Present Value & the Funding Ratio - Planning Forward!
May 23, 2021 by Chris
python
Inflation is the reason behind the erosion of the value of money over time. 1$ today is more valuable than 1$ after 5 years (given a positive Inflation rate). I will not get into the details of what inflation is, since there is lots of literature around that.
In this article I will showcase how to plan the financing of future expenses by including inflation, taxes on earnings and annual fees.
Let's see that with an example. Let's assume that a person decides a job break for 3 years. Estimates his/her monthly expenses to be 300$. What is the initial amount of money the person needs to have to be able to fulfill their goal?
The quick response to that is $! but that is not actually true due to the inflation. Let's (for the sake of the example) say that the inflation is 2% per year. That means that 1$ today will be valued at $ in one year from now.
Note: the notation where i is the annual inflation rate and n is the periods in years, is something we discussed in the Geometric Progression and the Compounding of the Returns article. Think of inflation as compounding with negative rate.
In other words we are asking for the Present Value of a future () 1$ () with a discount rate () of 2%. The formula is:
We call the discount factor.
So, for the first year, the person will pay 3600$ but the actual cost of 3600$ today is $. That means that there are $ that are missing and should be added to the initial amount. The second year the person will need to pay 3600 again, but the value today is $. The third year, the situation is the same and the calculated value for the 3600$ would today be $.
So far, the deficit for the first year is 72, for the second year is 145 and for the third year is 220. A total of 437$ needs to be added to our initial thought of 10800!
The example above can easily be extended to loan payments, house rent or even pension forecasting (the financial term for all these is annuities).
Let's get into code and replicate the scenario above.
Package Installation
%%capture import pandas as pd import numpy as np import ipywidgets as widgets from IPython.display import display %matplotlib inline
monthly_liabilities = 300 years = 3 annual_liabilities = 12 * monthly_liabilities inflation_rate = 0.02 liabilities = pd.Series(data=[annual_liabilities for i in range(years)], index=[i+1 for i in range(years)]) def liabilities_inflated(liabilities, inflation_rate): years = liabilities.index inflation_over_years = pd.Series(data=(1+inflation_rate)**years, index=years) return liabilities * inflation_over_years inflated_liabilities = liabilities_inflated(liabilities, inflation_rate) print(f"Liabilities per year: {list(liabilities.values)}, \ \nInflated liabilities: {list(inflated_liabilities.values)}, \ \nTotal amount needed today: {inflated_liabilities.sum():.2f}, \ \nNo inflation case: {liabilities.sum():.2f}")
Liabilities per year: [3600, 3600, 3600],
Inflated liabilities: [3672.0, 3745.44, 3820.3488000000007],
Total amount needed today: 11237.79,
No inflation case: 10800.00
Present Value - Balancing inflation through investing
Inflation is eroding the value of money when they stand still! However, a balanced situation can be achieved if the money is invested to something that yields a return equal (or even better, more) than the inflation rate. There are several options to that, from 0 risk saving accounts that pay an interest, to a type of bond that increases the risk a bit and gives a better return (or even inflation-adjusted bonds), to more risky assets like equity funds that might yield an even better return.
In any case, we should take into account the taxes on profits and the administration fees each investment scenario requires.
Let's go back to our example and say that we found a savings account that pays 2% annually in interest without fees and taxes. That would mean that we could invest 10800 to this account today and each year pay out the bill.
But what if there was a 10% tax on the profits from this interest? How much money would we need today to start with? Or else, what is the present value of the investment?
investment_annual_return = 0.02 investment_profit_tax = 0.1
def discount(t, r, earnings_tax): """ Computes the amount needed today which when invested for t periods with an r return per period and earning_tax on the profits, will equal to 1$ """ return (1 + r - r*earnings_tax)**-t def present_value(liabilities, return_annual, earnings_tax): """ Given a list of liabilities, an annual return and tax rate on earnings, returns the amount of money needed today. """ dates = liabilities.index discounts = discount(dates, return_annual, earnings_tax) return (discounts * liabilities).sum()
present_value(inflated_liabilities, investment_annual_return, investment_profit_tax)
10842.491757684631
Moving forward, lets say that instead of a savings account we invest in funds that have an annual return of 2%, a profit tax of 10% and an annual administration fee of 0.5% on the total invested amount.
What should the initial investment be?
annual_fee = .005 # percentage of total amount per year
def present_value_with_fees(liabilities, return_annual, earnings_tax, annual_fee): dates = liabilities.index discounts = discount(dates, return_annual, earnings_tax) liabilities_with_fees = [liabilities[len(dates) +1 - i] * (1-annual_fee)**-i for i in list(dates)[::-1]] return ((discounts * liabilities_with_fees).values).sum()
present_value_with_fees(inflated_liabilities, investment_annual_return, investment_profit_tax, annual_fee)
10951.755210394398
It becomes apparent that taxes and fees play an important role in calculating future cash flows.
The Funding Ratio
The funding ratio is a percentage that shows how much our current assets can contribute to future cash flows. A 100% funding ratio means that the initial amount of assets will be just enough to fulfill our goals. A ratio below 100% means that we will need more assets to cover for the future flows and a ratio more than 100% means that we are overfunded and as such we have a surplus of money we can use to fund other goals.
def funding_ratio(current_assets_value, liabilities, return_annual, earnings_tax, annual_fee): return 100*current_assets_value/present_value_with_fees(liabilities, return_annual, earnings_tax, annual_fee) def show_funding_ratio(monthly_liabilities, years, inflation_rate, current_assets_value, return_annual, earnings_tax, annual_fee): annual_liabilities = 12 * monthly_liabilities liabilities = pd.Series(data=[annual_liabilities for i in range(years)], index=[i+1 for i in range(years)]) inflated_liabilities = liabilities_inflated(liabilities, inflation_rate) fr = funding_ratio(current_assets_value, inflated_liabilities, return_annual, earnings_tax, annual_fee) print(f'{fr:.2f}')
funding_ratio(10951.76, inflated_liabilities, investment_annual_return, investment_profit_tax, annual_fee)
100.00004373368022
Below I have included a widget which helps me plan for my future goals. I will try to have it as an HTML widget at some point but for the ones who are interested try the colab representation of this article.
controls = widgets.interactive(show_funding_ratio, monthly_liabilities = widgets.IntSlider(min=100, max=100000, step=100, value=300), years = widgets.IntSlider(min=3, max=100, step=1, value=3), inflation_rate = widgets.FloatSlider(min=0.01, max=0.3, step=.01, value=.02), current_assets_value = widgets.IntSlider(min=10000, max=30000000, step=1000, value=10800), return_annual = widgets.FloatSlider(min=0.01, max=0.2, step=.01, value=.08), earnings_tax = widgets.FloatSlider(min=0.1, max=.5, step=.02, value=.3), annual_fee = widgets.FloatSlider(min=0.01, max=.05, step=.01, value=.03)) display(controls)
Enjoy!
