Question:

Assume a $200 two-asset portfolio with equal positions in both assets ($100 + $100). Asset #1 has volatility of 10%, Asset #2 has volatility of 14%. Their correlation is 20%. Our desired confidence is 99%.

(i) What is portfolio volatility?
(ii) What is portfolio VaR and diversified VaR and what is the difference?
(iii) What are the individual VaRs?
(iv) What is the incremental VaR?
(v) What are the component VaRs and percentage contributions?
(vi) If correlation is perfect (1.0), how will portfolio VaR compare to individual VaRs?

Answer:

Here is a link to the spreadsheet (XLS) with calculations.

(i)
Kudos if you can use matrix math to derive (see XLS). But also,
Variance = ($100^2)(10%^2)+($100^2)(14%^2)+(2)($100)($100)(0.20)(10%)(14%) = 352
Volatility = SQRT(352) = $18.8

Or, you can use percentage weights instead of dollar positions:
Volatility = SQRT[(50%^2)(10%^2)+(50%^2)(14%^2)+(2)(50%)(50%)(0.20)(10%)(14%)] = 9.38%
(9.38%)($200) = $18.8

(ii)
No difference!
With 99%, the normal deviate = NORMSINV(99%) = 2.33
Diversified VaR = (2.33)($18.8) = $43.6

This is a relative VaR. No expected returns are given and expected return is not netted, which would be an absolute VaR.

(iii)
($100)(10%)(2.33) = $23.26, and
($100)(14%)(2.33) = $32.57

(iv)
The incremental VaR can be approximated with: marginal VaR * trade.
Assuming the marginal VaR = 0.159 (see next), and the trade is +$10, then
approximate incremental VaR = (10)(0.159) = $1.59

But please note that using the marginal VaR is only an approximation. The true incremental VaR is given by the difference between Portfolio VaR (before trade) - Portfolio VaR (after trade)

(v)
First, we want the marginal VaR. Note in XLS we can get this two ways.
But, given that the betas are, respectively, 0.727 and 1.273,

Marginal VaR = Portfolio VaR/Portfolio Value * beta. In this case,

Asset #1 Marginal VaR = $43.6/$200 * 0.727 beta = 0.159
Asset #2 Marginal VaR = $43.6/$200 * 1.273 beta = 0.278

Component VaR = Position * Marginal VaR. In this case,

Asset #1 Component VaR = (0.159)($100) = $15.87
Asset #2 Component VaR = (0.278)($100) = $27.77

Percentage contributions are:

Asset #1 Component VaR % = $15.87/$43.6 = 36%
Asset #2 Component VaR = $27.77/43.6 = 64%

(vi)
If correlation = 1.0, sum of individual VaRs will EQUAL portfolio VaR.
But for imperfect correlation (rho < 1), sum of individual VaRs will be GREATER THAN portfolio VaR.
Sum of component VaRs, by definition, will be EQUAL to portfolio VaR.