Risk contribution and its application in asset and risk management for life insurance

IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2016,

Risk contribution and its

application in asset and risk

management for life insurance

JESPER SUNDIN

Risk contribution and its application in

asset and risk management for life

insurance

J E S P E R S U N D I N

Master’s Thesis in Mathematical Statistics (30 ECTS credits) Master Programme in Industrial Engineering and Management (120 credits)

Royal Institute of Technology year 2016 Supervisor at Skandia Liv: Andreas Lindell Supervisor at KTH: Boualem Djehiche Examiner: Boualem Djehiche

TRITA-MAT-E 2016:37 ISRN-KTH/MAT/E--16/37--SE

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

In risk management one important aspect is the allocation of total portfolio risk into its components. This can be done by measuring each components’ risk contribution relative to the total risk, taking into account the covariance between components. The measurement procedure is straightforward under assumptions of elliptical distributions but not under the commonly used multivariate log-normal distributions. Two portfolio strategies are consid- ered, the ”buy and hold” and the ”constant mix” strategy. The profits and losses of the components of a generic portfolio strategy are defined in order to enable a proper definition of risk contribution for the constant mix strategy. Then kernel estimation of risk contri- bution is performed for both portfolio strategies using Monte Carlo simulation. Further, applications for asset and risk management with risk contributions are discussed in the context of life insurance.

Keywords: risk contribution, capital allocation, Value-at-Risk, elliptical distri- bution, multivariate log-normal distribution, kernel estimation

Sammanfattning

En viktig aspekt inom riskhantering ¨ar tilldelning av total portf¨oljrisk till tillg˚angsportf¨oljens best˚andsdelar. Detta kan ˚astadkommas genom att m¨ata riskbidrag, som ¨aven kan ta h¨ansyn till beroenden mellan risktillg˚angar. Ber¨akning av riskbidrag ¨ar enkel vid antagande om elliptiska f¨ordelningar s˚asom multivariat normalf¨ordelning, men inte vid antagande om mul- tivariat log-normalf¨ordelning d¨ar analytiska formler saknas. Skillnaden mellan riskbidragen inom tv˚aportf¨oljstrategier unders¨oks. Dessa strategier ¨ar ”buy and hold” och ”constant mix” (konstant ombalansering). Tilldelning av resultaten hos de olika best˚andsdelarna med en generisk portf¨oljstrategi h¨arleds f¨or att kunna definiera riskbidrag f¨or ”constant mix” portf¨oljstrategin. ”Kernel estimering” anv¨ands f¨or att estimera riskbidrag genom simulering. Vidare diskuteras applikationer for tillg˚angs- och riskhantering inom ramen f¨or livf¨ors¨akringsbolag.

Contents

List of Figures iii

List of Definitions iv

1 Introduction 1

2 Theoretical Framework 3

2.1 Generic Portfolio Dynamics . . . 3

2.1.1 Buy and Hold strategy . . . 5

2.1.2 Constant Mix strategy . . . 5

2.2 Geometric Brownian Motion . . . 6

2.2.1 Dynamics of the BH and the CM strategy w.r.t. underlying GBM . . 6

2.2.2 Comparison of BH and CM under GBM price process . . . 7

2.3 Risk And Performance Measures . . . 8

2.3.1 Value-at-Risk . . . 9

2.3.2 Risk Contribution . . . 9

2.4 Optimal portfolio weights . . . 12

2.5 Optimal weight in risky assets, Kelly Crieterion . . . 14

3 Simulation study 16 4 Results 18 4.1 Risk contribution w.r.t BH and CM strategies . . . 18

4.2 Risk management considerations . . . 27

5 Summary and conclusions 29

List of Figures

2.1 Visualization of the log-utility function w.r.t the amount invested in a risk free asset. 1 (100%) corresponds to being fully invested in the risk free asset and 0 (0%) corresponds to being 100% invested in risky assets. Going to the negative domain is interpreted as investing in risky assets with leverage. . . . 15 4.1 Convergence of risk contribution with respect to the weight u₁. Risk contri-

bution of 0 (1) is interpreted as 0% (100%) of the VaR at the 0.05 quantile level. The reference distribution is N (0, I2). Simulation uses 10⁴samples for each estimation and the estimation are independently repeated 100 times to build the approximate confidence interval where the lower and upper bounds correspond to 5% and 95% empirical quantiles respectively. . . 19 4.2 Estimates of the risk contribution of the CM portfolio under the base case

according to definition 9. MMNA refers to the ”moment matching normal as- sumption” benchmark. Risk contribution of 0 corresponds to 0% of portfolio VaR and 1 corresponds to 100% of portfolio VaR. . . 20 4.3 Estimates of the risk contribution of the BH portfolio under the base case

according to definition 9. . . 21 4.4 Estimate of risk contribution given a fixed sample of S with respect to u₁and

changing rebalancing frequency. The figure is zoomed in to better visualize the different frequencies. . . 22 4.5 Estiamte of risk contributions for CM and BH with ρ = 0.7, everything else

the same as the base case. . . 23 4.6 Estimate of risk contribution for CM and BH with µ₁= 0.1. . . 24 4.7 Estimate of risk contribution for CM and BH with µ1= 0.1 and σ1= 0.4. . . 25 4.8 Estimate of risk contribution for CM and BH with T = 5. Rebalancing 12

times per year. . . 26 4.9 Efficient frontier for a constructed example investment universe of three assets

and a risk free investment. The dotted line corresponds to the one period investment strategy . . . 28

List of Definitions

BH - Buy and hold portfolio strategy.

CM - Constant mix portfolio strategy.

GBM - Geometric Brownian motion.

SCR - Solvency capital requirement, is the amount of funds that insurance and reinsurance undertakings are required to hold in the European Union.

SDE - Stochastic differential equation.

VaR - Value-at-Risk

1 Introduction

Risk management in asset portfolios is key to enable long term returns at an acceptable level of risk. For life insurance companies this means delivering high return while being able to meet its obligations to insurance takers with a high enough level of certainty. This level of certainty is governed by the Solvency II Directive and its associated measure of Solvency Capital Requirement as a minimum level of required capital. In this framework the SCR is measured by the Value-at-Risk with a 99.5% confidence level at a one year horizon. This measure is applied company wide on its whole operation but we will restrict our focus on the asset portfolio. The asset portfolio can be managed by different strategies giving rise to different properties of the risk and return of the portfolio. We will consider two portfolio strategies: buy and hold and constant mix. These strategies refer to how the allocation of initial wealth is divided amongst a fixed number of available stocks and managed throughout time. The buy and hold strategy decides an initial allocation of number of stocks in each available asset and keeps that number fixed as time evolves. The constant mix refer to the decision of an initial wealth allocation in proportion to the total wealth to be kept constant as time evolves. This is done by rebalancing the portfolio.

While the Value-at-Risk is measured portfolio wide it is important to be able to measure each components contribution to this value to enable better insights for risk management. Risk contribution is defined as the contribution of each component in the portfolio to the total VaR of the portfolio. These contributions sum to the total VaR and can be represented as the relative contribution in percentage to the total VaR. Such an allocation of risk is often referred to as allocation by the gradient or Euler allocation (McNeil et al, 2005), but is here referred to as risk contributions. The total VaR and the risk contributions of each component is dependent on the portfolio strategy as well as the underlying model of the price processes of the different stocks. Risk contributions can be expressed as conditional expectations when using the VaR measure. In the case of the buy and hold strategy with the underlying price process assumed to be an elliptical distribution for a fixed time length the calculation is straightforward. But not under the assumption of multivariate log-normal model and the constant mix strategy for which no analytical formula is available.

Estimation of risk contributions can be made with Monte Carlo simulation using a kernel estimator. But to be able to estimate the risk contribution for both portfolio strategies the profit and loss of the whole portfolio and each component must be defined clearly. This is done in the section 2.3.2 but a brief introduction is given here. For the whole portfolio the profit and loss is simply the value of the portfolio at a future time subtracted by the initial value. The profit and loss of each component in the buy and hold strategy is naturally

the profit and loss on each stock. We require that the profit and losses attributed to each component will sum to the total profit and loss which is clearly the case for the buy and hold strategy. For the constant mix strategy the profit and loss on each stock is the sum of the profit and losses measured between each rebalancing occasion. This is done to account for the fact that the number of shares held in each stock will vary over time. The interpretation of risk contribution as a conditional expectation of the profit and loss of each component given the portfolio wide profit and can be naively interpreted using the empirical VaR estimator. The empirical VaR estimate corresponds to a specific simulation where the profit and losses of the components will exactly sum to the portfolio VaR by definition. This can also be used as an estimate of the risk contributions but will converge extremely slowly (Epperlein and Smillie, 2006). The kernel estimator will estimate the risk contributions as a weighted average of the component profit and losses in the simulations near the portfolio empirical VaR with decreasing weights the further the distance from the empirical VaR. The kernel estimator will be used for both portfolio strategies for comparison of their properties.

The kernel estimates will also be compared to the analytical formula of risk contribution assuming a normal distribution.

The thesis is structured as follows. Section 2 will define the framework for evaluating risk contribution for different portfolio strategies. The key aspects of this framework is presented in Section 2.3.2. Section 3 presents the structure of the simulation study used to estimate risk contribution and in Section 4 the results of this study is presented.

2 Theoretical Framework

In this section we first define a general framework for evaluating portfolio strategies. No assumptions regarding the distribution of the underlying price processes are made in this generic setting. The framework is both presented with discrete and continuous time step.

Secondly the framework is applied when price processes are assumed to follow a geometric Brownian motion. Section 2.1 to 2.4 are useful in the context of risk contribution and its application and difficulties when elliptical distributions are not feasible options. Section 2.5 is mainly relevant for the understanding of optimal portfolio selection under different risk preferences.

2.1 Generic Portfolio Dynamics

We will base our framework notation on the works of Bj¨ork (2009, ch. 6). In the discrete case we model a financial market where the time is divided into periods periods of length

∆t with trades occurring at n∆t, n = 0, 1, . . . . With t = n∆t we refer to ”period t” as the interval [t, t + ∆t).

With d available stocks, hi(t) represents the number of shares held fixed in stock i during period t. The price of the stocks are represented by S_i(t) and the total value of the portfolio by V (t). At the start of period t the value of the portfolio will be determined by the development of the stock prices and the number of shares held from the previous period t − ∆t. We use the shorthand notation h(t − ∆t) = {hi(t − ∆t), i = 1, ..., d} and similarily for S(t). At the start of the period we observe the new prices S(t) and the value of the portfolio will be V (t) = h(t − ∆t)S(t) =P hi(t − ∆t)S(t) After observing the new prices we immediately choose a new allocation between the stocks while keeping the same total value of the portfolio as before. The reallocation or rebalancing is made consistent with a self financing portfolio where no exogenous infusion or withdrawal of money. The purchase of the new portfolio must be fully financed by selling the assets already in the portfolio. This discrete representation will prove useful later when profit and loss is defined for different portfolios in section 2.3.2.

For continuous time we have the following definition (Bj¨ork, 2009, ch. 6).

Definition 1 Let a d-dimensional price process {S(t)}t≥0 be given

1. d number of different stocks or assets

2. {h(t)}t≥0 represents the choice of a portfolio strategy. It is only allowed to depend on events up to time t

3. {V^h(t)}t≥0 is the value process of portfolio strategy h given by

V^h(t) =

d

X

i=1

hi(t)Si(t) (2.1)

4. h(t) = (h1(t) . . . hd(t))^T and S(t) = (S1(t) . . . Sd(t))^T are column vectors of the indi- vidual processes

5. u_i(t) are the relative portfolio weights defined by

u_i(t) = h_i(t)S_i(t)

V^h(t) (2.2)

where

d

X

i=1

ui(t) = 1

We are restricted to h(t) being an adapted process which states that we are dealing with portfolio strategies that are not allowed to ”look into the future”. The value of the portfolio is simply the number of stocks times their respective value. Definition 1 does not restrict to only stock prices. For example the price of an option could be incorporated in the portfolio but that will not be included in the scope of this thesis. Further we will find it useful to represent the dynamics of portfolio strategies in terms of differentials. This will allow for a representation of the portfolio value in terms of stochastic differential equations.

Definition 2 The dynamic of the value process for a self-financing portfolio h is described by

dV^h(t) =

d

X

i=1

hi(t)dSi(t) = h^T(t)dS(t) (2.3)

where dS(t) = (dS₁(t) . . . dS_d(t))^T is a column vector of the stochastic differentials for S(t)

We will normalize the initial value of the portfolio to 1 and the initial values of all the individual stocks to 1 without loss of generality. This leads to the initial values of the relative and absolute portfolio weights being the same. For this to work we need to make the assumption of being able to invest in fractions of stocks. The difference between the buy and hold and the constant mix strategies is that the number of stocks h(t) is constant

(implicitly forcing u(t) to vary) for the buy and hold and that u(t) is constant for the constant mix strategy (requiring h(t) to vary).

2.1.1 Buy and Hold strategy

The dynamics in the buy and hold strategy can be represented in terms of equation 2.3 with the requirement of h(t) = h(0). We fix the initial number of shares and keep them constant when time evolves.

dV^BH(t) = h^T(0)dS(t) V^BH(0) = h^T(0)S(0)

(2.4)

We keep the amount of shares held in each asset constant and equal to the initial amount throughout the whole investment period. The relative portfolio weights will vary over time according to equation 2.2.

2.1.2 Constant Mix strategy

In the constant mix strategy we keep the relative portfolio weights constant throughout the investment horizon. The instantaneous growth rate of the value process is equal to the weighted sum of the instantaneous growth rates of the individual assets. The value process follows the dynamics of

dV^CM(t) = V^CM(t)

d

X

i=1

ui(0)dSi(t) Si(t) V^CM(0) = h^T(0)S(0)

(2.5)

The constant mix strategy is expressed in continuous time which means that rebalancing is made in infinitesimal time steps. This is not practically feasible. A more realistic scenario involves rebalancing the investment holdings within the portfolio at predetermined time intervals and adjusting to the original allocation at a desired frequency. What we will use in the simulation study will range from daily to monthly frequencies. In this framework we assume that there are no transaction costs.

2.2 Geometric Brownian Motion

The SDE of the one-dimensional geometric Brownian motion (GBM) for stock i is on the form

dSi(t) = Si(t)µidt + Si(t)σidZi(t), i = 1, . . . , d (2.6) where Z_i(t) is a Brownian motion, possibly correlated with another BM. This commonly used model has the solution

Si(t) = Si(0) exp{(µi−σ²_i

2 )t + σiZi(t)} (2.7)

We want to express the possibly correlated Brownian motion Z in terms of a d-dimensional standard Brownian motion W , let

Z = CW (2.8)

where C is a matrix such that CC^T = ρ = (ρ_ij) is the correlation matrix of Z. We obtain C from the Cholesky decomposition of ρ. For this to be well defined we require that ρ is a positive-definite matrix. We have the following properties

• E[Zt] = E[CWt] = CE[Wt] = 0

• E[ZtZ_t^T] = E[CWt(CWt)^T] = CE[WtW_t^T]C^T = CIC^Tt = ρt

• E[σZt(σZt)^T] = σρσ^Tt = σρσt

where σ is a diagonal matrix with the volatilities σi on its diagonal.

The Z_i’s are from a multivariate normal distribution N_d(0, ρt) given the time t. Each Z_i has variance t (since diagonal elements of ρ are 1) but are potentially correlated with other Z_i’s (ρ_ij6= 0).

2.2.1 Dynamics of the BH and the CM strategy w.r.t. underlying GBM

The dynamics of the Buy and Hold were represented in differential form in Equation 2.4, but when the price process follows a GBM we actually have the explicit solution and can evaluate its value process directly according to equation 2.1. We have

V^BH(t) =

d

X

i=1

hi(0)Si(t) =

d

X

i=1

hi(0)Si(0) exp{(µi−σ²_i

2 )t + σiZi(t)} (2.9)

which is a sum of correlated log-normally distributed asset price processes.

For the value process of the CM strategy, given by 2.5, we do not actually have to solve the SDE of the geometric Brownian motion S to have a representation of the value process in terms of the BM Z. We do have to solve the SDE for the value process in terms of Z, which turns out to be a one-dimensional GBM. In the equations below we omit the CM notation and time notation on the value process to make it less messy, but keep in mind that V = V^CM(t).

dV

V = u^T(µdt + σdZ) =

d

X

i=1

u_i(µ_idt + σ_idZ_i)

Here u_i = u_i(0) is the constant proportion held throughout. To solve this let Y = ln(V ), then the Ito differential is

dY = dV

V −(dV )² 2V²

(dV V )²= (

d

X

i=1

uiµidt +

d

X

i=1

uiσidZi)²=

d

X

i=1 d

X

j=1

uiujσiσjρijdt = u^Tσρσudt

where we have used that (dt)²= (dt)(dZi) = 0 and dZdZ^T = ρdt. This gives us

dY = (u^Tµ −1

2u^Tσρσu)dt + u^TσdZ and then we integrate and solve for V gives us

V = e^Y = V (0) exp{(u^Tµ − 1

2u^Tσρσu)t + u^TσZ} (2.10)

To conclude this section the log-returns are in the case of the constant mix clearly normally distributed (see equation 2.10), the value process itself is a log-normal distribution. Compare this to the case with buy and hold (see equation 2.9) where the value process is a sum of log-normals, which lacks a simple distributional representation.

2.2.2 Comparison of BH and CM under GBM price process

Definition 3 Let R_t^h be the return of portfolio strategy h at time t by

R^h_t = V^h(t)

V^h(0)− 1 (2.11)

We are dealing with linear combinations of log-normal distributions which have the following expected values and variances.

E[R^BH_t ] = Pd

i=1h_i(0)S_i(0) exp{µ_it}

V^h(0) − 1 =

d

X

i=1

ui(0)e^µit− 1 (2.12)

E[R^CM_t ] = e^(uTµ)t− 1 (2.13)

V ar(R^BH_t ) =

d

X

i=1 d

X

j=1

u_iu_je^(µi+µj)t(e^tρijσiσj − 1) (2.14)

V ar(R^CM_t ) = e^2(uTµ)t(e^(uTσρσu)t− 1) (2.15)

There are two important aspect to acknowledge between the BH and CM strategies under the GBM assumption.

Proposition 1 Without further restriction on ui, µi and σρσ, we have

E[R^BH_t ] ≥ E[R^CM_t ] (2.16)

with equality if and only if µ₁= µ₂= · · · = µ_d.

Proof. By Jensen’s inequality (see Spinu, 2014)

Proposition 2 Assume µ1 = µ2 = · · · = µd = µ. Then E[R^BH_t ] = E[R^CM_t ] and V ar(R^BH_t ) ≥ V ar(R^CM_t ).

Proof. By Jensen’s inequality, (see Spinu, 2014)

2.3 Risk And Performance Measures

When talking about measuring risk there are many different measures available. Perhaps the most used measure in terms of stock price risk is the volatility measure. This can be measured as realized volatility based on historical prices using the standard deviation

measure. This measure of risk is the basis for the mean-variance optimization described for example in Hult et al (2012, ch. 4). In this framework the goal is not always to only minimize risk, but to also have an acceptable return.

The use of standard deviation as a measure of risk captures all aspects of the distributional risk under the assumption of normal distribution. When prices are marginally normally distributed they are completely determined by their location and scale parameters. And in the multivariate case also by their linear correlation structure. It is not often the case on the financial market that the prices are driven by normal distributions. Real (non-modelled) prices often exhibit skewness and higher kurtosis than under normality for example.

2.3.1 Value-at-Risk

Value at Risk (VaR) is a risk measure that can easily be represented in monetary units as the smallest amount needed to be invested in risk-free assets and added to the current portfolio to make the investment acceptable in terms of a decided risk level. For example in the Solvency II framework the Solvency Capital Requirement (SCR) can be seen as the amount of risk-free (solvency) capital insurers and reinsurers need to hold in order to meet their obligations to policy holders and beneficiaries over the following 12 months with a 99.5% probability. Here the risk level referred to is set to 99.5%.

Often VaR is calculated taking into account the risk free rate but in this thesis we will treat VaR as a proper quantile.

Definition 4 V aR_p(V ) = inf {m : P (V < m) ≤ p}.

2.3.2 Risk Contribution

Measuring VaR on a total portfolio will give you the total capital needed for the entire portfolio. Looking at the components of the portfolio (individual stocks) isolated from the rest, calculating the VaR for each component and then adding them all upp will most likely not give you the the same needed capital as when you calculated it together. Usually the sum of the individual VaR measures will be larger than the VaR measured on the entire portfolio. This is due to diversification effects. VaR is not generally a sub additive measure but it often is, as for the case of (multivariate) normal distributions.

By defining risk contribution we seek to achieve a measure that will separate the total risk measure, in this case the VaR, calculated on the entire portfolio into d (number of stocks)

components so that the sum of the risk contributions add up the VaR of the entire portfolio.

This procedure of separation is referred to as the Euler capital allocation principle in McNeil et al (2005, p. 258). For the case of separating VaR we interpret the risk contribution as the following; given the portfolio wide realized loss given at the decided risk level, how much is each component (stock) contributing to this loss. The sum of the components gives a full allocation meaning they sum to exactly the VaR. We will refer to the risk contribution from stock i as the percentage of the total VaR to which it contributes.

Before defining risk contribution we need a proper definition of the profit and loss of the portfolio as well as for the individual stocks under the different portfolio strategies.

Definition 5 LetPd

i=1P L_i= P L be the profit and loss of the portfolio (P L = V (T )−V (0)) on the time interval [0, T ]. This time interval contains τ periods with T = τ ∆t. The individual profit and losses determined by

P Li(T ) =

τ

X

k=1

hi((k − 1)∆t)(Si(k∆t) − Si((k − 1)∆t)) .

The definition of the portfolio wide profit and loss is straight forward, however for the individual components it is a bit trickier. For the case of the buy and hold portfolio the definition of the individual profit and losses reduce to h_i(S_i(T ) − S_i(0)) because the number of shares are held constant throughout. However for the constant mix portfolio the number of shares change each period. At the start of each period we can observe the price change and deduce the profit and loss from each component over that time period. After we have observed the price we immediately rebalance to the initial relative portfolio weight. In the next period we can observe the price change again and deduce the profit and loss for the next one. This process is repeated until we reach our time period of interest. For the constant mix we have

P Li(T ) = ui(0)

τ

X

k=1

V ((k − 1)∆t)

S_i(k − 1)∆t)(Si(k∆t) − Si((k − 1)∆t)

With the definition of the profit and losses for both portfolio strategies we define the risk contribution as

Definition 6 The risk contribution of component i = 1, . . . , d is defined as the conditional expected value of its profit and loss given that the portfolio wide profit and loss P L =P P Li

is equal to its Value-at-Risk

RCi= E[P Li|P L = V aRp(P L)] (2.17)

whereP RCi= V aR_p(P L).

Furthermore, the definition of risk contribution made in definition 6 may seem intuitive but there are some potential caveats. Firstly from an economic point of view it is of interest to have the risk contribution consistent with its usage in performance measures. Tasche (1999) shows that there is only one definition of risk contribution suitable for this purpose. This definition is to have risk contributions as the partial derivatives of the portfolio value with respect to their portfolio weights. Much like the mean-variance (see 2.4) framework relies heavily on partial derivatives parametrized by expect returns and portfolio wide variance to calculate the contribution of individual assets. Secondly, these partial derivatives might not always exist. Tasche (1999) provides a set of assumptions for when the risk contributions are well behaved. These are true for the case of the multivariate normally distributed stock prices. Further he shows that the partial derivatives under these assumptions are equivalent with definition 6.

In this thesis we use the conditional expectation approach when dealing with risk contri- butions. Mainly because it allows for easy computation in the Monte Carlo simulation framework. However the conditional expectations converge very slowly. To remedy this the a kernel estimation method is used to speed up convergence (see Epperlein and Smillie, 2006).

We will use N simulations of S(t) with τ number of discrete time steps to calculate the portfolio values V^BH and V^BH and their corresponding profit and loss P L. First we define the empirical V aRp and RCi estimators.

Definition 7 Given that the N simulations have been ordered in decreasing order {P L⁽¹⁾, ..., P L^{(N )}} where the first element is the largest, we have the empirical estimator of V aRp(P L) as

V aR˜ p(P L) = P L[N (1−p)]+1

where [x] is the integer part of x. This specific simulation outcome is referred to as the V aRp-outcome. For this specific outcome we have the risk contribution empirical estimator

RC˜i= P L[N (1−p)]+1]

i

wherePRC˜ i=V aR˜ p(P L) by definition.

This estimator will not converge fast enough and instead we use the weighted average of each P Li near the V aRp(P L)-outcome with the weights determined by the kernel function K(x; b) defined below. The bandwidth b of the estimator is chosen to give a low estimation error (root mean squared error for example). Here the bandwidth is chosen to be the same as in Epperlein and Smillie (2006). The triangular kernel represents the weights that are symmetrical around the empirical VaR estimate. This estimator is not in general going to satisfy the full allocation property where the sum of the risk contributions equal the Value-at-Risk. This property is forced by rescaling the estimator.

Definition 8 Let K(x; b) = max(1 − |^x_b|) be the triangular kernel function with bandwidth b. We have the (partial) kernel estimator for the risk contribution

RCˆ_i= PN

j=1K(P L^(j)−V aR˜ p; b)P L^(j)_i PN

j=1K(P L^(j)−V aR˜ _p; b)

where the word partial refers to this estimator not guaranteeing that PRCˆ_i=V aR˜ _p(P L).

The full allocation property is forced by rescaling the estimated risk contribution

RC_i^∗=RCˆi

V aR˜ p(P L)

PRCˆ_i (2.18)

whereP RC_i^∗=V aR˜ p(P L).

2.4 Optimal portfolio weights

This section describes the optimal portfolio weights for both portfolio strategies. This will be used as a reference point when comparing the risk contributions of the different portfolios for different weights.

Selecting portfolio weights is the selection of risk reward trade off. In the mean-variance one period model framework the optimization problem can for example be formulated as to maximize expected returns while not exceeding a certain level of variance. The solution to this problem can be split into two parts where (1) finding the optimal allocation between risky assets and (2) determine the level of leverage towards the risky assets. Given a set of risky investment opportunities there is one optimal portfolio. The efficient frontier is the linear combination of a risk free asset and this optimal risky portfolio (Hult et al, 2012, p.

92).

The continuous time problem allows for further complexity. It is a type of problem in the realm of stochastic control. For example the optimal portfolio weights are allowed to be time- and state dependent. In the framework of Merton (1969) the solution for the portfolio weights are constants only determined by the excess return over risk free assets, covariance matrix of the underlying price process and on the risk preference of the investor. This risk preference determines the leverage towards the risky assets. Very much like the one period model. The risk preference is determined by parameter selection of the utility function. In the case of Merton this utility function is used:

U (x) =







x^γ/γ γ ≤ 1or γ 6= 0 log(γ) γ = 0

Let the covariance matrix be

Ω = σρσ and the solution for the optimal weights:

u^∗= 1/(1 − γ)Ω⁻¹(µ − r)

where r is the risk free return. This optimal allocation is not dependent on time.

The weight for the risk free investment is u_r= 1 −P wi, which is the remainder to balance the money invested to 100%. This can be negative in the case of leverage. The optimal solutions at time zero coincide for both the discrete and the continuous time cases. This means that the initial capital allocation is optimal in both cases. Clearly it is better to rebalance to the optimal portfolio allocation in continuous time in this case. Thus the constant mix portfolio strategy is the optimal portfolio in continuous time.

The special case of the utility function where γ = 0 can be linked to the Kelly criterion (Øksendal, 2003, ch 11.2). The Kelly criterion maximizes long term growth rate and is the optimal share of capital to place on a single bet which can be repeated infinitely many times.

The kelly criterion corresponds to being a maximally aggressive investor when optimizing long term growth.

2.5 Optimal weight in risky assets, Kelly Crieterion

The Kelly cireterion can serve as a benchmark for the amount to invest in risky assets assuming that investors want to maximize long term growth over short term gains. This assumption is reasonable in the context of investing in pension funds with long term invest- ment horizons. Under the log-utlity function having too much volatility in the portfolio will decrease the long term growth rate. The growth rate refered to here can be described using a short example.

Consider Xi to be the payoff of a single game. The games are independent and identically distributed. With initial capital F0 and investing the proportion a to the current wealth Fn the wealth after n games can be written as Fn = F0e^{P log(1+aXi)}. The exponent is approximately nE[log(1+aX)] when n is large (Andersson and Lindholm, 2009, ch. 8). The expression within the expected value operator is the growth rate which can be maximized by a∗. The Kelly strategy for choosing the amount to invest in risky assets is maximally aggressive towards maximizing the expected growth rate. Investing more than the Kelly criterion leads to reduced growth rate at a higher risk. We will use the Kelly criterion to define the maximum aggressiveness of a risk averse investor, which the investors in pension funds are assumed to identify with. Going beyond the Kelly criterion in a risky investment is referred to being risk seeking. As can be seen in figure 2.1 investing more than the Kelly criterion in risky assets lead to a decline in long term growth rate. The log-utility function is interpreted as the long term growth rate. The region labeled as ”insane” corresponds to being risk seeking and preferring higher short term expected value in favor of long term growth.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

−8

−6

−4

−2 0 2 4 6

·10⁻²

Proportion riskfree

Expectedutility

Expected utility log(x)

Aggressive Insane

Figure 2.1: Visualization of the log-utility function w.r.t the amount invested in a risk free asset. 1 (100%) corresponds to being fully invested in the risk free asset and 0 (0%) corresponds to being 100% invested in risky assets. Going to the negative domain is interpreted as investing in risky assets with leverage.

3 Simulation study

The goal of the simulation study is to provide further insight into the risk contributions of the buy and hold and the constant mix portfolio strategies. Analytical formulas for risk contribution are well known for elliptical distributions (cf. McNeil et al (2005, ch. 6)). In this thesis we consider the geometric Brownian motion as the stochastic representation of our investment opportunities. The value of the BH and CM portfolio and its components are not elliptical distributions. They are linear combinations of log normally distributed random variables. The CM portfolio however has an analytical formula for the portfolio wide quantile w.r.t the underlying brownian motions, but the representation of the profit and loss vector (definition 5) is not analytically tractable for the analytical calculation of the risk contribution.

Risk contributions for the multivariate normal distribution will be used as a benchmark for the risk contributions of the BH and CM portfolios estimated by the simulations. This is because the risk contribution will be visualized with their It will also be used to validate the estimator of risk contribution in the (normal) case of known risk contributions. For an overview the steps of the simulation study are as follows:

1. Validity of the risk contribution estimator

(a) Simulate bivariate normal distributed variables

(b) Estimate by the kernel estimator and the empirical observation estimator from definition 8.

(c) Repeat estimation with independent simulation to construct confidence interval

(d) Compare to analytical results

2. Estimate risk contributions for BH and CM portfolios for different portfolio weights and different model parameters in the bivariate case according to:

(a) Base case µ1 = µ2 and σ1 = σ2 and zero correlation. 1 year horizon and rebal- ancing every month.

(b) Increasing rebalancing frequency

(c) Increasing correlation

(d) Different µ1

(e) Different σ₁

(f) Longer time horizon

3. Compare estimates with analytical formula under bivariate normal assumption with moments matching the lognormal

The analytical formulas for the risk contribution in the multivariate normal case can be found for example in McNeil et al (2005, p. 261).

Below is the definition of the simulation base case which will be used to study the differences between risk contributions with the two portfolio strategies.

Definition 9 The base case for which all simulations will build upon is defined by the fol- lowing bivariate case. µ₁ = µ₂= 0.05, σ₁ = σ₂= 0.2, ρ = 0 and relative portfolio weights u1and u2 vary between [0, 1] where u2= 1 − u1. Time horizon is 1 year with rebalancing at monthly timesteps. Simulation sample size is 10⁴ for each independent estimation. If lower and upper bounds are referred to they correspond to the 5% and 95% empirical quantiles respectively of 100 repeated independent estimations. When varying the portfolio weights u the same sample of S, the underlying price paths, is used. Thus the figures presented in the Results section are all conditioned on the same sample when no lower and upper bounds a presented in the figures. Risk contributions are evaluated at the 5% quantile level unless otherwise specified.

4 Results

4.1 Risk contribution w.r.t BH and CM strategies

First the convergence of the risk contribution estimator from definition 8 is tested and compared to analytical formulas for the bivaraite normal case. This is done only to verify the convergence of the estimator. From Figure 4.1 it is clear that the empirical observation estimator has not converged and is very volatile. The kernel estimator performs well in the sense that the confidence interval is narrow for all portfolio weights. The sample size of 10⁴is deemed sufficient for further analysis. From Figure 4.1 we can see that the risk contribution is symmetrical which we would expect since µ1= µ2 and σ1= σ2. The risk contribution is not linear in u since the variance is not a linear operator. The risk contributions are trivial in the case u1 = 0 and u1 = 1. The risk contribution in all figures are presented as their percentage of the total portfolio VaR (where 1 is 100%).

The base case and most of the extension cases will be compared with the analytical formula for risk contribution under bivariate normal assumption. This will serve mainly as a refer- ence point for the differences between the risk contributions in buy and hold and constant mix strategies. The estimated risk contributions in this section will often be presented with approximate confidence intervals, and to avoid cluttering the risk contributions of the two portfolios are shown in separate figures. The parameters used for expected value and covari- ance are chosen to match the moments of the returns of underlying price process S and are thus independent of portfolio strategy. These moments are determined by equations 2.12 and 2.14 with the exception of not depending on the relative weights in those equations.

This benchmark is referred to as the ”Matching Moment Normal Assumption” (MMNA).

This is motivated by the fact that the modeled log-returns of S are approximately equal to normal returns for small returns.

The risk contribution in the base case for constant mix can be seen in Figure 4.2. The risk contribution under CM corresponds almost identically to the simple normal (MMNA) case. For the BH portfolio in Figure 4.3 there is a slight deviation from the analytical value (MMNA) and thus also to the risk contribution in CM. The MMNA line in both figure 4.2 and 4.3 are the same and serve as a reference point when comparing CM and BH risk contributions. The risk contribution from component 1 in CM has a steeper decline than BH when u1 tends toward 0 starting in 0.5. This suggest that the loss from component 1 in CM becomes less influential on the overall portfolio loss faster than in BH. The optimal portfolio weights by section 2.4 are u1= u2= 0.5 in this case and the risk contributions are the same in this case.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

u1

Riskcontribution

Risk contribution for normal distribution

Kernel median Kernel l.b.

Kernel u.b.

Obs median Obs l.b.

Obs u.b.

Analytical formula

Figure 4.1: Convergence of risk contribution with respect to the weight u1. Risk contribution of 0 (1) is interpreted as 0% (100%) of the VaR at the 0.05 quantile level. The reference distribution is N (0, I2). Simulation uses 10⁴ samples for each estimation and the estimation are independently repeated 100 times to build the approximate confidence interval where the lower and upper bounds correspond to 5% and 95% empirical quantiles respec- tively.

By increasing the rebalancing frequency in steps we can observe how this affects the risk contribution. In Figure 4.4 we see that the risk contribution converge rather quickly (no noticable change going from monthly to daily to 8 times a day) when the rebalancing frequency is higher. This suggests that rebalancing does not have to be done very frequently (monthly) to have an impact on the risk contribution.

The effect on risk contribution by only changing the correlation should intuitively tend

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

u₁

Riskcontribution

Constant mix risk contribution

Kernel median Kernel l.b.

Kernel u.b.

MMNA

Figure 4.2: Estimates of the risk contribution of the CM portfo- lio under the base case according to definition 9. MMNA refers to the ”moment matching normal assumption” benchmark. Risk contribution of 0 corresponds to 0% of portfolio VaR and 1 cor- responds to 100% of portfolio VaR.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

u1

Riskcontribution

Buy and hold risk contribution

Kernel median Kernel l.b.

Kernel u.b.

MMNA

Figure 4.3: Estimates of the risk contribution of the BH portfolio under the base case according to definition 9.

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.6

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

u1

Riskcontribution

Risk contribution under different rebalancing frequencies

8 times a day daily

monthly semi-anually quarterly Buy and hold

Figure 4.4: Estimate of risk contribution given a fixed sample of S with respect to u1 and changing rebalancing frequency. The figure is zoomed in to better visualize the different frequencies.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u₁

Riskcontribution

Risk contribution

Kernel CM Kernel BH MMNA

Figure 4.5: Estiamte of risk contributions for CM and BH with ρ = 0.7, everything else the same as the base case.

towards a straight line when ρ → 1. This can be seen in Figure 4.5 when ρ = 0.7 the risk contributions are seemingly the same for BH, CM and MMNA.

Setting µ₁ = 0.1 yields a different shape of the risk contributions curve with respect to u1. However they are all relatively close to each other (see figure 4.6) especially near the optimal portfolio weight u^∗₁= ²₃. By keeping µ1 = 0.1 and also setting σ1= 0.4 shifts the risk contribution curve further and gives u^∗₁=¹₃. The risk contributions are relatively close near the optimal portfolio weight but diverge otherwise (see Figure 4.7).

The effect of increasing the time horizon further exaggerates the differences found in the base case. By setting the time horizon to 5 years (still rebalancing 12 times a year) we see the effects in figure 4.8. The risk contributions all coincide at u^∗₁ = 0.5 (the optimal portfolio weight) but diverge quickly from the MMNA when leaving the optimal portfolio allocation.

The assumption of multivariate normal instead of the actual multivariate lognormal dynam-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

u₁

Riskcontribution

Risk contribution

Kernel CM Kernel BH MMNA

Figure 4.6: Estimate of risk contribution for CM and BH with µ1= 0.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

u₁

Riskcontribution

Risk contribution

Kernel CM Kernel BH MMNA

Figure 4.7: Estimate of risk contribution for CM and BH with µ1= 0.1 and σ1= 0.4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

u1

Riskcontribution

Risk contribution

Kernel CM Kernel BH MMNA

Figure 4.8: Estimate of risk contribution for CM and BH with T = 5. Rebalancing 12 times per year.

ics present under the GBM model of price process can deviate by alot when measuring risk contribution compared to the actual value. However the approximation tends to perform better when the portfolio is optimally allocated by visual inspection of the cases studied in this section.

4.2 Risk management considerations

It was shown in section 2.5 that the optimal portfolio allocation in risky assets is dependent the tolerance to risk. This can also be seen in Figure 4.9 where the efficient frontier for the two strategies is presented for a fixed time horizon. Further we introduced the maximally aggressive long term growth maximizing strategy as the constant mix strategy with the percentage invested in risky assets according to the Kelly criterion. Going beyond this criterion lowers long term growth at the cost of higher volatility. This cannot be seen directly from the efficient frontier plot where returns increase linearly with volatility. However for life insurance companies the goal is to maximize return while being able to maintain its obligation to policy holders. These obligations are typically far into the future and long term growth is wanted. Having more invested in risky assets than the Kelly criterion is suboptimal with regard to their risk preference. On the other hand lowering the risky asset allocation to a level below the Kelly criteria is still optimal given a more risk averse utility function. Calculating the Kelly criteria based on historical prices is potentially prone to estimation errors. This motivates having less than the Kelly criterion invested in risky assets for long term investors.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0

0.2 0.4 0.6 0.8 1 1.2

Standard deviation (risk)

Expectedreturn

Risk-return profile

CM BH

Figure 4.9: Efficient frontier for a constructed example invest- ment universe of three assets and a risk free investment. The dotted line corresponds to the one period investment strategy

5 Summary and conclusions

In risk management one important aspect is the separation of total portfolio risk into its components. This enables measuring each components’ risk contribution relative to the total risk taking into account the covariance between components. The measurement procedure is simple and straightforward under assumptions of elliptical distributions but not under the commonly used assumption of multivariate log-normal distribution (geometric Brown- ian motion). In this thesis estimations of the risk contributions under geometric Brownian motion have been made for two different portfolio strategies, buy and hold and constant mix using Monte Carlo simulation. These estimations have been compared to approximations using analytical formulas provided by multivariate normal assumptions. These approxi- mations are relatively accurate when the portfolio allocation is near optimal (according to framework of Merton (1969)) and the time horizon is short (1 year). This is typically the case for large institutional investors where regulatory regimes such as Solvency II requires risk assessment on a one year horizon. Further the main difference in general between risk contributions under the buy and hold and constant mix portfolio strategies is the steeper decline of risk contribution in constant mix than in buy and hold when with respect to a reduction in that specific portfolio weight.

The constant mix strategy has better performance than the buy and hold strategy. This is because the the optimal portfolio strategy in continuous time is not dependent on time.

Moreover the risk tolerance of a life insurance company could be argued to be in corre- spondence with the long term investment horizon due to the long term obligations towards policy holders. This has a natural extension to the log utility function which maximizes long term growth rate. The amount to be invested in risky assets with such a utlilty func- tion corresponds to the maximum aggressiveness of an investor seeking to maximize long term growth. However due to estimation error in expected growth and volatility in avail- able investment opportunities it may be motivated to have slightly less invested in risky assets than dictated by the log utility. Having less invested in risky assets corresponds to being optimally invested (altough at a lower growth rate) while being less exposed to risk.

Having more invested in risky assets corresponds to reduced long term growth at a higher risk.

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McNeil, A., J., Frey, R., Embrechts, P., Risk Management Concepts Techniques Tools, Princeton series in finance, 2005

Merton, R. C., Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case, The Review of Economics and Statistics, 1969

Øksendal, B., Stochastic Differential Equations, Springer, 2003 Tasche, D., Risk contributions and performance measurement, 1999

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