Corner portfolio, trial and error??

CFA Level III Portfolio Management
#1

When you are asked to work out what weights of each corner portfolio will be used to make up the required return, is it trial and error to a certain extent? I feel like I’m missing something because it takes me a while to work it out using trial and error. For example…

Required return is 11.11% and two corner portfolios have expected returns of 12.79% and 10.54%:

So, 0.1111 = w (0.1279) + (1-w)(0.1054)

I realise the weights are 25% and 75% but only through trying different weights in my calculator. Am I missing something? I there a quicker way?! Thanks in advance!!

#2

Yep there is.

0.1111 = w(0.1279) + (1-w)(0.1054)

0.1111 = 0.1280w + 0.1054 - 0.1054w

0.1111-0.1054 = 0.0226w

w = 25.22%

#3

Oh wow. I’m actually embarrassed right now. I think I need to take a break. :confused: Haha thanks very much though!!

#4

Level 3 life…when you forget algebra haha.

Just to add…if there is no constraint on leverage then use the tangency portfolio (Highest sharpe) with the RF rate.

#5

If you write it as:

0.1111 = 0.1279_w_1 + 0.1054_w_2

Then,

w1 = (0.1111 – 0.1054) / (0.1279 – 0.1054) = 0.2533

w2 = (0.1111 – 0.1279) / (0.1054 – 0.1279) = 0.7467

More generally, if:

c = waa + wbb, where wa + wb = 1

then,

wa = (c – b) / (a – b)

wb = (c – a) / (b – a)

The symmetry is appealing.

If you draw it out on a number line, you’ll see that the denominator is distance between the known points (a & b) and the numerator is the distance between the unknown point (c) and the other point (i.e., for wa you use the distance from c to b, for wb you use the distance from c to a). If you think about it, it makes sense: if c is closer to a, then you want wa to be bigger than wb, so you use the _ bigger _ distance (c to b) in the numerator.

By the way, this works even if you’re extrapolating (e.g., figuring out the weights when you’re leveraging the tangency portfolio).