Approximations     and     Portfolio Evolution
a topic suggested by Ellis A.

In a neat article in the Journal of Financial Planning, Gobind Daryanani has a very interesting application of sensitivity analysis to problems in finance.

We consider a variable z which depends upon two random variables x and y, like so:

(1)     z = f(x,y)

We want to determine the statistical properties of z, given the statistical properties of x and y. In particular, we suppose:

• The Mean of the random variable x is M(x) = 0
• The Mean of the random variable y is M(y) = 0
• S(x), the Standard Deviation of random variable x, is known
• S(y), the Standard Deviation of random variable y, is known
and ask:   What are M(z) and S(z), the Mean and Standard Deviation of z ?

>I assume that M(this) is the Mean of this ... and S(that) is the Standard Deviation of that.
Yes.

We now suppose that the variables x and y vary little from their Mean, which is 0. Hence, if and when terms of third or higher order occur, such as x3 or x2y or x2y2 ... we'll ignore such terms.

>Can you do that?

Okay, so we expand the function f(x,y) in a Taylor series, about the Mean values: x = y = 0, like so:

z = f(0,0) + fx(0,0) x + fy(0,0) y + (1/2) fxx(0,0) x2 + fxy(0,0) x y + (1/2) fyy(0,0) y2 + ... or (ignoring those higher order terms), simply

(2)     z = f(0,0) + A x + B y + C x2 + D x y + E y2

where the constants A, B, etc. are partial derivatives of the function f, evaluated at x = y = 0, and the higher order terms are ignored.
Equation (2) is a Quadratic approximation to f(x,y). A Linear approximation would be z = f(0,0) + A x + B y.

If we let the random varables x and y vary (according to their statistical properties) and average the resultant z-values, we get the Mean of z, which we're calling M(z), by taking the Mean of equation (2). However, we first note that:

1. The Mean of a sum is the sum of the Means.
2. The Mean of a constant is the value of that constant.
3. S(u), the Standard Deviation of a random variable u, is given by:
S2(u) = Mean(u2) - Mean2(u)     (the average square) - (the square of the average)     See SD stuff
4. If Mean(u) = 0, then S2(u) = Mean(u2)     and we're calling this M(u2)
5. If the Mean of x and y are both 0, then Mean of xy is the Covariance of x and y:   M(xy) = COV(x,y).
6. r, the Pearson Correlation Coefficient (between random variables x and y), is given by r = COV(x,y) / {S(x) S(y)}
(See Pearson Correlation.)

Okay, we're now ready to calculate the Mean of z ... as given by the approximation in (2):
 M(z) = M(f(0,0)) + A M(x) + B M(y) + C M(x2) + D M(x y) + E M(y2)     using 1, above = f(0,0) + 0 + 0 + C S2(x) + D S(x) S(y) r + E S2(y)     using a bunch of stuff, above
Finally, then, we have:
 If x, y are random variables with Means: M(x) = M(y) = 0 and Standard Deviations S(x), S(y) and r is the (x,y) Pearson Correlation Coefficient and z = f(0,0) + A x + B y + C x2 + D x y + E y2 then M(z) = f(0,0) + C S2(x) + r D S(x) S(y) + E S2(y)

>Are you gonna talk about portfolio stuff? What good is ...?
Patience.

>And what about pictures? Don't you have any ...?
Patience. We're getting there. We now have to determine a formula for S(z), the Standard Deviation of z, using the approximation in (2), above.

Now S2(z) is the Average Squared Deviation of z from its Mean M(z), and we already have this Mean, above, so we need to square
z - M(z) and find the Average of that and then we'll ...

>So, just do it.
Okay.
 [z - M(z)]2 = [ { f(0,0) + A x + B y + C x2 + D x y + E y2 } - { f(0,0) + C S2(x) + D S(x) S(y) r + E S2(y) }]2 = [ A x + B y + C {x2 - S2(x)} + D {x y - r S(x) S(y)} + E {y2 - S2(y)} ]2 = A2 x2 + B2 y2 + 2 A B x y     ignoring higher order terms
Now we take the Mean and get:
 S2(z) = A2 M(x2) + B2 M(y2) + 2 A B M(x y) = A2 S2(x) + B2 S2(y) + 2 A B S(x) S(y) r
We give this a position of importance:
 If x, y are random variables with Means: M(x) = M(y) = 0 and Standard Deviations S(x), S(y) and r is the (x,y) Pearson Correlation Coefficient and z = f(0,0) + A x + B y + C x2 + D x y + E y2 then S2(z) = A2 S2(x) + B2 S2(y) + 2 r A B S(x) S(y)     approximately

Notice that, in these approximations, the Mean of z depends upon C, D and E, the coefficients of the second order terms whereas the Standard Deviation of z depends upon A and B, the coefficients of the first order terms.

>And what about that portfolio stuff?
If z is the value of a portfolio, and it depends upon annual returns x and y, each with some statistical distribution characterized by their Mean and Standard Deviation, then we can determine the statistical properties of this portfolio value ... in terms of the statistical properties of the returns x and y. The point here is that, whereas it'd be difficult to obtain the Mean and Standard Deviation of z = f(x,y) directly, it's pretty easy to get the Mean and SD of the quadratic expansion.

>Approximately! And the portfolio value depends upon just two returns, x and y? Are you kidding?
Well ... no. But we walk before we run.

>Then start running!
Okay, but in the meantime look at some pictures:

Figure 1

We look at f(x,y) = (1+x)(1+y) where x and y are annual returns, selected at random from some statistical distribution.
Figure 1 shows a typical distribution of (x,y) together with an indication of the correlation between f(x,y) and the Linear Approximation
f(0.08,0.12) + ∂f/∂x (x-0.08) + ∂f/∂y (y-0.12) where the derivatives ∂f/∂x and ∂f/∂x are evaluated at the Means: M(x) = 0.08, M(y) = 0.12.
The point where x=M(x), y=M(y) is shown in red, in the left chart ... and the light gray line in the right chart illustrates an exact match.

(For this example, a Quadratic approximation would be exact!)
Note that (x-0.08) and (y-0.12) replace the x and y we were considering earlier; these have Means which are 0.
Note, too, that the values of ∂f/∂x and ∂f/∂y give some indication of the sensitivity of f(x,y) to changes in x and y.

>That f = (1+x)(1+y) ... is that meaningful?
Uh ... not very.
>You're still walking?
Yes, but ...
>How about a real life example, say 25% bonds and 75% stocks and ...?
Okay. Suppose an initial \$1.00 portfolio is rebalanced yearly to maintain a 25% + 75% allocation, and we consider the portfolio growth over 5 years where the growth factor would be f(x,y) = (1+0.25 x + 0.75 y)5 where x and y denote bond and stock returns and their distributions are characterized by their Means and Standard Deviations with M(x) = 0.06, meaning 6%, and M(y) = ...
>A picture is worth a thousand words!
Okay. Here's a picture showing an (x,y) distribution and the resultant Linear and Quadratic approximations, L(x,y) and Q(x,y), respectively. The red dots show the deterministic result, when x and y are constant, equal to their Mean values: x = M(x) = 0.06 and y = M(y) = 0.10.

Figure 2

>So, when do you do something useful?
Patience.