Detour: correlation

So what drives diversification anyway?

If you wondered why our toy portfolio had lower volatility than any of the individual stocks it was because of correlation. While half the stocks were higher at the end of the period, the other half were lower, which means the correlations of many of the stocks with each other was negative. In addition, among the stocks that moved in the same direction, the amount they moved and the path they took was also sufficiently different to result in lower correlations (an example of the third nuance mentioned above).

Let’s build the intuition on how correlation helps diversification lower portfolio volatility over a range of correlations. We’ll start with a simple two stock portfolio, assuming both stocks have the same risk and return profiles and assuming an equal-weighting of those stocks. Then we’ll simulate the range of portfolio volatilities based on different correlation coefficients.

As the graph shows, when the two stocks are perfectly positively correlated, portfolio volatility equals stock volatility. When they’re perfectly negatively correlated, portfolio volatility is zero.³ It would, of course, be different than zero if the two stocks didn’t have exactly the same volatility and weren’t equal-weighted. We reversed the order of the x-axis to make the decline in volatility easier to read. Another important point is that the decline in volatility isn’t linear. It’s close to linear as correlation declines from 100% to zero, but then accelerates thereafter.

What happens if you add a third stock to the portfolio? The answer is it depends on the correlation of the initial stocks and how the third stock correlates with those stocks. It is difficlut to show this across a broad range of correlations without creating mathematically impossible scenarios. We’ll use three cases in which the stocks in the original portfolio have negative (-50%), zero, or positive correlation (50%). The stock we add will have a 50% correlation with the first stock in the two stock portfolio and a range of possible correlations with the second stock, as permitted by the math.⁴

The horizontal line represents the volatility of the two stock portfolio for its respective correlation. The sloping line is the portfolio volatility based on the correlation of the additonal stock to the second stock in the portfolio. Recall the correlation between the additional stock and the first stock in the portfolio is 50%.

A few takeaways from these graphs. The more positively correlated the stocks, the higher the risk. Adding another stock often lowers volatility, but not in all cases. But that addition also can cause the volatitlies of the portfolios to overlap, suggesting a wider range of risk. Another takeaway is that if you have a negatively correlated portfolio, adding a stock with positive correlation to one of the stocks only results in lower portfolio volatility if it is sufficiently negatively correlated with the other stock, around -40% or less. On the other hand, if the portfolio correlation is zero or positive, even adding a stock with postive correlation can lower portfolio volatility.

Based on this we can make some provisional generalizations. First, if the assets in your portfolio already exhibit negative correlation, adding a postively correlated asset with one of the the other assets will likely increase portfolio volatility unless it is negatively correlated with the other assets in the portfolio to almost the same magnitude as its positive correlation.

The second generalization is if there is little correlation among the assets of the portfolio, then adding another asset that is positively correlated to one of the former assets can lower total portfolio volatility so as long as its correlation to the remaining assets isn’t overly large. In this case, the correlation can’t be greater than 17%.

Third generalization. If the assets in the initial portfolio already exhibit positive correlation, adding another asset positively correlated to portion of the existing assets generally lowers portfolio volatility so long as its correlation with the other assets is not excessive; not greater than 85% in this case.

An easy way to see this in terms of likelihood that portfolio volatility will decline is shown below.

What’s the headline conclusion? When you combine stocks that aren’t perfectly positively correlated the overall volatility of your portfolio goes down. However, when you already own stocks that aren’t perfectly positively correlated and you add a new stock to that portfolio, how much volatility declines (if at all) depends on the magnitude and sign of the new stock’s correlation with the rest of the portfolio.

Why should you care about correlation? When constructing a portfolio you look at the expected risk and return of potential assets. The total risk of the portfolio is, of course, determined by the expected risk of each asset, modified by the correlation of each asset with every other asset, aggregated according to how much exposure you decide to have to each asset. For a three asset portfolio, there are three different correlations you need to know. For a four asset portfolio, there are six correlations, and for five asssets, 10 correlations. The amount of correlations you need to attend to increases significantly as you add more assets.

Understanding how a portfolio’s risk might change becomes relatively complex once you go beyond three assets. Why? Because the pairs of correlations increases. Recall the three stock portfolio. When we think of the interactions of those three stocks we only need to think about three pairs of correlations. That’s relatively easy to conceptualize like imagining which direction three balls might bounce if you dropped them from the second floor window of your house. Whichever direction they bounce in and what that direction is relative to the others is relatively easy to see your mind’s eye? Four balls it becomes more difficult. Four balls over multiple bounces? Maybe Rain Man could do it. Not sure about anyone else.

That returns us to our original question: is an investor better off being diversified if he or she doesn’t know in advance how a stock or group of stocks is likely to perform? The question implies not only not knowing how the stocks will perform, but also how correlated they’ll be with one another. This uncertainty surround correlation adds a further complication. Based on what we’ve shown in the previous post and what we’ve learned in this post about how correlation underpins diversification, our provisional vote in favor of diversification has weakened a bit. In some cases, adding a stock to a portfolio with the hope of lowering volatility coud have the opposite effect. How likely that is we’ll leave to another post where we’ll return to the original portfolio and run simulations with different correlation assumptions.

Until next time, here is the underlying code we used for this post.

# Load package
library(tidyquant)

# Create toy portfolio
set.seed(123)
mu <- seq(-.03/12,.08/12,.001)
sigma <- seq(0.02, 0.065, .005)

mat <- matrix(nrow = 60, ncol = 10)
for(i in 1:ncol(mat)){
  mu_samp <- sample(mu, 1, replace = FALSE)
  sig_samp <- sample(sigma, 1, replace = FALSE)
  mat[,i] <- rnorm(nrow(mat), mu_samp, sig_samp)
}

df <- as.data.frame(mat)
asset_names <- toupper(letters[1:10])
colnames(df) <- asset_names

# Cumulative returns 
df_comp <- rbind(rep(1,10), cumprod(df+1))

# Graph returns
df_comp %>% 
  mutate(date = 0:60) %>%
  gather(key, value, -date) %>%
  ggplot(aes(date, (value-1)*100, color = key)) +
  geom_line() +
  ylab("Return (%)") + xlab("Month") +
  ggtitle("Cumulative returns for random sample of stocks") + 
  theme(legend.position = "none")


# Create volatility date frame
vol <- df %>% summarise_all(., sd) %>% t() %>% as.numeric()
vol <- data.frame(asset = asset_names, vol = vol)


# Portfolio volatility
weights <- rep(0.1, 10)
port_vol <- sqrt(t(weights) %*% cov(df) %*% weights)
# round(port_vol*sqrt(12), 3)*100

# Graph volatility of assets vs portfolio
vol %>% 
  mutate(vol = vol*sqrt(12)*100) %>%
  ggplot(aes(reorder(asset,vol), vol)) + 
  geom_bar(stat = "identity", fill = "royalblue1") + 
  geom_hline(yintercept = port_vol*sqrt(12)*100, color = "red") + 
  labs(y = "Volatility (%)",
       x = "",
       title = "Stock volatility with portfolio volatility line") +
  theme(legend.position = "none") +
  annotate("text", x = "D", y = 10, 
           label = "Portfolio volatility", 
           color = "red")

# create simple return and volatility data
a_ret <- 0.07
b_ret <- 0.07

a_sd <- 0.14
b_sd <- 0.14

# Create std deviation, correlation, and covariance matrices
sds <- c(a_sd, b_sd)
sds_mat <- sds %*% t(sds)

cor_mat <- matrix(c(1,-1,-1,1),2)
cov_mat <- cor_mat*sds_mat

# Assign weights and calculate portfolio volatility
wt <- c(0.5,0.5)
port_sd <- sqrt(t(wt) %*% cov_mat %*% wt)

## Simulate portfolio volatilities across range of correlations

# Create correlation sequence and run for loop
cor_run <- seq(1,-1, -0.1)
port_run <- c()
for(i in 1:length(cor_run)){
  corMat <- matrix(c(1,cor_run[i], cor_run[i], 1), 2)
  covMat <- corMat * sds_mat
  port_run[i] <- sqrt(t(wt) %*% covMat %*%wt)
}

# create data frame
port_ex <- data.frame(corr = cor_run, risk = port_run)

# Graph portfolio volatility
port_ex %>%
  ggplot(aes(corr*100, risk*100)) +
  geom_line(color = "blue") +
  geom_hline(yintercept = a_sd*100, color = "red") +
  labs(x = "Correlation (%)",
       y = "Volatility (%)",
       title = "Portfolio risk relative to correlation") +
  annotate("text", x = -90, y = 13.5, label = "Stock volatility", color = "red" ) +
  scale_x_reverse()

## Add third stock

# Third stock risk/return
c_ret <- 0.07
c_sd <- 0.14

# Three stock variance matrix
sds_1 <- c(a_sd, b_sd, c_sd)
sds_mat_1 <- sds_1 %*% t(sds_1)

# Weighting
wt_1 <- c(1/3, 1/3, 1/3)

# Three scenarios
# Correlation a & b = -0.5, b & c = 0.5, a & c = range
cor_neg <- seq(0.5,-1, -1.5/20)
port_run_neg <- c()
for(i in 1:length(cor_neg)){
  corMat <- matrix(c(1, -0.5, 0.5, 
                     -0.5, 1, cor_neg[i], 
                      0.5, cor_neg[i], 1), 3, byrow = TRUE)
  covMat <- corMat * sds_mat_1
  port_run_neg[i] <- sqrt(t(wt_1) %*% covMat %*%wt_1)
}

# Correlation a & b = -0.5, b & c = 0.5, a & c = range
cor_zero <- seq(.85,-.85, -1.7/20)
port_run_zero <- c()
for(i in 1:length(cor_zero)){
  corMat <- matrix(c(1, 0, 0.5, 
                     0, 1, cor_zero[i], 
                     0.5, cor_zero[i], 1), 3, byrow = TRUE)
  covMat <- corMat * sds_mat_1
  port_run_zero[i] <- sqrt(t(wt_1) %*% covMat %*%wt_1)
}

# Correlation a & b = -0.5, b & c = 0.5, a & c = range
cor_pos <- seq(1,-0.5, -1.5/20)
port_run_pos <- c()
for(i in 1:length(cor_pos)){
  corMat <- matrix(c(1, 0.5, 0.5, 
                     0.5, 1, cor_pos[i], 
                     0.5, cor_pos[i], 1), 3, byrow = TRUE)
  covMat <- corMat * sds_mat_1
  port_run_pos[i] <- sqrt(t(wt_1) %*% covMat %*%wt_1)
}

# Create three stock portfolio correlation and risk
port_three <- data.frame(id = rep(1:3, each = 21), 
                        corr = c(cor_neg, cor_zero, cor_pos), 
                        risk = c(port_run_neg, port_run_zero, port_run_pos))

# Create original two stock correlation and risk ranges
cor <- c(-0.5, 0, 0.5)
port_two <- data.frame(id = c(1:3), corr = cor, risk = rep(NA,3))
for(i in 1:3){
  two_cor <- matrix(c(1,cor[i],cor[i],1),2)
  two_cov <- two_cor * sds_mat
  port_two[i,3] <- sqrt(t(wt) %*% two_cov %*% wt)
}

# Labels for faceting  
labels <- c("1" = "Negative correlation portfolio",
            "2" = "Zero correlation portfolio",
            "3" = "Positive correlaion portfolio")

anno <- data.frame(
  label = rep(c("Two stock \nportfolio"), 3),
  id = c(1, 2, 3),
  x = c(25, -50, 0),
  y = c(7.1, 10.0, 12.2)
)

# Graph three stock portfolios
port_three %>%
  ggplot() +
  geom_line(aes(corr*100, risk*100, color = factor(id))) +
  # geom_point(aes(corr*100, risk*100, color = factor(id)), data = port_two, size = 2) +
  geom_hline(aes(yintercept = port_two$risk[1]*100), 
             data = subset(port_three, id == 1),
             color = "red") +
  geom_hline(aes(yintercept = port_two$risk[2]*100), 
             data = subset(port_three, id == 2),
             color = "orange4") +
  geom_hline(aes(yintercept = port_two$risk[3]*100), 
             data = subset(port_three, id == 3),
             color = "olivedrab3") +
  facet_wrap(~id, labeller = as_labeller(labels)) +
  scale_color_manual("", values = c("blue", "purple", "black")) +
  geom_text(data = anno, 
            aes(x = x, y = y, label = label), color = c("red", "orange4", "olivedrab4")) +
  labs(x = "Correlation (%)",
       y = "Volatiilty (%)",
       title = "How a two stock portfolio's volatility changes by adding a third stock") +
  theme(legend.position = "none") 

# Cutoff points for lower volatility

neg_cut <- port_three %>% 
  filter(id == 1, risk <= port_two[port_two$id == 1, 3]) %>% 
  summarise(corr = round(max(corr),2)*100) %>%
  as.numeric()

zero_cut <- port_three %>% 
  filter(id == 2, risk <= port_two[port_two$id == 2, 3]) %>% 
  summarise(corr = round(max(corr),2)*100) %>%
  as.numeric()

pos_cut <- port_three %>% 
  filter(id == 3, risk <= port_two[port_two$id == 3, 3]) %>% 
  summarise(corr = round(max(corr),2)*100) %>%
  as.numeric()

correlations <- data.frame(port = c("neg_cor", "zero_cor", "pos_cor"),
                           prob = rep(NA,3))
for(i in 1:3){
  correlations[i,2] <- port_three %>%
    filter(id == i) %>%
    summarise(mean(risk < port_two[i,3])) %>%
    as.numeric()
}
correlations %>%
  mutate(port = case_when(
    port == "neg_cor" ~ "Negative correlation",
    port == "zero_cor" ~ "Zero correlation",
    port == "pos_cor" ~ "Positive correlation")) %>%
  ggplot(aes(reorder(port,prob), prob*100)) +
  geom_bar(stat = "identity", fill = "blue") +
  labs(x = "Portfolios",
       y = "Probability (%)",
       title = "Probability adding a third asset will lower volatility")