You have $12,000 to invest and would like to create a portfolio with an expected return of 9.75 percent. You can invest in Stock K with an expected return of 8.05 percent and Stock L with an expected return of 11.7 percent. How much will you invest in Stock K

The multiple choices are:

$5,589.04

$7,452.05

$4,890.41

$5,876.71

$6,410.96

Amount invested in K is $6,410.96  

Explanation:

L+K=12,000

from the return perspective

0.0975=K/12000*0.0805+L/12000*0.117

K=12000-L

Substitute for K in the second equation

0.0975=(12000-L)/12000*0.0805+L/12000*0.117

0.0975=(966-0.0805L)/12000+0.117L/12000

0.0975=(966-0.0805L+0.117L)/12000

12000*0.0975=966+0.0365 L

1170 -966=0.0365L

204=0.0365L

L=204/0.0365

L=$ 5,589.04  

K=$12,000-$ 5,589.04  

K=$6,410.96  

x = 6, y =3, and the minimum = 45

Step-by-step explanation:

There are plenty of ways to solve this problem.

The Lagranian multiplier method is chosen (it is simple to apply in this case).

Given: A = x^2 + y^2

Find: x and y that satisfy 4x + 2y = 30 or 4x + 2y - 30 = 0 and A is minimum.

This is the same as finding x, y, and lambda that make the following L minimum.

L= x^2 + y^2 + lambda(4x + 2y - 30)

To find x, y, and lambda, we take the derivative of L with respect to them, set the derivative to 0, and solve the equations.

L'(x) = 2x + 4(lambda) = 0    (1)

L'(y) = 2y + 2(lambda) = 0    (2)

L'(lambda) = 4x + 2y - 30 = 0   (3)

Multiply (2) by 2 and subtract to (1), we have:

4y - 2x = 0  (*)

Combine (*) with (3) we have:

4y - 2x = 0

4x + 2y - 30 = 0

Multiply the 1st equation by 2 and add to the 2nd equation:

=>  10y - 30 = 0

=> y = 3

=> x = 4y/2 = 4(3)/2 = 6 (as (*))

=> lambda = -y = -3  (as (2))

Substitute x = 6, y = 3, and lambda = -3 into L, or substitute x = 6 and y = 3 into A to get the minimum.

Both ways would lead to the same answer.

Check:

L = x^2 + y^2 + lambda(4x + 2y - 30)

  = 6^2 + 3^2 + (-3)[4(6) + 2(3) - 30]

  = 36 + 9 + (-3)(0)

  = 45

A = x^2 + y^2

   = 36 + 9

   = 45

Hope this helps!