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Math behind financial independence
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Early retirement extreme.ipynb
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"# Math behind early retirement scenarios\n",
"\n",
"This notebook provides easy way to personalize financial independence related calculations.\n",
"\n",
"## Acknowledgments\n",
"It's all thanks to Jacob Lund Fisker, the author of a book [_Early retirement extreme_](http://earlyretirementextreme.com/ere-book).\n",
"\n",
"Check out his blog at http://earlyretirementextreme.com/"
]
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"source": [
"from functools import partial\n",
"from math import log\n",
"import plotly.graph_objs as go\n",
"from plotly.offline import iplot, init_notebook_mode\n",
"\n",
"init_notebook_mode(connected=True)"
]
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{
"cell_type": "code",
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"p0 = 10000. # size of a fund\n",
"p = 1000. # withdrawn on the first day of year\n",
"i = 0.04 # the rate of investment\n",
"\n",
"INVESTMENT_RETURNS = [0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.15] # these are the i used in figures. You can change them here."
]
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{
"cell_type": "code",
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"source": [
"def amount_after_one_year(p0, p, i):\n",
" # p1 = (p0-p) + i*(p0-p)\n",
" # p1 = (p0-p) * (1 + i)\n",
" # p1 = p0 + i * p0 - p - i*p\n",
" p1 = p0 * (1+i) - p * (1+i)\n",
" return p1\n",
"\n",
"\n",
"def amount_after_N_years(p0, p, i, N):\n",
" pN = p0*(1+i)**N - p * ((1+i)*((1+i)**N-1))/i\n",
" return pN\n",
"\n",
"def how_long_porfolio_will_last(p0, p, i):\n",
" \"\"\"This is the key formula for extreme early retirement.\n",
"\n",
" It relates the number of retirement years to the rate of\n",
" return on your portfolio and the size of your portfolio.\n",
" \"\"\"\n",
" a = 1/(1-(p0/p)*(i/(1+i)))\n",
" N = log(a)/log(1+i)\n",
" return N"
]
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"source": [
"# if I saved 10000 and want to retrieve a 1000 each year, and my return is 4% annualy,\n",
"# how many years this fund will last?\n",
"how_long_porfolio_will_last(10000., 1000., 0.04)"
]
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{
"cell_type": "code",
"execution_count": 4,
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"source": [
"perpetual_i = (p/p0)/(1-p/p0) # this calculates the perpetuity interest\n",
"# after rearangment\n",
"p0 = (1+i) * p / i\n",
"\n",
"def required_fund_size(p, i):\n",
" \"\"\"Required fund size to withdraw p at the begining of each period\n",
" when interest is added at the end of this period.\"\"\"\n",
" return (1+i) * p / i"
]
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"source": [
"# If I want to live on 3000 a month, \n",
"# how much capital do I have to save (assuming 6% annual return)\n",
"# to live of it's returns?\n",
"required_fund_size(36000., 0.06)"
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"source": [
"def durability_dist_per_i(i):\n",
" data = []\n",
" for year in range(2, 50):\n",
" try:\n",
" dur = how_long_porfolio_will_last(float(year), 1., i)\n",
" except (ValueError, ZeroDivisionError):\n",
" # when i is so big that we enter perpetual mode, we\n",
" # assign an arbitrary durability of 101 years\n",
" dur = 101\n",
" finally:\n",
" data.append((year, dur))\n",
" return list(zip(*data))\n",
"\n",
"def prepare_data_for_multiple_i(y_func):\n",
" traces = []\n",
" for i in INVESTMENT_RETURNS:\n",
" data = y_func(i)\n",
" trace = go.Scatter(\n",
" x = data[0],\n",
" y = data[1],\n",
" mode = 'lines',\n",
" name = f'{i:.0%}'\n",
" )\n",
" traces.append(trace)\n",
" return traces\n",
"\n",
"data = prepare_data_for_multiple_i(durability_dist_per_i)\n",
"title = (\n",
" 'Size of fund given in years of expanses<br>'\n",
" 'versus<br>how long it will actually last before '\n",
" 'being depleted<br>given different investment return rates'\n",
" )\n",
"layout = dict(title = dict(text=title, xref='paper'),\n",
" xaxis = dict(title = 'Fund size (years)'),\n",
" yaxis = dict(title = 'Durability', range=[1,100]),\n",
" margin = {'t': 200}\n",
" )\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"iplot(fig, filename='line-mode')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The time to accumulate the fund P0/p can be calculated in a similar way...\n",
"\n",
"> P0/p = r / (1-r) \\* M\n",
"\n",
"where *r* is the savings rate and *M* is the number of years worked. Now, if the funds are invested at a rate *i* and allowed to compound we get\n",
"\n",
"> P0/p = r / (1-r) \\* ((1+i)**M - 1) / i\n",
"\n",
"In traditional personal finance planning the time invested M is the most important\n",
"factor, with typical values around 30 or 40 years. Some people will be clever enough\n",
"to achieve superior investment returns i. We are not counting on this. In the case of early\n",
"retirement, M will be small, and we will presume that i is of the usual range, maybe around 6%.\n",
"The main lever in this queation is thus r/(1-r). For a tradiitonal savings rate r=0.1\n",
"the lever is r/(1-r)=0.11, whereas for an extreme savings rate r=0.75, the lever\n",
"r/(1-r)=3, which is 27 times higher! Even with a lot of time to compound and a market\n",
"outperformance of a few percent, it's hard to beat a factor of 27."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"# The time to accumulate the fund P0/p can be calculated in a similar way...\n",
"# P0/p = r / (1-r) * M\n",
"# where r is the savings rate and M is the number of years worked\n",
"# Now, if the funds are invested at a rate i and allowed to compound we get\n",
"# P0/p = r / (1-r) * ((1+i)**M - 1) / i\n",
"\n",
"def accumulated_fund(r, M, i):\n",
" \"\"\"Returns number of years of accumulated fund (X axis in the figure above)\"\"\"\n",
" return r/(1-r) * ((1+i)**M - 1)/i\n",
"\n",
"# In traditional personal finance planning the time invested M is the most important\n",
"# factor, with typical values around 30 or 40 years. Some people will be clever enough\n",
"# to achieve superior investment returns i. We are not counting on this. In the case of early\n",
"# retirement, M will be small, and we will presume that i is of the usual range, maybe around 6%.\n",
"# The main lever in this queation is thus r/(1-r). For a tradiitonal savings rate r=0.1\n",
"# the lever is r/(1-r)=0.11, whereas for an extreme savings rate r=0.75, the lever\n",
"# r/(1-r)=3, which is 27 times higher! Even with a lot of time to compound and a market\n",
"# outperformance of a few percent, it's hard to beat a factor of 27."
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">>> accumulated_fund(0.2, 5, 0.06)\n",
"1.4092732400000012\n",
"\n",
"# it means that if we save 20% of our income for 5 years with return rate of 6%,\n",
"# we would accumulate capital for a 1.4 years of living"
]
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"# Now we want to ask how many years will it take to accumulate given fund?\n",
"\n",
"def time_to_accumulate_fund(p0, r, i):\n",
" # the formula is derived from the one used in `accumulated_fund` function\n",
" return log(1+i*(p0)*((1-r)/r))/log(1+i)\n",
"\n",
"# How many years to gather 30 years worth fund,\n",
"# given savings rate of 20% and investment return of 6%? \n",
"time_to_accumulate_fund(30, 0.2, 0.06)"
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"}\n",
"});</script><script type=\"text/javascript\">window.addEventListener(\"resize\", function(){if (document.getElementById(\"e3ddfbe7-0108-4907-be27-8d0b1b5c19b7\")) {window._Plotly.Plots.resize(document.getElementById(\"e3ddfbe7-0108-4907-be27-8d0b1b5c19b7\"));};})</script>"
]
},
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"output_type": "display_data"
}
],
"source": [
"p0 = 30 # number of fund's years to grow, play with this value if you want\n",
"\n",
"def working_years_dist_per_i(i, p0=30):\n",
" data = []\n",
" for r in range(1, 101, 1):\n",
" working_years = time_to_accumulate_fund(p0, r/100., i)\n",
" data.append((r, working_years))\n",
" return list(zip(*data))\n",
"\n",
"func = partial(working_years_dist_per_i, p0=p0)\n",
"\n",
"data = prepare_data_for_multiple_i(func)\n",
"title = (\n",
" f'The time it takes to grow the fund to {p0} years<br>'\n",
" 'as a function of savings rate'\n",
" )\n",
"layout = dict(title = dict(text=title, xref='paper'),\n",
" xaxis = dict(title = 'Savings rate (%)', range=[0, 100]),\n",
" yaxis = dict(title = 'Working years', range=[0,80]),\n",
" margin = {'t': 200}\n",
" )\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"iplot(fig, filename='line-mode')"
]
},
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"Assuming that the life expectancy is 100 years and one is either financially independent or working to become so for the last 80 years of that time span (adjust the numbers as you see fit), then\n",
"\n",
"> 80 = N + M\n",
"\n",
"where N is the number of years living off your money and M is the number of years accumulating it. \n",
"\n",
"We can now plot the working time M as a function of the savings rate r to find out how many years are needed to work to accumulate enough money for a given rate of return i. This is the most important figure in this chapter. Specifically, it shows that high savings rates lead to extremely early financial independence. Conversely, the traditionally recomended savings rates mean working for 40 years or more, and they're very dependent on the return rate i. It also shows the difference between a savings rate of say 35%, which most people would consider to be high, and a savings rate of 70%. It shows what your savings rate r should be to retire in M years."
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"# N + M = 80\n",
"\n",
"def frange(x, y, jump=1.0):\n",
" \"\"\"Range for floats.\"\"\"\n",
" # source: https://stackoverflow.com/questions/7267226/range-for-floats\n",
"\n",
" i = 0.0\n",
" x = float(x) # Prevent yielding integers.\n",
" x0 = x\n",
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" yield x # yield always first value\n",
" while x + epsilon < y:\n",
" i += 1.0\n",
" x = x0 + i * jump\n",
" yield x\n",
"\n",
"def calculate_years_of_provision(p0, i, r):\n",
" \"\"\"Given the amount of fund (in years) calculate\n",
" how many years will it take to accumulate it\n",
" and how many years it will last for.\n",
" \"\"\"\n",
" n = how_long_porfolio_will_last(p0, 1., i)\n",
" m = time_to_accumulate_fund(p0, r, i)\n",
" return n + m\n",
"\n",
"def find_p0(years, i, r):\n",
" \"\"\"Determine value of optimal P0 for `years` left.\n",
" \n",
" We do it numerically i.e brute forcefuly\"\"\"\n",
" for p in frange(1,years+1, 0.01):\n",
" try:\n",
" res = calculate_years_of_provision(p, i, r)\n",
" if res >= years:\n",
" return p\n",
" except (ValueError, ZeroDivisionError):\n",
" return p\n",
" raise ValueError(f\"p0 not found: i={i}, r={r}\")\n",
" "
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\"The time it takes to grow the fund provided 73 years till death<br>as a function of savings rate\", \"xref\": \"paper\"}, \"xaxis\": {\"range\": [0, 100], \"title\": {\"text\": \"Savings rate (%)\"}}, \"yaxis\": {\"range\": [0, 83], \"title\": {\"text\": \"Working years\"}}}, {\"showLink\": false, \"linkText\": \"Export to plot.ly\", \"plotlyServerURL\": \"https://plot.ly\"}); \n",
"}\n",
"});</script><script type=\"text/javascript\">window.addEventListener(\"resize\", function(){if (document.getElementById(\"01fd5811-e23f-4168-a7e3-a8f73299d8d5\")) {window._Plotly.Plots.resize(document.getElementById(\"01fd5811-e23f-4168-a7e3-a8f73299d8d5\"));};})</script>"
]
},
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}
],
"source": [
"years_till_death = 73 # number of years to live or N + M\n",
"\n",
"def working_years_dist_per_i(i, years):\n",
" data = []\n",
" for r in frange(1, 101):\n",
" p0 = find_p0(years, i, r/100.)\n",
" working_years = time_to_accumulate_fund(p0, r/100., i)\n",
" data.append((r, working_years))\n",
" return list(zip(*data))\n",
"\n",
"data = prepare_data_for_multiple_i(partial(working_years_dist_per_i, years=years_till_death))\n",
"title = (\n",
" f'The time it takes to grow the fund provided {years_till_death} years till death<br>'\n",
" 'as a function of savings rate'\n",
" )\n",
"layout = dict(title = dict(text=title, xref='paper'),\n",
" xaxis = dict(title = 'Savings rate (%)', range=[0, 100]),\n",
" yaxis = dict(title = 'Working years', range=[0,years_till_death + 10]),\n",
" margin = {'t': 200}\n",
" )\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"iplot(fig, filename='line-mode')"
]
}
],
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"language": "python",
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