## How Many Years To Financial Independence?

### Introduction

We define financial independence as the point at which active income is no longer necessary. This FI point represents a switching point between two phases of our life. In the pre-FI phase, we have to actively work to pay for our expenses and build our investments. In the post-FI phase, we can put our investments to work and use it's passive income to meet our expenses. Our Years-to-FI calculator shows how various factors influence the time required to reach FI. In this article, we present the math used to build the calculator.

We have two objectives:

• Calculate the time to reach the FI point as a function of its influencing factors.
• Predict the value of our portfolio over its lifetime.

Our portfolio value $$p$$ at any year $$n$$ can be represented by \begin{equation} p(n) = \begin{cases} p(n)\rvert_\text{preFI} &\text{if } n\leq n^*\\ p(n)\vert_\text{postFI} &\text{if } n\gt n^* \end{cases} \end{equation} where $$n^*$$ is the number of years required to reach the FI point. This represents the distinction between our pre-FI and post-FI phases. To meet our objectives, we will need to find the equation for $$n^*$$, and equations for $$p(n)\rvert_\text{preFI}$$ and $$p(n)\vert_\text{postFI}$$

#### Porfolio Scenarios

We will consider two scenarios:

• Forever Portfolio: We want our portfolio to last and grow forever.
• Finite Portfolio: We only need our portfolio to last for some definite number of years. We can run out of money at the end.

The forever portfolio is a safer and more robust choice, as it provides more protection from uncertainty.

The two scenarios exhibit the same portfolio growth during the years of active income, but they differ in the conditions at which FI is reached, and in their behavior after FI. Mathematically, they have the same $$p(n)\rvert_\text{preFI}$$, but different equations for $$n^*$$ and $$p(n)\vert_\text{postFI}$$

### Annual Portfolio Value Change

First, we will look at how our portfolio value, $$p$$, changes in a single year. (We could look at this on a monthly basis for more fidelity, but annual values are simpler to work with for now.) Initally, at $$n=0$$, our portfolio value has some value $$p(0)$$. At the end of 1 year, our portfolio value may be expressed as: \begin{equation} p(1) = p(0) + (\text{active income in year 1}) + (\text{passive income in year 1}) - (\text{expenses in year 1}) \label{eq:year-one-words} \end{equation} Active income is that brought in by your work, passive income is that generated by your existing assets, and expenses is any money spent and not saved. For any year $$n$$, let $$i_a(n)$$ = active income, $$i_p(n)$$ = passive income, and $$e(n)$$ = expenses. We can then rewrite Equation \eqref{eq:year-one-words} as, \begin{equation} p(1) = p(0) + i_a(1) + i_p(1) - e(1) \label{eq:year-one-general} \end{equation} In fact, this equation can be applied to any year $$n$$: \begin{equation} p(n) = p(n-1) + i_a(n) + i_p(n) - e(n) \label{eq:annual-change} \end{equation} This equation describes how a portfolio changes in value in a year. If we know incomes and expenses for every year $$n$$, we could simulate this out and see how our portfolio value changes over time. In real life, incomes and expenses vary every year, which makes financial prediction challenging. We will have to make some assumptions in order to make this problem tractable.

### Pre-FI Phase

#### Simplifying Assumptions

During the years of active income, both the Forever portfolio and Finite portfolio behave the same during the years of active income, and thus follow the same assumptions:

• Active income remains constant, adjusted for inflation: $$i_a(n) = i_a = \text{const.}$$
• Expenses remain constant, adjusted for inflation: $$e(n) = e = \text{const.}$$
• Passive income will be modeled as a function of the previous year's portfolio value: $$i_p(n) = p(n-1) r_{roi}$$ where $$r_{roi}$$ is a constant inflation-adjusted rate of return on investment.

Note that the constant active income assumption is conservative, as it is common for income to increase. Expenses are often a choice, and it could well be within your power to keep these constant. However, constant return on investment is undoubtedly an optimistic assumption. If your investment portfolio contains stocks, then consistency is impossible. If your investment portfolio includes real estate, then maybe your returns will be more predictable. This is definitely a big assumption, and one that requires awareness and greater discussion. In the meantime, here are some additional resources and opinions regarding market returns:

#### Portfolio Value Equation

Applying our assumptions to Equation \eqref{eq:annual-change}: \begin{align*} p(n) &= p(n-1) + i_a + p(n-1) r_{roi} - e\\ &= (1+r_{roi}) p(n-1) + (i_a - e)\\ \end{align*} Letting $$(i_a - e) = s$$, where $$s$$ represents annual savings: \begin{equation*} p(n) = (1+r_{roi}) p(n-1) + s \end{equation*} Thus, for any year $$n$$, our portfolio value can be calculated from the previous year's value. Because we know our initial value $$p(0)$$, we can step through each year and calculate the portfolio value. Let $$p(0) = p_0$$ After year 1: \begin{align*} p(1) &= (1+r_{roi}) p(0) + s\\ &= (1+r_{roi}) p_0 + s\\ \end{align*} For year 2: \begin{align*} p(2) &= (1+r_{roi})p(1) + s\\ &= (1+r_{roi})[(1+r_{roi}) p_0 + s] + s\\ &= (1+r_{roi})^2 p_0 + (1+r_{roi}) s + s\\ \end{align*} For year 3: \begin{align*} p(3) &= (1+r_{roi}) p(2) + s\\ &= (1+r_{roi}) [(1+r_{roi})^2 p_0 + (1+r_{roi}) s + s] + s\\ &= (1+r_{roi})^3 p_0 + (1+r_{roi})^2 s + (1+r_{roi}) s + s\\ \end{align*} For any year $$n$$ in the pre-FI phase: \begin{equation} p(n)\rvert_\text{preFI} = (1+r_{roi})^n p_0 + s\sum_{j=0}^{n-1} (1+r_{roi})^j \label{eq:year-n-sum} \end{equation} However, our finite sum simplifies to: \begin{align*} \sum_{j=0}^{n-1} (1+r_{roi})^j &= \frac{1 - (1+r_{roi})^n}{1 - (1+r_{roi})} \\ &= \frac{ (1+r_{roi})^n - 1}{r_{roi}} \end{align*} Equation \eqref{eq:year-n-sum} thus becomes: \begin{align} p(n)\rvert_\text{preFI} &= (1+r_{roi})^n p_0 + \frac{ (1+r_{roi})^n - 1}{r_{roi}} s\notag\\ &= (1+r_{roi})^n \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} \label{eq:portfolio-value-preFI}\\ \end{align} Equation \eqref{eq:portfolio-value-preFI} gives us the value of our portfolio at the end of any year $$n$$ during the active income phase. It depends on our initial portfolio value, our annual savings, and our rate of return on investment.

### Post-FI Phase

After FI, our assumptions regarding income and expenses change, and thus our portfolio value equation changes. Additionally, the time it takes to reach FI depends on the chosen portfolio scenario (Forever or Finite). In a forever portfolio, we want the portfolio to continue to grow forever. In a finite portfolio, we don't need the portfolio to grow, as long as we don't run out of money before the specified number of years. We will therefore apply different constraints to our portfolio value equation and define different FI conditions in order to calculate the times to FI.

#### Simplifying Assumptions

By definition, our active income disappears after FI, and we begin withdrawing directly from the portfolio to meet our expenses.

• Active income becomes zero: $$i_a(n) = 0$$
• Expenses remain the same as in pre-FI years: $$e(n) = e = \text{const.}$$
• Annual expenses are withdrawn at the start of the year.
• Passive income will be modeled as a function of the portfio value at the start of the year, after removing expenses: $$i_p(n) = (p(n-1) - e(n)) r_{roi}$$ where $$r_{roi}$$ is a constant inflation-adjusted rate of return on investment.

#### Portfolio Value Equation

Applying our assumptions to Equation \eqref{eq:annual-change}, our equation describing annual portfolio change becomes: \begin{align} p(n) &= p(n-1) + 0 + (p(n-1) - e) r_{roi} - e \notag\\ &= (1 + r_{roi})(p(n-1) - e) \label{eq:forever-response}\\ \end{align} This equation does not apply until after FI. Instead of beginning at year $$n=0$$, we can begin at the yet-unknown FI point $$n=n^*$$: $$p(n^*) = p_{n^*}$$ One year after FI: \begin{align*} p(n^* + 1) &= (1 + r_{roi})(p(n^*) - e) \\ &= (1 + r_{roi})(p_{n^*} - e) \\ \end{align*} Two years after FI: \begin{align*} p(n^* + 2) &= (1 + r_{roi})(p(n^* + 1) - e) \\ &= (1+r_{roi})\left[ (1 + r_{roi})(p_{n^*} - e) - e \right] \\ &= (1 + r_{roi})^2 (p_{n^*} - e) - (1+r_{roi}) e \\ \end{align*} Three years after FI: \begin{align*} p(n^* + 3) &= (1 + r_{roi}) (p(n^* + 2) - e) \\ &= (1+r_{roi}) \left[ (1 + r_{roi})^2 (p_{n^*} - e) - (1+r_{roi}) e - e \right] \\ &= (1 + r_{roi})^3 (p_{n^*} - e) - (1+r_{roi})^2 e - (1+r_{roi}) e \\ \end{align*} For any year $$m$$ after FI: \begin{equation} p(n^* + m) = (1 + r_{roi})^m (p_{n^*} - e) - e \sum_{j=1}^{m-1} (1+r_{roi}) \label{eq:year-m-sum} \end{equation} Simplifying our geometric series: \begin{align*} \sum_{j=1}^{m-1} (1+r_{roi}) &= \sum_{j=0}^{m-1} (1+r_{roi}) - 1\\ &= \frac{ 1 - (1+r_{roi})^m }{ 1 - (1+r_{roi}) } - 1\\ &= \frac{ (1+r_{roi})^m - 1 }{ r_{roi} } - 1\\ \end{align*} Equation \eqref{eq:year-m-sum} then becomes: \begin{align*} p(n^* + m) &= (1 + r_{roi})^m (p_{n^*} - e) - e \left[ \frac{ (1+r_{roi})^m - 1 }{ r_{roi} } - 1 \right]\\ &= (1 + r_{roi})^m \left( p_{n^*} - e - \frac{ e }{ r_{roi} } \right) + e \left( \frac{ 1 }{ r_{roi} } + 1 \right)\\ &= (1 + r_{roi})^m \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) + \frac{ 1 + r_{roi} }{ r_{roi} } e\\ \end{align*} Thus for any year $$n = n^* + m$$ during the post-FI phase: \begin{equation} p(n)\rvert_\text{postFI} = (1 + r_{roi})^{(n-n^*)} \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) + \frac{ 1 + r_{roi} }{ r_{roi} } e \label{eq:portfolio-value-forever-postFI} \end{equation} We can now write the combined equation for portfolio value for any year in a Portfolio's lifetime by combining Equations \eqref{eq:portfolio-value-preFI} and \eqref{eq:portfolio-value-forever-postFI} \begin{equation} p(n) = \begin{cases} (1+r_{roi})^n \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} &\text{if } n\leq n^*\\ (1 + r_{roi})^{(n-n^*)} \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) + \frac{ 1 + r_{roi} }{ r_{roi} } e &\text{if } n\gt n^* \end{cases} \end{equation}

#### Forever Portfolio

##### The Withdrawal Rate

It is common to use a withdrawal rate ($$r_w$$) to describe how much we take out of our portfolio every year: \begin{equation} e = r_w p_{n^*} \label{eq:withdrawal-rate-definition} \end{equation} That is, the amount we withdraw for expenses is some percentage of our portfolio value at the time of FI. For example, if your portfolio value is 1 million at the time of FI, and you use a withdrawal rate of 4%, then every year you would take out 40,000 to spend.

Let's introduce the withdrawal rate to our post-FI portfolio value equation \eqref{eq:portfolio-value-forever-postFI}: \begin{align} p(n)\rvert_\text{postFI} &= (1 + r_{roi})^{(n-n^*)} \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } r_w p_{n^*} \right) + \frac{ 1 + r_{roi} }{ r_{roi} } r_w p_{n^*} \notag\\ &= p_{n^*} \left[ (1 + r_{roi})^{(n-n^*)} \left( 1 - \frac{ 1 + r_{roi} }{ r_{roi} } r_w \right) + \frac{ 1 + r_{roi} }{ r_{roi} } r_w \right] \label{eq:portfolio-value-forever-postFI-rw}\\ \end{align}

##### Constraints

In order for our portfolio to last forever, we have to make sure that our portfolio value given by $$p(n)\rvert_\text{postFI}$$ is always positive. This requires that we set some constraints on Equation \eqref{eq:portfolio-value-forever-postFI-rw}.

First, our return on investment has to be positive: $$r_{roi} > 0$$ Otherwise, the factor $$(1 + r_{roi})^{(n-n^*)}$$ would cause our value to decay, and the $$\frac{ 1 + r_{roi} }{ r_{roi} } r_{w}$$ term could cause our portfolio value to go negative. Investment return is our only way to increase value during the post-FI phase, and it's hard to keep a portfolio forever if it never grows. (In reality, our return could very well be negative over some years, and we would need to have positive years that make up for it. Otherwise, we will have to pursue some active income.)

Secondly, we need \begin{align} \left( 1 - \frac{ 1 + r_{roi} }{ r_{roi} } r_w \right) &\geq 0 \notag\\ \implies 1 &\geq \frac{ 1 + r_{roi} }{ r_{roi} } r_w \notag\\ \implies r_w &\leq \frac{ r_{roi} }{ 1 + r_{roi} } \\ \end{align} to ensure our portfolio value remains positive. This puts an upper limit on what your withdrawal rate can be. Choosing a withdrawal rate less than this gives you a buffer against uncertainty in your return on investment. For example, if your $$r_{roi}$$ is 5%, then your withdawal rate must be less than $$0.05 / 1.05$$, or 4.76%. If you wanted to be a little more cautious, you could set a withdrawal rate of 4%, and withdraw less than your upper limit every year.

##### FI Condition

We assumed that we wanted to maintain our pre-FI expenses after FI. Using our withdrawal rate definition in Equation \eqref{eq:withdrawal-rate-definition}, we see that our required portfolio value at the FI point is: \begin{equation} p_{n^*} = \frac{e}{r_w} \label{eq:forever-condition} \end{equation} As an example, assume we have a withdrawal rate of 4%: \begin{equation*} p_{n^*} = \frac{e}{0.04} = 25e \end{equation*} When your portfolio value reaches 25 times your expenses, you have reached FI!

##### FI Point

Applying our FI condition from Equation \eqref{eq:forever-condition} to Equation \eqref{eq:portfolio-value-preFI} \begin{equation*} p_{n^*} = p(n^*) = (1+r_{roi})^{n^*} \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} = \frac{e}{r_w} \end{equation*} We can solve this for $$n^*$$: \begin{align} (1+r_{roi})^{n^*} \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} &= \frac{e}{r_w}\notag\\ (1+r_{roi})^{n^*} &= \frac{\frac{e}{r_w} + \frac{s}{r_{roi}}} {p_0 + \frac{s}{r_{roi}}}\notag\\ n^* &= \left. \log \left( \frac{ (e/r_w) + (s/r_{roi}) }{ p_0 + (s/r_{roi}) } \right) \middle/ \log (1+r_{roi}) \right. \label{eq:nstar-forever}\\ \end{align} This is the number of years it takes to get to FI for a Forever Portfolio. It depends on:

• annual savings, $$s$$
• annual expenses, $$e$$
• rate of return on investment, $$r_{roi}$$
• withdrawal rate, $$r_w$$
• inital portfolio value, $$p_0$$

Equation \eqref{eq:nstar-forever} can be rewritten in a variety of ways. For example, we can introduce a ratio of savings-to-expenses: \begin{equation*} s_e = \frac{s}{e} \end{equation*} and express the initial portfolio value $$p_0$$ as a multiple of expenses: \begin{equation*} p_{e,0} = \frac{p_0}{e} \end{equation*} Then we can rewrite Equation \eqref{eq:nstar-forever} as: \begin{equation} n^* = \left. \log \left( \frac{ (1 / r_w) + (s_e / r_{roi}) }{ p_{e,0} + (s_e / r_{roi}) } \right) \middle/ \log (1+r_{roi}) \right. \label{eq:nstar-forever-2} \end{equation} If we wanted to use the more intuitive ratio of savings-to-income ($$s_i$$), commonly called a savings rate, we could replace $$s_e$$ with: \begin{equation} s_e = \frac{s}{e} = \frac{s}{i_a - s} = \frac{i_a (s / i_a)}{i_a - (s/i_a) i_a} = \frac{i_a(s_i)}{i_a(1-s_i)} = \frac{s_i}{1-s_i} \label{eq:savings-rate} \end{equation} Equation \eqref{eq:nstar-forever-2} then becomes dependent on:

• savings rate, $$s_i$$
• rate of return on investment, $$r_{roi}$$
• withdrawal rate, $$r_w$$
• inital portfolio value as a multiple of expenses, $$p_{e,0}$$

This is form of the equation that we use to generate the graphs in our calculator, and the factors we include (we calculate $$p_{e,0}$$ behind the scenes).

#### Finite Portfolio

##### Constraints

Unlike in our Forever portfolio, the value of the Finite portfolio does not need to keep growing after FI, and we do not need to apply any constraints to Equation \eqref{eq:portfolio-value-forever-postFI}.

##### FI Condition

In the case of a Finite Portfolio, after we reach FI, our portfolio value will decline until it is empty. Therefore, we need to build up enough value before FI so that it doesn't drop to zero before the specified number of years. Therefore, we need to calculate $$n^*$$ such that: \begin{equation} p(n_f) \geq 0 \end{equation} where $$n_f$$ is the specified finite portfolio duration.

Note that we can't choose both the withdrawal rate and annual expenses for the finite portfolio, as this specifies $$p(n^*)$$ and doesn't allow for control over the portfolio duration.

##### FI Point

The FI point $$n^*$$ acts as a switching point between our pre-FI and post-FI portfolio value equations. In order to solve for $$n^*$$, we will apply our initial condition $$\left(p(0) = p_0\right)$$ and the final condition $$\left(p(n_f) = 0\right)$$, and utilize continuity at the switching point: \begin{equation} \left. p(n^*)\right\rvert_\text{preFI} = \left. p(n^*)\right\rvert_\text{postFI} \label{eq:fi-condition-finite} \end{equation} $$p(n^*)\rvert_\text{preFI}$$ is given directly by Equation \eqref{eq:portfolio-value-preFI}: \begin{equation} p(n^*)\rvert_\text{preFI} = (1+r_{roi})^{n^*} \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} \label{eq:p-nstar-preFI} \end{equation} $$p(n^*)\rvert_\text{postFI}$$ can be found by applying our final condition $$\left(p(n_f) = 0\right)$$ to Equation \eqref{eq:portfolio-value-forever-postFI}: \begin{equation*} p(n_f)\rvert_\text{postFI} = (1 + r_{roi})^{(n_f-n^*)} \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) + \frac{ 1 + r_{roi} }{ r_{roi} } e = 0 \end{equation*} and rearranging for $$p_{n^*}$$: \begin{align*} (1 + r_{roi})^{(n_f-n^*)} \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) &= - \frac{ 1 + r_{roi} }{ r_{roi} } e \\ \left( p_{n^*} - \frac{ 1 + r_{roi} }{ r_{roi} } e \right) &= - \frac{ (1 + r_{roi})^{(1+n^*-n_f)} }{ r_{roi} } e \\ p_{n^*} &= \frac{ 1 + r_{roi} }{ r_{roi} } e - \frac{ (1 + r_{roi})^{(1+n^*-n_f)} }{ r_{roi} } e \\ \end{align*} Thus: \begin{equation} p(n^*)\rvert_\text{postFI} = \frac{ 1 + r_{roi} }{ r_{roi} } e - \frac{ (1 + r_{roi})^{(1+n^*-n_f)} }{ r_{roi} } e \label{eq:p-nstar-postFI} \end{equation} We can now apply our continuity equation stated in Equation \eqref{eq:fi-condition-finite} and solve for $$n^*$$, using Equations \eqref{eq:p-nstar-preFI} and \eqref{eq:p-nstar-postFI}: \begin{align} (1+r_{roi})^{n^*} \left(p_0 + \frac{s}{r_{roi}}\right) - \frac{s}{r_{roi}} &= \frac{ 1 + r_{roi} }{ r_{roi} } e - \frac{ (1 + r_{roi})^{(1+n^*-n_f)} }{ r_{roi} } e \notag\\ (1+r_{roi})^{n^*} (p_0 r_{roi} + s) - s &= (1 + r_{roi}) e - (1 + r_{roi})^{(1+n^*-n_f)} e \notag\\ (1+r_{roi})^{n^*} (p_0 r_{roi} + s) + (1 + r_{roi})^{(1+n^*-n_f)} e &= (1 + r_{roi}) e + s \notag\\ (1+r_{roi})^{n^*} \left( p_0 r_{roi} + s + (1 + r_{roi})^{(1-n_f)} e \right) &= (1 + r_{roi}) e + s \notag\\ (1+r_{roi})^{n^*} &= \frac{(1 + r_{roi}) e + s}{ p_0 r_{roi} + s + (1 + r_{roi})^{(1-n_f)} e } \notag\\ n^* &= \left. \log \left( \frac{(1 + r_{roi}) e + s }{ p_0 r_{roi} + s + (1 + r_{roi})^{(1-n_f)} e } \right) \middle/ \log(1+r_{roi}) \right. \label{eq:nstar-finite}\\ \end{align} This is the number of years it takes to get to FI for a Finite Portfolio. It depends on:

• annual savings, $$s$$
• annual expenses, $$e$$
• inital portfolio value, $$p_0$$
• rate of return on investment, $$r_{roi}$$
• portfolio duration, $$n_f$$

Like we did for the Forever portfolio with Equation \eqref{eq:nstar-forever}, we can divide savings and initial portfolio value by expenses: \begin{align*} s_e &= \frac{s}{e} & p_{e,0} &= \frac{p_0}{e} \end{align*} and rewrite Equation \eqref{eq:nstar-finite} as: \begin{equation} n^* = \left. \log\left( \frac{1 + r_{roi} + s_e}{p_{e,0} r_{roi} + s_e + (1+r_{roi})^{1-n_f}} \right) \middle/ \log\left(1+r_{roi} \right) \right. \label{eq:nstar-finite-2} \end{equation} Again, we can use the savings rate $$s_i$$ (ratio of savings-to-income) to calculate $$s_e$$ by Equation \eqref{eq:savings-rate}: \begin{equation} s_e = \frac{s_i}{1-s_i} \end{equation} The time to reach FI for a Finite Portfolio then depends on:

• savings rate, $$s_i$$
• rate of return on investment, $$r_{roi}$$
• inital portfolio value as a multiple of expenses, $$p_{e,0}$$
• portfolio duration, $$n_f$$

This is the form we use to generate the graphs in our calculator, and the factors we include (we calculate $$p_{e,0}$$ behind the scenes).