Diversification
Kerry Back
Why diversify your investments?
- What is the risk of one coin flip for $10?
- The possible deviations from the mean are +10 and –10.
- What is the risk of two coin flips for $5 each?
- The possible deviations from the mean are
- +10 (prob 1/4),
- –10 (prob 1/4) and
- 0 (prob 1/2).
- The possible deviations from the mean are
More baskets is better
Variance and standard deviation
- Variance is expected squared deviation from mean
- Flipping once for $10 \(\Rightarrow\) variance = 100
- Flipping twice for $5 \(\Rightarrow\) variance = (1/4) \(\times\) 100 + (1/4) \(\times\) 100 = 50
- Standard deviation is square root of variance
- Flipping once for $10 \(\Rightarrow\) std dev = 10
- Flipping twice for $5 \(\Rightarrow\) std dev = 7.07
Portfolio Returns
- The return of a portfolio of two assets is \(r_p = w_1r_1 + w_2r_2\) where \(w_i\) is the fraction of the portfolio invested in asset \(i\).
- Example: $100,000 portfolio with 40% in asset 1 and 60% in asset 2.
- Suppose asset 1 \(\uparrow\) 20%. $40,000 \(\rightarrow\) $48,000
- Suppose asset 2 \(\uparrow\) 10%. $60,000 \(\rightarrow\) $66,000
- $100,000 \(\rightarrow\) $114,000
- So, up 14% = 0.4 \(\times\) 20% + 0.6 \(\times\) 10%
Expected portfolio return
- \(\bar{r}_1\) = expected return of asset 1
- \(\bar{r}_2\) = expected return of asset 2
- The expected portfolio return is
\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]
Portfolio variance
- The deviation of the portfolio return from its mean is
\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]
- This equals
\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]
The deviation of the portfolio return from its mean is
\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]
and \((a+b)^2 = a^2 + b^2 + 2ab\), so the squared deviation is sum of
- \(w_1^2(r_1-\bar{r}_1)^2\),
- \(w_2^2(r_2 - \bar{r}_2)^2\),
- \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\).
- Expected value of \(w_1^2(r_1-\bar{r}_1)^2\) is \(w_1^2 \text{var}(r_1)\).
- Expected value of \(w_2^2(r_2 - \bar{r}_2)^2\) is \(w_2^2 \text{var}(r_2)\).
- Expected value of \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\) is \(2w_1w_2\text{cov}(r_1,r_2)\).
- Portfolio variance is
\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]
Independent assets
- Suppose we start with $100
- There are two independent assets, each with possible returns of \(\pm\) 10%
- Put all $100 in a single asset \(\leftrightarrow\) flip once for $10
- Put $50 in each asset \(\leftrightarrow\) flip twice for $5
- std dev \(r_1 = 0.1\) \(\Rightarrow\) var \(r_1 = 0.01\)
- std dev \(r_2 = 0.1\) \(\Rightarrow\) var \(r_2 = 0.01\)
- assets are independent, so cov \(= 0\)
- put $50 in each asset \(\Rightarrow\) weights = 1/2
- portfolio variance is
\[\left(\frac{1}{2}\right)^2 (0.01) + \left(\frac{1}{2}\right)^2 (0.01) = 0.005\]
- portfolio std dev is \(\sqrt{0.005}= 0.0707\)
Correlation and covariance
- \(\text{cov} = \text{corr} \times \text{std}(r_1) \times \text{std}(r_2)\), so portfolio var is
\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2)\]
plus
\[2 w_1 w_2 \times \text{corr}(r_1, r_2) \times\text{std}(r_1)\times\text{std}(r_2)\]
Portfolio variance is lower when correlation is lower.
Correlation, covariance, and beta
The slope (beta) of the regression line of \(r_2\) on \(r_1\) is
\[\frac{\text{cov}(r_1, r_2)}{\text{var}(r_1)} = \text{corr}(r_1, r_2) \times \frac{\text{std}(r_2)}{\text{std}(r_1)}\]
Monthly returns of CVX and AAPL 2017–2021
std dev of CVX is 8.18%
std dev of AAPL is 8.50%
corr is 0.153
cov is 10.7%^2
beta is 0.16Monthly returns of CVX and XOM 2017–2021
std dev of CVX is 8.18%
std dev of XOM is 8.58%
corr is 0.889
cov is 62.4%^2
beta is 0.93CVX, AAPL, and XOM portfolios
- The volatilities (std devs) of AAPL and XOM were very similar over the 2017-2021 period.
- If you combined either with CVX, did you have similar volatilities over the period?
- AAPL is a much better diversification play than XOM for someone holding CVX.
Some portfolios
cvx = 0.0818 # std devs
aapl = 0.0850
xom = 0.0858
cvx_aapl = 0.153 # correlations
cvx_xom = 0.889
w1 = 0.5 # portfolio weights
w2 = 0.5
# 50% CVX and 50% AAPL
w1**2 * cvx**2 + w2**2 * aapl**2 + 2*w1*w2*cvx_aapl*cvx*aapl
# 50% CVX and 50% XOM
w1**2 * cvx**2 + w2**2 * xom**2 + 2*w1*w2*cvx_xom*cvx*xomResults
- std dev of CVX with XOM is 8.14%
- std dev of CVX with AAPL is 6.33%
- compared to
- std dev of CVX is 8.18%
- std dev of XOM is 8.58%
- std dev of AAPL is 8.50%