are cointegrated. As it is mentioned by Engle and Granger (1987), when a linear combination of two or more integrated variables has a mean-reverting behav-ior, these variables share a common stochastic trend driving the process. According to Granger representation theorem a set of cointegrated variables have an error correction representation. Thus, we use an error correction model to statistically model our inte-grated variables.
Firstly, we have to see how this model works. We consider a bivariate single-equation error correction model:
∆Yt= α0− α1(Yt−1− β1Xt−1) + β0∆Xt+ t (1.1)
where, Y and X are two time series and tis a stationary disturbance. According to this
equation, current changes in Y depend on current changes in X, the first lag level of Y and the first lag level of X. Specifically, Yt−1− β1Xt−1is known as the error correction
term, which expresses the degree to which the two series are outside of their equilibrium in the previous period. And if the series are cointegrated with cointegrating coefficient β1 (the long-term effect of X on Y ), then the error correction term will be stationary
(∼ I(0)). The parameter α1 expresses the rate at which the long-term effect occurs
and β0 expresses the contemporaneous effect of X on Y . Concerning multivariate time
series, when variables are I(1) we write the traditional Vector Auto Regression model 4
on first-differenced variables and thus we generate a stationary process:
yt= c∗+ Φ∗1yt−1+ ... + Φp∗yt−p+ t (1.2)
∆yt= c + Φ1∆yt−1+ ... + Φp∆yt−p+ t (1.3)
with Φi = − p+1 X j=i+1 ∆Φ∗j, i = 1, ..., p
where, p is the lag-length of the variables, Φi is the coefficient matrix for each of
the variables first differences and t ∼ i.i.d. N (0, Σ) is multivariate white noise with
V ar(t) = Σ. However, substantial predictive power of the model may be lost because
by following this structure we miss any information about price dynamics and the long-run associations between them. In order to consider the presence of long-long-run associations in asset prices we construct a generalized version of model (3) which is called Vector Error Correction model (VECM):
∆yt= c + Πyt−1+ Φ1∆yt−1+ ... + Φp∆yt−p+ t (1.4)
with Π = p+1 X i=1 Φ∗j− I = −Φ∗(1)
The matrix Π = αβ0 is considered to be the reduced rank matrix. Π cannot be a full rank matrix since this would mean that yt is stationary. If Π has zero rank, then the
first difference of yt does not depend on yt−1 but depends only on the first differences
of the yt−1, ..., yt−p, which means that there are no long-term associations between the
elements of yt−1 and as a result there is no cointegration. Since k represents the
num-ber of the time series, the rank of Π is r < k and it determines the numnum-ber of linear independent long-term equilibrium relationships. Hence, r is also called cointegrating rank. Furthermore, α and β are (k × r) matrices of rank r. Thus, according to Granger Representation Theorem, yt is cointegrated (yt ∼ I(1)), but r linear combinations of
βyt are stationary. Columns βi of β are cointegrating vectors and β
0
iy = 0 may be
in-terpreted as the long-run equilibrium relations of the variables. The product of βi0yt
displays mean-reversion and cannot drift too far away from its zero mean, so α reflects the speed to the long-term equilibrium and elements of are the adjustment parameters in the VEC model.
If we call wt = α
0
⊥
Pt
s=1s a (k − r)-vector random walk wt = wt−1 + α
0
⊥t. Then
yt= y0+ Awt+ xt− x0, where xta stationary linear process and A = β⊥[α
0
⊥Φ(1)β⊥]−1
So k time series in ytare driven by k − r random walks or common trends.
Augmented Dickey-Fuller test (ADF)
In order to proceed with cointegration analysis, we should first test whether our time series are stationary using Augmented Dickey-Fuller test (ADF). ADF test is a test for a unit root in an autoregressive process of order p(AR(p)). Testing on a stochastic trend considers three specifications to detect the presence of a unit root. Two of them contain two sub-cases. In particular, model with neither constant nor trend (c = µdt = 0)
corresponds to case 1. When dt= 1, and so µdt= µ, Granger representation takes the
following form yt= y0+ C t X s=1 (µ + s) + xt− x0 = y0+ Cµt + t X s=1 s+ xt− x0 (1.5)
6 Eleni Orfanou — Asset allocation under cointegration approach
Cµ 6= 0 implies that intercept in error correction model induces a linear trend in yt which corresponds to case 3. When Cµ = 0, α0⊥µ = 0 or µ = αc0 and we have no trend,
so the model takes the following form ∆yt= α(β
0
yt−1+ c0) + Φ1∆yt−1+ ... + Φp∆yt−p+ t (1.6)
which implies c0 = −E(β
0
yt), meaning that we have an equilibrium relation with
non-zero intercept. The aforementioned model form corresponds to case 2. When dt = (1, t)
0
, so µdt = µ0 + µ1t, there is a quadratic trend and it corresponds to
case 5. But if Cµ1 = 0 that implies α ⊥
0
µ1 = 0 or µ1 = αc0, the model takes the
following form
∆yt= µ0+ α(β
0
yt−1+ c0t) + Φ1∆yt−1+ ... + Φp∆yt−p+ t (1.7)
It corresponds to case 4 and implies a linear trend in cointegration relation. Johansen’s cointegration test
Once we ensure cointegration relationship between our time series, then we should identify the nature and the number of common trends driving the process. Johansen’s method helps us determine all r cointegrating relationships, because it permits more than one cointegrating relationship. In order to describe Johansen’s cointegration test, we suppose that dt= (1, t)0, so that
∆yt= µdt+ αβ
0
yt−1+ Φ1∆yt−1+ ... + Φp∆yt−p+ t (1.8)
where t = p + 1, ..., T . Johansen concentrates on the likelihood function with respect to the deterministic trend and the stationary effects. If we defineubtandbvtas the residuals of OLS regressions of ∆yt and yt−1on dt, ∆yt−1, ..., ∆yt−p, and the sample covariance
matrices S00= 1 (T − p) T X t=p+1 b utub 0 t, S11= 1 (T − p) T X t=p+1 b vtbv 0 t, S01= 1 (T − p) T X t=p+1 b utbv 0 t
then we solve the eigenvalue problem |λS11−S10S10−1S01| = 0, with (bλi, ei) the eigenvalue
and eigenvector pairs and bλ1 > bλ2 > · · · > bλr. The maximum likelihood estimate of the
cointegrating vector β is β = [e1, , em] and satisfies bβ
0 S11β = Ib r and I( bβ) = c −(T − p) 2 log |S00| − T − p 2 r X i=1 log(1 − bλi) (1.9)
We consider the hypotheses H0:Rank Π = r and H1:Rank Π > r. Johansen (1988)
proposed the likelihood ratio (LR) statistic
LRr= −(T − p) k
X
i=r+1
log(1 − bλi) (1.10)
We should mention that the asymptotic null distribution is not χ2, but it is a standard Brownian motions function because of the presence of unit roots. In order to determine
the cointegrating rank r we repeat the test mentioned above via simulation. The process to define the cointegrating rank r is as follows:
Johansen cointegration test first starts with i = 0 and compares LRiwith critical value.
In case Hi is rejected and i is smaller than k − 1, then it increases i by 1 and it repeats
the test. If Hi is not rejected, then the result isbr = i. If Hk−1 is rejected thenbr = k. In case all hypotheses are rejected, stationarity is implied and br = k.
Chapter 2
Data Modeling
As it has been mentioned in previous chapters, we follow the data-set of Hoevenaars et al. (2008) for purposes of comparison. In our analysis we include stocks, government bonds, corporate bonds, T-bills, listed real estate and commodities. We exclude hedge funds from Hoevenaars’ data-set because of data unavailability. More specifically, as a government bond we use the 10-year US treasury constant maturity yield, and for T-bills we use the 3-month US Treasury bill, both of which have been obtained from the FRED website1. We also use FRED to obtain the seasonally adjusted consumer price index for all urban consumers and all items in order to picture inflation, and the Moody’s Seasoned Baa Corporate Bond Yield in order to derive credit spread. Concern-ing stock prices, we use the Irrational Exuberance data of Shiller2 which is based on S&P Composite Index. As regards commodities and real estate, as already mentioned in Hoevenaars et al. (2008), we use the GSCI Index and NAREIT North America return index respectively. For credits, we use the BofA Merrill Lynch US Corporate Master Index instead of Salomon Brothers long-term high-grade corporate bond index, which is used in Hoevenaars because of lack of data accessibility. BofA Merrill Lynch US Corpo-rate Master Index tracks the performance of US dollar denominated investment grade rated corporate debt, publically issued in the US domestic market. These series have been obtained from FRED website.
Furthermore, we include three state variables as introduced by Heovennars which are: inflation rate as the log-return of the Consumer Price Index, the credit spread as the difference between the log Baa corporate yield and the log 10-years bond, and the nominal short-term interest rate (realized T-bill rate). We should note that according to Campbell and Viceira (2002), the risk free rate is known one period in advance at time t and is realized at time t+1. Regarding liabilities, we adopt Hoevenaars’ approach, which was first described by Campbell and Viceira (2002), where liabilities series is a proxy to a constant maturity index-linked bond and is calculated using the real yield of the 10-year constant maturity US bond as follows,
rL,t=
1
4rrt+1− DL(rrt+1− rrt) (2.1)
where the rrt is the 10-year US real interest rate, obtained from FRED website, and
DLindicates the liabilities duration. It is assumed that the fund is in a stationary state
and pays full indexation, which means that the distribution of the age cohorts and the build-up pension rights per cohort should remain constant through time. In addition, an essential assumption set up by Hoevenaars is that the contributions inflow is equal to the present value of the new liabilities and also to the current payments.
1
http://research.stlouisfed.org/fred2/
2http://aida.econ.yale.edu/ shiller/data.htm
Figure 2.1: Plausible common trend between commodities, real estate and stocks. Log-prices of commodities, real estate and stocks are presented through time.
Figure 2.2: Plausible common trend between government bond and corporate bond. Log-returns of 10-year US government bond and corporate bond are presented through time.
All time series are transformed to natural logarithmic scale because of the exponential trends that financial time series usually exhibit and are on a quarterly basis. In addition, the period of time chosen for our analysis is set between 1973:I and 2013:III. Two decades is enough to capture the trends of our time series. Besides that, we want to highlight the recent global financial situation which is driving the upcoming financial events. A look into the above two graphs where some common trends seem to appear between our time series provides us with the first evidence of the presence of cointegration relationships among our data.
Now, let us explain the notation we use:
• Pi,t is the price index for asset class i at time t.
• pi,t is the log price index for asset class i at time t, i.e. pi,t = log(Pi,t). For the
10 Eleni Orfanou — Asset allocation under cointegration approach
• ri,t is the logarithmic return for asset class i at time t, i.e. ri,t= pi,t− pi,t−1.
• si,t is the logarithmic value of the i state variable at time t.
As regards liabilities, PL,t= PL,t−1(1 + rL,t) and thus pL,t= log(1 + PL,t).
In addition, vector yt consists of all times series and ∆yt is their first differences:
yt= pb,t pc,t pcr,t pre,t ps,t pSinf,t pScs,t pSnom,t pL,t (2.2)
It is worth mentioning that we set the log series of 1973:I equal to 1 and therefore the log data is rescaled. Because of how we rescale the log series, the first differences of the elements of ytreflect their log-returns. In particular, sinf,trepresents the inflation. Thus,
∆yt can be expressed as follows:
∆yt= rb,t rc,t rcr,t rre,t rs,t sinf,t scs,t snom,t rL,t (2.3)
In order to form a statistically adequate model we have first to check whether our data is considered stationary. So, we perform the Augmented Dickey Fuller (ADF) unit root test. The results of ADF test are illustrated in Table 2.1. The critical value is equal to −4.02 for 1% confidence level and −3.44 for 5% confidence level and therefore we do not reject the null hypothesis according to which there is a unit root in the level series. In other words, we do not reject the presence of stochastic trends in our time series. Furthermore, we also perform the ADF test on the first differences of our time series in order to ensure that they are not integrated of order 2. The critical value is equal to −3.47 for 1% confidence level and −2.88 for 5% confidence level and we reject the null hypothesis that there is a unit root for six of the variables. In particular, the 10-year government bond, the CPI index and the 3-month T-bill display unit roots in their first differences. However, for simplicity reasons we prefer to overcome this issue and assume that all the variables present stationary first differences. Hence, we proceed to our coin-tegration analysis by testing for coincoin-tegration and then we estimate a VEC model for the level time series.
Lag order
Since we have confirmed that our data is non-stationary, we perform the Johansen trace test in order to test for the presence of cointegration in the time series. First, we should determine the lag order using the Schwarz information criterion (SIC), which is more consistent to our analysis. Other information criteria offered in literature are Akaike information criterion (AIC) and Akaike’s Final Prediction Error Criterion (FPE) and Hannan-Quinn information criterion (HQ). Nevertheless, we prefer SIC criterion since
Table 2.1: Augmented Dickey Fuller test statistics for both level and first differenced series
Level series
First differenced series
Series
Statistic
P-value
Statistic
P-value
Bonds
-0.10184
0.9945
-0.62547
0.8604
Commodities
-1.45825
0.8396
-5.54829
0.0000
Credits
-1.28090
0.8889
-12.5093
0.0000
Real estate
-2.60643
0.2781
-5.26415
0.0000
Stocks
-2.23428
0.4670
-11.0148
0.0000
Inflation
-2.96650
0.1451
-1.51958
0.5211
Credit spread
-0.09591
0.9946
-4.80809
0.0001
Nominal interest rate
-0.64372
0.9996
-1.52366
0.5191
Liabilities
-2.83986
0.1854
-6.03108
0.0000
Table 2.2: Johansen Trace test for two model specifications. The first one considers the absence of linear trend in the cointegrating equations and the second one considers the presence of linear trend in the cointegrating equations
Hypothesized No.
Trace Statistic
Prob.
Trace Statistic
Prob
of CE(s)
(No trend in CE)
(Trend in CE)
None *
293.6128
0.0000
341.2871
0.0001
At most 1 *
209.4157
0.0000
256.0768
0.0000
At most 2 *
159.9458
0.0001
184.7629
0.0001
At most 3 *
118.3195
0.0006
135.9667
0.0021
At most 4 *
77.40873
0.0109
94.36481
0.0187
At most 5
43.66428
0.1172
59.91806
0.1029
At most 6
22.02506
0.2971
33.75971
0.2996
At most 7
9.965061
0.2834
16.67904
0.4393
At most 8
2.817988
0.0932
6.168256
0.4392
AIC and FPE are more suitable for small samples in selecting the true model (L¨utkepohl, 2006), which is not the case here. The aforementioned criteria are defined as follows:
AIC(p) = log(deteΣ(p)) +
2pK2
T (2.4)
SIC(p) = log(deteΣ(p)) + log(T )
pK2
T (2.5)
HQ(p) = log(deteΣ(p)) + 2 log log(T )
pK2
T (2.6)
F P E(p) = deteΣ(p) +
(T + pK + 1)K
(T − pK − 1)K (2.7)
where eΣ(p) represents the estimation of the residual covariance matrix of the VAR(p)
model, T is the number of observations and K is the dimension of the VAR(p) model. Concerning the lag order, AIC and FPE criteria indicate a lag order equal to three, but
12 Eleni Orfanou — Asset allocation under cointegration approach
HQ and SIC criteria indicate a lag order equal to two. We use a lag order of two in or-der to examine for cointegration associations among our data, using Johansen trace test. Johansen trace test
We continue to Johansen trace test, or else LR test, in order to check whether there are cointegrating associations among our time series and if there are, how many. Trace test is operationalized for two model specifications. There are several specifications we can take into account which enable us to include constants and trends into the cointegrat-ing relationships, such as the cases of; neither constant nor trend, a constant but no trend, both a constant and a linear trend. In our case we examine the cases of constant but no trend and both of them in the cointegrating relationships. We have excluded other specifications since the chosen ones are more relevant for series with growth, e.g. log-asset prices. From now on we call them as the case 3 (equation 1.5) and the case 4 (equation 1.7) according to EViews. So, case 3 considers the absence of linear trend in the cointegrating equations and case 4 considers the presence of linear trend in the cointegrating equations. The test for cointegration is calculated based on the rank of Π matrix through its eigenvalues. The rank of Π is determined by the number of eigenval-ues which define its characteristic roots not equal to zero. In our case, Johansen trace test indicates that there are five cointegrating equations under a 5% confidence level, which means that there are five common trends driving the process.
Apart from Johansen trace test results, we observe graphically that some of the asset classes display a common trend (Figure 2.1 and Figure 2.2). Particularly, the 10-year US bond and the Baa Corporate bond, which is included in credit spread, share a common trend. The co-movements of commodities and stocks and potentially of real estate are obvious as well.
Estimation Results
In this section we report the results of our model’s estimation. Table 2.3 presents the estimates of the parameters and the t-statistics below them in parenthesis. The last column presents the R2, which expresses the predictive power of each variable in the
cointegration analysis. We highlight t-statistics which are larger than 1.5 in light grey color and t-statistics, which are larger than 2.0 in dark gray color. We run the model considering the Case 3 model specification rather than Case 4 because of its simplicity to interpret.
As regards Table 2.3, bond prices, inflation, credit spread, T-bill and liabilities signif-icantly drive bond returns (R2 = 99.94%). The coefficient of bond returns on its own
lag indicates that bond prices are persistent. The Johansen trace test exhibits the high persistency of bonds. In particular, it has shown that 10-year US government bonds are cointegrated of order 2. But for simplicity reasons we overlook this issue, as it was mentioned earlier on Johansen trace test results. In addition, T-bill is a mean-reverting mechanism to bond prices, since their correlation is positive while the significant coef-ficient of T-bill to 10-year bond is negative. Their positive correlation implies that a positive innovation of T-bills has a positive effect on contemporaneous bond returns, whilst T-bill predicts that the next period bond returns decline.
The 10-year US bond is a strong predictor for stock returns. Apart from the 10-year bond, real estate, T-bill, credit spread and liabilities have a significant positive effect on next period stock returns. The bond returns have mean-reverting properties for stocks since their contemporaneous correlation is positive, while bond returns predict that the
next period stock returns decline. The fit for the stock returns is quite good with a R2 equal to 56.7%.
Table 2.3: Parameter estimates for the VEC model with the following variables: 10-year government bonds, stocks, commodities, real estate, credits, credit spread, nominal interest rate, and liabilities. The t-statistics are given in parenthesis.
rb,t rc,t rcr,t rre,t rs,t sinf,t scs,t snom,t rL,t R2
rb,t+1 0.93 0.00 0.00 0.00 0.00 -0.01 -0.07 -0.01 -0.02 99.94 (46.69) (0.81) (1.90) (-0.64) (0.00) (-2.4) (-3.07) (-2.01) (-38.75) rc,t+1 -9.04 0.10 -0.44 0.03 0.07 2.49 -3.16 9.91 0.26 17.05 (-0.76) (1.30) (-0.85) (0.29) (0.60) (1.61) (-0.23) (2.65) (0.75) rcr,t+1 13.66 -0.02 -0.22 -0.04 -0.02 -2.01 3.05 -1.67 -0.29 27.80 (3.58) (-0.95) (-1.36) (-1.17) (-0.44) (-4.07) (0.70) (-1.4) (-2.64) rre,t+1 11.78 0.16 -0.18 -0.06 -0.15 -3.35 25.98 3.44 -0.37 28.94 (1.30) (2.64) (-0.45) (-0.74) (-1.64) (-2.85) (2.50) (1.21) (-1.40) rs,t+1 -16.7 -0.02 -0.12 0.54 -0.03 -0.39 12.99 6.32 0.42 56.70 (-2.71) (-0.45) (-0.43) (9.24) (-0.40) (-0.49) (1.84) (3.28) (2.34) sinf,t+1 2.96 -0.01 -0.07 0.01 0.00 0.07 0.60 0.62 -0.11 70.88 (5.31) (-2.65) (-3.06) (2.05) (0.42) (1.01) (0.94) (3.54) (-6.63) scs,t+1 -0.04 0.00 0.03 0.00 0.00 0.04 0.57 -0.05 0.00 87.81 (-0.49) (-4.60) (9.11) (-5.57) (-1.67) (3.47) (6.37) (-2.09) (-0.78) snom,t+1 -0.10 0.00 -0.02 0.00 0.00 0.02 -0.24 0.65 0.01 95.13 (-0.51) (1.94) (-2.27) (1.70) (0.58) (0.57) (-1.04) (10.19) (1.21) rL,t+1 8.68 -0.09 1.07 -0.15 -0.03 1.15 2.65 -1.81 -0.16 80.53 (2.62) (-4.11) (7.45) (-4.72) (-0.80) (2.68) (0.70) (-1.74) (-1.69)
Inflation is well explained (R2 = 70.88%) by bond returns, real estate and nominal inter-est rate with a positive effect and by commodities, credits and liabilities with a negative effect. However, government bond, real estate, T-bill and liabilities are mean-reversion mechanisms for inflation. Shocks on commodities have a poor fit (R2 = 17.05%) and therefore it is difficult to explain their prices movements. It seems that commodities are positively driven by T-bill and inflation. Although inflation is a mean-reverting factor for commodities, its effect is limited since it is significant for a 90% confidence inter-val. Shocks on real estate prices could be satisfactorily explained (R2 = 28.94%) by commodities and credit spread with a positive effect and by stocks and inflation with a negative effect.
Table 2.4: Error covariance matrix. Diagonal elements are standard deviations and off-diagonal are correlations.
rb,t rc,t rcr,t rs,t rre,t rs,t sinf,t snom,t rL,t
rb,t 0.02 rc,t -9.52 10.82 rcr,t 23.82 -7.56 3.46 rs,t 6.19 19.05 36.53 8.22 rre,t -4.51 7.07 -0.39 11.44 5.59 rs,t -80.33 -3.19 -29.83 -15.04 1.19 0.50 sinf,t -28.98 -12.58 -23.96 -38.16 -24.86 34.84 0.07 snom,t 20.6 6.38 12.71 22.03 8.41 -15.44 -35.71 0.19 rL,t -7.71 -13.54 -20.33 -45.38 -30.25 11.25 71.26 -39.31 3.00
Credits predictability is similar to real estate’s (R2 = 27.8%). The 10-year government bond, inflation and liabilities significantly drive credits series’ movements. Liabilities are very well explained (R2 = 80.53%) by 10-year government bonds, credits and inflation with a positive effect and by commodities, real estate, T-bills and its own lag with a negative effect.
14 Eleni Orfanou — Asset allocation under cointegration approach
Table 2.5: Adjustment parameters estimates (adjustment parameter matrix α) of the VEC model for each cointegration relationship, with the following variables: 10-year gov-ernment bond, commodities, credits, real estate, stocks, inflation, credit spread, nominal interest rate and liabilities. The t-statistics are given in parenthesis.
CE1
CE2
CE3
CE4
CE5
10-year US Bond
-0.00
-0.00
-0.00
6.37
0.00
(-1.48)
(-3.28)
(-1.83)
( 0.09)
( 1.31)
Commodities
-0.24
-0.18
0.45
-0.02
-0.18
(-0.27)
(-4.14)
( 0.98)
(-0.65)
(-3.93)
Credits
-0.05
-0.00
-0.01
-0.01
-0.01
(-0.16)
(-0.12)
(-0.08)
(-0.85)
(-1.24)
Real Estate
-0.19
0.09
0.80
-0.05
-0.07
(-0.28)
( 2.76)
( 2.27)
(-1.92)
(-2.10)
Stocks
0.98
-0.02
0.31
0.02
-0.06
( 2.10)
(-1.02)
( 1.29)
( 1.00)
(-2.79)
CPI
-0.04
0.00
0.04
-0.00
-0.00
(-0.90)
( 3.98)
( 2.21)
(-3.44)
(-2.94)
Credit Spread
-0.01
1.43
-0.00
0.00
0.00
(-1.46)
( 0.04)
(-2.13)
( 0.82)
( 4.67)
3-month US T-Bill
0.07
-0.00
-0.02
0.00
0.00
( 4.25)
(-0.45)
(-3.03)
( 3.71)
( 1.10)
Liabilities
-0.33
0.00
0.45
0.01
0.02
(-1.33)
( 0.02)
( 3.49)
( 1.05)
( 1.64)
Besides our findings on variables parameter estimates and the correlations between them, we should also examine the findings in the cointegrating equations. The inter-pretation of the cointegration equations matrix is of great importance since the αi,j
parameters determine which prices ”equilibrium adjust” and which do not, something which means that they provide an adjustment to previous disequilibria between prices in different asset classes, state variables and liabilities. It is worthwhile to refer to liabil-ities series, four out of seven significant variables of which are negative in the predicting equation. However, the two significant adjustment parameters (Table 2.5) are positive. Hence, the long-term dynamics partly offset the short-term dynamics, which makes the volatility reduce. This offsetting effect holds for commodities as well. Specifically, com-modities returns are significantly explained by inflation and T-bill with a positive effect while the two significant adjustment parameters are negative. However, the offsetting effect is less discernible for stocks since the adjustment parameters keep a balance.
Table 2.6: Cointegration relationships estimates (parameter matrix β) of the VEC model for each cointegration relationship, with the following variables: 10-year government bond, commodities, credits, real estate, stocks, inflation, credit spread, nominal interest rate and liabilities. The t-statistics are given in parenthesis.
CE1
CE2
CE3
CE4
CE5
10-year US Bond
1.00
0.00
0.00
0.00
0.00
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Commodities
0.00
1.00
0.00
0.00
0.00
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Credits
0.00
0.00
1.00
0.00
0.00
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Real Estate
0.00
0.00
0.00
1.00
0.00
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Stocks
0.00
0.00
0.00
0.00
1.00
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
CPI
7.72
-64.72
5.09
-190.88
103.78
( 1.67)
(-1.60)
( 1.51)
(-1.70)
( 1.71)
Credit Spread
-7.89
62.41
-6.06
174.11
-96.18
(-2.35)
( 2.13)
(-2.48)
( 2.13)
(-2.19)
3-month US T-Bill
-2.64
9.58
-1.68
40.19
-22.94
(-0.71)
( 0.29)
(-0.62)
( 0.44)
(-0.47)
Liabilities
-7.01
63.36
-5.53
167.86
-93.11
(-1.85)
( 1.91)
(-2.00)
( 1.82)
(-1.87)
Chapter 3
Term structure of assets and
liabilities
In this part of the thesis, we discuss the term structure of risk implied by VEC model, the risk diversification properties and the inflation and real interest rate hedging qualities of the different asset classes. We use the method introduced by Campbell and Viceira (2005b) in order to project the implied volatilities and correlations between the asset classes and liabilities. In addition, the same method determines the projected correla-tions between the asset classes and Consumer Price Index in order to obtain evidence for the inflation hedging qualities of each asset class.
3.1
Term structure of risk
A longer investment time horizon usually implies a greater portfolio proportion allo-cated to risky assets. However, Harlow (1991), Leibowitz and Krasker (1988), and Bodie (1995) have shown that long investment horizons imply higher risk for risky assets and therefore cause doubts about time diversification strategy. In this section we analyze the time diversification properties within each asset class. The structure of risk assists us to estimate how volatile an asset class could be in a holding period of 25 years, so as to build an as less volatile portfolio as possible depending on investor’s risk appetite. Our results focus on an asset-liability investor and show the difference of cointegration analysis against the traditional VAR specification.
Figure 3.1 illustrates the conditional standard deviation of cumulative returns of all asset classes for a 25-year investment horizon. As far as we can tell, the 3-month T-bill and the 10-year government bond are pictured as the less volatile assets in our port-folio. Government bonds are backed by the creditworthiness of U.S. government and therefore considered to have no credit risk. Regarding Treasury bills, there are always investors eager to invest in them, which therefore become the most liquid investment asset. The 3-month T-bill indicates low volatility comparing to the other asset classes, starting at the level of 0.37% and climbing from 1.02% at the end of the second year to 1.44% in the fifteenth year at which level it remains relatively stable for the rest of the holding period. Regarding the 10-year government bond, the volatility starts at 14% and it climbs gradually to 26%, to climax at 28% after 8 years of investment. Both the 10-year bond and the 3- month T-bill face the risk of reinvestment. In fact, all bonds have more risk when interest rates are rising.
Liabilities risk exhibits significant horizon effects. Specifically, their volatility starts ris-ing from 6.01% to 7.55% durris-ing the first year, then it displays a downward trend from 6.13% at the end of the second year to 3.37% at the end of the investment period. As it has been mentioned in the previous chapter, four significant variables are negative for liabilities series, whereas two significant adjustment parameters are positive. As a result, the long-term dynamics partially offset the short-term ones and therefore the volatility reduces in the long run.
Credits exhibit very limited horizon effects. Their volatility starts at 6.92% and it slightly 16
drops to 6.75% in the fifth year, from which point and on we observe a fluctuation at around 6.73% until the end of the investment horizon. In general, credits term structure displays a minor downward trend. Significant drivers in the credits returns equation have mean-averting properties.
Figure 3.1: Time diversification. Standard deviations across different investment hori-zons.
Commodity returns do not exhibit time diversification. Instead, commodities risk rises from 21.64% to 30.46% in the third year and then it gradually increases at 34.38% for the entire investment horizon. The risk term structure of commodities is not flat. In fact, it seems that commodities volatility rises through time which supports the latest findings for large fluctuations in international commodity prices.
Stock returns do not exhibit horizon effects, since their volatility slowly increases through time. They seem to be riskier in the long run. Jorion (2003) claims that previous studies on time diversification properties of stocks have been based on limited data and thus their statistical power is poor. His results support that there is no evidence of mean re-version on stock returns. However, according to our results there are minor fluctuations in stocks volatility through time.
Real estate series exhibit minor horizon effects. Their risk term structure has a slight downward trend. There is a balance between the significant parameters in predicting equation and the adjustment parameters. In particular, real estate returns volatility starts at 16.44% and it falls to 15.30% in the seventh year and then it decreases steadily to 15.24% at the end of the investment horizon. It is worth mentioning that although stocks risk term structure slightly increases through time, their returns are less volatile than real estate returns.
18 Eleni Orfanou — Asset allocation under cointegration approach
Comparing to Hoevenaars et al. (2008), our results indicate that T-bills are on average 2.3 times riskier than bonds, while Hoevenaars shows that the duration mismatch of T-bills makes them as risky as bonds. However, the volatilities which we obtain are lower than Hoevenaars. Regarding liability risk, our findings indicate that bonds are a mean-reverting mechanism, whereas inflation exhibits mean aversion. Thus, the risk term structure of liabilities has the same behavior as the one obtained by Hoevenaars. In contrast, stocks are presented less volatile at Hoevenaars et al. (2008), while our findings provide evidence of no time diversification properties at all.
3.2
Risk diversification properties
When many investors add an additional investment to their portfolio, they pay atten-tion only to the individual risk of the added asset and do not consider how the asset could probably mitigate the overall portfolio’s risk. In fact, by adding risky assets to the investment portfolio might make it safer. Usually investors prefer to split their portfolio between stocks and bonds with a proportion depending on their risk appetite. If they are more conservative, they opt for a portfolio of 80% bonds and 20% stocks, whereas if they are more aggressive they may prefer a proportion of 20% bonds and 80% stocks. The correlation between stocks and bonds is of great importance from a portfolio man-agement perspective. However, including more asset classes into our portfolio we take advantage of the diversification effects and benefit of greater return potential. Besides that, ”safe” investment assets such as bonds do not provide protection against infla-tion in the long run. If investors wish to protect their investments purchasing power, they should consider riskier assets such as commodities, real estate, credits etc. In fact, correlations between the different asset classes at different horizons may determine how well-diversified a portfolio can be. In our particular analysis, we explore the correlations of stocks and bonds with the rest of the asset classes.
Based on our findings, the correlation between stocks and 10-year government bond is an average of 2.83% during the first year of investment, with a peak at 17.60% and then it starts declining to -29.94% in the second year. A sharp drop to -42.42% is projected in the tenth year with minor fluctuations around -43% to -44% until the end of the investment horizon. Hence, a portfolio of stocks and bonds offers diversification proper-ties in the long run. Stocks and bonds move to different directions. In particular, during periods of intense economic activity and growth, companies perform well and increase their prices. Governments tend to slow that growth in order to avoid inflationary effects by raising interest rates and therefore lower bond prices. Thus, while stock market out-performs, bond prices decline. During economic uncertainty periods, stock prices tend to decrease, while investors turn their attention to safer investments such as government bonds, which in turn make the bond prices increase.
Regarding 3-month T-bill, it offers diversification properties but only as regards to stocks, since it moves to a different direction from them, while it exhibits high correla-tions with government bonds at the entire investment horizon. The correlation between T-bill and stocks displays an ascending trend during the first year, after which it starts descending to -28% on average in the intermediate-term horizon, to finally reach -32% at the end of the investing period. T-bills usually behave like other government bonds. When the economy grows the investors are more risk averse and thus less willing to buy treasury bills, as they prefer riskier asset classes such as stocks. Therefore stock prices tend to rise. On the other hand, when economy straggles, investors resort to safer and probably more liquid investments such as T-bills. Concerning the association between T-bills and bonds, we observe that the diversification qualities are poor at the beginning
of the investment period when correlation equals to 20%. It sharply rises to 76% in the fifth year and then it displays a negligible fluctuation at around 80% until the end of the investment horizon.
Surprisingly, commodities correlation with bonds features a negative sign in the short term period while it gradually climbs to positive values in the intermediate and long term horizons approaching 79%. Commodities move towards the same direction as the 10-year government bond, especially in the long run. However, the opposite effect holds for stocks. Particularly, commodities exhibit poor association with stocks especially in the short run; 10% on average with a peak at 14,22%. During the intermediate-term the correlation drops to -36%, declining further to -39% at the end of the investment period. Commodities prices rise at periods of uncertainty, while stock prices fall. A small portfolio proportion could have great risk mitigating benefits.
Real estate is observed to have a significant correlation with stocks, which supports the fact that real estate movements often appear similar to stocks. The correlation displays a slashing increase reaching 40% during the first year and then it declines with the investment horizon approaching 32%. The stock market may precede or follow real estate price movements. Especially during bubbles real estate and stock market affect each other. As regards the correlation of real estate with bonds, it starts at 6,2% and climbs rapidly to 32% during the first year. In fact, the shape of real estate returns correlation with government bonds exhibits major fluctuations in the short-term. However, it fluctuates around 29% at the intermediate and long term horizons.
Figure 3.2: Risk diversification. Correlations between different asset classes across dif-ferent investment horizons. Solid lines indicate correlation of stocks with the rest of the asset classes. Dashed lines represent correlation of 10-year government bond with the rest of the asset classes.
Concerning the association between credits and bonds, it seems that they are signifi-cantly correlated in the short run, but it instantly plunges to -20% in the sixth year
20 Eleni Orfanou — Asset allocation under cointegration approach
while it firmly fluctuates around -25% for horizons beyond fifteen years, providing great diversification properties. The correlation between stocks and credits displays similar structure to the correlation between stocks and bonds. Although the correlation starts at 0% at the beginning of the investment period, it climbs to 50% in the sixth year and it leads to 56% in the long run. As Hoevenaars explains, probability of default moves with equity prices. If stock market is on downturn, the probability of default increases.
3.3
Inflation hedging potential
In this section we examine the inflation hedging qualities of our different asset classes for a holding period of 25 years. In contrast to Hoevenaars et al. (2008), we have in-cluded CPI index in our model and therefore we are able to obtain inflation properties directly. Hoevenaars constructs inflation properties from the difference between nomi-nal interest rate and real interest rate since CPI index is not included in the VAR model. Investors can be certainly affected by inflation. Thus they have to construct their portfo-lios accordingly. Figure 3.3 displays correlation between Consumer Price Index returns and the return on various asset classes and the liabilities. In general, correlations to inflation through time are low; however, trends and patterns are clear enough to have a better perspective of correlations behavior during the entire investing period.
According to our results, stocks offer poor hedging properties. In fact the correlation between stock returns and inflation is very low at the entire investment horizon. In times of rapidly rising prices, assets classes other than stock market have been more attracting to investors such as real asset investments, including metals and real estate. Moreover, in periods when inflation exceeded 6%, stocks did not perform as well as other asset classes. Bonds offer good inflation hedging properties within the entire investment period except for the first year. Inflation makes interest rates rise, resulting in higher current yields on bonds. This does not help investors who have invested in lower yield bonds, but the high interest rates will lead investors to fixed-income investments such as government bonds. Long term constant maturity government bonds conceal accumulated inflation, which turns them a good inflation hedge for the whole investment period.
Real estate is a traditional cure for inflation risk. In fact, increasing inflation has im-pact on real estate prices, which increase during inflationary periods. Increasing inflation devalues the currency, forcing assets prices to move upwards. Commodities have great hedging qualities both in the short and in the long run. Commodity prices rise in periods of uncertainty as well as inflation does. This is innate since the Consumer Price Index has been obtained by the movement of the expenditure on a set of consumer goods and therefore it is strongly associated to commodities prices movements.
Regarding credits, from an inflation hedging perspective, it is obvious that a long posi-tion is not recommended neither in the short nor in the long run. This is intuitive since inflation is negatively linked to real economic growth, as it has already been mentioned by Hoevenaars. Higher returns on high-yield bond funds are positively related to real economic growth, and therefore they appear as an inappropriate inflation hedge invest-ment. T-bill is an attractive investment from an inflation hedge point of view. Treasury bills appear to have significant downside risk when inflation moves upwards. Reinvesting makes T-bill incorporate inflation movements. T-bills are not completely risk free when we consider inflation, but they seem to catch up with inflation changes quickly.
Figure 3.3: Inflation hedge properties. Correlations between inflation and asset returns for different investment horizons.
3.4
Real interest rate hedging properties
As it is mentioned in Hoevenaars et al. (2008), insurance and pension funds liabilities correspond to the present value of future outflows discounted at a real interest rate. In this section we analyze the real interest rate hedge potential of the various asset classes for different investment horizons. Figure 3.4 assists us to examine the hedging qualities of the asset classes, as regards to real interest rate. In order to illustrate our results, we use the same method as the one implemented in the previous section, where the covariances are projected in a 25-year investment horizon.
Credits and stocks present the best liabilities hedging properties. Specifically, stocks do not offer any liabilities hedging potential in the very short run, whilst in the second year, the correlation between stocks and real interest rate improves significantly. Stocks perform as a good hedge for liabilities in the intermediate and long run. Credits act as a good liabilities hedge within the entire investment period. Particularly, credits correlations with liabilities are higher in the short run, while they move downwards in the long run.
22 Eleni Orfanou — Asset allocation under cointegration approach
Figure 3.4: Real interest rate hedge properties. Correlations between assets and liabili-ties.
As far as T-bills are concerned, they do not offer any liabilities hedging properties. Their correlations with liabilities are negative during the whole investment horizon. An explanation is the duration mismatch. Moreover, inflation risk also affects the correla-tion between T-bills and liabilities. Government bonds, in contrast to Hoevenaars et al. (2008), provide poor liabilities hedging potential. Even if their correlation with lia-bilities is positive during the investment horizon, it climaxes at levels below 10%. This is intuitive since liabilities can be expressed as a bond that has to be paid off at the maturity. Hence, an investor with liabilities in the long run is like an investor who goes short on a long-term bond.
Regarding real estate liabilities hedging qualities, they seem to be quite poor in the short run, but there is a recover in the long run at which the liability shocks lead to positive shocks to future real estate returns. Concerning commodities, they present low correlations with liabilities. The pattern of the correlation between commodities and liabilities appears approximately flat.
Mean Variance Portfolio
Efficient Frontier
Mean-variance theory is an important model for investments based on decision theory. It was developed in the 50’s and 60’s by Markowitz, Tobin, Sharpe, and Lintner, among others. While no longer the most modern model, mean-variance theory continues to be the main workhorse on which analytical portfolio management is based.
Mean-variance theory is characterized by its simplicity. By assuming that preferences depend only on the mean and variance of payoffs and not on other features, we obtain a number of robust results. In mean variance analysis, we consider only the first two moments of the portfolio model. The investor makes decisions taking into account the trade-off between expected return and risk, namely volatility. Mean variance analysis can be considered as a way to allocate assets in a system of risk and return.
In this chapter we construct a mean variance portfolio for an asset-liability investor by obtaining optimal weights for each asset class and then constructing the efficient frontier. But first things first, we should know what the efficient frontier is. In order to understand the meaning of efficient frontier, we should recall that investors are keen on achieving the best possible trade-off between risk and return. While considering a fixed set of assets, portfolio composition determines the risk/return profile. Thus, we obtain asset mixes that maximize expected portfolio return and minimize its volatility at the same time. The aforementioned asset mixes determine the optimal portfolios that can be illustrated by a line in risk/return plane, called efficient frontier. By setting expected re-turns equal to their long-term sample means, while considering the variance-covariance of returns changing across investment horizons, we obtain a set of efficient frontiers. We use the method introduced by Hoevenaars et al. (2008), where the asset-liability man-agement is approached from a funding ratio return perspective. Campbell and Viceira approach concerns the asset-only investor, who measures their returns against inflation. Yet, as regards our approach based on the asset-liability investor, we measure the return in excess of liabilities.
Funding ratio F is defined as the ratio of assets to liabilities. Thus, the log-return of funding ratio rF is the assets return minus the liabilities return, presented as follows:
rF = rA− rL (4.1)
where rA is the asset portfolio return and rL is the liabilities return. Following the
findings of the previous chapters, we build our portfolio. According to Hoevenaars et al.(2008), through term structure of risk, which has already been estimated in previous chapters, we are able to have a better perspective of the relative importance of different assets for an asset-liability investor. The investor chooses a portfolio at of the risky
assets and invests the remainder fraction of wealth 1 − ι0atin T-bills, where ι is a (6 × 1)
vector of 1’s. Each risky asset i has an ai,t weight, such that at is the (6 × 1) vector that
contains the portfolio weights ai,t’s.—
Now, in (4.1), we substitute rAwith the log-linear approximation to the portfolio return
24 Eleni Orfanou — Asset allocation under cointegration approach
that Campbell and Viceira (2002, 2005a,b) present, rA,t+1= rtb,t+1+ a 0 t(xA,t+1+ 1 2σ 2 A) − 1 2α 0 tΣAAat (4.2)
where rtb is the risk free log-return, σA2 is the vector with the diagonal elements of ΣAA
which represents the variance-covariance matrix of the different asset classes.
According to Campbell and Viceira (2005a, b), we take into account that the gross return on the wealth portfolio is given by
RA= Rtb+ a(R − Rtb) (4.3)
and we consider the mean log-returns as adjusted by one-half their variance so that they reflect mean gross returns,
E[RA] ≈ E[rA] + 1 2σ 2 A (4.4) V ar[rA] ≈ V ar[RA] = α2σ2 (4.5) and that RA≈ rA.
Thus, by substituting (4.3) in (4.4) we obtain E[rA] + 1 2σ 2 A= rtb+ a(E[r] + 1 2σ 2 A− rtb) (4.6)
and by substituting (4.5) to (4.6) while taking into account that σA2 = a2σ2 we obtain, E[rA] = rtb+ (µ − rtb) −
1 2α
2σ2 (4.7)
where µ = E[r] + 0.5σ2.
Campbell and Viceira (2002, 2005) solution to the mean-variance problem concerns the ”asset-only” investor, who invests in equity, government bonds and cash. We have ex-tended the solution in order to meet ”asset-liability” investor’s preferences.
From (4.1) we obtain the expected return on funding ratio,
E[rF] = E[rA] − E[rL] (4.8)
Now, by substituting (4.7) into (4.8) we result in
E[rF] = rtb+ a(µ − rtb) − 0.5α2σ2− E[rL] (4.9)
Funding ratio return variance can be found as follows:
V ar[rF] = V ar[rA− rL] = V ar[rA] + V ar[rL] − 2Cov(rA, rL) = σA2 + σL2− 2Cov(rA, rL)
(4.10) And by substituting (4.3) and (4.5) into (4.10), we result in the variance of the funding ratio return,
σF2 = α2σ2+ σ2L− 2Cov(rA, rL) (4.11)
Hoevenaars et al. (2008) defines the volatility of the funding ratio return as the mis-match risk. Assuming CRRA preferences on funding ratio and considering investor utility as Van Binsbergen and Brandt (2008) do, we solve the following investor utility maximization problem with risk aversion γ:
max
a E[(1 − γ)
which, by assuming normality of the excess returns according to Hoevenaars et al.(2008), is reduced to max a E[rF] + 0.5(1 − γ)σ 2 F (4.13)
suggesting that we have a mean-variance optimal investment strategy as Hoevenaars et al.(2008) has already proved. We can rewrite (4.13), by inserting (4.9) and (4.11), in the following form
max
a a(µ − rtb) − 0.5α
2σ2+ 0.5(1 − γ)(α2σ2− 2Cov(r
A, rL)) (4.14)
The solution to our maximization problem obtained by differentiating (4.14) with re-spect to our portfolio weights, is
a∗= 1 γ (µ − rtb) σ2 + (1 − 1 γ) (Cov(rA, rL)) σ2 (4.15)
As far as we can tell, the optimal a∗’s contain two parts where the first part represents the mean-variance optimal portfolio and the second part forms a correction for covari-ance between assets returns and liabilities returns, representing the liability hedging demand. Since we have a multiple assets problem we replace σ2 by Σ−1, where
Σ =
ΣAA σAL
σAL σL
and Cov(rA, rL) by σAL. We should note that represents the vector of the expected
returns µ = µA µL (4.16)
4.1
Optimal Portfolio
The results of our mean-variance problem solution are illustrated in this section. We discuss the portfolio mix for investment horizons of 1, 5, 10 and 20 years. The optimal weights a’s are presented for risk aversion parameters γ = 5 and 20. Table 4.1 shows the portfolio mix for different investment horizons and each risk aversion parameter. The portfolio is well diversified during the whole investment period. Except for bonds, stocks and T-bills, the other asset classes have significant roles in the portfolio as well. Table 4.1: Optimal Portfolio Weights. Weights of different investment horizons and different risk appetite. Weights may not add up to one due to rounding.
Horizon (years)
γ = 5
γ = 20
1
5
10
20
1
5
10
20
Bonds
0.89
0.49
0.47
0.46
1.04
0.52
0.48
0.46
Commodities
-0.01
0.07
0.07
0.07
-0.12
0.00
0.03
0.04
Credits
0.15
0.15
0.15
0.16
0.26
0.24
0.21
0.21
Real estate
-0.01
0.12
0.15
0.13
-0.06
0.10
0.14
0.14
Stocks
-0.06
0.11
0.11
0.11
-0.09
0.14
0.14
0.14
T-bills
0.04
0.06
0.06
0.06
-0.03
0.00
0.00
0.00
As long as risk aversion increases, the investment policy becomes more conservative since the investor seeks to invest their wealth in less volatile assets classes. A risk-loving investor usually prefers to invest in risky asset classes in order to benefit from the high expected return, even if the respective portfolio is too volatile. On the other hand, a
26 Eleni Orfanou — Asset allocation under cointegration approach
risk-averse investor aims to keep more cash than invest it all in risky assets.
As far as our portfolio is concerned, the optimal weights for the government bonds through time do not change. The government bonds are the major investment asset for both risk aversion parameters. Credits, stocks and real estate have significant propor-tion for both risk aversion parameters. We observe that alternative asset classes gain more proportion to the portfolio as time passes. Moreover, during the first investment year, the investor goes short to commodities, real estate and stocks for γ equal to 5, whilst T-bills are gone short for γ equal to 20 among commodities, real estate and stocks.
As risk aversion parameter increases, T-bills have no value for a long-term portfolio due to re-investment risk and due to the fact that they have the worst liabilities hedge potential. This is not the case for a less risk-averse investor who keeps a fixed proportion of T-bills in their portfolio through the entire investment period. Bonds have the major weight in portfolio during the whole investment horizon due the low volatilities at the entire investment period and due to the inflation and real interest rate hedging qualities.
Credits have a considerable weight in the portfolio, especially as the risk aversion param-eter increases. This is due to the fact that credits offer great liabilities hedging qualities and the fact that they keep small volatilities during the investment period. Regarding commodities, they have important inflation hedging qualities and good diversification properties against stocks. This is the main reason that they appear in the portfolio with a minor proportion, which remains fixed in the intermediate-term and the long-term horizon.
Stocks are presented as less attractive compared to other assets, mainly due to their poor inflation hedging qualities. However, they remain in the portfolio because of the real interest rate hedging potential. Besides, they are a good diversifier to the other asset classes. Their proportion remains stable in the medium-term and in the long-term investment horizon. Concerning real estate, it has the same behavior as stocks in the portfolio, however it keeps larger proportion. The main reason of its significant allocation in the portfolio is the fact that real estate offers both inflation and liabilities hedging properties.
4.2
Efficient Frontier
In this section we present the efficient frontier constructed as it is shown in (4.9) and (4.11). Figure 4.1 represents the set of optimal portfolios and their development through time, namely for t = 0, 1, 5, 10, 20 years. The graph below shows the largest possible expected funding ratio for each level of risk presented as the standard deviation of the portfolios return.
