Suppose the risk measure \(R\) is \(VaR(\alpha)\) for some \(\alpha\). Let \(P_1\) and \(P_2\) be two portfolios whose returns have a joint normal distribution with means \(\mu_1\) and \(\mu_2\), standard deviations \(\sigma_1\) and \(\sigma_2\), and correlation \(\rho\). Suppose the initial investments are \(S_1\) and \(S_2\). Show that \(R(P_1+P_2) \leq R(P_1)+R(P_2)\).
We know for correlated normal random variables \(P_1\) and \(P_2\), their summation is also a normal distribution, which has mean equal to \(P_1\) + \(P_2\), standard deviation equal to \(\sigma=\sqrt{ \sigma_{1}^2 + \sigma_{2}^2 + 2\rho\sigma_1 \sigma_2}\).
We also know the inverse cumulative distribution function of a normal distribution N(\(\mu,\sigma\)) is
\[\begin{align*} F^{-1}(p) &= \mu + \sigma \sqrt{2} erf^{-1}(2p-1), p\in (0,1) \end{align*}\]
So we have \[\begin{align*} R(P_1+P_2) &= VaR(\alpha)\\ &= S_1 + S_2 + \sqrt{ \sigma_{1}^2 + \sigma_{2}^2 + 2\rho\sigma_1 \sigma_2} \sqrt{2} erf^{-1}(2\alpha-1)\\ \end{align*}\]
where \(1-\alpha\) is the confidence level, erf(x) is defined as \(erf(x)=\frac{2}{\sqrt \pi}\int_{0}^{x}e^{-t^2}dt\).
We also have
\[\begin{align*} R(P_1)+R(P_2) &= VaR_1(\alpha)+VaR_2(\alpha)\\ &= [S_1 + \sigma_1 \sqrt{2} erf^{-1}(2\alpha-1)] + [S_2 + \sigma_2 \sqrt{2} erf^{-1}(2\alpha-1)]\\ &= S_1 + S_2 + (\sigma_1 + \sigma_2) \sqrt{2} erf^{-1}(2\alpha-1)\\ &= S_1 + S_2 + \sqrt{ \sigma_{1}^2 + \sigma_{2}^2 + 2\sigma_1 \sigma_2} \sqrt{2} erf^{-1}(2\alpha-1)\\ \end{align*}\]
Because \(\rho \in [0,1]\), comparing the two formula above, we can draw a conculsion \[\begin{align*} R(P_1+P_2) \leq R(P_1)+R(P_2) \end{align*}\]
Reference:
Consider daily stock price data in the file Stock_FX_Bond.csv. Use only the first 500 prices on each stock. The following R code reads the data and extracts the first 500 prices for five stocks. “AC” in the variables’ names means “adjusted closing” price.
dat = read.csv("Rlab9_Stock_FX_Bond.csv" ,header=T)
prices = as.matrix(dat[1:500,c(3,5,7,9,11)])(a) What are the sample mean vector and sample covariance matrix of the 499 returns on these stocks?
r=diff(prices,1)
smv = colMeans(r)
ecm = var(r)
cat("sample mean vector is as follows:")## sample mean vector is as follows:smv## GM_AC F_AC UTX_AC CAT_AC MRK_AC
## 0.0107615230 0.0043486974 -0.0004809619 0.0042284569 0.0033667335cat("sample covariance matrix is as follows:")## sample covariance matrix is as follows:ecm## GM_AC F_AC UTX_AC CAT_AC MRK_AC
## GM_AC 0.063996608 0.013006923 0.008462415 0.014741352 0.012889800
## F_AC 0.013006923 0.005888481 0.002474184 0.004779366 0.004250591
## UTX_AC 0.008462415 0.002474184 0.004432700 0.003946014 0.003240779
## CAT_AC 0.014741352 0.004779366 0.003946014 0.011947747 0.006157623
## MRK_AC 0.012889800 0.004250591 0.003240779 0.006157623 0.009130008(b) How many shares of each stock should one buy to invest $50 million in an equally weighted portfolio? Use the prices at the end of the series, e.g., prices[ 500,].
To invest $50 million in an equally weighted portfolio means to invest $10 million to each stocks. So the respective share number for each stock is as follows:
S=50000000
floor(S/5/prices[500,])## GM_AC F_AC UTX_AC CAT_AC MRK_AC
## 594530 2232142 2923976 1821493 1754385(c) What is the one-day VaR(0.1) for this equally weighted portfolio? Use a parametric VaR assuming normality.
w=matrix(rep(0.2,5))
m=smv %*% w
sigma=sqrt(t(w) %*% ecm %*% w)
# -S*(m+qnorm(0.1) *sigma) # the same result
var1day=-S*qnorm(0.1, mean = m, sd = sigma)
var1day## [1] 6098984(d) What is the five-day VaR(0.1) for this portfolio? Use a parametric VaR assuming normality. You can assume that the daily returns are uncorrelated.
var5day=-S*qnorm(0.1, mean = 5*m, sd = sqrt(5)*sigma)
var5day## [1] 13023473