Monte Carlo and Sequence of Return Risk

Lump Sum Investment vs. Dollar-Cost Averaging

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Wealth Wisdom: Your Path to FIRE

7 min readMar 13, 2023

Regarding sequence of return risk, in the article “What is Sequence of Return Risk?”, it was pointed out that the impact of this risk is not just a simple risk as commonly believed, but rather an opportunity to accumulate assets faster. In this article, I will further explain the relationship between the two most commonly used investment methods by investors: lump sum investment and dollar cost averaging, and sequence risk, and try to help you understand why investing in a lump sum is a better way to overcome this risk.

The Reason for Sequence Risk — Market Fluctuations

First, let’s take a simple example.

Assuming that the future market annualized return is 8%, a total market ETF with a current price of 100 yuan will undergo the following changes in assets each year as shown below.

The stock price changes annually at a fixed rate of 8%, which is a common assumption about stock price changes among the public. However, the average return obtained from mathematical calculations does not represent that the market will maintain the same growth rate every year.

But in this fixed return growth process, let’s consider the following question.

Should investors make a lump sum investment or invest in installments?

It is clear that in this situation, making a lump sum investment is better than investing in installments.

The reason is that the cost of a lump sum purchase is only $100 per share, while the average cost of installment purchases will be greater than $100 due to the continuous rise in stock prices. Therefore, with the same principal, a lump sum investment with lower costs will have more shares and thus more assets.

Therefore, in a market trend that always rises, making a lump sum investment is a better choice.

However, the real market environment is not always rising. Stock prices fluctuate, sometimes rising and sometimes falling, and the price changes caused by fluctuations are the source of sequence risk.

Market Fluctuations Create Volatility in Stock Prices

In the previous article introducing sequence of return risk, VT was used as an example of the ups and downs from 2008–2021. The price changes were examined after reversing the sequence of returns.

Even if we adjust the sequence of returns, the starting and ending prices remain the same, indicating that the total return over the same period is the same.

In addition, although the order of price changes in the middle is different, they are all composed of the same set of numbers.

In other words, the risks (volatility) that these two types of returns bear are the same.

Therefore, if we view stock price volatility (mathematically defined as standard deviation) as a risk and a fixed value, the greater the value, the greater the volatility of stock prices; the smaller the value, the smaller the volatility.

Therefore, when we try to evaluate long-term market trends mathematically, due to the relationship between standard deviation and volatility, the stock price appears to be a rising trend, but on the other hand, it is accompanied by ups and downs in the process, just like the return trend of VT.

In other words, it is because of volatility that market stock prices are not simply rising, but the process also includes the possibility of falling, and because of the relationship between volatility (standard deviation), sometimes the fluctuations are large, and sometimes they are small, and the size of the change depends on probability.

In addition, since the market price will continue to break new highs in the long run, the overall price increase speed depends on the annualized return we set. The higher the value, the faster the increase; the opposite is true.

Lump Sum Investment or Dollar Cost Averaging in a Long-Term Bull Market with Fluctuations?

Now we are very clear that to see the future market trends, we can simulate based on two factors.

Annualized return rate
Annualized volatility

Once these two factors are thrown out, you may immediately realize that this is none other than the famous Monte Carlo simulation method.

Correct.

The Monte Carlo simulation uses these two variables to simulate the investment process that investors may experience in a fluctuating market. This method is more realistic than the previous single linear (non-fluctuating) one, but please remember that this is only a simulation and cannot be used to predict the future.

Therefore, next, I will use the Monte Carlo simulation to try to understand the relationship between the final returns obtained from lump sum investment vs dollar cost averaging under different annualized return rates and annualized standard deviations.

Monte Carlo Simulation: Lump Sum Investment vs Dollar Cost Averaging

For most investors, the investment period can range from a few days to several decades, but short-term investments have high volatility and a high probability of loss, and are not a suitable investment period.

The best investment period is actually forever, with continuous investment without interruption.

Therefore, in the following simulation test, the investment period I will use is 30 years. However, as annualized return rates and annualized standard deviations can be combined into many combinations, I will use a 5% annualized return rate and 10% annualized standard deviation as an example to provide a reference, respectively examining the range of returns and the volatility of returns that can be obtained between lump sum investment and dollar cost averaging.

5% Annualized Return Rate + 10% Annualized Volatility

Dollar cost averaging:

Highest: 833.5%
Lowest: -37.21%
Average return: 149.09%
Return standard deviation: 173.35%

Lump sum investment:

Highest: 2751.62%
Lowest: -51.6%
Average return: 404.86%
Return standard deviation: 387.97%

Under this investment environment, the return range for dollar cost averaging is -37.21% to 833.5%, compared to -51.6% to 2751.62% for lump sum investment. While dollar cost averaging loses less money, the upper limit of earning is also reduced, resulting in an average return that is half of that of lump sum investment. However, the volatility (standard deviation) of returns for dollar cost averaging is smaller, making it appear more stable than the large fluctuations of lump sum investment.

Although these data give us an understanding of the potential for both dollar cost averaging and lump sum investment to win in a fluctuating environment, how likely it is to happen in reality is not clear.

Therefore, next, by executing a certain number of Monte Carlo simulations, the winning rate of returns for both methods will be calculated.

In this chart, the lowest winning rate for lump sum investment is 86.8%
The return of the long-term stock market is about 8%, and the volatility is 15%. Based on this data, lump sum investment has an extremely high winning rate of 97.8%
If the volatility caused by annualized standard deviation is large enough, dollar cost averaging has a chance to win
If your investment period is long enough, choosing lump sum investment is a rational choice.

Conclusion

From the simulation results, we can see that in a long-term bull market with fluctuations, lump sum investment has a higher probability of higher returns than dollar cost averaging. This is because the market trend of the stock price is generally upward, and the impact of the sequence risk is reduced over time. Therefore, choosing lump sum investment can overcome the sequence risk and obtain higher returns.

However, it should be noted that whether to choose lump sum investment or dollar cost averaging depends on individual investment goals, investment period, and risk tolerance. For short-term investments, dollar cost averaging may be a better choice to reduce volatility and risk. For long-term investments, lump sum investment may be a better choice to achieve higher returns.

In addition, investors need to be aware that the Monte Carlo simulation is only a theoretical model and cannot predict the future accurately. Therefore, it is recommended that investors conduct their own research and analysis before making investment decisions.

Finally, it is important to note that while lump sum investment may be a better choice for overcoming sequence of return risk, it is not without risk. Stock prices can still fluctuate significantly, and investors should be prepared for potential losses. Therefore, it is important to diversify investments and have a long-term investment strategy.