How can you generate passive income from return on stock market investments or dividends*Dividends are of course also a kind of investment income. I just wanted to mention them again because they play a special role in this article. and when does that make sense? In addition to explanations and anecdotes, I also present mathematical formulas to answer this question🥳*That's a smiley celebrating the existence of math in this article..

Before I continue to write about a financial topic, I would like to point out that I am not a financial expert at all. I am only a private person who is interested in their own retirement provision and other financial topics, therefore amassing solid half-knowledge. Especially, the tax related stuff represents my understanding of the situation in Germany.

The original text of this post is in German. The translation was a collaboration between man and machine. The man was me and the machine was Google Translate.

Dividend

Some stock corporations, i.e. companies whose shares can be bought on the stock market, pay out a certain amount in order to let their shareholders participate in their profits. These shared profits are called dividends. The share price falls by the corresponding amount after the dividend has been paid out. From an private investor’s point of view, the payment of dividends equals the sale of shares*That is not entirely true. Capital gains tax is payable on dividends. Taxes are only incurred on the sale of shares if they are realized capital gains. In return, transaction costs are often due for sales, which are probably negligible with the current offers. Moreover, selling shares diminishes the weight of your vote at the stock corporation's general assembly..

Passive income

By passive income I mean an income that is only indirectly linked to my working hours. I’ll try to illustrate this with a few examples.

Passive income is drawn when you …

  • ... wrote a sci-fi novel and earned money on its sale.
  • ... take capital gains, e.g. in the form of dividends.
  • ... are the owner of a service company and bill and squeeze out employees on an hourly basis, but you earn from the company's profits*Even without squeezing out, the income would be passive..

Active income is drawn when you …

  • ... are employed and receive a salary of 100,000 euros per year for 35 hours of work per week *I have deliberately chosen a salary that is significantly above average for Germany with a below average number of hours as an example. Of course I also consider that to be an active income*Incidentally, there is an interesting application by the Federal Statistical Office for salary comparison for employees in Germany, which, as far as I can remember, is significantly more accurate than everyone else I've seen so far..
  • ... are a freelance clown with a hourly rate for your performance.

These examples make hopefully clear that passive income does not necessarily imply the lazy spring.

Retirement provision by investing in the global economy

With my first salary, I also had my first interview with a representative of Deutsche Vermögensberatung a few years ago. The products on offer didn’t convince me at the time. I did not want to feed the insurance industry without receiving adequate value in return. In the absence of alternatives, I then deposited money into a de facto non-interest-bearing money market account*Tagesgeldkonto in German every month for private retirement provision. At that time I only dealt very superficially with the stock market. I was very suspicious of things like high-frequency trading and overpaid fund managers. A few years later I still hadn’t found an alternative to my money market account. Instead I came across a book*Souverän investieren mit Indexfonds und ETFs: Wie Privatanleger das Spiel gegen die Finanzbranche gewinnen, Gerd Kommer, 2015 about ETFs. In short, ETFs are, among other things, a financial instrument that allows private investors to easily make a broadly diversified investment in the global stock market. In simplified terms, the return corresponds to the growth of the global economy. However, it took a few more years before I became familiar with the content of the book and started using ETFs to organize my private retirement provision.

Dividends as passive income

I recently exchanged text messages with a friend who is pursuing a similar private pension strategy. In addition to his savings plan in the global stock market, he has now invested money in stocks that pay out a particularly large amount of dividends in order to generate passive income now and possibly have to work less. An additional stream of income that you don’t have to work for is sure to feel good.

My first thought was that it doesn’t make any sense to invest a certain amount in the stock market as a retirement plan every month and at the same time withdraw another amount in terms of dividends. Receiving dividends is identical to selling shares. So you can simply lower the savings rate. Then, as with passive income, you have more money available every month. In addition, you have to tax dividends above the exemption order, although the diligent saver wants every Euro to generate as much return as possible before it is taxed. A passive income from investment income is still an economically sensible option if you are no longer in the savings phase.

Let us assume that you have left the savings phase and the savings plan rate is accordingly 0 monetary units. In order to generate passive income with dividends, you have to limit yourself to those companies that pay dividends. But it doesn’t have to be the companies whose share price is performing best. For example, a company can pursue the strategy of re-investing all profits in its own growth and future. As I understand it, it’s called a growth strategy. Many of the most successful companies of the past decade have pursued such a growth strategy. So you may have opportunity costs if you rely on a dividend strategy. These are not real costs, but missed returns. Instead of drawing dividends, with transaction costs that are negligible these days, you can take capital from your own stock portfolio by regularly selling shares. In a blog post *accessed on April 4, 2021 Gerd Kommer et. al. mentions that private investors classify high dividend yields as being particularly desirable emotionally. Instead they should be rationally focusing on the sole relevant total return of a share. This is one of the many cognitive investor errors that the research area Behavioral Finance has documented over the past decades. Interestingly, the behavioral finance field has also dealt with the fear of missing out on returns (Fomo). So if I succumb to the temptation of dividends, my Fomo is likely to be eligible. I understand that an income stream from dividends feels good, even with economically better decisions at hand.

If I wanted to generate passive income from stock market investment, I would do the following.

  1. Think about how high the income should be and how long it would like to be drawn, e.g. 1000 euros per month for the rest of your life.
  2. Save as quickly as possible through 100% re-investment of all dividends enough capital to be able to pay out the income over the desired period of time. Do not pay attention to shares that pay particularly high dividends in order to avoid opportunity costs.
  3. After reaching the required amount, withdraw the desired amount from the stock portfolio every month. Whether this should happen through the distribution of dividends or the sale of shares plays a subordinate role*Receiving dividends is probably easier than selling shares since you usually cannot sell fractions of shares. I have not heard of a broker offering an automated withdrawal plan.. Even now, do not switch to high-dividend shares so that there are no opportunity costs.

Now you could think to yourself: “Great tips. Unfortunately, they do not help. How do I know how much capital I need for which monthly payout amount, you fool?” In order to find an answer to this question, I have done a little math and made a calculator so that the willing reader can enter their personal numbers at the end of the article.

Passive income after the savings phase

In this section I derive an answer to the following question.

How much capital do you have to save in order to be able to generate an income stream of a certain amount over a certain period of time during the withdrawal phase?

If you don’t feel like pushing formulas, you can scroll down to the calculators and enter specific numbers. I repeat that I am not a financial expert and the following has sprung from my more or less common sense. I could certainly have looked it up in any book or script. Feel free to report errors to me.

Calculation

Let $W_{i\tau}\in\mathbb{R_{\geq 0}}$ be the market value of the $i$-th security in the portfolio at time $\tau$. The amount of a security at time $\tau$ is denoted by $x_{i\tau}\in \mathbb{R_{\gt 0}}.$ Further, let $p_\tau=\frac{\sum_i W_{i\tau}x_{i\tau}}{\sum_j x_{j\tau}}$ be the weighted mean of the securities’ market value.

Let $a$ be the amount that you want to withdraw each month and $K_0=\sum_iW_{i0}x_{i0}$ the total value of the stock portfolio at the start of the withdrawal phase. Further, let $w_t=\frac{p_t}{p_{t-1}}$ for $t\gt 0$ denote the relative market value at the end of month $t$ and $w_0 = 1$. For example, $w_1 = 2$ if the market value doubled in month $ 1 $. The first withdrawal takes place after one month at the time $ t = 1 $ with a relative market value $ w_1 $. To determine the capital after the first withdrawal $ K_1 $, we calculate the total market value $ K_0w_1 $ corresponding to the relative market value at time $ t = 1 $ and subtract the withdrawal amount $ a $ from this. Accordingly, we get $ K_1 = K_0w_1-a. $ This can be continued with $ K_2 = K_1w_2-a $ and $ K_3 = K_2w_3-a $. Inserting $ K_1 $ in $ K_2 $ and $ K_2 $ in $ K_3 $ results in $$ K_3 = ((K_0w_1-a) w_2-a) w_3-a $$ $$ = K_0w_1w_2w_3-a (w_2w_3 + w_3 + 1) $$ which is generalized to $$ K_m = K_0 \prod_ {t = 1} ^ mw_t-a (\sum_ {k = 2}^m \prod_ {t = k}^mw_t + 1) $$ where $ K_m $ corresponds to the value of the stock portfolio after $ m $ months. Further, we have $$K_0=\frac{K_m}{\prod_{t=1}^mw_t}+\frac{a(\sum_{k=2}^m\prod_{t=k}^mw_t+1)}{\prod_{t=1}^m w_t}$$ $$=\frac{K_m}{\prod_{t=1}^mw_t}+\sum_{k=2}^m\frac{a}{\prod_{t=1}^{k-1}w_t}+\frac{a}{\prod_{t=1}^m w_t}.$$ We distinguish two cases.

  1. Capital is preserverd
    We have $K_0=K_m$ and hence $$K_0=\frac{K_0}{\prod_{t=1}^mw_t}+\sum_{k=2}^m\frac{a}{\prod_{t=1}^{k-1}w_t}+\frac{a}{\prod_{t=1}^m w_t}.$$ This is equivalent to $$K_0=(\sum_{k=2}^m\frac{a}{\prod_{t=1}^{k-1}w_t}+\frac{a}{\prod_{t=1}^m w_t})/(1-\frac{1}{\prod_{t=1}^mw_t}).$$ Note that $m\rightarrow \infty$ and $w_t>1$ for almost all $t$ implies $(1-\frac{1}{\prod_{t=1}^mw_t})\rightarrow 1$.
  2. Capital is consumed
    In the case $K_m=0$, you need a starting capital of $$ K_0 = \sum_ {k = 2} ^ m \frac {a} {\prod_ {t = 1} ^ {k-1} w_t} + \frac {a} {\prod_ {t = 1 } ^ m w_t}.$$ The following Elm calculation tool uses this formula. Modulo bugs.

Calculator in Elm

Instead of using a true share price trend, the tool calculates savings plans in a simplified manner based on annual returns. From this, a price trend is simulated that can be used with the above formula. See the following plot for an example with 10% annual return. Note how the exponential growth has made the price more than fifteenfold in 30 years. In comparison, the price multiplies by a factor of approx. 4.32 after 30 years with an annual return of 5%.

In addition to the savings plan calculator, I have also included a savings plan calculator. In this way you can also check out how long you have to save for your passive income. If you want to draw the passive income forever, you can enter a large number of years. The upper limit of years is 999.

I have described some technical details of the implementation of the savings plan calculator in a previous article. The code is on Github.