The reality is that most people fundamentally work in life because they need money. Without it, it’s impossible to be independent and have much in the way of an enjoyable, fulfilling lifestyle. There also (ideally) comes a point where enough is stowed away such that work is no longer necessary and other pursuits can be explored, commonly known as retirement.

For many, retirement planning is usually undertaken through the means of an employer-based system, a 401(k), IRA, personal brokerage accounts, third-party investors, and often a combination.

The goal, then, is to find out how much should be saved/invested and for long to have enough to satisfy one’s standard of living during retirement.

There are a few main variables in play – let’s call them the 7 basics:

- initial sum invested
- how much will be invested periodically
- annualized expected return
- how the amount is compounded (e.g., continuously, daily, weekly, monthly, quarterly, annually)
- the number of years over which the investments are held, and
- any leverage used in the investment account
- how much will be taken out (preferably none before retirement)

The scenario where an initial sum is invested and left until retirement can be readily modeled via an Excel spreadsheet. The more likely scenario where a certain amount of money is put away periodically and built up over time is more unwieldy to model through Excel. It requires integrals (or more tedious modeling), which simply isn’t the best use of Excel, but can be readily done in a statistical computing software, such as R.

In modeling the first scenario (one big lump sum invested and taken out at retirement), I chose the following figures:

- Initial sum invested = $100,000
- Leverage factor = 33% (1.5:1 leverage factor)
- Regular contribution = $1,000 per month
- Annualized return = 6% (9% with leverage)
- Number of years held = 40 (480 months)

This can be solved via the following formula:

- Total sum = Initial sum invested * 1/Leverage factor * (1 + Annualized return)^(# of years held)

With these inputs, that initial $100,000 would be worth about $4.3 million

So if $100,000 was invested at age 25, in a portfolio with a 1.5:1 leverage margin, earned an average of a 6% annualized return (9% with leverage), and was withdrawn at age 65, the individual would have a little over $4 million before taxes were applied.

Let’s assume that after taxes are applied the individual would have $3 million (this is in today’s dollars), and spent $100,000 per year as part of his or her lifestyle, that would provide for up to 30 years of retirement wealth.

Is this enough?

Assuming the individual doesn’t live beyond the age of 95, this would be sufficient, but there’s always the risk of outliving one’s savings, which can be a scary thought.

However, if one spends $75,000 per year, that’s 40 years (to age 105).

If $70,000 per year in spending, that’s 46 years (to age 111).

If $60,000 per year, that’s 50 years (to age 115).

Now there also might be some objections to the figures we used as part of this.

First, very few 25-year-olds have $100,000 to invest toward retirement planning or the discipline to leave it sit for 40 years. This is true, but assumptions need to be made for purposes of projection.

Second, is the amount of leverage involved.

Leverage involves borrowing money from a broker to purchase more stocks than you would be able to do otherwise.

With a leverage factor of 33%, this means that 67% of your own money goes into buying the securities, while the other 33% is borrowed on margin at a fee. This means that with a 40% drawdown in the market, the account would be down 60%.

This type of drawdown will be rare, especially with some small amount of diversification into things like fixed income, smaller amounts of commodities, etc.

A 20% drawdown is considered a bear market, and with central banks in place ready to buoy up a market when things turn even remotely ugly, a 50% drawdown can easily happen if concentrated in one or a few assets or sectors of the stock market (or any asset class), but would be difficult if well-diversified.

Also, with proper investment picking, risk management, hedging, etc., any investment strategy should aim to beat the overall market, which is a general assortment of equities to choose from.

Moreover, leverage can also be pared down over time to limit risk, but kept high early in life to robustly grow the account despite relatively low exposure. Moreover, some might argue that something like 2:1 margin early in life actually lowers risk.

Instead of black-and-white statements like “no leverage is good, any leverage is bad” a well-diversified, prudently leveraged portfolio is better than a concentrated, unleveraged one.

And third, some might squabble with a 6% annualized return given the earnings yields on stocks and bonds. Since February 1971, the main market indices in the US have returned roughly the following on an annualized basis:

- Dow Jones Industrial Average: 7%
- S&P 500: 7%
- NASDAQ: 9%

(These figures can be calculated through measuringworth.com.)

Naturally, one should expect about a 7% return on equities per year over the long run with some notable bumps along the way. The NASDAQ returns about 9%, but is more tech-focused.

These stocks have higher betas and are hence riskier as a result, which should cause larger drawdowns but theoretically higher long-run returns.

However, when considering an entire index, there are bound to be several stocks that are bad apples at any given moment and aren’t worth the investment.

Some may have flat or negative growth projections over the next few years. And some might simply be overvalued by several valuation metrics.

Whatever the case, beating an index is not necessarily *too* difficult so long as you make sound investing decisions, rebalance the portfolio at least quarterly, and keep emotions out of play.

There have been tens of thousands of books written regarding the content of the previous paragraph as it pertains to making sound investment decisions, so I will leave that topic for now.

But beating 7% annualized can be done (and with less risk), and investors with long-term time horizons (e.g., retirement planners) might actually have an advantage over short-term investors who may not strictly invest with value in mind.

However, for comparison’s sake, we also did some sensitivity analysis to compare how this initial invested sum would compare if it achieved an annualized return 3% above or below the amount entered in the spreadsheet.

For instance, if 6% annualized is projected (9% with leverage), the sensitivity analysis will look at a return ranging from 3%-12% annualized (4.5%-15% with leverage).

At 6%, we forecasted just over $4 million before taxes. However, if we go all the way down to 3% (4.5% with leverage), our sum comes to just about $1.3 million before taxes.

On the flip side, at 9% annualized (15% with leverage), this balloons to over $23 million before taxes.

A one-percent difference between 12% and 13% comes to about an $8 million gain or loss.

Due to the power of compounding, a small 1% difference (e.g., 6%-7%) manages to generate an exponential effect over time.

For example, going from 6% to 7% pre-leverage (9% to 10.5% with leverage) causes a 50% difference in outcomes ($6.4 million vs. $4.3 million).

## The importance of contributing regularly

Now let’s take a look at the alternative scenario where an individual invests a set amount in periodic even intervals over time.

Let’s assume the individual invests $10,000 per year and let’s say they have the following mix going on:

- 2:1 leverage
- annualized return of 10%, and
- a 40-year investment period (last $10,000 investment – in 2022 dollars – made in year 39, one year before withdrawal)

This isn’t very fun to do in Excel. We can basically model this situation using a mathematical equation and then find the total return by finding the area under the curve, or the mathematical function of an integral.

Our equation would look like this:

- Total sum = Amount invested per period * 1/Leverage factor * (1 + Annualized return)^(# of years held)

We can fill in amounts for the first three variables on the right side of the equation:

- Total sum = $10,000 * 1/0.50 * (1 + 0.10)^(# of years held)

“Total sum” can essentially serve as our *y* variable, while “# of years held” can represent our *x*.

Simplified:

- y = 20,000*(1.1)^x

If we integrate with respect to x from 0 to 40 (the years), we have:

- 0 40 ∫ $20,000*(1.1)^x dx = $9,287,414

For the sake of ease, I performed this integral on R software using the following code:

` integrand <- function(x) {20000*(1.1)^x}`

integrate(integrand, lower=0, upper=40)

Investing $10,000 per year from age 25 through 64 actually has the chance to outpace the figure obtained from investing a lump $100,000 sum.

The main difference is that in the annual investment scenario, a grand total of $400,000 is invested over the years, or four times the amount as the first scenario.

This illustrates the importance of investing early in life, as the funds invested earliest will likely have the greatest impact on terminal value earnings.

## Conclusion

How much is needed for retirement depends on your lifestyle, personal spending habits, and other factors. This determines time-related factors as well, such as whether you can retire early.

When it comes to your 401k and retirement investing, it’s important to start investing as early as possible, contribute regularly, and use all advantages available at your disposal, such as a tax-deferred retirement plan.

Due to the power of compounding, it all adds up over time.