Here. Someone sent me a fabulous follow up email to "The Average Return Myth," but in the heat of the PDC, I didn't get a chance to read it and now that the PDC is over, I can't find it again! All I remember is that the person's email address started with "cat" (I think). Can you please send it again, Mr. Cat? Thanks!

Here. Gurley sums up nicely the tendency of the IT industry to eat other industries and itself, leading to nice things for consumers, but not industries.

Here. The recent economic conditions caused me to become very interested in not just making money (I've always been a capitalist : ), but also in keeping a careful eye on it. In the past, I've let most of the "keeping an eye on it" be done by a financial advisor (I've had one since I was 22). These days, I have a stack of financial books that I'm reading [1] and it's been a fabulous educational experience. Given the supreme importance of money in our society, I think everyone should know the basics of money management. I'm very disappointed that I made it all the way through high school, let alone college, with no more instruction then how to fill in a check (something we want folks to do less of, not more!). Towards that end, I really love Bloomberg University’s no-frills financial tutorials, including one on the basics of a 401(k) and another on the basics of investing. They provide just the tip of the iceberg and I could recommend a lot of other reading, but this is a good place to start for folks that want to actually keep the money they make. [1] http://www.sellsbrothers.com/money

Let’s say that you have $1000 to invest. The first year, you invest it and get a 25% return, so you leave your money invested. The next year, the market doesn’t do as well and your return is -15%. What’s the average rate of return over the two years? You way think that it’s 5%, that is, (25% + (-15%))/2. Let’s do the math for 25% and -15%:

• Using simple interest, after 1 year, $1000 + $1000 * 25% = $1250
• After 2 years, $1250 + $1250 * -15% = $1062.50

Here we’re using Interest = Principle * Rate * Time calculation for yearly aka simple interest (I = PRT and Time is 1 year). Taking the numbers the other way, i.e. -15% the first year and 25% the next year, yields the same result:

• After 1 year, $1000 + $1000 * -15% = $850
• After 2 years, $850 + $850 * 25% = $1062.50

In fact, the result is the same no matter in which order that the rates come or how many there are:

Table 1: From Good to Bad

year	return	total
0	0	$1,000.00
1	25%	$1,250.00
2	15%	$1,437.50
3	5%	$1,509.38
4	-5%	$1,433.91
5	-15%	$1,218.82

Table 2: Starting Bad to Good

year	return	total
0	0	$1,000.00
1	-15%	$   850.00
2	-5%	$   807.50
3	5%	$   847.88
4	15%	$   975.06
5	25%	$1,218.82

Table 3: A Mixed Bag

year	return	total
0	0	$1,000.00
1	25%	$1,250.00
2	-15%	$1,062.50
3	5%	$1,115.63
4	15%	$1,282.97
5	-5%	$1,218.82

This result surprised me. I found it unintuitive that no matter how the rates vary over time, it doesn’t matter if they come first, last or in between. I expected large losses up front to swamp later gains or early gains to make up for late losses, but the change of the underlying principle amount evens things out, e.g. a smaller percentage drop later is against a larger principle if there have been early gains.

When you figure it as a single formula, Future Value = Principle * (1+Rate1) * (1+Rate2) * (1+Rate3) * (1+Rate4) * (1+Rate5) or F = P*(1+R1)*(1+R2)*(1+R3)*(1+R4)*(1+R5), the independence of the order makes more sense, since multiplication is commutative, i.e. it doesn’t matter in what order you do it:

f = p*    (1+r1)* (1+r2)* (1+r3)*(1+r4)* (1+r5)
f = $1000*(1+25%)*(1+15%)*(1+5%)*(1-5%)* (1-15%) = $1218.82
f = $1000*(1-15%)*(1-5%)* (1+5%)*(1+15%)*(1+25%) = $1218.82
f = $1000*(1+25%)*(1-15%)*(1+5%)*(1+15%)*(1-5%)  = $1218.82

Having varied rates like in Tables 1-3 in a stock or stock mutual fund investment isn’t uncommon (as we’ve just seen during and after the Internet bubble). On the other hand, if you compare this a fixed yield (like a bond) with our “average rate of return” of 5%, you’ll see a different result:

Table 4: Small But Fixed Rate of Return

year	return	total
0	0	$1,000.00
1	5%	$1,050.00
2	5%	$1,102.50
3	5%	$1,157.63
4	5%	$1,215.51
5	5%	$1,276.28

With a fixed interest rate, we can simply our calculations somewhat using Future value = Principle * (1 + Rate)^Number of compounds. So, $1000 at 5% for 5 years is:

f = p*(1+r)^n
f = $1000 * (1 + 5%)^5
f = $1000 * (1.05)^5
f = $1276.28

Any way you calculate it, not only does the boring, fixed interest rate bond out-perform the variable rate even for the same average rate of return, but clearly our average rate of return calculation isn’t very useful. We’re not really getting 5% year to year on our varied stock rates of return, or they’d show the same results as the bond. Instead, if you reverse the formula for Rate, we get:

r = (f/p)^(1/n) - 1
r = ($1218.82/$1000)^(1/5) -1
r = 1.21882^(1/5) -1
r = 4.037%

This gives us a annualized rate of return:

Table 5: Annualized Rate of Return

year	return	total
0	0	$1,000.00
1	4.037%	$1,040.37
2	4.037%	$1,082.37
3	4.037%	$1,126.07
4	4.037%	$1,171.52
5	4.037%	$1,218.82

Taking this further, because the future value of an investment is the same whether you consider a fixed rate of return or a variable rate of return, for any principle, you can calculate the fixed rate of return by deriving from this formula (assuming r0 is the fixed rate of return and r1-r5 are the variable rates of return):

p*(1+r0)^n = p*(1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5)

Further, because principle plays the same role on each side of the equation, you can remove it:

(1+r0)^n = (1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5)

Solving for the annualized rate of return from the variable rates of return gives you this:

r0 = ((1+r1)*(1+r2)*(1+r3)*(1+r4)*(1+r5))^(1/n) - 1

Applying it in our example:

r0 = ((1+25%)*(1+15%)*(1+5%)*(1-5%)*(1-15%))^(1/5) - 1
r0 = (1.25*1.15*1.05*0.95*0.85)^(1/5) - 1
r0 = 4.037%

So what happens when we increase the variability, but leave the average rate of return the same? The variability adjusted return gets smaller:

Table 6: Extended Variability

year	return	total
0	0%	$1,000.00
1	25%	$1,250.00
2	-15%	$1,062.50
3	5%	$1,115.63
4	15%	$1,282.97
5	-5%	$1,218.82
6	5%	$1,279.76
7	-25%	$   959.82
8	40%	$1,343.75
9	-30%	$   940.62
10	35%	$1,269.84

Notice that we’re still got an average rate of return of 5%, but increasing the variability gives us an annualized rate of return of 2.42%.

On the other hand, extending the same average without increasing the variability looks like this:

Table 7: Extended Time, Variability Unchanged

year	return	total
0	0%	$1,000.00
1	25%	$1,250.00
2	-15%	$1,062.50
3	5%	$1,115.63
4	15%	$1,282.97
5	-5%	$1,218.82
6	25%	$1,523.53
7	-15%	$1,295.00
8	5%	$1,359.75
9	15%	$1,563.71
10	-5%	$1,485.52

In this case, when the variability remains unchanged, the annualized rate of return remains unchanged at 4.037%. In other words, as the variability increases, the annualized rate of return gets further away and lower than the simple average rate. On the other hand, as the variability decreases, the annualized rate approaches the maximum value of a fixed rate of return, i.e. zero variability.

So, while it’s comforting that so long as your investment doesn’t go to zero, it doesn’t matter when the highs and lows come, it’s somewhat unintuitive that the average rate of return is not what you want to use to calculate the rate of return that you’re actually getting. In fact, the annualized rate of return will always be lower than the average rate as variability increases.

Chris’s Notes on The Motley Fool Investment Guide: How The Fool Beats Wall Streets Wise Men And How You Can Too, David Gardner, Tom Gardner, Fireside, 1997. I don’t recommend this book for anything but an example of the hubris that was rampant during the Internet bubble and a few chapters that hold the reader’s hand making the markets seem approachable.

In general, I recommend Peter Lynch’s Learn to Earn for the hand holding instead.

Chris’s Notes on The Instant Millionaire: A Tale of Wisdom and Wealth, Mark Fisher, New World Library, 1991. I recommend this book for folks that are willing to believe that the act of willing something to happen hard enough will make it happen.

On the one hand, I’ve solved many an engineering problem by simply letting my subconscious know that it needed solving. On the other hand, I’ve never been able to successfully wish for something to happen. The question is whether you view this book as channeling your subconscious energies into finding ways of making your dreams of financial independence come true or whether you view it as a fairy tale.

Chris’s Notes on Stock Options For Dummies, Alan R. Simon, Wiley Publishing, Inc., 2001. I recommend this book for folks that need to know the details of their company’s stock options, although the details aren’t interesting enough or important enough to warrant a book of this size, especially given how worthless most company’s stock options are these days.

I bought this book when I was too stupid to recognize it as a book about a company’s stock options instead of about publicly traded options. In general, the strategy that I’ve developed if I ever get the opportunity to exercise company stock options is as follows:

1. Exercise stock options as soon as possible, paying taxes as appropriate
2. Hold stocks for at least one year to reduce taxes on gains from normal income tax rates, e.g. ~30%, to the long-term capital gain tax rate, 15%, using 0 as the basis and subtracting the taxes already paid in step 1

Doesn’t seem like enough for an entire book…

Chris’s Notes on All the Math You Need to Get Rich: Thinking with Numbers for Financial Success, Robert L. Hershey, Open Court Pub Co, 2001. I recommend this book for the basics it covers, the slim size, the exercises and the approachable text.

I graduated from high school with all kinds of wonderful math grades (I finished all of my high school math a year early and had to attend a calculus course at a nearby college in my senior year), but managed to get out without a firm grasp of some simple ideas. Specifically, I never learned why it is that when I make a fixed house payment, the actual amount that goes to the interest and to the principle varies each month. The reason this is (as I’m sure all of my readers already know) is because the interest is only paid against the outstanding principle that remains on the total loan each month. It’s just like compound interest in reverse (an idea that I always did understand).

I haven’t been reading this book cover to cover (I already know how fractions, scientific notation and fractions work), but this one fact alone makes this book worth the price. Plus, it makes a wonderful reference (this is where I got F=P(1+R)^N used in The “Average Return” Myth). The world needs more short, focused, well-written books.