partii:evaluating managementactions€¦ · volatility of variance 0 0.2 0.4 0.6 0.8 1 sharpe ratio...

November 2015

PART II: EVALUATINGMANAGEMENT ACTIONS

In the first article of this series, ”Part I: Management Actions in a SolvencyII World”, we presented the Solvency II regulatory requirements an un-dertaking need to fulfil in order to be able to take management actionsinto consideration when calculating their capital requirement. The articlediscussed the difference between static and dynamic management actionsand concluded that dynamic management actions aimed at reducing thesolvency capital requirement (SCR) in practice require a partial or fullinternal model.

In this article, we will focus on how management actions used in an in-ternal SCR model can be evaluated and validated. This will be done froma perspective of both risk and return. An investment strategy monitoringrealised volatility levels will be used as an example of a management actionthat can be used within an internal model to reduce the risk and thereforealso the SCR of an undertaking.

PERFORMANCE MEASURES

So what type of measures are suitable when evaluating and validating amanagement action aimed at reducing the SCR? We are obviously inter-ested in the effect on the 99.5 % Value-at-Risk (VaR) which is the definitionof SCR according to Solvency II. In addition to this we will also study theSharpe ratio, a widely used measure of risk-adjusted return in our analysis.The Sharpe ratio (Sharpe, 1966, 1994) is defined as:

Sp =E[Rp −Rf ]

σ(Rp),

and relates the expected portfolio return less the risk free return, E[Rp −Rf ],to the riskiness of the portfolio returns expressed in terms of its standarddeviation, σ(Rp).

VOLATILITY CONTROL EXPLAINED

In light of Solvency II coming into force in January 2016, one managementaction approach used for capital savings appearing to gain traction in the in-surance industry is an investment strategy previously applied within hedge

Part II: Evaluating Management Actions 1

November 2015

funds and by asset managers, commonly known as Volatility Control (VC).VC aims to keep the volatility of a portfolio constant at a pre-defined levelby rebalancing between a risky asset and cash invested in an asset yieldinga rate considered risk free. The risky asset is often, but not always, equity.If the realised volatility of the underlying portfolio falls below the targetedlevel, risk is increased by borrowing money and investing it in the risky as-set (henceforth the equity). On the contrary, should the realised volatilityof the portfolio increase, the allocation to equity is decreased in favour ofthe risk free asset (henceforth, cash). (Redington, 2013)

The VC execution approach is summarised in Figure 1 below. The feed-back loop illustrated in the flow chart indicates that VC is carried out iter-atively, i.e. that rebalancing is needed. It is the rebalancing aspect of VCthat makes it a dynamic strategy, and only applicable as a SCR reducingmanagement action under a partial or full internal model.

Figure 1: Flow chart illustrating how volatility control is executed.

To demonstrate the effects of applying VC, daily equity returns are simu-lated for a portfolio with and without VC, with a volatility target level cor-responding to 10 % using daily rebalancing. A Stochastic Volatility JumpDiffusion (SVJD) model, more on this later, was used to simulate the eq-uity process. The result of the simulation can be seen in Figure 2, wherethe simulated cumulative equity returns have been plotted against the re-alised volatility, where the latter has been calculated as an exponentiallyweighted moving average:


November 2015

σ̂yearly,t =

√252

nσ̂2daily,t

σ̂2daily,t = λS σ̂

2daily,t−1 + (1− λS)

[ln

(St

St−n

)]2for t > T0.

The returns are accumulated over 252 trading days resulting in yearly re-turns. In order to estimate the realised volatility λ = 0.96 has been appliedto the daily returns. Notice how the returns where VC have been appliedare centred on a realised volatility level of 10 %, and that the distributionof the VC-strategy exhibits a lower VaR than the buy-and-hold strategy.

Realised volatility0 0.2 0.4 0.6 0.8 1 1.2

Ret

urn

(%

)

-100

-50

0

50

100

150SVJD with and without Vol Control

Return without VCVaR without VCReturn with VCVaR with VC

Figure 2: Daily equity returns plotted against realised volatility with and without VC. Thehorizontal lines highlight the 1-year VaR at a 95 % confidence level.

Given the encouraging results in Figure 2, one might ask how VC wouldhave performed historically over time. In Figure 3 we can see daily returnsof S&P 500 from the last 10 years with and without the VC-strategy applied(blue and red line respectively).


November 2015

Figure 3: Daily historical returns for S&P 500 for the last 10 years with (blue) and without(red) VC. Target vol equals 10 %.

A target level corresponding to 10 %with daily rebalancing was used. Study-ing Figure 3, we observe that:

⋄ During periods of low volatility, e.g. from 2006 to 2007, both indicesperformed similarly. This is due to that the VC portfolio was closedto 100 % invested in the underlying stock index.

⋄ In late 2008 when the credit crisis escalated, the volatility of the un-derlying index rapidly increased whereas the volatility controlled in-dex de-leveraged quickly and maintained low levels of volatility. Theunderlying index continued to show periods of significantly increasevolatility until 2012, while the rebalanced index was kept almost con-stant.

⋄ During 2013 to 2015, when the stock market performed relativelywell, the volatility controlled index has underperformed in compari-son to the underlying S&P 500 index.

The Sharpe ratio for the period 2006 to 2015 is 0.51 for the S&P 500 index.The number for our VC approach equalled 0.67, an increase of slightlymore than 30 % in risk-adjusted return. We have also previously seen thatVC was capable of lowering VaR (or SCR).

VALIDATION OF VOLATILITY CONTROL

Up until now, we have focused on how VC can be applied as an investmentstrategy and we have seen some indications of how that strategy performs.


November 2015

As a next step we will investigate how certain model assumptions affectsthis performance. For example, by looking at the historical volatility levelsin Figure 3, one might draw the conclusion that the assumption of con-stant volatility is not an entirely realistic one. We will therefore start byconsidering this particular aspect when choosing a suitable model for fur-ther analysis of VC.

CONSTANT VS. STOCHASTIC VOLATILITY

Since equity return distributions tend to demonstrate fat tails not capturedby e.g. a generalised brownian motion model (GBM), we need to dig fur-ther into the model toolbox. One way of capturing this is by incorporatingstochastic volatility and jumps into our model for equity returns, by usingthe already briefly stated SVJD model (Bates, 1996). The SVJD model isdescribed by the following set of stochastic differential equations (SDE):

dSt = St

((r − d)dt+

√V (t)dWS(t) + (J − 1)dN(t)

)dV (t) = κ(θ − V (t))dt+ ϵ

√V (t)dWV (t)

where the Wiener processes WS(t) and WV (t) are correlated with ρ. Theparameters r and d are the risk neutral rate and the dividend yield respec-tively, whereas ϵ denotes the volatility of variance. N(t) is the number ofrandom jumps over the interval [0, t] determined by a Poisson process withintensity λ, and J is the random jump size:

J = µJe− 1

2σ2J+σJZ , Z ∼ N (0, 1)

where µJ ∈ R the average jump size and σJ > 0 the jump volatility. Wediscretise the stock price process with the Quadratic Exponential (QE)scheme (Andersen, 2008) extended to fit our process. Figure 4 comparessimulated annual returns for SVJD and GBM where VC has been applied inboth cases.

As Figure 4 shows, the GBM process’s returns are all close to a volatil-ity level of 10 %, whereas the volatility level of the returns from the SVJDmodel tend to vary more. We also notice returns with high volatility levelseven though VC has been applied for the SVJD model.


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Realised volatility0 0.2 0.4 0.6 0.8 1 1.2

Ret

urn

(%

)

-100

-50

0

50

100

150SVJD and GBM with Vol Control

SVJD ReturnSVJD 5.0% VaRGBM ReturnGBM 5.0% VaR

Figure 4: Comparison between GBM and SVJD both applying VC.

EXTREME MARKET EVENTS

That the SVJD model exhibits outcomes with higher levels of volatility ismainly due to the jump process incorporated into the model. This processallows themodel to capture unexpected extreme, negative, market returns.It is therefore of interest to further investigate how the VC portfolio per-forms for different levels of jump intensity λ. This is illustrated in Figure5a and Figure 5b where Sharpe ratios and 1-year VaR levels are plottedagainst different levels of λ.


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Jump intensity0 0.2 0.4 0.6 0.8 1

Sh

arp

e ra

tio

0.45

0.5

0.55

0.6

0.65

0.7

0.75Sharpe ratio vs jump intensity

Sharpe without Vol controlSharpe with Vol control

(a) Sharpe ratio.

Jump intensity0 0.2 0.4 0.6 0.8 1

VaR

5%

-35

-30

-25

-20

-15

-10VaR 5% vs jump intensity

VaR without Vol controlVaR with Vol control

(b) 1-year VaR.

Figure 5: Varying jump frequency λ.

Looking at Figure 5a and Figure 5b, we see that the Sharpe Ratio andVaR measures of the VC-strategy both decrease as the jump intensity in-creases1. From Figure 5a, we can also see that the risk-adjusted return islower for the VC-strategy than for the underlying index when jumps occurmore frequently. A possible explanation for this might be that the jumpsoccur more frequently than the rebalancing of the portfolio when λ ap-proaches one.

VOLATILITY OF VARIANCE

As seen in the previous section, the VC portfolio’s risk-adjusted return wasdrastically reduced when we increased the jump intensity, and was outper-formed by the underlying index portfolio for high levels of λ. It is thereforenecessary to further investigate how the VC portfolio performs when thevolatility of variance is increased. Figure 6a and Figure 6b shows simulatedresults for our portfolios with target volatility level equalling 10 %.

As we can see in Figure 6a the risk-adjusted return increases significantlyfaster for the VC-strategy than for the 100 % index portfolio when the volatil-ity of variance ϵ increases. However, in Figure 6b we see that the relativeadvantage from a VaR perspective of running VC decreases as ϵ increases.This might be due to that the VC-strategy overcompensates its equity/cashreallocation when the volatility shifts from low to high and vice versa. As aresult, the VaR measure decreases for the VC portfolio when the volatilityof variance is high.

1Note that the VaR decreasing means that it becomes more negative, indicating a largerloss.


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Volatility of variance0 0.2 0.4 0.6 0.8 1

Sh

arp

e ra

tio

0.55

0.6

0.65

0.7

0.75

0.8Sharpe ratio vs volatility of variance

VaR without Vol controlVaR with Vol control

(a) Sharpe ratio.

Volatility of variance0 0.2 0.4 0.6 0.8 1

VaR

5%

-28

-26

-24

-22

-20

-18

-16

-14

-12VaR 5% vs volatility of variance

Sharpe without Vol controlSharpe with Vol control

(b) 1-year VaR.

Figure 6: Varying volatility of variance ϵ.

REBALANCING EFFECTS

Last but not least, we choose to examine how the VC portfolio performswhen the number of rebalancing frequency is varied. One year is assumedto have 252 trading days. In Figure 7a and Figure 7b we see the Sharperatio and 1-year VaR vs. number of rebalances per year.

No. of rebalances per year0 50 100 150 200 250

Sh

arp

e ra

tio

0.59

0.6

0.61

0.62

0.63

0.64

0.65Sharpe ratio vs no. of rebalances per year

(a) Sharpe ratio.

No of rebalances per year0 50 100 150 200 250

VaR

5%

-20

-19

-18

-17

-16

-15

-14

-13VaR 5% vs no. of rebalances per year

(b) 1-year VaR.

Figure 7: Varying the number of rebalances per year.

By looking at the plots, we can see that the VaR of the VC-strategy improves(i.e. becomes less negative) as the rebalancing frequency increases. How-ever, the marginal effect decreases as the frequency becomes higher.

CONCLUSION AND LIMITATIONS

From our evaluation of VC we conclude that:


November 2015

⋄ Management actions can be used to reduce an undertakings SCRwhen a partial or full internal model is used. This can be seen fromthe VC portfolio’s simulated VaR results.

⋄ It is hard to draw any general conclusions for VC with respect to riskadjusted return.

⋄ Simulation is a useful tool for evaluating management actions. How-ever, the main challenge when doing so is to model volatility andextreme market events in an efficient way.

We would also like to point out some limitations in our evaluation:

⋄ We have assumed that it is always possible to lend or invest moneyat the risk free rate.

⋄ We have assumed that there are no transaction costs. Introducingtransaction costs will probably effect how often we want to rebalanceour portfolio.

SUMMARY

In this article, we have presented an approach for howmanagement actionsaimed at reducing SCR can be evaluated. This has been done by usingVolatility Control as an example of a management action. The evaluationpresented in this article has been based on simulated daily returns with themain emphasis on the investigation of how a VC portfolio performs whenstochastic volatility and jumps are introduced. Rebalancing effects havealso been analysed.


November 2015

References

Andersen, Leif. 2008. Simple and efficient simulation of the Hestonstochastic volatility model. Journal of Computational Finance, 11(3),1–42.

Bates, David S. 1996. Jumps and Stochastic Volatility: Exchange Rate Pro-cesses Implicit in Deutsche Mark Options. Review of Financial Studies,9(1), 69–107.

Redington. 2013 (March). Risk-Controlled Investment Strategies: Volatil-ity Control. Tech. rept. Red Views, London.

Sharpe, William F. 1966. Mutual Fund Performance. The Journal of Busi-ness, 39(1), 119–138.

Sharpe, William F. 1994. The Sharpe Ratio. The Journal of Portfolio Man-agement, 21(1), 49–58.


November 2015

CONTACT US

Do you want to know more about Kidbrooke Advisory?Please contact us for more information.

e. [email protected]

a. Kidbrooke Advisory AktiebolagBirger Jarlsgatan 20SE-114 34 StockholmSweden

EDVARD SJÖGRENPartner+46(0)733 182 [email protected]

DAG UTTERBORNAnalyst+46(0)761 171 [email protected]

DISCLAIMER

Copyright 2015 Kidbrooke Advisory Aktiebolag; All rights reserved. Reproduction in whole or in part isprohibited except by prior written permission of Kidbrooke Advisory Aktiebolag registered in Sweden.The information in this document is believed to be correct but cannot be guaranteed. All opinions andestimates included in this document constitute our judgement as of the date included and are subjectof change without notice. Any opinions expressed do not constitute any form of advice (includinglegal, tax, and or investment advice). This document is intended for information purposes only and isnot intended as an offer or recommendation to buy or sell securities. Kidbrooke Advisory Aktiebolagexcludes all liability howsoever arising (other than liability which may not otherwise be limited orexcluded under applicable law) to any party for any loss resulting from any action taken as a result ofthe information provided in this document. The Kidbrooke Advisory Aktiebolag, its clients and officersmay have a position or engage in transactions in any of the securities mentioned.


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