If you want to be happy, think of something to be grateful about.

I had a deja vu moment this week. Yahoo had an article on how a large amount of manganese nodules have been found on the Atlantic Ocean’s seafloor (Figure 1). Back in the 1970s, I remember reading about Howard Hughes building the Glomar Explorer to mine manganese nodules from the bottom of the ocean. It turned out this story was a CIA cover story for Project Azorian, but that is another story.

Figure 1: Manganese Nodules on the Atlantic’s Seafloor.

While these nodules exist in all the world’s oceans, the bulk have been found in the Pacific Ocean.

Figure 2 gives you an idea of the size of these nodules.

I hope that after I die, people will say of me: “That guy sure owed me a lot of money.”

Introduction

Figure 1: Percentages of People with No Retirement Savings By Age Group (Data from Federal Reserve).

I have had a number of posts recently on saving for retirement. These posts ALWAYS follow discussions I have had with my family on the importance of starting to invest early – both for retirement and general financial health. I do not want my family members to get to retirement without an investment portfolio (Figure 1), a situation that I have seen and is not pretty.

Yesterday, my wife was told at an investment seminar that we should have 11 times our annual income in retirement savings. The same day, I saw an article on the Fidelity web site that recommended 8 times our annual income in retirement savings. As for me, I routinely tell my sons that they will need at least 20 times their annual incomes. Each of these rules is correct in the proper circumstances.

There is no mystery here – as I always say, the answer to all life’s interesting questions is “It depends …”.

Background

Objective

I will demonstrate how to derive these retirement multiples for yourself using some basic financial mathematics. This knowledge will allow you to change the assumptions as you require for your situation and you can make your own retirement multiple rule of thumb.

Analysis Caveats

My focus here will be on 401K retirement plans because I want to make sure that my sons take advantage of their employers matching plans. Since 401K plans use deferred income, this income is subject to taxes. My analysis assumes that your retirement tax rate is the same as your working tax rate. That may or may not be true. For example, many retirees have paid off their home mortgage and do not have mortgage deductions anymore.

I do not model the effect of inflation once you have retired. We now see people retired for twenty to thirty years and inflation will have an impact.

Young people need to decide if they think Social Security will be there for them. I sometimes wonder if it will be there for me. My sons should plan as if it will not be there.

Definitions

There are a some terms that will be important in the analysis to follow.

Retirement Multiple (Symbol: M)

The amount of retirement savings needed at the time of your retirement expressed as a multiple of your working income at the time you retire. So if you make $100K annually at the time you retire and you have saved $800K, your retirement multiple is 8.

Replacement Income Percentage (Symbol: K_{R})

The percentage of your working income that you will require during retirement. For the discussion here, I will assume that you need 85% of your working income while in retirement. I base this number on my assumption that at the time your retire you will be putting 15% of your working income into retirement investments and these investments will cease when you retire.

Number of Annual Payments (Symbol: N)

The number of investment payments made per year. I will assume one payment month or 12 per year.

Number of Years in Retirement (Symbol: Y)

For this calculation, I will assume my pension investment must provide a fixed income for 25 years, say from age 67 to age 92 − no one lives forever.

Annual Rate of Return (Symbol: R_{A})

This is the rate of return I expect on my retirement investments annually. For my analysis here, I will assume R_{A} =6.0%, which is a commonly used value.

Rate of Return Per Payment (Symbol: R)

This is the rate of return per payment period, which is given by the formula .

Inflation Factor (Symbol: K_{I})

This represents the loss of purchasing power per dollar from the time of retirement planning to retirement. I will assume that you will plan your retirement at age 30 and retire at age 67 − the case for one of my sons. I will also assume that you will average 2.3% annual inflation over your work life. Some R_{M} values assume inflation while others do not.

Non-Investment Income Percentage (Symbol: NI)

Many retirees will have Social Security income or pensions. This income will reduce the amount of money you need to get from your investments. Note that Social Security payments are adjusted for inflation and pension payments are usually not. Some R_{M} values assume non-investment income and others do not.

Analysis

I will be using Mathcad’s Present Value (PV) function for the following work, however, Excel has the same function and you can easily perform the following analysis using Excel’s PV and goal seek functions.

Formula Derivation

Figure 2 shows how to derive the key relationship for determining your retirement multiple, which really is just the standard present value formula for an annuity due. The formula I will be using is highlighted in yellow.

Figure 2: Derivation of Retirement Planning Formula.

Derive 8x Rule

We can compute R_{M} = ~8 by assuming a nominal non-investment income (NI=25%) and no inflation (K_{I}=1). We can substitute these values into the formula of Figure 2 as follows, which gives us a retirement multiple of ~8.

Figure 3: Calculation of 8x Retirement Multiple.

An M=8 is appropriate for people near retirement who expect Social Security and pensions. Since these folks are near retirement, inflation will not have time to significantly impact your planning.

Since few people receive more than $2K per month in Social Security, this calculation means that your retirement income is capped around $96K per year, which is a lot for a retiree.

Derive 11x Rule

We can compute R_{M} = 11 by ignoring non-investment income (NI=0) and inflation (K_{I}=1). We can substitute these values into the formula of Figure 2 as follows, which gives us a retirement multiple of ~11.

Figure 4: Calculation of 11x Retirement Multiple.

M=11 is appropriate for people who are near retirement and do not want to count on any Social Security income. It is a conservative approach to retirement planning, but can be justified.

Derive 20x Rule

We can compute R_{M} = 20 by assuming a nominal non-investment income (NI=25%) and inflation (). We can substitute these values into the formula of Figure 2 as follows, which gives us a retirement multiple of ~20.

Figure 5: Calculation of the 20x Retirement Multiple.

M=20 is appropriate for young people that will be investing for many years and will experience significant inflation. This number also assumes Social Security will be there for them, which may not be the case.

Conclusion

This quick post summarizes discussions that i had with my wife and sons on retirement planning. I hope it prods some of you into looking at your retirement plans.

“Life is like a snowball and all you need is some wet snow and a really long hill.”

Figure 1: Ronald Read (left), Superinvestor.

Because I am an annoying father who wants to encourage his sons to invest, one of the things I like to collect on this blog are stories of successful, long-term, investors. I loved this story about a janitor, Ronald Read (Figure 1), who recently died at 92 and left an estate worth nearly $8 million dollars.

He earned his fortune the old-fashioned way through conservative investing practices and living a frugal life. As far as his portfolio went, Newsmax reported that

Read’s investments included shares of AT&T, Bank of America, CVS, Deere, General Electric and General Motors. “He only invested in what he knew and what paid dividends. That was important to him,” his lawyer, Laurie Rowell, told CNBC.

Through regular contributions over a long time, he used the power of compound interest to amass an amazing amount of money. His investment approach epitomized the “snowball” approach that Warren Buffet advocates (see quote that leads this post). To illustrate that even small amounts of money regularly invested over a long-period of time can add up, Newsmax and CNBC both stated that

For example, to reach Read’s $8 million fortune, Hogan [a financial analyst] calculated that investors would have to invest about $300 a month at an 8 percent interest rate over 65 years.

I illustrate this compound interest calculation below using Mathcad’s Future Value (FV) function – Excel has a similar FV function.

“When a revolutionary succeeds, he should be given five years then shot, or otherwise removed.”

Introduction

Figure 1: Typical Bathroom Ventilation Scenario.

I was so happy with my previous fan installation that I am considering replacing some old fans with new, higher throughput, and quieter fans. The installations will be similar to that shown in Figure 1.

I have been using a nomograph (Figure 2) for my home HVAC calculations (example). I have decided that I am now living in the 21st century and I should figure out the formula that this graph represents. In this post, I will generate part of this nomograph to verify that I have put together the correct formula.

Background

Figure 2 shows the nomograph that I want to calculate for myself using the Darcy-Weisbach formula.

Figure 2: Air Flow Versus Static Pressure Drop.

The chart makes certain assumptions.

100 foot duct length

Fixed temperature, altitude, and humidity values that are unstated

No duct roughness factor

I will make some reasonable guesses and compute my version of the nomograph.

Analysis

I wanted a formula that would allow me to change the temperature, humidity, and air density. Figure 1 shows my calculation setup and how I determined the Reynolds number of the air. The model in Figure 1 for air density and specific humidity is rough. I show an alternative form in Appendix A that appears to be a bit more accurate. It does not make a huge difference in my final result.

Figure 3: Calculation Setup and Reynolds Number Determination.

Now that I have the Reynolds number, I can compute the Darcy friction factor and define my Darcy-Weisbach formula (Figure 4).

Figure 4: Creation of Functions for Graphing.

Now that I have my flow volume and velocity formulas defined, I can plot them (Figure 6) in the same manner as Figure 2. I did not plot every case shown in Figure 2 because it would take too long. However, I am quite confident that I have a formula that is close to the same one that was used to compute the graph in Figure 2.

Figure 6: Generate My Version of Figure 2.

Conclusion

I now have a formula that I can use in place of the graph shown in Figure 2. This will be more convenient for me in the analysis work that I have coming up.

Appendix A: Alternative Approach for Computing the Density and Specific Humidity of Moist Air

I worked the moist air density and specific humidity problems a couple of ways and this is one alternative (Figure 7) that I investigated. They do not give exactly the same answers, but the results are similar.

Figure 7: Alternative Approach for Computing Density and Specific Humidity of Moist Air.

“I’ve missed more than 9000 shots in my career. I’ve lost almost 300 games. 26 times, I’ve been trusted to take the game winning shot and missed. I’ve failed over and over and over again in my life. And that is why I succeed.”

I saw an interesting article on Yahoo Autos about 9 car models in which zero drivers died during the time period from 2009 to 2012. I thought this was interesting information – it sure stimulated some discussion in my family. I show the car models in the table below.

“Early in life I had to choose between honest arrogance and hypocritical humility. I chose the former and have seen no reason to change.”

Figure 1: Frank Lloyd Wright.

Because my wife and I are designing a cabin to replace our hunting shack in northern Minnesota, we have been looking at various house designs. Many of the designs we have looked at show the influence of Frank Lloyd Wright, who was the most famous member of the Prairie School of architecture. I know that Frank Lloyd Wright (Figure 1) is considered America’s greatest architect (according to American Institute of Architects, 1991), but I do not think I would have wanted to work for him.

I have read many stories of his extreme arrogance. Arrogance can be very difficult to tolerate. I consider myself to be fairly tolerant of arrogance − nature usually provides a healthy dose of humility in due time. For example, I once had a boss that believed that he could overcome theoretically unsolvable problems (e.g. metastability). Reality soon corrected his erroneous belief in a rather nasty way.

I believe that managers should take more than their share of the blame and less than their share of the credit. Such was not the case with Frank Lloyd Wright. Consider the following quote from Givers and Takers.

Wright success is described as being helped often by apprentices, yet rarely giving them any credit. He required his apprentices to put his name on any work they completed to insure all recognition would be allocated to him. At several points in his career he was abandoned by the architectural community and went years without work. The book cites these challenging intervals a result of his unwillingness to share the spotlight and recognize those who contributed to his success.

His family didn’t even like him. Consider the following quote from the Orlando Sentinel.

An unloving father, Wright was often estranged from his children. “I have had the father feeling for a building, but I never had it for my children,” Wright once remarked. Grandson Tim Wright calls him “an embarrassing relative” and “a torment” to the family.

Wright was legendary for not paying his bills − he was called “Slow Pay Frank” (Source).

But they are shunned by neighbours outraged in equal parts by their living in sin and “Slow Pay Frank’s” perennial refusal to honour his debts. As one cook explains to Mamah as she tenders her resignation: “It’s sinful, that’s what it is. And sin and pay is one thing, but sin and no pay I just can’t abide.”

He even refused to pay his family members (Source).

While he worked for his father on the Imperial Hotel in Tokyo (1920), Frank Lloyd Wright’s son, John, an architect, was instructed by his father to complete a set of six drawings, present them to the client, Viscount Inouye, and collect the fee for his father. John did as he was asked, kept his unpaid back pay, and sent the balance to his father, who was then in the United States. On receipt of the balance Frank Lloyd Wright sent his son a cable-wireless, firing him.

In addition to firing his son, he also “presented him with a list of the total amount of money that John had cost him over his entire life” (Source).

I always tell my sons that most of life’s management lessons are negative − as when we see a manager doing something that we swear we will never do. Frank has provided me a few more negative lessons.

“The only thing worse than training employees and losing them is to not train them and keep them.”

Introduction

Figure 1: An Active LED Backlight.

I was reading an application note by Texas Instruments yesterday on how to design an LED backlight for an LCD display (Figure 1). The article was interesting, but it did bother me because it presented a rather involved formula and did not provided any motivation for the formula, a derivation, or even definitions of the parameters used in the formula.

Equation 1 shows the formula that bothered me. I will provide a derivation, definitions for all the parameters, and a worked example.

Eq. 1

The principles behind designing LED backlight systems are fairly straightforward. However, Equation 1 looks rather complicated. Using basic physics, we can derive Equation 1 using a few basic principles. In the process of working through the derivation, we will acquire some insight into critical factors behind LED backlight designs.

Background

Backlight Basics

At the risk of grossly oversimplifying LED backlights, there are two basic types:

Lit from the side

LEDs are mounted on the side of the display, which the advantage of giving you a very thin display. Unfortunately, the side backlighting can be uneven. Figure 2 shows an example of how this type of display is constructed.

Figure 2: Side-Mounted LEDs.

Lit from the back

The LEDs are mounted directly behind the display. This approach gives more even lighting, but requires a thicker display. Figure 3 shows an example of how this type of display is constructed.

Figure 3: Lit from Behind Display Construction.

Approach

Like most engineering calculations, backlight design is a form of budget calculation − so much light in, so much light out minus any losses. The process for this calculation is as follow:

Determine the amount of backlight you need from the display for your application.

Estimate the amount of loss the light will experience between the display and LEDs.

Compute the total amount of light you need from the LEDs.

Knowing the total amount of light needed, you can estimate the number of LEDs needed based on the type of LED you are using.

Definitions

Aspect Ratio (AR)

The aspect ratio of an image describes the relationship between its width and its height.It is commonly expressed as two numbers separated by a colon, as in 16:9. Sometimes the ratio is expressed in terms of pixel counts. For example, standard HDTV’s aspect ratio could be expressed as 1920:1080 instead of 16:9 (Wikipedia).

Luminous Flux (Φ_{V})

In photometry, luminous flux or luminous power is the measure of the perceived power of light (Wikipedia).

Display Size (L_{S})

Displays are usually specified in terms of their diagonal measurement. To obtain an actual length and width, you need both the display size and the aspect ratio.

V_{Disp}

The fraction of light lost between the LEDs and the display.

M_{V}

The illuminance of the display. You can think of the illuminance as the luminous flux density.

Analysis

Intuitive Viewpoint

I derive a simple relationship for the number of LEDs required to generate a specified level of screen brightness in Figure 4.

Figure 4: Intuitive Derivation of the Screen’s Brightness.

You can think of the the formula for the required number of LEDs as taking the total amount of light needed (A·M_{V}) and dividing it by the effective amount of light per LED (K_{X}·Φ_{V}).

Screen Area

Display area is usually computed in terms of the display’s diagonal length and its aspect area, which is not as intuitive as using the display’s height and width. Figure 5 shows the formula for computing display area in terms of aspect ratio and diagonal length.

Figure 5: Derivation of Screen Area Formula.

LED Quantity Formula

Figure 6 shows the form of the display brightness formula in terms of the variables that I think in.

Figure 6: My Formula for the Number of LEDs Required

Equation 2 shows this equation using more conventional notation.

Eq. 2

where

L_{S} is the diagonal length of the screen.

M_{V} is the required display luminosity.

Φ_{V} is the luminous flux from a single LED.

K_{X} is fraction of light that makes it from the LED to the display.

My Equation 2 is the same as Equation 1 given the proper substitutions, which I show in Figure 7.

Figure 7: Converting My Version to Equation 1 Form.

Equation 1 also has a constant called K that simply converts inches squared to meters squared . I just let Mathcad handle my unit conversions.

Simple Worked Example

Figure 8 shows a worked example using common parameter values.

Figure 8: Worked Example.

Conclusion

When I first saw Equation 1, I could not understand it. Now that I have gone through the derivation, I understand every term. Since I find the form of Equation 1 non-intuitive, I will work with Equation 2.