Fixed income + S&P 500 index investment: flexible vs. static asset allocation

We will analyze simulated 10-year returns from a combination of fixed income and investment in S&P 500 index.

This is a fundamental and well-studied topic covered in many sources such as:

Given a certain risk-free return R_f and properties of S&P 500 returns' distribution, there are many ways to determine allocation w from [0, 1] for R_f (leveraged portfolio is out of consideration), for instance using indifference curve tangent to the capital allocation line. In the current analysis we will focus on overall 10-year return and use a simple condition to describe a risk tolerance instead of utility function: probability of negative overall returns should be very low (<0.1%). Such condition was motivated by the following reasons:

it is highly likely that investment has a target date (retirement, education), thus losses may inflict severe life-style limitations,
obviously any person could have limited amount of 10-year investment cycles and it is not easy to cover losses from the first 10-year period in the next periods (to cover 50% loss 100% returns are required).

Historical data

We will use S&P 500 index data from 1990-01-02 to 2024-07-19. This is how it looks.

First of all, let's notice that historical 10-year returns from S&P 500 in the selected period range from -50% to +400%. The rate of negative returns is around 10%.

The distribution of 10-year returns in the histogram above is very far from belonging to a common statistical class, it suffers from a limited amount of global events (crises) in the selected period, meanwhile annual returns distribution has a more 'robust' profile.

The goal of the current article is to compare static and flexible approaches to asset allocation, but not to precisely characterize stock market return's distribution which is an unsolvable problem. For convenience and generalization instead of using historical 10-year returns we will use simulated 10-year returns obtained by picking 10 random points from the adjusted distribution of S&P 500's annual returns. We recognize that it is significant simplification since annual returns of the next year are (historically) not statistically independent from the previous years returns, nevertheless, on the whole any approach exploiting statistical properties of the past to simulate future returns should be taken with a grain of salt.

Once we initially performed such simulations with non-adjusted historical annual returns we observed statistical bias: historical average of 10-year returns was 114.7% while simulated average was 150.89%, also the rate of negative returns was 6.7% instead of 10.2%. To compensate this bias we applied linear transformation to annual returns adjusted_annual_return_% = annual_return_% * 0.97 - 1.4

We will utilize 100k samples, this is how our simulated distribution looks in comparison to actual 10-year returns.

Here is a summary of the main statistical properties of historical vs. simulated distributions.

	10Y Returns %				Loss rate %
	min	median	mean	max
Type
historical	-45.41	100.01	114.76	390.45	10.27
simulated	-83.99	98.01	114.97	1597.00	10.18

Static investment strategy

Let's consider the following strategy of 10-year investment:

Each year W% of total assets are allocated to a fixed-income tool with RF% annual returns and the rest (100-W)% goes to S&P 500 index;
For simplicity we will assume that RF remains constant for the whole 10 year period;
Each year returns from S&P 500 index are sampled from adjusted historical distribution of annual returns described above;

Given a certain RF% returns of a fixed income tool our goal is to select W% that maximizes median overall 10-year returns while maintaining very low risk of losses. We will perform simulations for RF in a range [1%, 10%] with 1% step.

Let's define 'very low risk of losses' as the rate of positive returns above 99.9%. We assume that a fixed income tool is risk-free. Median is selected for maximization instead of mean since it is not feasible to 'repeat' 10-year investment cycles many times. Although W% is static, the actual amount relocated to fixed income each year is not static: for instance if W=50% and after the first year overall returns were 20%, then half of 120%=60% from initial funds is allocated to a fixed income tool during the second year.

Results for the static strategy

Table below describes the results of simulations.

		10Y Returns %				Loss rate %
		min	median	mean	max
RF %	Optimal W %
1	90	-7.52	18.32	18.25	42.04	0.09
2	82	-13.07	35.25	35.32	89.47	0.10
3	76	-15.06	50.49	50.76	137.57	0.08
4	70	-24.41	65.06	65.75	192.70	0.10
5	65	-20.69	78.01	79.02	224.84	0.08
6	62	-25.91	90.38	91.90	269.76	0.09
7	58	-24.20	102.04	104.03	306.21	0.08
8	94	89.47	115.97	115.84	144.99	0.00
9	100	136.74	136.74	136.74	136.74	0.00
10	100	159.37	159.37	159.37	159.37	0.00

Here are some important implications of the results.

Median 10-year returns of S&P 500 index are around 100% with 10% risk of losses. In order to have the same median returns but <0.1% risk of losses 7% fixed income tool is required.
Fixed income of 3% allows to achieve median 10-year returns of 50% without losses.
The portion of funds allocated to S&P 500 index grows with the fixed income rate until 7% and then rapidly drops to 0.

Flexible allocation strategy

Instead of static W% let's give it flexibility to be adjusted according to actual annual results of S&P 500 investment portion. The frequency of allocation adjustment remains the same as in the static approach: once per year.

In such a setting, the optimization task becomes much more sophisticated. Let's utilize AI to find the quasi-optimal strategy.

It is necessary to emphasize that the main effort of optimization is devoted to maximizing median returns. In case of flexible strategy the difference of optimizing mean and median is much higher than for the static strategy: mean maximization is usually achieved via increased positive tail of distribution with median far from being optimal.

Results for the flexible strategy

Table below summarizes the results of simulations.

		10Y Returns %				Loss rate %
		min	median	mean	max
RF %	Starting W %
1	87	-9.55	20.01	20.19	59.97	0.10
2	78	1.56	37.64	41.85	245.82	0.00
3	61	2.05	60.18	61.38	213.46	0.00
4	50	-6.14	75.90	75.68	236.69	0.00
5	40	-2.19	98.56	83.28	211.40	0.01
6	41	-5.07	107.59	92.61	256.17	0.01
7	44	-13.12	111.60	100.74	198.24	0.03
8	44	-2.48	130.04	115.66	209.13	0.00
9	32	-4.98	150.51	132.01	248.50	0.00
10	67	-5.69	166.72	152.31	234.23	0.00

Let's compare flexible vs. static strategy results:

Median returns from flexible strategy are higher for all values of RF%.
Starting rate W% is always lower than in the static approach, it does not go to 100% for high fixed income.
Fixed income of 5% is enough to achieve almost 100% median returns using the flexible strategy.

The graph below summarizes the comparison.

Impact of taxation

Let's include tax effect into the flexible strategy optimization:

Assume 20% tax rate for both income from a fixed-income tool and realized capital gains from S&P 500 index;
Capital losses are deductible without restrictions and could be accumulated;
Taxes are not imposed on the final returns of the last year.

For clarification: we do not just include tax effects on top of previously optimized algorithm, we train a new model that makes tax-aware decisions. As could be observed from the results, the effect of tax significantly reduces median returns for cases of high fixed-income rate.

		10Y Returns (tax-aware) %				Loss rate %
		min	median	mean	max
RF %	Starting W %
1	84	-4.42	19.27	20.15	77.78	0.10
2	75	0.36	34.97	40.57	251.71	0.00
3	59	0.36	53.18	61.23	328.02	0.00
4	46	0.92	66.89	76.03	428.89	0.00
5	38	0.02	77.38	85.45	494.03	0.00
6	35	-1.40	84.99	93.88	520.06	0.00
7	37	1.70	95.23	94.40	327.13	0.00
8	50	-3.14	106.49	95.60	258.17	0.00
9	55	-1.91	117.04	105.96	272.41	0.00
10	62	-20.84	128.04	120.62	188.77	0.01

Return estimates for the flexible strategy performance are subject of random sampling thus could slightly vary.

Conclusion

We considered static and flexible asset allocation strategy which minimizes the risk of loss using a combination of a fixed income tool and S&P 500 index. Simulations showed that the flexible strategy with adaptive W% allocation of funds allows better median returns than the static approach with constant W%. Simulated estimates of returns and optimal percentages were based on certain assumptions and simplifications, which could be surely compromised:

Method of sampling annual stock market's returns from empirical distribution is a severe simplification;
Rate of fixed income was constant over 10 year period, which is also a simplification;
We did not consider impact of inflation;
We required risk of negative 10-year returns to be less than 1/1000, which might be considered overcautious.

Publication date: 31/08/2024