Okay, we're pretty sure that (in the past at least, and for certain stock+bond portfolios and over certain periods of time) the Volatility of our stock+bond portfolio (as measured by the Standard Deviation) will increase (almost) linearly as the fraction devoted to stocks increases - eventually - for larger stock fractions. >Pretty sure? Almost? Eventually? Can't you ...? Fig. 1 shows the asymptote for a mix of S&P500 + a BondIndex Mutual Fund (which mimics the Lehman Brothers Aggregate Bond Index, composed of government and corporate bonds) ... for ten years, ending in Dec/99. The green curve is hyperbolic. >You plot SD as a fraction yet you label the ends of the graph with SD as a percentage. That's inconsistent. If there's one thing I hate, it's ...

Fig. 1

Fig. 2

Further, if we plot the 10-year Gain (from Jan 1/90 to Dec 31/99) vs the Percentage of Stock in our portfolio (with monthly rebalancing to maintain the prescribed percentage), we get Fig. 2. It also shows a (roughly) linear relationship. >Very rough. In what follows, we stare at Fig. 1 and assume that: (1)        SD2 = P2x2 + Q2(1-x)2    the equation for an hyperbola where P and Q are certain constants which depend upon the Standard Deviations of the stocks and the bonds and the number of months and ... >And the weather in Bermuda? Sure, why not?

Equation (1) gives the Standard Deviation for uncorrelated portfolio components. (For more highly correlated portfolio components, see Efficient Frontier.) Okay, continuing ...
>Before you continue, do you have a formula for the minimum, in Fig. 1?
Yes, it's:

You can play with this calculator:

P = Standard Deviation of Stocks =%     Q = Standard Deviation of Bonds =% Percentage of Stocks = %

Continuing ...

>Wait! What's a typical value for P/Q?
For P/Q = 2 (that's the case for our S&P & BondIndex investments), that'd mean:

Continuing ...

>Wait! What about S&P 500 and, say, T-bills?

Okay, let's consider the annual returns for the S&P 500 and Treasuries from 1928 to 2000. The Standard Deviations of these annual returns are shown in the table

S&P 500 20.1% T-bills 3.2%

1. We'll assume a particular percentage of S&P 500 and T-bills, pick a year at random, select the returns for that year and apply it to a $1000 portfolio.
2. We'll repeat this 40 times (to get a forty year evolution of our portfolio) and we'll note the volatility of the 40 annual returns for our portfolio.
3. Then we'll repeat steps 1 and 2 a thousand times, each time noting the portfolio volatility.
4. Then we'll average the volatilities of these 1000 portfolios.
5. Then we'll select another percentage for the S&P and T-bills and repeat the above steps.
6. We end up with a set of portfolio volatilities associated with a particular S&P 500 percentage.

>Okay. That's for two asset classes ... stocks and bonds. What about having three asset classes, like maybe large cap, small cap and bonds or maybe large cap, bonds and foreign equities, or maybe ...?

Okay, suppose we have three asset classes with corresponding constants P, Q and R. Suppose, further, that we maintain fractions x, y and 1-x-y of each in our portfolio. Then, if we again ignore the cross-product "garbage" terms (as we did in here, in Part 1) and consider SD² = P²x² + Q²y² + R²(1-x-y)² the minimum volatility is achieved for:

For P, Q and R in the ratio 3:2:1 we'd get: x = (1/3)2/{ (1/3)2+(1/2)2+(1/1)2} = 0.082 or 8.2%, and y = (1/2)2/{ (1/3)2+(1/2)2+(1/1)2} = 0.134 or 13.4%, and, for the last asset class 1 - x - y = 1 - .082 - .134 = 0.784 or 78.4%. On the other hand, if all three had the same volatility, we'd minimize the volatility (that is, SD = Standard Deviation) of our portfolio by putting equal amounts (approximately, since we're ignoring certain "garbage", meaning we're assuming little correlation) in each class. Illustrated here: is a chart of z = P2x2 + Q2y2 + R2(1-x-y)2

Fig. 2a

where P = Q = R = 1 and, when x, y and 1-x-y are all between 0 and 1, then z is the Standard Deviation of our portfolio. It has a minimum when equal weights are given to each component.

>Wait! Are you saying that P, Q and R are volatilities? How did you ...?
Patience, we'll get to that in a minute.

>Have you ever tried it - that formula? Well, yes. Suppose we split our portfolio between Foreign, US Small Cap and US Large Cap, for the 20-year period 1978 to 1998. The Standard Deviations of annual returns are: EAFE = P = 20.8% Small Cap = Q = 17.7% Large Cap = R = 12.7% and the magic formulas give the appropriate - and estimated/approximate - fractions for each asset class. The results are shown here: For this example, the SD of this "Mix" is clearly smallest.

Fig. 2b

Let's continue. We stare at Fig. 2 and write the N-month Gain (N=120 in Fig. 2) as a linear relation, namely:

(2)        G = Rx + S(1-x) the equation for a straight line

Now, some initial observations pertaining to equations (1) and (2)
(which, when the smoke has cleared, will allow us to determine the various constants and do some plotting of graphs):

• x varies between 0 and 1 (The portfolio fraction devoted to stocks)

• For x = 0, SD = N-month Standard Deviation of the bonds = Q.
• For x = 1, SD = N-month Standard Deviation of the stocks = P. (See Fig. 1)

• For x = 0, G = N-month Gain Factor for the bonds = S.
• For x = 1, G = N-month Gain Factor for the stocks = R.

>Why don't you use "S" for the gain in "S"tocks? It seems to me ...
Pay attention. We now want to determine some scheme for dividing our portfolio between stocks and bonds in the "best" possible way, so that "something" is minimized - as small as possible ... perhaps our "volatility per unit of gain" (where, here, we're talking Gain Factor over N months).

>Is that a good criterion?
Well, if we invest in stocks or bonds with greater gain, we expect a greater volatility. So it's reasonable to consider minimizing the volatility per unit of gain. Anyway, let's start with that.

Here's the problem:

(3)        Minimize SD/G = {P²x² + Q²(1-x)²}^1/2/ {Rx + S(1-x)}     x lying in the interval [0,1]

>Aha! Now I see why you chose P, Q, R and ...
For an all-bond portfolio (x = 0), the Ratio is Q/S = Volatility(bonds)/GainFactor(bonds).
For an all-stock portfolio (x = 1), the Ratio is P/R = Volatility(stocks)/GainFactor(stocks).

>What's the "Gain Factor", again, please?
Gain Factor is the total gain over umpteen months. Here, we're considering 120 months or ten years. In fact, it's what $1.00 grows to, after umpteen years. If $1.00 grows to $2.34 the Gain factor is 2.34 and if it grows to ...
>Okay, continue ... please.

>If (3) is the criterion, then we go 100% bonds or 100% stocks, whichever has the smaller "volatility per unit gain", right?
So, what's smaller?
For a portfolio that has a stock component which mimics the S&P500 and a bond component which mimics our Bond Index Mutual Fund, we get:

Q/S = 0.0104 (for 100% bonds)        P/R = 0.0085 (for 100% stocks) >So an all-stock portfolio wins! Actually, no. A portfolio with 40% stock wins. (See the yellow dot?) >Are you using the actual, historical values or some invented ...? I'm using the actual Volatilities and Gains (over 120 months) in order to compute P, Q, R and S. Indeed, if we plot the curve SD/G = {P2x2 + Q2(1-x)2}1/2/ {Rx + S(1-x)}     we get

Fig. 3

>I assume you have a formula for that minimum. Absolutely. The "optimal" stock fraction (in order to minimize the Volatility per unit GainFactor) is: (4)      >I assume you have a chart for the ... Absolutely. The "optimal" stock fraction, and its dependence upon the Standard Deviation (or Volatility) and N-month Gain Factors for the stock & bond components of our portfolio ... it's like so:

Fig. 4

The yellow dot represents the situation illustrated in Fig. 3 (P/Q = 2.01 and R/S = 2.47), with an "optimal" stock percentage of about 40%.

>I notice that the percentage devoted to stocks increases if the ratio R/S increases. That means pick a greater stock fraction if the stock gain is greater. Right?
Right.

>And the percentage devoted to stocks decreases if the ratio P/Q increases. That means pick a smaller stock fraction if the stock volatility is greater. But it depends upon so many things. You're using a particular time interval and particular stock and bond investments and ... and ... are you saying ... I mean ... does it make any sense?
It makes about as much sense as:   "The percentage of Stock should be 100 - (your age)".
>What's the technical name for that ... that formula, equation (4)?
The gummy fraction.
>You're kidding, right?
Would I kid you?

I'm told (thank you Boomerbucks) that some aggressive advisors suggest "120 - (your age)" which, at age 80, means 40% stocks.

>Sounds good to me, but I think the gummy fraction is a little ... uh, gruesome.
Can't you generate a simple "Rule of Thumb", like 100 - (your age)?

Absolutely! We'll inundate ourselves with mathematically sophisticated assumptions typical of the financial genre, such as ...

>Financial genre?

... such as: We assume that the S&P500 and our BondIndexFund are representative of any stock+bond portfolio. We assume that the gains for both stocks and bonds have a Normal distribution. We assume that both P and Q (the Standard Deviations of stocks and bonds) increase with time as t1/2 ... which makes the ratio P/Q a contant. We'll take it as P/Q = 2 (like the S&P and BondIndex). The annualized gains for stocks and bonds ("s" and "b") are 10% and 5% respectively and don't change materially over time. Two thirds of monthly returns lie within one Standard Deviation of their Mean. The Santa Claus Rally knows who's naughty or nice. RRSPs are ALWAYS the best way to invest. You'll live & invest to age 90. Fill in the blank: >And the weather in ...? Then the gummy fraction of equation (4) becomes: Stock Fraction = 1/[ 1+4{(1+b)/(1+s)}90-Age] = 1/[ 1+4{(1.05)/(1.1)}90-Age]     whose graph looks like Fig. 5a (the blue guy). and if we then stick in a straight line fit we get the red guy in Fig. 5b

Fig. 5a Fig. 5b

and the equation of that red line is:

Stock Fraction = 1.1 - 0.01(Age)

so, in percentage terms, we have ... voila!

for Part III: Stocks vs Bonds and Risk