#### Required savings to investment goal

Given our investment goal and time horizon, how much money we
should stock away to reach our goals? While web is full of
"investment" calculators, it makes sense to do our own analysis. This
ensures that we understand the assumptions behind the calculations and
are comfortable with the inherent risks in predicting the future.
[Side note: Some retirement calculations are just crazy. For example, a major
investment firm from my old employer sends me quarterly reports that assume
retirement age of 68 and "expire" date of 75 years! This is not
exactly in line with my expectations. Read the fine print and do your own
analysis!]

Fortunately, retirement calculations are not
exactly a rocket science. Assuming rate of return *r*
for our investments, starting balance *B*0, and annual savings rate *S*,
the portfolio balance after *N* years is

Equation
(1) can be solved for the annual savings *S*:
Table
1 below illustrates the use of Equation (2). As an example, we desire to fund
sustainable retirement expenses totaling $2,000/month ($24,000/year)
in __today's dollars__. Based on the 4% rule, the desired
retirement portfolio is 25 times the annual expenses which equals $600,000. The
rate of return is *r*=4%
which corresponds to the historical __inflation
adjusted return__ for
SP500.

Case
1: With zero starting balance and 30 year investment horizon, the spreadsheet
gives savings rate of $892.

Case
2: With the starting balance of B0=$100,000 and 30 year investment horizon, the
spreadsheet gives savings rate of $410. This is less than half of the Case 1
showing the importance of building a nest egg as early as possible. The
compound interest helps those who have capital saved! Lesson learned: start saving early.

Case 3: The 10 year investment horizon corresponds either to a late start or early retirement. With only a short time available, the compound interest does not help much. The savings rate is over $4,000/month (double the desired retirement expenses)!

Case 4: Assuming annual return of 6% (adjusted for inflation) that beats the index investing by 2% points, the savings rate drops a little in comparison to Case 1. Be careful with your assumptions, however, as consistently beating index by 2% points requires effort. It is therefore better to start with a conservative *r* = 4% and save more especially in the beginning as the early nest egg is so important (see Case 2).

Case 5: An ultra conservative investor might put all the money in TIPS that usually yield around 2% after inflation. The outcome is pretty guaranteed but the desired savings rate is correspondingly higher.

Case 6: This is the optimist case that assumes bubble returns on stock market (think from 1980 to 2000) and no inflation. It is amazing how many still believe that index investing delivers this type of returns!

*Table 1: Google spreadsheet for calculating the savings rate for given financial goal.*

#### Savings rate calculator

Savings rate

A link for an editable spreadsheet for the above analysis is below (to edit use 'Save as' to make a copy of the file):

#### Required savings relative to income

Instead on focusing on absolute dollar amounts, it is often more meaningful to calculate the savings rate relative to income. In other words, we want to know how many percents of our income we should save every month in order to sustain our lifestyle in retirement.

We start by noting that our goal is have an investment balance BN after N years that is M times our yearly expenses E:

For example, if we want to cover our current expenses with investment income alone, the for 4% rule tells that the multiple should be M = 25.

Given our income I, the savings S is what is left after our expenses E (savings must be a positive number!):

By combining Equations (1), (3), and (4), we can calculate our required monthly savings in order to suppor our lifestyle in retirement.

Table 2 below illustrates the use of Equation (5). As an example, we consider a middle class person with an __after tax income__ of $60,000/year. The
rate of return is *r*=4%
which corresponds to the historical __inflation
adjusted return__ for
SP500.

Case
1: The expenses are kept constant while entering retirement in 40 years. Thus, we choose M = 25 so we can support our current lifestyle without additional income. As the spreadsheet shows, __to maintain constant lifestyle, one should save 21% of the after tax income for 40 years. __

Case
2: If one can make with lower expenses in retirement than during working life (for example, mortage paid off), the multiplier M can be lowered. In case 2, we assume that our test person has retirement expenses that are 80% of the current expenses which gives M = 0.8*25=20. The required savings rate has dropped to 17%.

Case
3: Assuming that one needs to replace only 60% of expenses (perhaps social security will be there after all!), the multiplier is M = 0.6*25 = 15 and the savings rate drops to 14% relative to the income.

Case
4: Assuming that the person has some capital saved and still 40 years to retirement, the savinges rate drops drastically to 7% while still supporting constant expenses through life. Capitalism is great for those that have capital!

Case 5: In this case, we assume a late start with only 30 years to retirement. Oops, the person should save 31% relative to the income. It only gets worse the longer one prograstinates.

Case 6: Doubling the inflation adjusted rate of return to r = 8% would allow significantly lower savings rate. This is a good motivator for learnig investing but constantly beating average stock market performance can be a challenge....

*Table 2: Google spreadsheet for calculating the savings rate relative to the income.*

#### Retirement savings calculator

Retirement savings calculator

A link for an editable spreadsheet for the above analysis is below (to edit use 'Save as' to make a copy of the file):

https://spreadsheets.google.com/spreadsheet/ccc?key=0AlyBMfBlh5aodEgyVjNaT0UtNkdyMVZqc2YtV0tXamc&hl=en_US#### Bottom line

Start early and save at least 15% of your income. Anything less sets you up for sharp adjustment in retirement.