Answered: You are allocating your wealth between…

Essentials Of Investments

11th Edition

ISBN:9781260013924

Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Chapter1: Investments: Background And Issues

Section: Chapter Questions

Problem 1PS

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Question

You are allocating your wealth between two shares, Tinkle.com and Circumbendibus Wheels. Tinkle.com has volatility 35.10%, while Circumbendibus Wheels has volatility 58.30%. The correlation beteween the two shares' returns is 0.40. What percentage of your wealth should you allocate to Tinkle.com to minimise your portfolio's volatility?

Expert Solution

Step 1

Minimum variance portfolio of two stocks

In order to determine minimum variance portfolio, the weights of stocks are determined that causes the minimum variance for the portfolio.

With volatility or standard deviation of stock T (SD_T), volatility or standard deviation of stock C (SD_C), weight of stock T (W_T), weight of stock C (W_C) and correlation between two stocks (Cor), the variance of portfolio is calculated as shown below.

$\begin{array}{rcl} V a r i a n c e o f p o r t f o l i o \\ = & {(S D_{T} \times W_{T})}^{2} + {(S D_{C} \times W_{C})}^{2} + 2 \times W_{T} \times W_{C} \times S D_{T} \times S D_{C} \times C o r \\ = & {(S D_{T} \times W_{T})}^{2} + {(S D_{C} \times (1 - W_{T}))}^{2} + 2 \times W_{T} \times (1 - W_{T}) \times S D_{T} \times S D_{C} \times C o r \\ H e r e, \\ W_{T} + W_{C} & = & 1 \\ W_{C} & = & 1 - W_{T} \end{array}$

In order to determine minimum variance portfolio, the portfolio should be differentiated with respect to W_T and equate the result to zero. Then, whatever the value for comes W_T, it represents the weights of stocks in the portfolio, resulting the minimum portfolio variance as shown below.

$\begin{array}{rcl} V a r i a n c e o f p o r t f o l i o & = & {(S D_{T} \times W_{T})}^{2} + {(S D_{C} \times (1 - W_{T}))}^{2} + 2 \times W_{T} \times (1 - W_{T}) \times S D_{T} \times S D_{C} \times C o r \\ V a r i a n c e o f p o r t f o l i o & = & S {D_{T}}^{2} \times {W_{T}}^{2} + S {D_{C}}^{2} - 2 \times W_{T} \times S {D_{C}}^{2} + S {D_{C}}^{2} \times {W_{T}}^{2} + 2 \times W_{T} \times (1 - W_{T}) \times S D_{T} \times S D_{C} \times C o r \\ D i f f e r e n t i a t i n g b o t h s i d e s w i t h W_{T} a n d e q u a t i n g w i t h z e r o, w e g e t \\ 2 \times S {D_{T}}^{2} \times W_{T} + 0 - 2 \times S {D_{C}}^{2} + 2 \times S {D_{C}}^{2} \times W_{T} + 2 \times S D_{T} \times S D_{C} \times C o r - 4 \times W_{T} \times S D_{T} \times S D_{C} \times C o r & = & 0 \\ 2 \times W_{T} \times (S {D_{T}}^{2} + S {D_{C}}^{2}) - 4 \times W_{T} \times S D_{T} \times S D_{C} \times C o r & = & 2 \times S {D_{C}}^{2} - 2 \times S D_{T} \times S D_{C} \times C o r \end{array}$

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