Question

What value of w makes the mean of R as large as possible? What is the SD of R for this value of w?

What is the value of w that minimizes the SD of R?

Answer 1

$1 in a stock yeilds Rs, $1 in a bond yeilds Rb

Rs is random with a mean .08 and SD of .07

Rb is random with a mean .05 and SD of .04

Correlation of Rs and Rb is .25

If you place a fraction w of your money in the stock fund and the rest, 1-w, in the bond fund, then your return on you investment is

R=wRs+(1-w)Rb

SD(R)=sqrt(0.0051((w)^2)-0.0018W+0.0016)

E(R) = 0.05 + 0.03w

Answer 2

w makes the mean of R as large as possible?
I think that is the same as maximize the expected value E(R) = 0.05 + 0.03w
If that equation is correct, then w=1 maximizes. it

Answer 3

Yeah that's the correct answer! I can see why, intuitively. Can you explain why taking the derivative doesn't bring us to that, though? Kind of confused. Wouldn't taking the derivative and setting = 0 maximize? Or am I misremembering my calculus hahaha

Answer 4

E(R) = 0.05 + 0.03w
is an equation of a line
y= mx + b
y = 0.03x + b
the derivative dy/dx = 0.03 gives you the slope of the line
*if* a curve has a changing slope e.g. \( \cap\) , then perhaps there is a point where the slope is zero, and that signals a possible min, max or inflection pt.

that idea does not work for a line.

the other criteria for a min/max is to evaluate any boundary conditions
in this case, w has values from 0 to 1. test those values.

Answer 5

Okay sweet thanks for the clarification.

For what is the value of w that minimizes the SD of R test 0,1 and then see if there's any critical points w/ derivative = 0, right?

Answer 6

sounds good.

Answer 7

Mind checking my answer when I get it? Give me a second. :-)

Answer 8

Well the answer is x=3/17, not sure the easiest way to come to that though. Is there an easier way than taking the derivative and =0? The derivative is a little ugly

Answer 9

the derivative is
\[ \frac{2aw+b}{2\sqrt{aw^2+bw+c} }=0\]
clearing the denominator:
2aw+b=0
\[ w= -\frac{b}{2a} \]
where a= 0.0051, b= -0.0018, c=0.0016
thus you get
\[ w= \frac{0.0018}{0.0102} = \frac{18}{102} = \frac{3}{17} \]