By Jay Vadiveloo and Zhiguo Wang

**Motivation**

For the millions of retirees, financial advisers, and investment managers, there is one pervasive problem—how to balance performance against the safety of retirement assets. Life annuities can provide the guaranteed income that could satisfy the safety need, while an investment strategy could then help provide the performance requirement. It is clear that a combined investment and annuity strategy is required; what is unclear is how adding annuities impacts the safety and performance of retirement assets for the retiree.

The first step in this article is to demonstrate what adding
annuities to an investment strategy accomplishes with regards to safety and
performance of the integrated strategy (i.e., investments plus annuities). The
article then develops an **optimal** strategy that demonstrates for a given
annuity allocation how an integrated strategy can be constructed to provide
more effective performance and safety outcomes for the retiree. To develop this
optimal strategy, we use a two-step process:

- Define a safety-performance metric that allows us to compare two different investment strategies using a similar criterion.
- Using this criterion, calculate performance measures to explicitly demonstrate the benefits of incorporating annuities within an investment strategy.

**Impact of combining investments with annuities**

To simplify this illustration, we are going to combine a single premium immediate annuity (SPIA) with a pure investment strategy. Consider $1 million of retirement assets that are invested in a portfolio that generates an annual mean return of 8% with a standard deviation (SD) of 10%. We will also assume an annual mortality rate of the retiree age 65 from standard actuarial tables and a risk-free interest rate of 4% to discount cash flows. Fix the annual level of spending at $60,000 and perform the following calculations:

- Generate 1,000 scenarios of projected investment returns with a mean of 8% and SD of 10%.
- Fix an investment return scenario and duration and calculate the present value of the annual payoff, which equals the present value of the annual spending of $60,000 and the remaining value of the investment portfolio at that duration. Use the risk-free interest rate of 4% to calculate the present value.
- Multiply by the probability of dying in that
duration to get the
**expected**present value of the annual payoff for that duration and investment return scenario. - Sum up the expected present value across all durations to generate the expected payoff for the given investment return scenario.
- Repeat the above for all interest rate scenarios
to generate the
**distribution**of the expected payoff across all investment return scenarios. - Calculate the mean and standard deviation of the expected payoff distribution.

**The Goldenson Center Safety Performance (GCSP) metric**

The Goldenson Center Safety Performance (GCSP) metric is a single number to evaluate an investment strategy. It equals the ratio of mean/SD of the expected payoff distribution; it has similarities with the Coefficient of Variation used to measure investment performance of an investment portfolio. Similar to the Coefficient of Variation, the larger the GCSP metric, the better the return-to-risk performance of the investment. However, unlike the Coefficient of Variation, the GCSP incorporates the mortality of the retiree and measures investment performance over a time horizon and not just a point in time.

**The GCSP for the integrated strategy**

Now repeat the above steps but allocating 20% (i.e. $200,000) of retirement assets into a SPIA and the remaining $800,000 into the risky investment with mean 8% and SD of 10%. We will call this the integrated strategy in contrast to the pure investment strategy.

Table 1 compares the percentiles of both these strategies and the corresponding GCSP metric. We can draw the following conclusions:

- The integrated strategy generates a
**higher**GCSP than the pure investment strategy—i.e., it has a higher return-to-risk performance. - The lower percentiles of the integrated strategy are generally higher than the same percentiles for the pure investment strategy.
- The upper percentiles of the integrated strategy are generally lower than the pure investment strategy.

From this illustration, we can see that incorporating
annuities within an investment strategy **reduces** the volatility of the
expected payoff distribution and **also reduces** the mean of the payoff
distribution. However, the reduction in volatility is greater than the
reduction in the mean, which results in a higher GCSP for the integrated
strategy compared to the pure investment strategy. This illustration has
inspired the development of the **optimal **investment strategy to adopt for
a given allocation of annuities into an investment portfolio.

**Optimal integrated strategy**

Given that the inclusion of annuities reduces the payoff
volatility, we have developed the optimal integrated strategy by **increasing
both **the level of risk and return for the investment component of the
integrated strategy. We increase the investment risk and return sufficiently
such that the GCSP for the integrated strategy **equals** the GCSP for the
pure investment strategy. We use this criterion to ensure that the integrated
strategy has the same return-risk profile as the pure investment strategy. We
call this the **investment-modified **integrated strategy.

Using the same example as before, we now have the following:

- The mean and SD for the investment component of the integrated strategy is 8.48% and 10.72%, while the mean and SD for the pure investment strategy remains the same at 8% and 10%.
- With this investment adjustment to the integrated strategy, the GCSP of both strategies are the same at 2.31.

Table 2 compares the percentiles of the pure investment strategy against the investment-modified integrated strategy. We can draw the following conclusions:

- For the same return-risk profile, the
investment-modified integrated strategy consistently
**outperforms**the pure investment strategy at all percentiles except the 50^{th}percentile. - The investment-modified integrated strategy provides better performance in a bad market, maintains performance in an average market, and shows improved performance in a good market.

There are other performance measures to illustrate the superiority of the investment-modified integrated strategy over the pure investment strategy.

**Annual spending criteria for a fixed ruin probability**

To simplify the model, we will remove the mortality
component by assuming the retiree is certain to live to age 95. For both the
pure investment strategy and investment-modified integrated strategy, we solve
for the **maximum **annual spending under each strategy such that the ruin
probability is 5%—i.e., 50 out of the 1,000 investment scenarios end up with a
negative fund value at 95.

For the pure investment strategy, the maximum annual spending is $50,247 while for the investment-modified integrated strategy, the maximum annual spending is $53,827, which represents an increase of 7% in annual spending.

**Comparing ruin probabilities for a fixed level of annual
spending**

We now fix the annual spending at $60,000 for both strategies. Then the ruin probability for the pure investment strategy is 17.7%, while the ruin probability for the investment-modified integrated strategy is much lower at 13%.

**Conclusions**

For the first time, an actuarial model has been developed to rigorously demonstrate the added value of incorporating life annuities with investments in retirement planning. In the process, we have introduced the GCSP metric—a safety-performance metric to evaluate a retirement planning strategy that is stochastic and incorporates mortality and an investment time horizon. The examples in this article have only focused on combining an investment strategy with a single premium immediate annuity (SPIA). But the modeling framework can easily be extended as follows:

- Incorporating deferred immediate annuities (DIAs), variable and fixed annuities, fixed indexed annuities, or multiyear guaranteed annuities;
- Incorporating a health component to the retiree—average, below average or above average mortality;
- Modeling different investment mixes for the pure investment strategy and generating the investment-modified integrated strategy using this investment mix; and/or
- Extending the investment-modified integrated model for the accumulation and near-retirement phases of an individual.

In all of the examples we have tested, the results show that combining life annuities with investments **always **improves financial performance in all phases of retirement planning. Using our model findings, insurers now have a compelling case to argue in marketing life annuities—namely, that an integrated strategy incorporating investments and life annuities will **always **improve financial performance compared to a pure investment strategy.

I will like to end with a quote from British author and personal finance writer Jonathan Clements that reflects this article’s philosophy.

“Retirement is like a long vacation in Las Vegas. The goal is to enjoy it to the fullest, but not so fully that you run out of money.”

By incorporating annuities within a retirement planning strategy, you can enjoy retirement life more fully and also reduce the risk of running out of money.

**JAY VADIVELOO, MAAA, FSA, Ph.D., is director of the
Goldenson Center for Actuarial Research at the University of Connecticut. ZHIGUO
WANG, Ph.D., is assistant director at the Goldenson Center.**