We're considering a variable z = f(x,y) which depends upon random variables x and y which have known statistical properties: The Mean of the random variable x is M(x) The Mean of the random variable y is M(y) S(x) is the Standard Deviation of random variable x S(y) is the Standard Deviation of random variable y >What happened to Mean = 0, for x and y? We made that assumption in Part I, for sanitary reasons. Here, we expand f(x,y) in a Taylor series about the Mean values M(x) and M(y), hence a power series in the variables u = x - M(x) and v = y - M(y), and these do have Mean values equal to 0 ... so we can use the results obtained earlier

If x and y are random variables with Means: M(x) and M(y) Standard Deviations: S(x) and S(y) r = Pearson Correlation Coefficient = Cov(x,y)/{S(x) S(y)} and we set u = x - M(x) and v = y - M(y) and f0 = f(M(x), M(y)) ... that is, f(x,y) evaluated at the Mean values and assume: z = f0+A u+B v+C u2+D u v+E v2 then M(z) = f0 + C S2(x) + r D S(x) S(y) + E S2(y) and S2(z) = A2 S2(x) + B2 S2(y) + 2r A B S(x)S(y)     approximately

Figure 1

As an example, we'll again consider a 25% Bonds + 75% Stocks portfolio with 5-year growth of $1.00 according to: z = f(x,y) = (1 + 0.25 x + 0.75 y)5 and we'll select, at random, 500 (x,y) points and chart the distribution of the Actual gain factor f(x,y) and compare to the distribution for the Linear and Quadratic approximations ... with fewer points. >Fewer points? Why fewer points? To see if there is some saving in computations if we use the approximation. If there isn't, then why wouldn't we just use the Actual gain? >So? So here's a typical result, in Figure 1 where (as before) the Bond and Stock Means are 6% and 10% and Standard Deviations 10% and 20%.

Our primary goal is to determine the following:

How sensitive is f(x,y) to changes in x and y ?

>How do you pick these random values, for x and y?
Well, I have to admit that I've assumed they're Normally distributed so I can use the Excel function NORMINV(RAND(),Mean,SD).

>And what about the correlation. Did you ...?
Actually, the approximations L(x,y) and z = Q(x,y) don't depend upon the correlation, only upon the rates of change of f(x,y). In actual fact, however, the correlation is pretty small - usually less than about 5% when I use Excel to generate 500 points (x,y).

>Why not pick a correlation? Maybe you'll get a better approximation.
Better approximation? I might get better Mean and Standard Deviation estimates, since they DO depend upon the correlation. We'll see ...

>Don't tell me. You're not running, yet, eh?
Yeah ... still walking, but let's consider withdrawing from a portfolio ...

In Investing we developed a formula which described annual withdrawals from a portfolio, like so:

Initial Portfolio = $B Annual Withdrawals = $P, increasing with inflation u = 1+R where R is the annual return on your portfolio (for an 8.9% return, put R = 0.089) v = 1+i where i is the annual inflation (for 2.3% inflation, put i = 0.023) N is the number of years which have passed Then our portfolio balance, after N years, is:     f(u,v) = BuN - Pv(uN - vN)/(u - v) or, for a 2-asset portfolio with a fraction a devoted to asset A (with constant annual return x) and the balance, (1-a), devoted to asset B (with constant return y), then: f(x,y) = B uN - P(1+i ) { (1+u)N - (1+i )N } / (u - (1+i) )   with u = 1 + a x + (1-a) y

>When you got this formula, $P was the annual amount invested, not withdrawn.
Yes, so we stuck it in, above, with a negative sign in front. In what follows, you can put in a negative P-value and get the formula for investing (instead of withdrawing).

Anyway, we could use the formula for f(u,v) directly or we could use ...
>An approximation!
Exactly.
>And why would that be better?
We'll see. In the meantime, you can see the result of applying the above formula, where everything remains constant:

Initial Portfolio B = $ Annual Return on Investments R = %   expressed a percentage Annual Inflation Rate i = %   expressed a percentage Annual Withdrawal Rate W = %   this identifies P = W% of B Number of Years N = Portfolio f(u,v) = $     after N years

Starting with $100K earning 9% and extravagant withdrawals of 7% (increasing with 3% inflation) you wind up with lots of money after 30 years. That's misleading.

We need to consider random variations in our portfolio return. As before, we'll consider our portfolio to be part Bonds and part Stocks and these have returns x and y (assumed constant, for the moment) and we want to determine how a final portfolio (after N years) depends upon x and y. We could, as well, consider the dependence upon the inflation rate ... but let's keep it simple.

So what we do is replace f(x,y), our final portfolio, with an approximation and analyze that approximation.
We can use either L(x,y), the Linear or Q(x,y), the Quadratic approximation:

    L(x,y) = f₀ + f_x {x - M(x)} + f_y {y - M(y)}
or
    Q(x,y) = L(x,y) + (1/2) f_xx {x - M(x)}² + f_xy {x - M(x)}{y - M(y)} + (1/2) f_yy {y - M(y)}²
where the various derivatives (like f_x, f_y, etc.) are evaluated at the Mean values M(x), M(y).

We need the rates of change with respect to x and y, however, we can estimate these by evaluating f(x,y) at six points (when x and y change by small amounts dx and dy), like so:

fx = {f(x+dx,y) - f(x,y)}/dx fy = {f(x,y+dy) - f(x,y)}/dy fxx = {f(x+2dx,y) - 2f(x+dx,y) + f(x,y)}/dx2 fyy = {f(x,y+2dy) - 2f(x,y+dy) + f(x,y)}/dy2 fxy = {f(x+dx,y+dy) - f(x,y+dy) - f(x+dx,y) + f(x,y)}/(dx dy)

>That's simple enough. Better than using calculus, at least.
Well, yes, but the above prescriptions are approximate. The get closer and closer to the actual derivatives as dx and dy get closer to zero.

Okay. Now we consider (again!) our 25% Bond + 75% Stock portfolio, where the Bonds return 6% and Stocks return 10% and we assume 3% inflation and 5% annual withdrawal and ... >Picture? Look at Figure 2 where we examine the final value of a $1000 portfolio after N years using the Linear and Quadratic approximations with the Bond and Stock returns 2% off their mean values. >Huh? The final portfolio depends upon what returns we assume. The approximations are right on the money if we use the Mean values ... so we use values somewhat different, namely Bond return = 8% and Stock return = 12%, to see how good (or bad) the approximations are. For the exact result, we use ...

Figure 2

>So what're the formulas you're using ... for the Quadratic and ...?
Okay. For the exact value we use:

f(x,y) = 1000 (1+0.25 x + 0.75 y)^N - (0.05)(1000)(1.03){(1+0.25 x + 0.75 y)^N - 1.03^N} / (0.25 x + 0.75 y - 0.03).

Now let's look at the formulas for the approximations (where the partial derivatives are calculated according to the scheme shown above, with dx = dy = 0.01% or, equivalently, dx = dy = 0.0001):

Years	Q(x,y)
N	fo+fx (x-0.06) + fy (y-0.10) + (1/2)fxx (x-0.06)2 + fxy (x-0.06)(y-0.10) + (1/2)fyy (y-0.10)2
5	$1213+$1607(x-0.06) + $4821(y-0.10) + $772(x-0.06)2 + $4634(x-0.06)(y-0.10) + $6951(y-0.10)2
10	$1489+$4430(x-0.06) + $13293(y-0.10) + $4963(x-0.06)2 + $29782(x-0.06)(y-0.10) + $44681(y-0.10)2
20	$2344+$17231(x-0.06) + $51718(y-0.10) + $43297(x-0.06)2 + $259896(x-0.06)(y-0.10) + $390018(y-0.10)2
30	$3963+$51723(x-0.06) + $155295(y-0.10) + $208112(x-0.06)2 + $1249555(x-0.06)(y-0.10) + $1875660(y-0.10)2
40	$7250+$141568(x-0.06) + $425181(y-0.10) + $794487(x-0.06)2 + $4771596(x-0.06)(y-0.10) + $7164414(y-0.10)2

>That doesn't seem quite right. I mean ...
The 25%+75% portfolio with bond return at 6% and stocks at 10% gives a combination return of 25% x 6% + 75% x 10% or 9%.
If you stick this 9% return into the calculator above, starting with a $1000 portfolio (and 5% withdrawal and 3% inflation) you'll get the numbers f_o in the table, above, and if you ...

>No! I mean, the numbers out front, in Q(x,y), like that last one, $7164414. That's over 7 million dollars!
Yes. You should see it when N = 100! When the number of years is large (like N = 40) and the deviations of x and y from the Mean Returns of 0.06 and 0.10 are more than a schnitzel, then ...
>A schnitzel?
... the large coefficients in the Quadratic approximation get multiplied by these deviations and since the rates of change of f(x,y) ... they give the coefficients f_x , f_xx , etc. ... since these are rather sensitive to changes in the number of years, remembering that portfolios tend to grow exponentially with N, then you'd expect dramatic changes. In fact ...

>Portfolios grow exponentially? You're kidding, eh? Mine, it decays exponentially. I mean ...
Pay attention.
Look at the coefficient of (y-0.10) in the 40-year Q(x,y), namely $425,181. Even though the value of f(x,y) is only $7250 when x = 0.60 and y = 0.10, small changes in the returns x and y produce large changes in f. For example, if y changed from 0.1 to 0.105, meaning 10% to 10.5%, so y-0.10 is 0.005, then f(x,y) will change by more than 425181(0.005) = $2126. A 0.5% change in the return for asset B causes a change in our portfolio of about 30%. Now that's sensitivity, eh?

>Yeah, yeah, but I think it's time to run.
Of course, you'd expect the approximations to be pretty good for smallish deviations from the Mean Returns M(x) and M(y). That'd be the case if the Standard Deviations were small, say less than 5%. However, even for larger Volatilities, if the number of years is small, say N less than 5, then the approximations are okay. Try a 0.5% in the asset B return for N = 5. You'll find that ...
>Why don't you just supply some examples, since a picture is worth ...
Here are some examples:

Figure 3a Volatilities = 10% and 20%, but N is only 5

Figure 3b N = 30, but Volatilities are only 3%

And, it you'd like to play, trying various combinations of N and Standard Deviations and staring at pictures of how good (or bad) the Linear and Quadratic are, then here's a spreadsheet where you fill in the RED boxes, press the CALCULATE APPROXIMATION button (to get the coefficients in the approximations, called AA, BB, etc. in the spreadsheet) then, each time you press F9 to recalculate, you get a set of random returns for x and y (selected from a Normal distribution with your prescribed Mean and Standard Deviation) and a plot of the Actual f(x,y) as well as the approximations Q(x,y) and L(x,y):

To download the .ZIP'd spreadsheet, RIGHT-click on picture and Save Target or Save Link.
>I assume that for large N and large volatilities the approximations are pretty lousy, eh?
Try it.
>I see there's a correlation to enter, called rr.
Yes, but that only affects the Mean and Standard Deviation of the Quadratic approximation (shown with a sort of pink background).
>The chart says "Final Portfolio vs 50 random Returns". That's for 50 years, eh?
No! It just means that we're comparing

f(x,y) = 1000 (1+0.25 x + 0.75 y)^N - (0.05)(1000) (1.03) {(1+0.25 x + 0.75 y)^N - 1.03^N} / (0.25 x + 0.75 y - 0.03).

where we start with $1000 and invest in two assets with returns of x and y, withdrawing 0.05 (that's 5%) each year and inflation is 0.03 (that's 3%) ... we're comparing with the L and Q approximations, using 50 random values for x and y ...

>I'd like to see the result of 30 or 40 or 50 random annual returns, to see the portfolio growth, to see ...
First we have to be convinced that the approximations are good ... and under what conditions.
>Still walking, eh?

for Part III