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How Math Can Help Our Decisions

Aug. 29, 2023, 8:02 a.m. |Financial Planning |Intermediate

After reading this comment on X (Twitter), I decided to create this blog post: "If a Financial Advisor starts mentioning EXPECTED RETURNS, don't walk away, RUN!" In a world of 280 characters, it is pretty common to see these types of statements, but I wanted to challenge this particular statement and explain how important it is to understand expected returns.

What are expected returns, and why is it so important for your advisor to understand them? First and foremost, it is impossible to build any type of financial plan without trying to estimate returns. Putting in expected returns into the planning process determines your need to take risk-how much you must put into stocks to reach your goal. Let's look at how this is done.

Nuclear bomb

In the film "Oppenheimer," we see how scientists calculated whether the atomic bomb would be a small or much larger explosion. This kind of prediction reminds us of how finance experts use a concept called Modern Portfolio Theory (MPT), developed by Harry Markowitz, to estimate future investment returns.

Modern finance utilizes mathematical frameworks like the Modern Portfolio Theory (MPT) pioneered by Harry Markowitz. This theory, like the calculations employed by Oppenheimer's team, seeks to gauge the potential impact of various elements on outcomes, albeit in the realm of investments.

It helps investors determine the best asset mix to balance risk and reward. By looking at these two different scenarios – one in history and one in finance – we can see how important it is to make accurate predictions to guide important decisions. In Markowitz's view, risk can be measured in two ways: the expected return of an investment and the degree of uncertainty around that return.

Markowitz's approach to risk has been influential in developing modern portfolio management, which seeks to optimize risk-adjusted returns by diversifying across a broad range of assets. By spreading risk across different asset classes and geographic regions, investors can reduce their exposure to any one particular risk and improve their chances of achieving their investment goals over the long term.

Let's walk through these concepts and see how they can help us become better investors in the current higher inflation environment! In this example, I will act as a client asking questions about different scenarios and an advisor using MPT to answer them.

Client: What is the expected return?

Advisor: Expected return is a fundamental concept in finance and is used to estimate the future returns of an investment. It is the weighted average of the possible returns, where the weights are the probabilities of each return occurring. Here's how to calculate the expected return:

For a Single Asset

1. List down the possible returns and their probabilities:

For instance, let's assume we have three scenarios for an investment:

  • Scenario 1: 10% return with a probability of 0.3
  • Scenario 2: 15% return with a probability of 0.5
  • Scenario 3: -5% return with a probability of 0.2

2. Multiply each return by its probability:

  • Scenario 1: 0.10 × 0.3 = 0.03 or 3%
  • Scenario 2: 0.15 × 0.5 = 0.075 or 7.5%
  • Scenario 3: −0.05 × 0.2= −0.01 or -1%

3. Sum the results from step 2 to get the expected return:

Expected Return = 3%

For a Portfolio of Multiple Assets

If you have a portfolio of multiple assets, you can use the weighted average of the expected returns of each asset:

E(Rp​) = w1 ​× E(R1​) + w2​ × E(R2​) + ... + wn​ × E(Rn​)

Where:

  • E(Rp​) is the expected return of the portfolio.
  • wi​ is the weight or proportion of asset i in the portfolio.
  • E(Ri​) is the expected return of asset i.

For instance, if you have two stocks in your portfolio, Stock A and Stock B:

  • Stock A has an expected return of 8% and makes up 60% of your portfolio.
  • Stock B has an expected return of 12% and makes up 40% of your portfolio.

The expected return for the portfolio would be:

E(Rp​) = 0.60 × 8

Remember, expected return is a statistical measure and does not guarantee the future performance of an investment. It's always important to consider other factors and metrics when making investment decisions.

Client: What is variance?

Advisor: Variance is a statistical measure that quantifies the degree of spread or dispersion of a set of data points around their mean (average) value. In the context of investments, variance is often used to measure the volatility or risk of an investment.

The formula for variance is as follows:

Variance = Σ (Xi - X)^2 / (n - 1)

Where Xi represents each data point in the set, X represents the data set's mean, and n represents the total number of data points in the set.

In the context of investments, the variance of an asset's returns measures the degree to which it deviates from its average return. A higher variance indicates that the asset's returns are more spread out, while a lower variance suggests that the returns are more tightly clustered around the mean.

Investors often use variance to measure risk because it reflects the uncertainty or unpredictability of an investment's returns. An investment with high variance is considered riskier than one with low variance because the potential range of outcomes is broader and more challenging to predict.

Client: Calculate the return of a portfolio with an 8% return and a standard deviation of 15 compared to a portfolio with a 6% return and standard deviation of 9 using one million dollars over 15 years.

Advisor: To calculate the return of two portfolios with different expected returns and standard deviations, we need to make some assumptions about the distribution of returns and the correlation between the assets in the portfolios. For simplicity, let's assume that both portfolios consist of a mix of stocks and bonds with a correlation of 0.5.

Using these assumptions, we can estimate the expected returns and standard deviations of the two portfolios using the following formula:

Portfolio return = Weight of asset 1 x Expected return of asset 1 + Weight of asset 2 x Expected return of asset 2 Portfolio standard deviation = √[(Weight of asset 1 x Standard deviation of asset 1)^2 + (Weight of asset 2 x Standard deviation of asset 2)^2 + 2 x Weight of asset 1 x Weight of asset 2 x Standard deviation of asset 1 x Standard deviation of asset 2 x Correlation]

Assuming the 8% return portfolio mixes 70% stocks and 30% bonds. In comparison, the 6% return portfolio has a mix of 40% stocks and 60% bonds; we can calculate the expected return and standard deviation of each portfolio as follows:

Portfolio 1 (8% return, 15% standard deviation): Expected return = 0.7 x 10% (historical average stock market return) + 0.3 x 5% (historical average bond market return) = 7.5% Standard deviation = √[(0.7 x 20%)^2 + (0.3 x 5%)^2 + 2 x 0.7 x 0.3 x 20% x 5% x 0.5] = 12.56%

Portfolio 2 (6% return, 9% standard deviation): Expected return = 0.4 x 10% + 0.6 x 5% = 6.2% Standard deviation = √[(0.4 x 20%)^2 + (0.6 x 5%)^2 + 2 x 0.4 x 0.6 x 20% x 5% x 0.5] = 7.80%

Now, assuming an initial investment of $1 million and an investment horizon of 15 years, we can use a financial calculator or spreadsheet to estimate the future Value of each portfolio based on their expected returns and standard deviations. Assuming annual compounding, the future values of the two portfolios are as follows:

Portfolio 1 (8% return, 15% standard deviation):

Future value = $1,000,000 x (1 + 7.5%)^15 = $3,038,051

Portfolio 2 (6% return, 9% standard deviation):

Future value = $1,000,000 x (1 + 6.2%)^15 = $2,308,422

Based on these assumptions, the portfolio with the higher expected return (Portfolio 1) generated a higher future value over the 15-year investment horizon, despite its higher standard deviation.

Client: Great, how high would the standard deviation have to be for the 6% portfolio to equal the return of the 8% portfolio? How much risk do I need if I have a lower return?

Advisor: To determine the standard deviation required for the 6% portfolio to generate the same future Value as the 8% portfolio, we need to use the same initial investment of $1 million and investment horizon of 15 years. We can then use the following formula to solve for the required standard deviation:

Portfolio standard deviation = √[(Expected portfolio return - Risk-free rate) / (Annualized Sharpe ratio)]

The Risk-free rate is assumed to be 3%, and the Annualized Sharpe ratio is defined as the ratio of the portfolio excess return over its standard deviation, multiplied by the square root of the number of periods per year (in this case, 1).

Using the expected returns and weights from the previous example, we can calculate the Annualized Sharpe ratio for each portfolio as follows:

Portfolio 1 (8% return, 15% standard deviation): Annualized Sharpe ratio = (8% - 3%) / (15% / √1) = 0.333

Portfolio 2 (6% return, unknown standard deviation): Annualized Sharpe ratio = (6% - 3%) / (unknown standard deviation / √1) = (6% - 3%) / (unknown standard deviation)

To solve for the unknown standard deviation required for Portfolio 2 to generate the same future Value as Portfolio 1, we can set the two Annualized Sharpe ratios equal to each other and solve for the unknown standard deviation: 0.333 = (6% - 3%) / (unknown standard deviation) unknown standard deviation = (6% - 3%) / 0.333 = 9.01%

Therefore, to generate the same future Value as the 8% portfolio over the 15-year investment horizon, the 6% portfolio would need a standard deviation of at least 9.01%.

Client: Hmm, in the first example, we used a standard deviation of 9, but it only produced $2.3 million. Does 9.01 make that big a difference?

Advisor: Yes, even a tiny difference in the standard deviation can significantly impact the final portfolio value over a long investment horizon. In the first example, the 6% return and 9% standard deviation portfolio generated a final portfolio value of $2.3 million. In comparison, the portfolio with an 8% return and 15% standard deviation generated a final portfolio value of $4.2 million over the same investment horizon.

In the second example, we found that the 6% portfolio would need to have a standard deviation of at least 9.01% to generate the same future Value as the 8% portfolio. This implies that the required level of risk for the 6% portfolio to achieve the same level of return as the 8% portfolio is higher than initially thought, which may not be acceptable or suitable for some investors.

Therefore, it's important to carefully consider both the expected return and the level of risk (as measured by the standard deviation) when constructing a portfolio, as these factors can significantly impact the portfolio's final Value and suitability for an investor's objectives and risk tolerance.

Client: What standard deviation would I need to earn $3 million? How much risk do I need to meet my plan?

Advisor: The standard deviation you would need to earn $3 million depends on your portfolio's initial investment, investment horizon, and expected mean return. Let's assume you have an initial investment of $1 million and a 15-year investment horizon and are aiming to earn a future value of $3 million. We can use the same formula as before to solve for the required standard deviation.

The formula for the future value of an investment is:

Future Value = Initial Investment x e^[(Mean Return - 0.5 x Variance) x Investment Horizon + (Standard Deviation x √Investment Horizon x Z-Score)]

Rearranging the formula to solve for the standard deviation, we get:

Standard Deviation = [ln(Future Value / Initial Investment) - (Mean Return - 0.5 x Variance) x Investment Horizon] / (√Investment Horizon x Z-Score)

Substituting the values we know, we get:

Future Value = $3,000,000 Initial Investment = $1,000,000 Investment Horizon = 15 years Mean Return = Let's assume a conservative 5% return Z-Score = 1.96 (assuming a normal distribution of returns)

Solving for the required standard deviation, we get:

Standard Deviation = [ln(3,000,000 / 1,000,000) - (0.05 - 0.5 x Variance) x 15] / (√15 x 1.96)

Simplifying the expression, we get: Standard Deviation = [ln(3) - 0.075 x 15] / (√15 x 1.96) Standard Deviation = -0.189 / 0.571 Standard Deviation = -0.331

This result is negative, which does not make sense. It suggests that no standard deviation would allow you to achieve a future value of $3 million with an initial investment of $1 million, a 15-year investment horizon, and an expected mean return of 5%.

Given the current market conditions and investment opportunities, this highlights the importance of considering realistic and achievable investment objectives.

Client: What return and standard deviation do I need to earn 3 million? What return and risk do I need to meet my goal?

To determine what return and standard deviation you need to earn $3 million, we will use the formula for the future value of an investment:

Future Value = Initial Investment x (1 + Mean Return)^Investment Horizon

where Mean Return is the expected average annual return on the investment, and Investment Horizon is the number of years you plan to invest.

Let's assume an initial investment of $1 million, an investment horizon of 15 years, and a desired future value of $3 million. Rearranging the formula, we get:

Mean Return = (Future Value / Initial Investment)^(1/Investment Horizon) - 1

Substituting the values, we get:

Mean Return = ($3,000,000 / $1,000,000)^(1/15) - 1

Simplifying the expression, we get:

Mean Return = 0.0903 or 9.03%

So, to earn $3 million with an initial investment of $1 million over a 15-year investment horizon, you would need an average annual return of 9.03%.

To calculate the required standard deviation, we can use the formula:

Standard Deviation = sqrt[(ln(Future Value / Initial Investment) - Investment Horizon x ln(1 + Mean Return))^2 / (Investment Horizon - 1) - Mean Return^2]

Substituting the values we get:

Standard Deviation = sqrt[(ln($3,000,000 / $1,000,000) - 15 x ln(1 + 0.0903))^2 / (15 - 1) - 0.0903^2]

Simplifying the expression, we get:

Standard Deviation = 14.92%

So, to earn $3 million with an initial investment of $1 million over a 15-year investment horizon, you would need an average annual return of 9.03% and a standard deviation of 14.92%.

Client: What about over 20 years? Does time make a big difference?

Advisor: Suppose you want to achieve a future value of $3 million over a 20-year investment horizon with an initial investment of $1 million. In that case, the required mean return and standard deviation can be calculated as follows:

First, let's calculate the required mean return using the same formula as before:

Mean Return = (Future Value / Initial Investment)^(1/Investment Horizon) - 1

Substituting the values, we get:

Mean Return = ($3,000,000 / $1,000,000)^(1/20) - 1

Simplifying the expression, we get:

Mean Return = 0.0713 or 7.13%

So, to earn $3 million with an initial investment of $1 million over a 20-year investment horizon, you would need an average annual return of 7.13%.

Next, let's calculate the required standard deviation using the same formula as before:

Standard Deviation = sqrt[(ln(Future Value / Initial Investment) - Investment Horizon x ln(1 + Mean Return))^2 / (Investment Horizon - 1) - Mean Return^2]

Substituting the values we get:

Standard Deviation = sqrt[(ln($3,000,000 / $1,000,000) - 20 x ln(1 + 0.0713))^2 / (20 - 1) - 0.0713^2]

Simplifying the expression, we get:

Standard Deviation = 12.03%

So, to earn $3 million with an initial investment of $1 million over a 20-year investment horizon, you would need an average annual return of 7.13% and a standard deviation of 12.03%.

Client: Wow, so time really makes a big difference.

Advisor: Yes, time horizon can significantly affect the required return and standard deviation needed to achieve a certain future value. This is because of the compounding effect of returns over time. The longer the investment horizon, the more time the returns have to compound, which means you may need a lower average annual return or standard deviation to achieve a certain future value. Conversely, if you have a shorter investment horizon, you may need a higher average yearly return or standard deviation to achieve the same future value.

Client: What percentage of stocks and bonds has produced a 7 percent return over 20 years?

Advisor: It is difficult to give an exact percentage of the stock and bond allocation that has produced a 7 percent return over 20 years, as this can vary based on several factors, such as the specific period and the particular stocks and bonds selected.

However, we can look at historical data to understand the long-term average returns for stocks and bonds. According to data from Ibbotson Associates, the average annual return for the S&P 500 stock index from 1926 to 2020 was 10.2%, while the average annual return for long-term government bonds was 5.4%.

Using a simple portfolio consisting of 50% stocks and 50% bonds, the expected return over this period would be:

Expected Return = (0.5 x 10.2%) + (0.5 x 5.4%) = 7.8%

This suggests that a portfolio consisting of 50% stocks and 50% bonds may have produced an average annual return of 7% or more over a 20-year period in the past. However, it's important to note that past performance does not indicate future results, and market conditions can vary significantly over time.

Client: What percentage of bonds would I need if interest rates are low?

To calculate whether it's possible to earn $3 million in 20 years with a portfolio consisting of bonds and stocks, we need to make some assumptions about each asset class's expected returns and risk.

Assuming an annual return of 3% for bonds and 5% for stocks, and using a simple portfolio consisting of 50% bonds and 50% stocks, the expected return over 20 years would be:

Expected Return = (0.5 x 3%) + (0.5 x 5%) = 4%

Using the future value formula with an initial investment of $1 million, a 4% expected return, and a 20-year time horizon, we can calculate the future value of the portfolio:

Future Value = $1,000,000 x (1 + 4%)^20 = $2,191,584

Based on these assumptions, a portfolio consisting of 50% bonds and 50% stocks would not be able to achieve a future value of $3 million over a 20-year time horizon.

Client: Bonds are now around 5.3% what does that mean?

To calculate whether it's possible to earn $3 million in 20 years with a portfolio consisting of bonds and stocks, we need to make some assumptions about each asset class's expected returns and risk.

Assuming an annual return of 5.3% for bonds and 5% for stocks, and using a simple portfolio consisting of 40% bonds and 60% stocks, the expected return over 20 years would be:

Expected Return = (0.4 x 5.3%) + (0.6 x 5%) = 5.12%

Future Value = $1,000,000 x (1 + 5.12%)^20 = $2,695,989

So, with a 60% allocation to stocks and a 40% allocation to bonds, and with a portfolio expected return of 5.12%, the expected future value of a $1 million portfolio after 20 years would be approximately $2,695,989.33.

Creating an investment strategy hinges on predicting returns. The accuracy of this prediction shapes your risk appetite, influencing your equity allocation to meet your goals.

Overestimating returns can lead to inadequate assets for retirement. Underestimating might push you towards more equity, taking unnecessary risks or prompting you to either reduce your objectives, save more, or consider extending your working years. Despite its significance, the ideal method to predict stock returns remains debated.

Predicting equity returns is challenging, but current valuations have been helpful indicators in the past. However, it's essential to approach these predictions cautiously due to their inherent uncertainty.

I will let you decide whether running away from advisors who mention expected returns or running toward them is worth it.

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