# Using Neural Networks to Price and Hedge Variable Annuity Guarantees

^{1}

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## Abstract

**:**

## 1. Introduction

_{1}-n

_{2}-n

_{3}architecture refers to a neural network with n

_{1}inputs, n

_{2}nodes in the hidden layer, and n

_{3}nodes in the output layer. (Note that all neural networks utilized here employ a feedforward architecture with a single, dense hidden layer.) For an introduction to neural networks and their structure, as well as the associated terminology and their use in deep learning problems, see Nielsen (2015) or Bishop (2006).

## 2. Methods and Models

#### 2.1. Assumptions and Data

_{t}is the stock price at time t, σ

_{s}is the volatility of the stock, Z

_{s}(t) is a standard Brownian motion process, and r

_{t}is the short rate at time t, as defined by the Hull-White model (Hull and White 1990):

_{r}is the volatility of the short rate, and Z

_{r}(t) is a standard Brownian motion process. We assumed that the two standard Brownian motion processes Z

_{s}(t) and Z

_{r}(t) are independent.

_{t}and is assumed to follow a similar process to the stock price:

_{s}using the 5-year rolling historical volatility of this index. For interest rates, we used daily Treasury Yield rate curves from 1990 to 1999 (U.S. Department of the Treasury 2018). For the parameters in the Hull-White model, we used values of a = 0.35 and σ

_{r}= 0.02, and the value of θ(t) was determined from that day’s Treasury yield curve.

#### 2.2. VA Guarantee Pricing Using MC Simulations

_{20}(t) is the present value of a 20-year annuity-due that begins payment at time T.

_{n}is the benefit base, ratcheted in proportion to the account value less withdrawals.

#### 2.3. Delta-Rho Hedging

#### 2.4. Neural Network Pricing Methodology

- Time until maturity of the contract;
- Spot rate of the yield curve;
- S&P 500 opening price;
- Price of a zero-coupon bond maturing at the end of the contract;
- Volatility of the S&P 500;
- The first three components of a principal component analysis of the yield curve, which are generally interpreted as the height, slope, and curvature of the yield curve;
- Fee for the guarantee;
- “Moneyness” (M) of the guarantee, defined as M = (G − V)/G, where V is the account value. For the GMWB, G is the amount that the policyholder is allowed to withdraw in a given year. For the other guarantee types, G is the amount guaranteed to the policyholder on their death, survival, or annuitization, respectively. Because this amount ratchets up on policy anniversaries, the guarantee amount represents the highest value seen to date at a policy anniversary.

#### 2.5. Neural Network Hedging Methodology

#### 2.6. Neural Network Training

## 3. Results

#### 3.1. Pricing

^{2}, a measure has been used by other researchers to measure neural network accuracy, as in, e.g., Hattab et al. (2013). There are, however, several other options for neural network performance metrics; a few such possibilities include correlation (Chen and Sutcliffe 2012), relative error (Hejazi and Jackson 2016), and mean square prediction error (Garcia and Gençay 2000). We expect that any of these metrics would have yielded substantially the same results. Table 2 gives the R

^{2}values calculated when comparing the neural networks values to the corresponding values calculated by MC simulations.

^{2}measure, we can see that the neural network performs quite well for all four types of guarantees, though the performance for the GMWB is considerably worse than for the other three guarantees; this is primarily due to the presence of a few outliers. A scatterplot of the results for the GMDB is shown in Figure 3; from this figure, we can see a high degree of accuracy and no evidence of systematic biases or patterns. The scatterplots for the other guarantees show similar results.

#### 3.2. Hedging

_{1}from Cotter and Hanly (2006) that uses standard deviation instead of variance:

_{H}is the standard deviation of the portfolio’s returns with hedging and SD

_{U}is the standard deviation of the portfolio’s returns without hedging. Using the standard deviation instead of the variance provides a more conservative metric for the degree to which a hedging strategy reduces the portfolio’s risk. For this metric, a value close to 1 is good because hedging reduces the standard deviation of returns almost completely. A value close to 0 or negative is bad, because that means that hedging either produces similar or worse standard deviations of returns than leaving the portfolio unhedged.

_{i}and L

_{i}are the values of the assets and liabilities on day i, respectively, and D is the total number of days under consideration for the contract. Assets are the stocks and bonds used to hedge a given guarantee and liability is the guarantee itself. Graphs of the total value and the percentage changes of the hedged position produced from this error function for the GMDB are shown in Figure 4 below.

_{1}for different values of k. For values of k equal to 0.01 and 0.02, the network reduces the total level of assets used to hedge the guarantee, while still providing a significant decrease in the standard deviation of returns, as was intended. However, once k increases to 0.03 and 0.04, the network converges to a solution that almost exactly matches the actual account value of the GMDB account, while also using more assets than the previous strategies. For all values of k, the network generalizes well from training to testing and outperforms the delta-rho strategy considerably, in terms of its ability to reduce the standard deviation of returns, while providing a significant time savings. (Note that the delta-rho strategy does not require the data to be split into training and testing sets; the values of HE

_{1}reported for this strategy apply to the entire dataset).

## 4. Conclusions

_{1}metric) and the cost of capital required for the hedging assets. Of course, whether this tradeoff is worthwhile will depend on the cost of capital relative to the benefit of the more efficient calculations, cost of computing resources, and other considerations.

_{1}metric. In all cases, the neural networks exhibited promising results.

_{1}hedging metric we used here seems to be a reasonable method for capturing the performance of the hedge for these guarantees, many other metrics are possible and have been used by various researchers. It would be instructive to repeat this type of study using some of the other measures. In addition, it may be helpful to generalize the results obtained here by applying this methodology to other time periods. The time period used here was chosen arbitrarily in order to demonstrate the methodology, but a deeper look into the relative performance of the neural network over different times periods spanning a broad range of interest rate and stock market conditions would seem to be warranted. Further, while our model accounts for mortality, it does not incorporate elements of policyholder behavior, such as lapses or dynamic withdrawal behavior. As neural networks are very flexible, they could certainly be trained to incorporate these important aspects of variable annuities. Finally, one of the most significant benefits of using the neural network to hedge is the flexibility of choice of hedging assets. While we used only two assets here, namely a stock index and bond, the neural network approach could be used with virtually any number and/or type of assets. There is an opportunity to explore this idea and determine which combinations of asset types are most effective for hedging the various types of VA guarantees.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sample neural network. This neural network is an example of a feedforward neural network with a single hidden layer of fully connected neurons. The architecture of this network can be described as 8-10-3.

**Figure 3.**Scatterplot of the results of using a neural network to approximate the MC pricing methodology for the guaranteed minimum death benefit (GMDB). For this guarantee, the neural network (NN) achieves an R

^{2}better than 0.997.

**Figure 4.**Results for hedging for GMDB using squared error function: (

**a**) The daily position value over the 10-year life of the variable annuities (VA) contract; (

**b**) the daily percentage change in the position over the life of the contract, including both training and testing.

**Figure 5.**Results for hedging for GMDB using k = 0.02 error function: (

**a**) The daily position value over the 10-year life of the VA contract; (

**b**) the daily percentage change in the position over the life of the contract, including both training and testing.

GMDB | GMAB | GMIB | GMWB |
---|---|---|---|

33 | 93 | 2.6 | 95 |

GMDB | GMAB | GMIB | GMWB |
---|---|---|---|

0.9972 | 0.9952 | 0.9986 | 0.9870 |

Hedging | Delta-Rho | k = 0 | k = 0.01 | k = 0.02 | k = 0.03 | k = 0.04 |
---|---|---|---|---|---|---|

Training HE_{1} | 0.8805 | 0.9965 | 0.9964 | 0.9958 | 0.9932 | 0.9895 |

Testing HE_{1} | 0.9962 | 0.9961 | 0.9955 | 0.9929 | 0.9891 | |

Time (s) | ~4200 | 178 | 134 | 139 | 63 | 51 |

**Table 4.**Hedging metrics for different hedging strategies and error functions for guaranteed minimum accumulation benefit (GMAB).

Hedging | Delta-Rho | k = 0 | k = 0.0025 | k = 0.005 | k = 0.0075 | k = 0.01 |
---|---|---|---|---|---|---|

Training HE_{1} | 0.9956 | 0.9991 | 0.9967 | 0.9961 | 0.9957 | 0.9957 |

Testing HE_{1} | 0.9984 | 0.9945 | 0.9937 | 0.9931 | 0.9930 | |

Time (s) | ~4200 | 207 | 70 | 76 | 65 | 67 |

**Table 5.**Hedging metrics for different hedging strategies and error functions for guaranteed minimum income benefit (GMIB).

Hedging | Delta-Rho | k = 0 | k = 0.001 | k = 0.002 | k = 0.003 | k = 0.004 |
---|---|---|---|---|---|---|

Training HE_{1} | 0.9648 | 0.9966 | 0.9938 | 0.9937 | 0.9937 | 0.9937 |

Testing HE_{1} | 0.9981 | 0.9647 | 0.9645 | 0.9645 | 0.9645 | |

Time (s) | ~4200 | 429 | 63 | 27 | 26 | 27 |

**Table 6.**Hedging metrics for different hedging strategies and error functions for guaranteed minimum withdrawal benefit (GMWB).

Hedging | Delta-Rho | k = 0 | k = 0.01 | k = 0.02 | k = 0.03 | k = 0.04 |
---|---|---|---|---|---|---|

Training HE_{1} | 0.9717 | 0.9991 | 0.9966 | 0.9961 | 0.9960 | 0.9955 |

Testing HE_{1} | 0.9978 | 0.9896 | 0.9869 | 0.9863 | 0.9828 | |

Time (s) | ~4200 | 400 | 55 | 46 | 49 | 47 |

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**MDPI and ACS Style**

Doyle, D.; Groendyke, C. Using Neural Networks to Price and Hedge Variable Annuity Guarantees. *Risks* **2019**, *7*, 1.
https://doi.org/10.3390/risks7010001

**AMA Style**

Doyle D, Groendyke C. Using Neural Networks to Price and Hedge Variable Annuity Guarantees. *Risks*. 2019; 7(1):1.
https://doi.org/10.3390/risks7010001

**Chicago/Turabian Style**

Doyle, Daniel, and Chris Groendyke. 2019. "Using Neural Networks to Price and Hedge Variable Annuity Guarantees" *Risks* 7, no. 1: 1.
https://doi.org/10.3390/risks7010001