Portfolio management for insurers and pension funds and COVID-19: targeting volatility for equity, balanced, and target-date funds with leverage constraints

Abstract

Insurers and pension funds face the challenges of historically low-interest rates and high volatility in equity markets, that have been accentuated due to the COVID-19 pandemic. Recent advances in equity portfolio management with a target volatility have been shown to deliver improved on average risk-adjusted return, after transaction costs. This paper studies these targeted volatility portfolios in applications to equity, balanced, and target-date funds with varying constraints on leverage. Conservative leverage constraints are particularly relevant to pension funds and insurance companies, with more aggressive leverage levels appropriate for alternative investments. We show substantial improvements in fund performance for differing leverage levels, and of most interest to insurers and pension funds, we show that the highest Sharpe ratios and smallest drawdowns are in targeted volatility-balanced portfolios with equity and bond allocations. Furthermore, we demonstrate the outperformance of targeted volatility portfolios during major stock market crashes, including the crash from the COVID-19 pandemic.

1. Introduction

The current low-interest rate environment and high equity market volatility, that have been accentuated due to the COVID-19 pandemic, are a challenge for insurers and pension funds. Although equity investments offer the potential for higher returns, the increased volatility of these investments must be taken into account. We analyze whether insurers and pension funds with constraints on the degree of leverage can benefit from targeted constant volatility portfolios for equity funds, balanced funds, and target-date funds with declining glide paths of reduced equity exposure as the target-date approaches. We consider these strategies with a range of leverage constraints from conservative levels, as seen commonly in pension funds and insurance companies, to more aggressive levels often associated with alternative investments in hedge funds.

The findings in this paper directly relate to industry applications of targeted volatility portfolios that have been increasingly occurring in recent years with univariate time series forecasting methods. Targeted volatility portfolios are now widely applied in the provision of variable annuity products in the insurance industry. In the Actuary magazine article https://theactuarymagazine.org/tag/target-volatility-funds/, there is a discussion of targeted volatility funds. The statement is made that “the volatility-target concept as the new benchmark in variable annuity guarantees. Transamerica, Lincoln, AIG, Ameriprise, Nationwide – in fact, all carriers except Jackson National, Pacific Life, and Principal Life – have since launched versions of managed volatility funds.” Recent research into the application of targeted volatility portfolios with univariate time series forecasting methods to retirement income products can be found in Li et al. (2022) and Olivieri et al. (2022).

The outperformance of targeted constant market volatility portfolios is driven from the well-known negative relationship between equity market returns and conditional volatility, see Hocquard et al. (2013), Moreira & Muir (2017) and Doan et al. (2018). This relationship is primarily explained by the volatility feedback effect (see Poterba & Summers, 1986) where higher (lower) volatility results in a stock market price fall (rise) as the required rate of return on the stock market increases (decreases).¹

The approach to targeting a constant market volatility that was developed in Doan et al. (2018) showed how a simple univariate time series method could generate higher risk-adjusted returns, relative to the existing literature.² Doan et al. (2018) apply their methodology without leverage constraints, and an important contribution in this paper is that we extend these results to include constraints on leverage. We focus on the U.S. market. The results in Doan et al. (2018) cover a broad range of equity markets so that the results extend to those other markets. Results for equity portfolios demonstrate the outperformance of constant volatility portfolios over the different leverage constraints and that there are small increases in the Sharpe ratio as leverage increases.

In addition, we also extend the target volatility analysis to traditional balanced portfolios, with 65:35 and 55:45 splits between equity and bonds. We find that the highest levels of the Sharpe ratio and smallest drawdowns are in targeted volatility-balanced portfolios. Applications to target-date funds over a range of investment life cycles (35, 25, and 15 years) are also studied. Three glide paths of declining equity exposure are examined; aggressive, moderate, and conservative. Furthermore, three leverage constraints are examined; no leverage, conservative leverage, and aggressive leverage. Outperformance is found in constant volatility target-date portfolios with both conservative and aggressive leverage constraints, where more aggressive leverage leads to higher average investment outcomes with higher variability.

Of significant current interest, we examine investment performance during major stock market crashes including COVID-19. We study the largest U.S. stock market crash in the last 100 years which is associated with the Great Depression in the 1930s. We also study the largest U.S. stock market crash in the last 50 years which is associated with the Global Credit Crisis of 2008. And finally, we study the recent stock market crash from the COVID-19 crisis. Targeted volatility typically results in substantial limitation to portfolio drawdowns during these crashes. For example, in the COVID-19 crisis, the U.S. CRSP value-weighted index, including dividends fell 34.27% from 19 February 2020 to 23 March 2020, whereas the corresponding targeted constant average stock market volatility investment portfolio fell only 18.61%. Over this same time period, the balanced and targeted volatility-balanced portfolios fell 23.52% and 12.64 %, respectively. The minimization of portfolio drawdowns is an important criteria for many types of investments, including that in pension and insurance companies. In addition, drawdown minimization substantially contributes to the long-run return outperformance of targeted volatility stock index portfolios and targeted volatility-balanced portfolios. Targeted volatility results in volatility more consistent with historical average market volatility, avoiding the high volatility during and following market crashes, and also provides higher returns for periods including the crashes.

This paper is organized as follows. The next section describes the volatility forecasting, trading strategy, and data. The third section is on average returns and risk for equity, balanced, and target-date funds, over a range of leverage constraints. The fourth section is on investment performance during major stock market crashes, and the final two sections are on discussion and conclusions.

2. Volatility Forecasting and Trading Strategy

We begin by describing the asset class and portfolios that are analyzed. This is followed by a discussion of the volatility forecasting approach, the trading strategy, and portfolio performance statistics, similar to Doan et al. (2018) and repeated here for completeness.

We implement the strategy for three different portfolios; equity, balanced. and target date. Our equity portfolios are market portfolios tracking the market index, like that of an equity index ETF. These portfolios are a substantial component of assets under management by insurers and pension funds. Our balanced portfolios contain allocations of 65% to equity and 35% to bonds, and 55% to equity and 45% to bonds. The 65:35 split is a standard asset allocation for a balanced portfolio of an insurer or pension fund, whereas the 55:45 split is a more conservative asset allocation. In addition, we also study target-date portfolios due to their importance for defined-contribution pension plans. These portfolios have declining equity and risk exposure as the target-date approaches, with rebalancing to conservative assets, typically bonds. Our target-date portfolios invest in the equity market index and bond index over life cycles of 35, 25, and 15 years.

2.1 Volatility forecasting

One-day-ahead forecasts of portfolio return volatility are generated from an outlier-corrected GARCH(1,1) model, as the GARCH(1,1) model is the most widely utilized return volatility forecasting model in settings where daily returns are the highest frequency readily available.³ Bond volatility is not used in the volatility calculations as it is relatively small when compared to equity volatility. The GARCH model was originally proposed by Bollerslev (1986), though it has been shown to have biased parameter estimates and forecasts when outliers are present in the return series, see Gregory & Reeves (2010), Carnero et al. (2006), and Harvey (2013). To overcome this problem, we follow the approach in Doan et al.(2018), by first winsorizing returns at a specified level, $r_{\max}$ of 4%, such that the return series over the estimation period is in the range between $-r_{\max}$ and $r_{\max}$ . The estimation then follows standard Gaussian quasi-maximum likelihood estimation of the GARCH(1,1) model on the winsorized return series. Further details on the properties of these estimations for sample sizes of 1,000 observations can be found in Doan et al. (2018), which also includes Monte Carlo results.

The weighted daily equity return (in percentage) at date t is given by;

(1) \begin{equation} r_t = \varepsilon _t, \end{equation}

where $r_t$ is winsorized at $\pm 4\%$ , $\varepsilon _t$ is i.i.d $(0,\sigma ^2_t)$ , and the conditional variance $\sigma ^2_t$ follows the GARCH(1,1) process;

(2) \begin{equation} \sigma ^2_t = \alpha _0 + \alpha _1\varepsilon ^2_{t-1} +\alpha _2\sigma ^2_{t-1} \end{equation}

with the following parameter constraints $\alpha _0 \gt 0$ , $\alpha _1 \geq 0$ , $\alpha _2 \geq 0$ , and $\alpha _1 + \alpha _2 \lt 1$ . The starting values for $\hat{\varepsilon }^2_0$ and $\hat{\sigma }^2_0$ are the unconditional sample variance.

Given the estimated parameter set $\{\hat{\alpha }_{0},\hat{\alpha }_{1}, \hat{\alpha }_{2}\}$ for each estimation window of 1,000 observations, we compute the one-day ahead volatility forecast using the following equation;

(3) \begin{equation} \hat{\sigma }^2_{t+1} = \hat{\alpha }_0 + \hat{\alpha }_1\hat{\varepsilon }^2_t + \hat{\alpha }_2\hat{\sigma }^2_t \end{equation}

2.2 Trading strategy

To implement our targeted volatility approach, we first determine the desired absolute daily target volatility level of equity returns, which is constant over the period of analysis. We also introduce a daily participation ratio which is the weight $w_t$ invested in the market equity portfolio:

(4) \begin{equation} w_t=\frac{target \text{ } volatility}{\hat{\sigma }_t}, \end{equation}

where $\hat{\sigma }_t$ is the volatility forecast for trading day $t$ . To control risk in leveraging the equity portfolio, we set different levels of the maximum participation ratio, namely 1, 1.5, 2, and unrestricted value.

The intuition underlying the dynamic trading strategy follows Doan et al. (2018). When the forecast volatility for a given trading day is greater than the target volatility, we shift away from equity markets by selling futures contracts on the equity market, leading to a decrease in the portfolio volatility. When the forecast volatility is less than the target volatility, we purchase futures contracts on the equity market, in order to increase our equity exposure, leading to an increase in the portfolio volatility.

In implementation, we also set a threshold weight change ( $\delta$ ), to minimize excessive turnover, in that we only change our market exposure when the new participation ratio differs from the prior by an absolute amount greater than $\delta$ . For large institutional investors such as insurers and pension funds, the transaction costs of the futures overlay strategy for the U.S. equity market are negligible relative to the return performance of the portfolio. Doan et al. (2018) conduct a transaction cost analysis for running the futures overlay strategy on a $\$$ 100 million U.S. market portfolio based on a $\delta$ value of 0.2 and find the average annual cost to be less than 10 basis points, under a conservative assumption of relatively high brokerage fees ( $\$$ 10 per futures contract) and also in an environment of no limit on the maximum participation ratio. In their paper, they also table results on the percentage of days when the participation ratio changes over a range of target volatility levels and $\delta$ ’s. They find that when targeting the average level of U.S. market return volatility, the percentage of days since the 1980s that required changes to the futures positions was approximately 5% with a $\delta$ value of 0.2, and 12.5% with a $\delta$ value of 0.1. In our current environment with limits on the maximum participation ratio and with brokerage fees typically less than $\$$ 5 per futures contract, average annual transaction costs are unlikely to exceed 10 basis points.

In the case where we take positions in the futures market, the daily return at date $t$ of the trading strategy on the equity portfolio is computed as;

(5) \begin{equation} r_{equity,t}=(w_t - 1)r_{futures,t} + r_{market,t} \end{equation}

where $r_{futures,t}$ is the index futures return at date $t$ .

During the period around the stock market crash associated with the Great Depression, futures contracts on the stock index were not available so we instead compute the daily return at date $t$ of the trading strategy on the targeted volatility equity portfolio as;

(6) \begin{equation} r_{equity,t}=w_t\left(1+r_{market,t}\right) - (w_t-1)\left(1+r_{f,t}\right) -1 \end{equation}

where $r_{f,t}$ is the borrowing and lending rate at date $t$ . This is based on approximating future returns by subtracting the risk-free rate from spot returns.

For the balanced portfolio, the daily returns at day $t$ of the trading strategy are computed as;

(7) \begin{equation} r_{balanced,t}= 0.65r_{equity,t} + 0.35r_{bond,t} \end{equation}

where $r_{bond,t}$ is the daily bond return at date $t$ .

With respect to the target-date portfolio, we consider a worker who contributes 9% of his/her salary to a target-date fund at the end of each year over his/her career. His/her initial annual wage is 20,000 USD, which grows by 4% in nominal terms every year.⁴ The target-date fund equity contributions are reset at the end of each year, without considering tax implications or transaction costs. To determine the glide path of the target-date fund, we collect the asset allocation for aggressive, moderate, and conservative strategies from the Morningstar Lifetime Allocation Indexes as of June 2017, when we collected this data. The aggressive strategy has the highest levels of equity exposure, while the conservative strategy has the lowest levels of equity exposure. Given our focus on equity and bond investment, we aggregate the contribution of non-equity securities into the bond asset class. We also linearly interpolate the quinquennial values provided by Morningstar to obtain yearly data. The percentage of equity invested in the portfolio is presented in Fig. 1. The daily returns of equity and bond components on day $t$ are given by $r_{equity,t}$ and $r_{bond,t}$ , respectively.

Figure 1 Equity contribution to target-date portfolio. This figure displays the equity contribution to the target-date fund from Morningstar Lifetime Allocation Indexes.

2.3 Summary statistics

We define (in percentages) the annualized returns, $\mu$ , and annualized standard deviation, $\sigma$ , of a given strategy as;

(8) \begin{equation} \mu = 100\left[(1+\hat{r})^{252} - 1\right] \text{ with } \hat{r} = Y^\frac{1}{n} - 1 \end{equation}

(9) \begin{equation} \sigma = 100\sqrt{252\frac{\sum ^{n}_{t=1} (r_t - \bar{r})^2}{n - 1}}, \end{equation}

where $n$ is the total number of trading days in the investment sample period, $r_t$ is the daily return of the equity portfolio or balanced portfolio, $\bar{r}$ and $Y$ are the average daily return and cumulative amount from $1$ initial value in the investment period, respectively. The annualized Sharpe Ratio (SR) is calculated as;

(10) \begin{equation} \text{SR} = \frac{\mu _{ER}}{\sigma }, \end{equation}

where $\mu _{ER} = 100[(1+\hat{r}_{ER})^{252} - 1]$ with $\hat{r}_{ER}$ being the geometric average of daily portfolio excess returns $r_t - r_{ft}$ . Summary statistics on portfolio daily returns are also calculated, in particular, the daily mean return, the 25th, 50th, 75th percentiles of daily returns, the maximum (max) and minimum daily return (min). We also calculate the worst cumulative returns in 1, 5, and 10 years, where the worst cumulative returns over T years are defined as;

(11) \begin{equation} \text{min T-year return} = minimum \{R_{j}, j=1,\ldots,n-T\times 252+1\}, \end{equation}

where $R_{j}=\prod _{t=j}^{T\times 252+j-1}(1+r_{t})-1$ .

For the results on target-date portfolios, we calculate the ending market value of a target-date fund that starts to invest at every trading day in the sample and present their summary statistics of average, standard deviation, minimum and maximum values. We also report the internal rate of return (IRR) of the average target-date ending value by solving the following equation for IRR;

(12) \begin{equation} \text{Average ending value} = \sum ^{T}_{t=1} C_t (1 + IRR)^{T-t} \end{equation}

where $C_t$ is the end of year $t$ contribution and $T$ is the number of years in the target-date fund life.

2.4 Data

The data for this study are from the following sources. The U.S. market index returns are the CRSP value-weighted market returns, as these are commonly used to represent a very well diversified broadly based U.S. equity portfolio. The series is adjusted to account for dividend re-investment and runs from 9 May 1978 to 30 June 2020. The same series over the period 1 July 1926 to 30 July 1937 is also used to study the performance of the target volatility strategy during the stock market crash in the Great Depression. We obtain daily settlement price series of the most liquid futures contracts on the S&P500 from Datastream. The daily returns of futures contracts start on 23 April 1982. For the bond data, we collect the U.S. bond return index that invests in a wide set of government and corporate bonds, provided by Barclays (mnemonic: LHAGGBD). The bond returns start at the same time as equity returns and the risk-free rate is sourced from the Kenneth French Data Library.

3. Average Returns and Risk with Leverage Constraints

3.1 Equity portfolios

Fig. 2 displays the participation ratio when there is no restriction on the amount of leverage over our sample period starting 26 April 1982 and finishing 30 June 2020, from the strategy targeting a daily 1% standard deviation of equity market returns. Daily threshold weight changes ( $\delta$ ) of 0.05, 0.1, and 0.2 are considered. Often the participation ratio is above 1 and sometimes above 1.5. On a few occasions, it exceeds 2. Leverage is utilized during periods when the participation ratio is greater than 1. In our study, we analyze the no restriction on leverage case and three cases where there is a leverage restriction. These restrictions lead to three levels of the maximum participation ratio; 1, 1.5, and 2. The 1 (no leverage) and 1.5 levels correspond to settings common in pension funds and insurance companies, whereas the 2 level is more applicable to alternative investments, for example in hedge funds.

Figure 2 The daily participation ratio. This figure displays the participation ratio without a leverage restriction over the sample period of 26 April 1982 to 30 June 2020 from the strategy targeting a daily 1% standard deviation of equity market returns, with a daily threshold weight change of 0.05, 0.1, and 0.2.

Summary statistics on U.S. equity market portfolio performance with a threshold weight change ( $\delta$ ) of 0.05 are displayed in Table 1. Over our sample period, starting 26 April 1982 and finishing 30 June 2020, the annualized average return of the market portfolio is 11.59%, with an annualized standard deviation of 17.70%. For all four maximum participation ratio levels, all targeted volatility portfolios outperform the market portfolio. These targeted volatility portfolios all have Sharpe ratios exceeding that of the market portfolio.

Table 1. Equity portfolio performance statistics with $\delta =0.05$ . This table presents the summary statistics of equity portfolios over the sample period of 26 April 1982 to 30 June 2020. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model. The statistics include the annualized average return in percentage ( $\mu$ ), annualized standard deviation in percentage ( $\sigma$ ), annualized Sharpe’s ratio (SR), daily mean return, maximum daily drawdown (min), 25th, 50th, 75th percentiles of daily returns, the maximum daily return (max), and the worst cumulative returns in 1, 5, and 10 years (min 1y to min 10y). The daily target constant volatility portfolios have a threshold weight change of 0.05, and the panels present the results of different levels of maximum participation ratios of 1, 1.5, 2, and unrestricted. In each panel, the rows present the results from the trading strategy that targets a constant level of daily market volatility.

When targeting the average daily market volatility⁵ (a constant daily return volatility of 1%) with GARCH(1,1), the annualized average return rises to 12.58%, with a maximum participation ratio of 1.5. The Sharpe ratio is 0.52, compared with 0.43 for the market portfolio. Furthermore, the volatility targeting reduces the tail values of the daily return distribution. This reduction in tail risk is also present over longer investment horizons, with the worst 1-year return being –32.91%, compared with –46.85% for the market portfolio. Over a 10-year investment horizon, the worst return was –8.35%, compared with –28.43% for the market portfolio.

When the maximum participation ratio rises to 2, the performance in terms of the Sharpe ratio increases slightly from 0.52 to 0.53, while the characteristics of the daily return distributions and the tail risk over longer investment horizons remain mostly unchanged. When the participation ratio is unrestricted, there is a further small increase in the Sharpe ratio from 0.53 to 0.54, with similar levels of tail risk to when the maximum participation ratio is set at 2.

Even though the Sharpe ratio is not changing substantially when the maximum participation ratio is ranging from 1.5 to unrestricted, there is still an impact on average returns and volatility from leverage. Approximately proportional changes in average returns and volatility are resulting in the Sharpe ratio having small changes when the maximum participation ratio is changed from 1.5 to 2 and then from 2 to unrestricted. However, the changes in average returns are more substantial over these varying leverage levels. For example, when targeting the average daily market volatility with GARCH(1,1) and a maximum participation ratio of 1.5, the annualized average return is 12.58%, compared to 13.21% when the maximum participation ratio is 2.

Table 2 displays the summary statistics on portfolio performance when the $\delta$ is 0.1, with results being very similar to Table 1. This favors the 0.1 threshold which is associated with lower transaction costs than the 0.05 threshold. When the $\delta$ is set at 0.2 (Table 3), there are occasional small deteriorations in portfolio performance, with small declines in the Sharpe ratio, compared with $\delta$ set at 0.1. For example, when targeting the average daily market volatility with GARCH(1,1) and a maximum participation ratio of 1.5, the Sharpe ratio is 0.51, compared to 0.52 when $\delta$ is 0.1. The remainder of the analysis in this paper is conducted with $\delta$ set at 0.1.

Table 2. Equity portfolio performance statistics with $\delta =0.1$ . This table presents the summary statistics of equity portfolios over the sample period of 26 April 1982 to 30 June 2020. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model. The statistics include the annualized average return in percentage ( $\mu$ ), annualized standard deviation in percentage ( $\sigma$ ), annualized Sharpe’s ratio (SR), daily mean return, maximum daily drawdown (min), 25th, 50th, 75th percentiles of daily returns, the maximum daily return (max), and the worst cumulative returns in 1, 5, and 10 years (min 1y to min 10y). The daily target constant volatility portfolios have a threshold weight change of 0.1, and the panels present the results of different levels of maximum participation ratios of 1, 1.5, 2, and unrestricted. In each panel, the rows present the results from the trading strategy that targets a constant level of daily market volatility.

Table 3. Equity portfolio performance statistics with $\delta =0.2$ . This table presents the summary statistics of equity portfolios over the sample period of 26 April 1982 to 30 June 2020. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model. The statistics include the annualized average return in percentage ( $\mu$ ), annualized standard deviation in percentage ( $\sigma$ ), annualized Sharpe’s ratio (SR), daily mean return, maximum daily drawdown (min), 25th, 50th, 75th percentiles of daily returns, the maximum daily return (max), and the worst cumulative returns in 1, 5, and 10 years (min 1y to min 10y). The daily target constant volatility portfolios have a threshold weight change of 0.2, and the panels present the results of different levels of maximum participation ratios of 1, 1.5, 2, and unrestricted. In each panel, the rows present the results from the trading strategy that targets a constant level of daily market volatility.

3.2 Balanced portfolios

Summary statistics of U.S. balanced (65:35) portfolio performance are displayed in Table 4. The annualized average return of the balanced portfolio is 11.94% and exceeds that of the equity market portfolio, before transaction costs with daily rebalancing, over the same sample period. It has an annualized standard deviation of 11.74%, compared to 17.70% for the equity market portfolio. This results in the balanced portfolio’s Sharpe ratio (SR = 0.68) being substantially higher than the equity market portfolio’s Sharpe ratio (SR = 0.43). In addition, the maximum daily drawdown of the balanced portfolio is –11.32%, compared with –17.41% for the equity market portfolio. Furthermore, the maximum yearly drawdown of the balanced portfolio is –31.45%, compared with –46.85% for the equity market portfolio.

Table 4. The 65-35 balanced portfolio performance statistics with $\delta =0.1$ . This table presents the summary statistics of balanced portfolios over the sample period of 26 April 1982 to 30 June 2020. The balanced portfolio invests 65% in equity markets and 35% in bond markets. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model. The statistics include the annualized average return in percentage ( $\mu$ ), annualized standard deviation in percentage ( $\sigma$ ), annualized Sharpe’s ratio (SR), daily mean return, maximum daily drawdown (min), 25th, 50th, 75th percentiles of daily returns, the maximum daily return (max), and the worst cumulative returns in 1, 5, and 10 years (min 1y to min 10y). The daily target constant volatility portfolios have a threshold weight change of 0.1, and the panels present the results of different levels of maximum participation ratios of 1, 1.5, 2, and unrestricted. In each panel, the rows present the results from the trading strategy that targets a constant level of daily market volatility for the equity component.

Overall performance of the balanced portfolio dominates that of the equity market portfolio. This result highlights how the conventional approach to mitigating the volatility of equity market returns through balancing with bonds, is very effective in generating strong long-run portfolio performance. In particular, the bond component of balanced portfolios provides a support to balanced portfolio values during periods of large drawdowns from falls in the equity market, that leads to strong growth in cumulative portfolio value over time.

Outperformance of balanced portfolios is extended further when the equity component of the portfolio has targeted volatility. Sharpe ratios rise through targeting volatility. There is a monotonically increasing rise in SR as the target level of equity return volatility is lowered. This occurs for all levels of the maximum participation ratio. When targeting the average daily market return volatility with GARCH(1,1), the annualized average return rises to 12.53% and the maximum daily drawdown is –9.16%, with the maximum participation ratio set at 1.5. The maximum yearly drawdown is –20.11%, compared with –31.45% for the balanced portfolio. In addition, the worst 10-year return is 22.08%, compared with 5.63% for the balanced portfolio. Over the different target levels of volatility for the balanced portfolios, the SR, maximum daily, and yearly drawdowns and other characteristics of the tail in the return distributions, are relatively constant with respect to the leverage levels.

In addition, we also report results for a more conservative U.S. balanced portfolio where the equity:bond ratio is 55:45, and these are displayed in Table 5. The annualized average return of this balanced portfolio is 11.94% which is the same as that of the 65:35 balanced portfolio. However, the annualized standard deviation of this more conservative portfolio is lower at 10.26%, leading to a higher Sharpe ratio (SR = 0.78). Furthermore, the maximum daily drawdown of the more conservative balanced portfolio is –9.58%, compared with –11.32% for the 65:35 balanced portfolio. Over a one-year investment horizon, the worst return is –26.93% for the 55:45 balanced portfolio, compared with –31.45% for the 65:35 balanced portfolio. Over a 10-year investment horizon, the worst return is 16.56% for the 55:45 balanced portfolio, compared with 5.63% for the 65:35 balanced portfolio.

Table 5. The 55-45 balanced portfolio performance statistics with $\delta =0.1$ . This table presents the summary statistics of balanced portfolios over the sample period of 26 April 1982 to 30 June 2020. The balanced portfolio invests 55% in equity markets and 45% in bond markets. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model. The statistics include the annualized average return in percentage ( $\mu$ ), annualized standard deviation in percentage ( $\sigma$ ), annualized Sharpe’s ratio (SR), daily mean return, maximum daily drawdown (min), 25th, 50th, 75th percentiles of daily returns, the maximum daily return (max), and the worst cumulative returns in 1, 5, and 10 years (min 1y to min 10y). The daily target constant volatility portfolios have a threshold weight change of 0.1, and the panels present the results of different levels of maximum participation ratios of 1, 1.5, 2, and unrestricted. In each panel, the rows present the results from the trading strategy that targets a constant level of daily market volatility for the equity component.

Table 6. 35-year target-date fund performance statistics. This table presents summary statistics on cumulative amounts and the internal rate of return (IRR) at the end of target-date funds over 35-year life cycles in the U.S. The period of analysis is from 26 April 1982 to 30 June 2020. The columns present the results under aggressive, moderate, and conservative strategies. Panel A presents results without volatility targeting and Panel B presents results with targeting daily market volatility at 1%, with maximum participation ratios of 1, 1.5, and 2. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model, and the threshold weight change is 0.1. The 35-year cumulative amount that starts at each trading day in the period of analysis is calculated, where the equity contribution is sourced from Morningstar Lifetime Allocation Indexes. It is assumed that the worker (investor) starts with a salary of 20,000 USD, which grows by 4% in nominal terms each year during a 35-year career. The worker contributes 9% of his/her salary to target-date fund at the end of each year for their 35 years of work. The target-date fund rebalances daily without considering tax implications or transaction costs.

Outperformance of balanced portfolios is again extended further when the equity component of the portfolio has targeted volatility. When targeting the average daily market return volatility, the annualized average return is 12.43% (SR = 0.86) and the maximum daily drawdown is –7.75%, with the maximum participation ratio set at 1.5. The maximum yearly drawdown is –17.09%, compared with –26.93% for the 55:45 balanced portfolio. In addition, the worst 10-year return is 30.99%, compared with 16.56% for the 55:45 balanced portfolio. Overall, the 55:45 balanced portfolios outperform relative to the 65:35 balanced portfolios, both with and without the equity component of the portfolio having a volatility target, and over all the leverage levels studied.

3.3 Target-date portfolios

We next move to analysis of target-date funds over life cycles of 35, 25, and 15 years. Table 6 displays results for the 35-year life cycle, with cumulative amounts calculated over the 35-year investments for the period of analysis. Panel A contains results when there is no targeting of a constant volatility for the equity component. The investment strategies considered are aggressive, moderate, and conservative. The mean cumulative amount increases as the investment strategy has a greater equity exposure. These means are $\$$ 581,898, $\$$ 641,232, and $\$$ 692,538 for the conservative, moderate, and aggressive strategies, respectively. The variability in the cumulative amounts is not too different, between the three investment strategies, when considering standard deviation and differences between the maximum and minimum cumulative amounts.

Results from targeting the sample volatility of daily equity returns (1%) with GARCH forecasts are displayed in Panel B. Cumulative amounts rise in all summary statistics on investment performance, relative to results with no volatility targeting on the equity component when the maximum participation ratio is 1.5 and 2. The mean cumulative amounts, with a maximum participation ratio of 1.5, are $\$$ 652,772, $\$$ 736,941, and $\$$ 809,166 for the conservative, moderate, and aggressive strategies, respectively. These mean cumulative amounts rise further to $\$$ 690,271, $\$$ 784,743, and $\$$ 862,591 with a maximum participation ratio of 2. Furthermore, the variability of cumulative amounts with the GARCH volatility targeting rises when the maximum participation ratio is 1.5 and 2, relative to no volatility targeting. Volatility targeting with no leverage (maximum participation ratio of 1) results in cumulative amount means and variabilities reducing, relative to no volatility targeting.

Internal rate of return (IRR) of mean cumulative amounts rises from volatility targeting with leverage. For example, an aggressive strategy with no volatility targeting has an IRR of 9.78%, whereas with volatility targeting and a maximum participation ratio of 2, this rises to 10.85%. The minimum cumulative amounts also rise from volatility targeting with leverage, in-line with drawdown mitigation from volatility targeting. For example, an aggressive strategy with no volatility targeting has a minimum cumulative amount of $\$$ 525,889, whereas with volatility targeting and a maximum participation ratio of 2, this rises to $\$$ 714,500.

Tables 7 and 8 display results for the 25- and 15-year life cycles, respectively. Similar patterns to Table 6 are also observed in these tables, though cumulative amounts reduce as the life cycle years reduce. As an approximate summary, for every 5-year reduction in the life cycle, the cumulative amount reduces by half. The lowest average investment outcomes are from the 15-year target date with a conservative glide path. In this case, with no volatility targeting, the mean cumulative amount is only $\$$ 71,633. Volatility targeting with a maximum participation ratio of 2 results in this mean cumulative amount rising to $\$$ 76,076. This also highlights the importance of a sufficiently long investment period.

Table 7. 25-year target-date fund performance statistics. This table presents summary statistics on cumulative amounts and the internal rate of return (IRR) at the end of target-date funds over 25-year life cycles in the U.S. The period of analysis is from 26 April 1982 to 30 June 2020. The columns present the results under aggressive, moderate, and conservative strategies. Panel A presents results without volatility targeting and Panel B presents results with targeting daily market volatility at 1%, with maximum participation ratios of 1, 1.5, and 2. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model, and the threshold weight change is 0.1. The 25-year cumulative amount that starts at each trading day in the period of analysis is calculated, where the equity contribution is sourced from Morningstar Lifetime Allocation Indexes. It is assumed that the worker (investor) starts with a salary of 20,000 USD, which grows by 4% in nominal terms each year during a 25-year career. The worker contributes 9% of his/her salary to target-date fund at the end of each year for their 25 years of work. The target-date fund rebalances daily without considering tax implications or transaction costs.

Table 8. 15-year target-date fund performance statistics. This table presents summary statistics on cumulative amounts and the internal rate of return (IRR) at the end of target-date funds over 15-year life cycles in the U.S. The period of analysis is from 26 April 1982 to 30 June 2020. The columns present the results under aggressive, moderate, and conservative strategies. Panel A presents results without volatility targeting, and Panel B presents results with targeting daily market volatility at 1%, with maximum participation ratios of 1, 1.5, and 2. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model, and the threshold weight change is 0.1. The 15-year cumulative amount that starts at each trading day in the period of analysis is calculated, where the equity contribution is sourced from Morningstar Lifetime Allocation Indexes. It is assumed that the worker (investor) starts with a salary of 20,000 USD, which grows by 4% in nominal terms each year during a 15-year career. The worker contributes 9% of his/her salary to target-date fund at the end of each year for their 15 years of work. The target-date fund rebalances daily without considering tax implications or transaction costs.

4. Investment Performance During Major Stock Market Crashes

In this section, we report results of the performance of targeted volatility portfolios during major U.S. stock market crashes. These include the crash from 3 September 1929 to 8 July 1932 which is associated with the Great Depression in the 1930s and is the largest U.S. stock market crash in the last 100 years. We also study the crash from 9 October 2007 to 9 March 2009 which is associated with the Global Credit Crisis and is the largest U.S. stock market crash in the last 50 years. Finally, we also report results on the targeted volatility portfolios during the recent COVID-19 pandemic stock market crash.

Tables 9 and 10 report results for the crashes associated with the Great Depression and the Global Credit Crisis. Table 9 contains holding period returns for the one year prior to the crash, the crash period, the five years post-crash, and the period starting one year prior to the crash and ending five years after the crash. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% or 0.8% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%.

Table 9. Holding period returns pre-, during, and post-Great Depression and Global Credit Crisis crashes. This table presents the holding period returns (in percent) for the one year prior to the crash, the crash period, the five years post-crash, and the period starting one year prior to the crash and ending five years after the crash. The annualized returns are presented in square brackets. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% or 0.8% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%.

Table 10. Annualized volatility pre-, during and post-Great Depression and Global Credit Crisis crashes. This table presents the annualized volatility of returns (in percent) for the one year prior to the crash, the crash period, the five years post-crash, and the period starting one year prior to the crash and ending five years after the crash. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% or 0.8% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%.

The crash from 3 September 1929 to 8 July 1932 resulted in the CRSP value-weighted index, including dividends, falling 83.84%. During this crash, the targeted volatility strategy resulted in a substantial mitigation of the drawdown. The corresponding CRSP portfolio that targeted average daily volatility at 1% had a reduced drawdown at 70.01%, and when targeting volatility at 0.8% the drawdown was 60.36%. Over the period starting one year prior to the crash and ending five years after the crash, the CRSP index had a negative return of 1.62%, whereas, the targeted 1% and 0.8% volatility portfolios had positive returns of 14.87% and 20.84%, respectively. Fig. 3 displays the cumulative outperformance of the targeted 1% volatility portfolio, against the market portfolio over this period.

Figure 3 Crash from great depression. This figure displays the cumulative value from a one-dollar investment starting one year prior to the U.S. crash 3 September 1929 to 8 July 1932. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model.

The crash associated with the Global Credit Crisis resulted in the CRSP value-weighted index, including dividends, falling 54.32% from 9 October 2007 to 9 March 2009. The corresponding CRSP portfolios that targeted 1% and 0.8% daily volatility had much less of a drawdown at 38.40% and 30.28%, respectively, over the same period. Over the period starting one year prior to the crash and ending five years after the crash, the CRSP index had a return of 71.06%, whereas, the targeted 1% and 0.8% volatility portfolios had returns of 88.23% and 73.84%, respectively. Fig. 4 displays the cumulative outperformance of the targeted 1% volatility portfolio, against the market portfolio over this period.

Figure 4 Crash from Global Credit Crisis. This figure displays the cumulative value from a one-dollar investment starting one year prior to the U.S. crash 9 October 2007 to 9 March 2009. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model.

Table 10 presents the annualized volatility of returns for the same portfolios and over the same periods as in Table 9. During the Great Depression and Global Credit Crisis crashes, the annualized volatility of returns in the value-weighted CRSP index was 33.76% and 37.11%, respectively. The targeted volatility portfolios had substantially less volatility, as expected. The 1% volatility portfolios over the Great Depression and Global Credit Crisis crashes had an annualized volatility of 20.02% and 20.91%, respectively. Over the periods starting one year prior to the crash and ending five years after the crash, the target volatility portfolios displayed volatility close to their volatility target.

Tables 11 and 12 report returns and volatilities for the equity and balanced portfolios and corresponding target 1% and 0.8% volatility portfolios that apply the targeted volatility strategy to the equity component with maximum leverages of 50% and 100%, over the first half of 2020 which includes the COVID-19 crash. In this episode, the pre-crash period is 1 January 2020 to 19 February 2020, the crash period is 19 February 2020 to 23 March 2020, the post-crash period is 23 March 2020 to 30 June 2020 and the full period is 1 January 2020 to 30 June 2020. The market index is again the value-weighted CRSP index, including dividends and the balanced portfolio invests 65% in equity markets and 35% in bond markets.

Table 11. Holding period returns pre-, during and post-COVID-19 crash. This table presents the holding period returns (in percent) for the pre-crash period (1 January 2020 to 19 February 2020), the crash period (19 February 2020 to 23 March 2020), the post-crash period (23 March 2020 to 30 June 2020) and the full period (1 January 2020 to 30 June 2020). The annualized returns are presented in square brackets. The market index is the value-weighted CRSP index, including dividends and the balanced portfolio invests 65% in equity markets and 35% in bond markets. The target volatility strategy targets a daily 1% or 0.8% standard deviation of market returns, with a daily threshold weight change of 0.1.

Table 12. Annualized volatility pre-, during, and post-COVID-19 crash. This table presents the annualized volatility of returns (in percent) for the pre-crash period (1 January 2020 to 19 February 2020), the crash period (19 February2020 to 23 March 2020), the post-crash period (23 March 2020 to 30 June 2020) and the full period (1 January 2020 to 30 June 2020). The market index is the value-weighted CRSP index, including dividends and the balanced portfolio invests 65% in equity markets and 35% in bond markets. The target volatility strategy targets a daily 1% or 0.8% standard deviation of market returns, with a daily threshold weight change of 0.1.

Figure 5 Equity portfolios over the COVID-19 pandemic. This figure displays the cumulative value in equity portfolios from a one-dollar investment starting 1 January 2020 and ending 30 June 2020. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy targets a daily 1% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model.

During this crash period the CRSP index, fell 34.27% from 19 February 2020 to 23 March 2020. Over the same period, the targeted volatility strategy again resulted in a substantial mitigation of the drawdown. The corresponding CRSP portfolio that targeted daily volatility at 1% (with maximum leverage of 50%) had much less of a drawdown at 18.61% over this period. Fig. 5 displays these two portfolios from the beginning of January 2020 to the end of June 2020. Over the months following the crash, the stock index recovered substantially with the targeted volatility portfolio continuing to show outperformance. The CRSP index was down 1.89% over the period from 1 January 2020 to 30 June 2020, whereas the targeted 1% volatility portfolio was up 1.77 %.

Outperformance in the balanced portfolio with targeted volatility on the equity component also occurred during the pandemic. The 65:35 equity bond split in the balanced portfolio resulted in a portfolio decline of 23.52% from 19 February 2020 to 23 March 2020. While targeting a daily 1% volatility on the equity component (with maximum leverage of 50%) in the balanced portfolio, resulted in a decline of only 12.64%. Outperformance from targeting volatility continued over the extended period to the 30 June 2020. From 1 January 2020 to 30 June 2020 the balanced portfolio gained 2.26%, while the balanced portfolio with targeted volatility at 1% on the equity component, gained 3.80%. The performance of these portfolios is displayed in Fig. 6 from the beginning of January 2020 to the end of June 2020.

Figure 6 Balanced portfolios over the COVID-19 pandemic. This figure displays the cumulative value in balanced portfolios from a one-dollar investment starting 1 January 2020 and ending 30 June 2020. The portfolios invest 65% in equity markets and 35% in bond markets. The market index is the value-weighted CRSP index, including dividends. The target volatility strategy in the equity component targets a daily 1% standard deviation of market returns, with a daily threshold weight change of 0.1 and with maximum leverage of 50%. The volatility forecasts are computed from the outlier-corrected GARCH(1,1) model.

Table 12 presents the annualized volatility of returns for these portfolios. The COVID-19 crash was extremely volatile with annualized volatility of the CRSP index during the crash being 79.28%. The corresponding CRSP target 1% and 0.8% volatility portfolios (with maximum leverage of 50%) had substantially less volatility at 38.93% and 31.92%, respectively. The equity volatility also created substantial volatility in the balanced portfolio, which was 50.47% during the crash. Over the full period from 1 January 2020 to 30 June 2020, the volatility in the balanced portfolio with the equity component having a 1% target volatility, was approximately half that of the balanced portfolio without the targeted volatility strategy. Results with maximum leverage of 50% and 100% were very similar as during these periods leverage was not high due to high volatility.

Overall, the investment performance of the targeted volatility strategies during and after these major stock market crashes has demonstrated important benefits of the approach. Firstly, the control of volatility and the mitigation of drawdowns can be substantial, which is often very important to insurers and pension funds. In particular, during the extreme volatility of the COVID-19 crash, the CRSP equity portfolio that targeted daily volatility at 1% (with maximum leverage of 50%) had a lower drawdown of 18.61%, relative to the 65:35 balanced portfolio without the targeted volatility strategy that had a drawdown of 23.52%. Secondly, and related to the drawdown mitigation, is the return outperformance of the targeted volatility investment portfolios relative to benchmarks, over evaluation periods that include the crash and also the post-crash period.

5. Discussion

Our focus in this paper has been on the targeted volatility portfolio strategies which in recent years have been increasingly implemented in practice with univariate time series models for volatility. We have studied a long span of historical data, back to the Great Depression, to analyze performance with actual market data including time periods with extreme market scenarios.

The research and industry applications on targeted volatility portfolios with univariate time series forecasting methods have a number of potential areas for future development. These include forecasting methods for volatility, portfolio construction methodologies, applications to a variety of insurance and pension products, and the backtesting of these methods with historical data over a range of periods (including periods in which targeted volatility portfolios have return outperformance and underperformance) from a variety of countries. An example of a period of return underperformance could be during a rapid market rebound after a crash, such as the COVID-19 crash. Parts of the methodologies could also be tested with simulated data.

An important benefit of targeted volatility portfolios is in the mitigation of substantial drawdowns in the equity component of investments. This is a particularly desirable feature for many insurance and pension products and is a primary reason for the increasing use of targeted volatility portfolios in retirement products such as variable annuities. Targeted volatility portfolios reduce their equity exposure when volatility rises which mitigates drawdowns which typically occur during volatile periods. The targeting of volatility also results in a more constant level of volatility exposure with less extremes in volatility, which is also an appealing feature for investment options offered by variable annuity providers, and other types of retirement products.

The other key benefit from targeted volatility portfolios is the potential for return outperformance, leading to higher levels of income for example in individual pension accounts. The management of account-based pensions is becoming increasingly important, for example, in Australia, assets under management in the superannuation (pension) industry were $\$$ 3.4 trillion at the end of the March 2022 quarter and a large portion of these funds were in account-based pensions. Under the Retirement Income Covenant in Australia there is increased interest in retirement income products with equity exposure as well as managing longevity risk.

Return outperformance in targeted volatility portfolios often occurs both in high and low market volatility periods. During high volatility periods, outperformance is typically generated by reducing equity exposure which tends to mitigate drawdowns, relative to overall market drawdowns. In low volatility periods, outperformance is often generated from increasing equity exposure in periods of relatively high market returns.

6. Conclusion

This paper has analyzed targeted volatility investment portfolios over a range of leverage constraints; from no leverage, to conservative and aggressive leverage, to unconstrained leverage. Results are presented for equity, balanced and target-date portfolios. Significantly, substantial risk-adjusted return outperformance is shown for the targeted volatility portfolios with leverage constraints. The highest Sharpe ratios and smallest drawdowns are found in targeted volatility-balanced portfolios with equity and bond allocations. We show how different constraints, result in relatively small changes in the Sharpe ratio for the targeted volatility portfolios. We also show the extent to which for target-date funds with targeted volatility, less conservative leverage constraints result in higher average investment outcomes.

Of significant current interest, the paper has also demonstrated outperformance of targeted volatility investment portfolios during and after major stock market crashes. A primary generator of outperformance comes from the mitigation of the drawdown during the crash, particularly in highly volatile crashes. For example, during the extreme volatility in the recent COVID-19 crash the targeted volatility strategy approximately halved the drawdown, relative to the benchmark stock index.