Here I continue my review of "The Investor's Manifesto" by Bernstein with my notes from Chapter 2, which could be subtitled "Here comes the math!"

I'm not naturally math-minded (I count on my fingers) but I've worked
through each mathematical example in this chapter using either pencil
and paper or a calculator. I **strongly** suggest you do the same. I
know in my case, I'll just skim the math parts if I don't make myself
actually do the math.

### Historical Returns

Diversification among different kinds of stock asset classes works well over the years and decades, but often quite poorly over weeks and months.

Investors who can earn an 8 percent annualized return will multiply their wealth tenfold over the course of 30 years, and if they have half a brain, they will care little that many days, or even years, along the way their portfolios will suffer significant losses.

In the world of finance, the only black swans are the history that investors have not read. Anyone familiar with the events of 1929-1932, which saw stocks fall by almost 90 percent in value (let alone what happened to the St. Petersburg exchange after 1914), would not have been dumbfounded by the recent market decline.

Historical returns are valuable to prepare you for the scope of boom and bust that you may experience in your own life, but they are dangerous when used to estimate future returns.

Bernstein talks about two relatively recent examples where depending on past returns to estimate future ones went very badly. In the years following World War II, inflation gradually accelerated, zooming to the hyperinflation of the 1970s and 1980s.

During the 30 years between 1952 and 1981, long-term Treasury bonds returned just 2.33 percent on an annualized basis during a period in which inflation averaged 4.31 percent per year. Thus, on average, bond investors lost 2 percent of their purchasing power each year.

Since companies can raise their prices when faced with inflation, stocks did much better than bonds during the same period, with the S&P 500 returning 9.89% per year.

Yet, anyone who could add and subtract was presented with this calculus: On September 30, 1981, the government sold 20-year Treasury bonds yielding 15.78 percent interest, while during the previous five years inflation averaged “only” 10.11 percent. Further, by that point, the Federal Reserve, under the greatest of its chairmen, Paul Volcker, had dramatically tightened the money supply. By 1981, this had the desired effect of lowering the inflation rate; over the next five years, it fell to just 3.42 percent.

To summarize: By the beginning of 1982, bond investors had been hammered. However, a dispassionate look at bond interest yields and a reasonable estimate of future inflation suggested high returns ahead—at least 5 percent on a real basis. This hard-headed assessment was indeed borne out: Over the 20-year period beginning in 1982, the real return of the long Treasury bond was in fact 8.66 percent.

Bernstein points to a remarkable opportunity. In 1982, the interest rate was 3.42%, and bonds were offering returns significantly higher than that. Yet few people wanted to invest in bonds because bond investors had lost money for the last 30 years. Investors with the insight to see the opportunity and the fortitude to take advantage of it made a killing.

### A Bond Yield Digression

I'm going to take a moment here to expand on some math that was not intuitively obvious to me: "If an investor buys a 30-year bond yielding an initial 5 percent coupon, and long-term interest rates subsequently increase to 10 percent, the value of that bond temporarily falls by nearly half."

Let's say I invested one year ago in a 10-year bond giving me a 5% return on my $1,000 investment. I get $50 interest every year (5% * $1,000). Since then, interest rates have increased to 10%. Other investors can buy a $1,000 bond today yielding 10%. If I try to sell my $1,000, 5% bond today, nobody will buy it -- I'll have to reduce the price to get another investor to purchase.

If I sell my \(1,000 bond yielding 5% for only $500, then another investor would make the deal. The bond still yields 5% on $1000 (\)50) but since the new investor bought in for $500, effectively he's getting 10% return on his investment. (10% of $500 is $50).

This is why raising interest rates decimates bond value.

### My Explanation of Expected Return

Since we can not predict future returns, the best we can do is produce an "expected return." Rather than an exuberant or morose guess, it's an analysis based on hard financial figures -- but it's still a guess, just a more informed one.

Expected return is a familiar concept to anyone who has studied gambling odds. (I used to play poker.) Bernstein uses a roulette wheel as an example. But I'm a simple sort of guy, and I've never played roulette. I have flipped coins, though.

Imagine that you're gambling with someone by flipping coins for money. For every time you win, you get $1. For every time you lose, you give up $1.

The formula is (amount you get if you win * odds of winning) - (amount you risk * odds of losing)

For the coin-flipping game as I described it, you get $1 if you win, and your odds of winning are 1/2, so your expected return is 0:

(1 * 1/2) - (1 * 1/2)

However, if you have a foolish opponent who is willing to bet $2 for every $1 you bet, your expected return is $0.50 per coin toss:

(2 * 1/2) - (1 * 1/2)

In that case, you can expect to make $0.50 cents for every toss. However, you'll never actually earn $0.50 on a single toss -- you'll get $2 sometimes, and lose $1 sometimes, and you'll average out to $0.50 per toss if you play long enough.

I just played this game against my five-year-old nephew. (When the odds
are this much out of balance, it's good to have an opponent who doesn't
know math.) And yet he took me for a ride! The tosses were
H-H-T-T-T-H-T-T-T-T, meaning he owed me $6 for the three heads, and I
owed him $7 for the 7 tails, with a net win of $1 for my nephew. This
gives a key insight into expected return -- it is **not** guaranteed!

If we had played the game for longer, say 100 tosses, my actual return would have been much closer to the expected return. And thus one of the key rules of this chapter -- you need to keep your money in for the long-term and ignore short-term results.

One more note about coin tosses and then we'll move on -- in a coin
toss, you know the odds -- you know how many sides the coin has. With
stocks, you're guessing what you'll earn if you win **and** the odds of
winning. This makes calculating the expected return more difficult, but
still possible, and it's still a very valuable exercise to perform.

Oddly enough, after working through it I think there's an error in Bernstein's Roulette odds. He gives the formula as 35/37 - 1, which comes out to -\(0.54, but it's actually 35*(1/38) - 1* (37/38), which gives the [correct expected return of -\)0.53](http://en.wikipedia.org/wiki/Roulette#Bet_odds_table).

I'm going to stop here, and continue tomorrow with how to estimate expected return for bonds and stocks (and you can follow along as I actually do it, with tomorrow's numbers.)