Math for Investing

Math for investing.

Savings rate to 25X expenses:

| Savings Rate(%) | Time in years |

5 | 66 |
10 | 51 |
15 | 43 |
20 | 37 |
25 | 32 |
30 | 28 |
35 | 25 |
40 | 22 |
45 | 19 |
50 | 17 |
55 | 14.5 |
60 | 12.5 |
65 | 10.5 |
70 | 8.5 |
75 | 7 |
80 | 5.5 |
85 | 4 |
90 | 3 |
95 | 2 |
100 | 0 |

source Assumptions: 5% real CAGR, 25X expenses

In Reddit format:

Savings Rate: Time in years

  • 100%: 0 years
  • 95%: 2 years
  • 90%: 3 years
  • 85%: 4 years
  • 80%: 5.5 years
  • 75%: 7 years
  • 70% 8.5 years
  • 65% 10.5 years
  • 60%: 12.5 years
  • 55% 14.5 years
  • 50% 17 years
  • 45%: 19 years
  • 40%: 22 years
  • 35%: 25 years
  • 30%: 28 years
  • 25%: 32 years
  • 20%: 37 years
  • 15%: 43 years
  • 10%: 51 years
  • 5%: 66 years

Expenses by withdrawal rate(WR):

The math: 1 / %WR * 100

4% WR: 1/4*100 = 25

Example in the table below is for $30,000/yr in retirement expenses

| Withdrawal Rate(%) | Expense Multiple | Example($30k)|

2% | 50 | $1,500,000 |
2.5% | 40 | $1,200,000 |
3% | 33.4 | $1,002,000 |
3.5% | 28.6 | $858,000 |
4% | 25 | $750,000 |

4%WR has historically been fine for US based retirements w/ a 60/40 AA for a 30 year long retirement, see Trinity Study & William Bengen’s work.

Equities vs Bonds

Since 1794 to today over various time periods, equities have a X% chance of beating bonds[0]:

  • 1 year: 56.2%
  • 5 years: 61.7%
  • 10 years: 65.0%
  • 20 years: 65.7%
  • 30 years: 68.0%
  • 50 years: 68.4%

source: McQuarrie, Edward F., Stocks for the Long Run? Sometimes Yes. Sometimes No. (July 23, 2021). Available at SSRN: https://ssrn.com/abstract=3805927 or http://dx.doi.org/10.2139/ssrn.3805927

Calculating rate of return

use Modified Dietz

Expected Returns

A “normal” view, not including countries that lost a major world war, etc:

A quick look at one of the Credit Suisse Yearbooks yielded the list of ten countries below — showing real (inflation-adjusted), geometric average returns from 1900 to 2016:

|Stocks|Bonds

Australia|6.8%|1.7% Canada|5.7%|2.2% Ireland|4.4%|1.6% New Zealand|6.2%|2.1% Norway|4.3%|1.8% Spain|3.6%|1.8% Sweden|5.9%|2.7% Switzerland|4.4%|2.3% United Kingdom|5.5%|1.8% United States|6.4%|2.0% AVERAGE|5.3%|2.0%

So expecting 4-5%/yr real in equities and about half that in bonds seems reasonable. Or they way I say it, expect a touch above inflation for bonds and about 2X more in equities.

Personal Inflation Rate

I think calculating a PIR is interesting and can be useful for planning purposes in retirement.

Here is how I do it:

  • Add up expenses from last year, labeled: last_year_expenses
  • Add up expenses from this year, labeled: this_year_expenses
  • Calculate the dollar amount difference you spent between the two years: this_year_expenses - last_year_expenses labeled: diff
  • PIR = diff / this_year_expenses

taxable account optimization

Arguably not worth the effort. https://www.bogleheads.org/forum/viewtopic.php?p=7483189#p7483189

Calculating ER’s

Example:

Given an AA OF:

  • 70% FCTDX
  • 20% FUSIX
  • 10% FGOMX

FCTDX 0.70 X 0.0031=0.0027 FUSIX 0.20 X 0.0038=0.00076 FGOMX 0.10 X 0.0036=0.00036 Blended Fund Expense Ratio is 0.382% Advisor Fee is 0.75%

All in costs are 1.132% a year with fee waivers and 1.382% annually without fee waivers.

source: https://www.bogleheads.org/forum/viewtopic.php?p=8021280#p8021280