Insurers need to monitor dynamic policyholder behaviour to prevent major risk exposure in the case of mass migration of the policyholders. To mitigate the risk, the company should take management actions. Among all the management action, in this article, we study dynamic hedging. Furthermore, we investigate the effects of this type of management actions on reducing Solvency capital requirement stated in Solvency II regime.

## Hedging

Hedging is a strategy that aims to eliminate or reduce the risk caused by the adverse change in underlying asset/liability value. Consequently, the volatility of the portfolio can be reduced.

In this article we consider Delta hedging, i.e., neutralization of the portfolio with regard to the changes of the price of the underlying. Here we go through it briefly.

### Delta Hedging

Delta-hedging of any financial instrument states the relation between its price and the underlying asset. In other word, delta-hedging is the reflected change in the instrument price in case of change in underlying asset. Furthermore, delta (hedging ratio) is the rate of change of the derivative value \( P \) with respect to the underlying asset price, i.e., the delta hedge ratio of a derivative with price \( P \) is defined as

$$\Delta = \frac{\partial P}{\partial S}.$$

Moreover, a delta-neutral portfolio is hedged such that its value remains unchanged when small changes occurs in the underlying price and we aim to set and maintain such a portfolio and make the delta of portfolio as close to zero as possible. Also it is needed to re-balance the portfolio periodically in order to keep delta neutrality and have a perfect hedging.

### Dynamic Hedging

In general, hedging mitigates the volatility of earnings and losses. The impact of dynamic hedging on VaR and consequently decreasing capital requirement is particularly of our interest here.

We deem it necessary to mention that the transaction cost may have a negative effect on hedging performance and for this reason we shall not consider a more frequent approach than a monthly or bi-monthly hedging policy in which the transaction cost might be negligible compared to the advantages of re-balancing.

Considering dynamic hedging, we face the following challenges:

- Simulation of step-by-step economic scenarios for realistic market risk factors
- Fast evaluation of the sensitivities (greeks) to these market risk factors of the liabilities

### Confronting the challenges

The first challenge is a consequence of the dynamic nature of the hedging strategy; we have to simulate risk factors step-by-step over the desired time intervals. Hence, we need simulated risk factor paths as opposed to just one year returns of the risk factors.

To tackle this issue, we apply the LMM and SVJD models to model interest rate and the volatility of stock prices, respectively. As we have already discussed in our earlier notes, LMM is an adequate model in terms of capturing market consistent prices in economic scenarios for interest rates (see our post about Libor market model for more detailed information here). Moreover, one of the most important obstacles that one may face in analysis of the market is that the volatility and jumps are not easily observable. Stochastic volatility jump diffusion is a parametric approach that estimates volatility and jumps as possible state variables with computational method like Monte Carlo method. SVJD can be used for the modelling of the level as well as the volatility of stock prices. We refer the readers to our previous article series for studying SVJD model in depth. series. It is worth noting that using a simpler approach such as the geometric Brownian motion (GBM) underestimates the variation of the volatility and consequently undervalues the SCR. In contrast, using the SVJD model we capture many of the desired stylized facts of the historical data.

**Nested Stochastic Simulations**

Before moving on to the analysis of the hedging itself, we note that facing the problem in the first stage requires a setup of nested stochastic scenarios. The outer scenarios are then the real world scenarios and the inner ones the risk-neutral scenarios to value the liabilities along each scenario such that the values are consistent to the market prices of the outer real-world scenarios.

The very large number of scenarios makes it almost impossible to evaluate the greeks in this way. A better choice is to implement the so-called proxy technique Least Squares Monte Carlo (LSMC) in order to calibrate a polynomial approximating these greeks.

**Proxy Functions and Least Square Monte Carlo method**

The LSMC method is a combination of Monte Carlo simulation and a regression model, and is based on fitting a set of functions (proxy functions) to a given data set. Here, we use second degree polynomials for the equity return index and the ten-year spot rate where the coefficients are found using Lasso linear regression. Once the proxy functions are calibrated, the greeks can be obtained by finding partial derivatives of the proxy function. Applying this method, the required number of inner scenarios and time steps is reduced substantially and hence we have secured the desired reduction in run-time making the analysis feasible.

**Case Study**

**Unit-Linked Insurance Product**

A unit-linked insurance product is a life type insurance that some part of the premium is invested in mutual funds or other investment strategies in the market.

It is usually considered with an extra guarantee feature that assures a minimum payment with different claims.

**Related Risks**

Because of the nature of the product, the related risk consist of two different parts:

- Insurance Risks (e.g., Longevity, surrender, ...)
- Financial Market Risks (e.g., Interest rate risk, equity risk, ...)

We aim to analyse a unit linked insurance product together with a guarantee element. The insurance contracts are ﬁctive and created to replicate the contracts that are available at the Swedish insurance market. We generate the return path for the end client and also hedge the guarantee factor. The liability of the insurance company is modelled using a policyholder population consisted of one hundred of individuals with mortality being a function of dates of birth and gender. The different states a single policyholder may belong to during the insurance period is modelled using the states diagram in the following figure:

The different states are premium paying (PP), pension (PO), surrender (SU) and death (DE). The diagram demonstrates the possible transitions between different states for the policyholders. It is worth noting that the starting capital and premiums varies for the different age groups. To provide a better understanding of the problem, we illustrate the future monthly cash flows for a policyholder that is expected to retire in 210 months (where we see a significant increase in the graphs that indicates a change of state from paying premiums to receiving pension payments), simulated under ten different scenarios.

The following chart, also illustrates the evolution of the various states for a single policyholder over the full time horizon

As this figure depicts, there is an instant transition, from paying premiums to receiving pension payments, when the policyholder retires. Moreover, a significant increase in the probability of death for the policyholder over the time horizon is noticeable in the above area plot.

Specially, drawing our attention to the grantee element, many of the guarantees are written when the the interests rate were high. Hence they are deeply in-the-money and we need to hedge away the interest rate risk.

Moreover, in this case study, the underlying asset portfolio consists of 50 % equity and 50 % zero coupon bonds with a maturity of ten years.

**Hedging Efficiency**

Hedging effectiveness is a measurement of how much hedging strategy actually reduces the risk. There are different approaches to define hedging performance. One way is to measure the difference between the Sharpe ratio of the hedged and un-hedged portfolios, that we have already investigated in our previous article series. (See our preceding article for more information.)

Based on the situation and our purpose, we consider the effectiveness of the hedging strategy as follows

$$1-\frac{\text{Net}\,\, 99.5 % \,\, \text{VaR}}{\text{Gross}\,\, 99.5 % \,\, \text{VaR}}.$$

We summarize our results for different hedging frequencies in the following table:

\begin{array}{|c|c|} \hline \hbox{Hedging Frequency} & \hbox{Hedging Effectiveness %} \\\hline \hbox{Monthly} & 65 \hbox{%} \\\hline \hbox{Bi-monthly} & 63 \hbox{%} \\\hline Quarterly & 62 \hbox{%} \\\hline \hbox{Semi-annual} & 56 \hbox{%} \\\hline \hbox{Annual} & 53 \hbox{%} \\\hline \end{array}

As we observe, a more frequent hedging means a better performance. We again remark that transaction costs are not included in this case study for the sake of simplicity.

**Sensitivity of P&L to Hedging Frequency**

We aim at looking into how hedging market implied volatility can reduce the volatility of P&L. We apply a delta hedging scheme in different time frequency. Our results show that the volatility of the profit and loss (P&L) decreases as we hedge more frequently. The following diagram indicates the relationship between P&L-variation and hedging frequency.

Especially directing our attention to the 99.5 % percentile which is equivalent to the SCR under Solvency II, we conclude that SCR can be reduced quite drastically in case of a more frequent hedging as is the case for the monthly and bi-monthly schemes.

**Conclusion**

Implementing dynamic hedging as part of the management actions is an effective technique to reduce the uncertainty of future P&L. As a consequence, VaR can be potentially reduced and hence we have a decrease in the SCR for the insurance company. Furthermore, one may consider applying dynamic hedging strategies for the other key financial measures, e.g., return on risk-adjusted capital (RoRAC).