# Finding weights of portfolio when return given

Hi all,

Would anyone be able to help me with the following calc?

If the return of a two asset portfolio is given, what is the calculation for finding the weights as in the following practice problem from reading 42:

A portfolio consists of 2 securities. Security 1 has an expected return of 16% and Security 2 has an expected return of 12%.

If the portfolio has an expected return of 15%, what are the proportions in each security?

I could answer the question by the process of elimination however I would like to learn the actual calc

The part I dont understand is how this

Rp = w1 x 16% + (1 - w1 ) x 12%

gets to

15% = 0.75(16%) + 0.25(12%)

All help much appreciated!

Cheers

When I would get these questions I would trial and error but what you could also do is take the absolute values of the stock returns from the portfolio return:

Stock A = 16% return, Stock B = 12% return, portfolio return = 15%

Stock A absolute deviation from port return is 16 - 15 = 1

Stock B absolute dev = 15 - 12 = 3

Sum of absolute values = 1 + 3 = 4

So now the wordy part:

weight of Stock A = (Sum of absolute values - Stock A absolute dev) / Sum of absolute values

weight of Stock B = (Sum of absolute values - Stock B absolute dev) / Sum of absolute values

so…

The weight of Stock A (4- 1)/4 which equals 3/4 = .75

the weight of Stock B (4-3/4) which is 1/4 = .25

DPB’s solution seems to be right, but not too intuitive. In the end, its “just” algebra:

Rp = w1 x 16% + (1 - w1 ) x 12%

You know that Rp = 15% (=expected return of the portfolio).

This leads to the following equation: 15% = w1 x 16% + (1-w1) x 12%

Some rearranging leads to… 0,15 = 0,16w1 + 0,12 - 0,12w1

…and finally to… 0,03 = 0,04w1 w1 = 3/4 = 0,75 => w2 = 1 - 0,75 = 0,25

Now you could substitute w1, w2 and Rp and you will get

15% = 0.75(16%) + 0.25(12%)

I hope this helps!

Rp = w1 x 16 + (1 - w1 ) x 12

15 = w1 x 16 + (1 - w1) x 12

15 = w1 x 4 + 12

3 = w1 x4

w1 = 3/4

Algebra!!!

Consider two assets with the following:
µ1 = 0.10, σ1 = 0.40
µ2 = 0.03; σ2 = 0
Calculate the weights of the portfolio, (w1, w2) such that the expected return,
µw = 16. Calculate the variance of this portfolio.

Consider a security with the stock prices
S(1) =



80 with probability 1
8
90 with probability 2
8
100 with probability 3
8
110 with probability 2
8
(a) What is the current price of the stock for which the expected return
would be 12%?
(b) What is the current price of the stock for which the standard deviation
would be 18%