Defining Portfolio Loss for VaR with Contributions and Rebalancing

# Expected loss of investment over 30 years


# Expected loss of investment over 30 years












Let $V_{30}$ denote the value of my portfolio after 30 years. Each year, I add 1000\$ and rebalance my portfolio such that invest $c_1$ in a risky asset and $c_2$ in a risk-free bond. Denote the return of the risky asset over the $k$ year as $R_{k+1}$ and that the risk-free interest is $r$. The value process looks like this:

\begin{equation} \begin{split} V_0 &= 0 \\ V_{k+1} &= (V_k + 1000)(c_1R_{k+1} + c_2 e^r) \end{split} \end{equation}

Now I want to calculate the empirical Value at Risk over 30 years. How do I define the "loss" $X$?

In my textbook, over 1 period, we set $X = V_1 - V_0 R_0$ and $L = -X/L_0$. Given a sample of losses $\{L_1, \dots, L_n \}$, the empirical estimate of $\text{VaR}_p(X)$ is given by $\hat{\text{VaR}_p} = L_{[np]+1,n}$, where $L_{1,n} \geq L{2,n} \geq \dots \geq L_{n,n}$ is the ordered sample.

My question is, from a sample $\{V_{30}^1, \dots V_{30}^n \}$, define the loss $X$?

regards

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/77431

The amounts that you add or withdraw are not part of your trading profit and loss, and shouldn't affect the calculation.

Assuming that the cost of rebalancing is immaterial,

If you have some amount $V_l$, and invest $c_{l,1}$ in the risky asset and $c_{l,2}$ in the riskless asset, and the market behaved from time $l$ until time $l+1$ like it did in the historical period from time $k$ until time $k+1$, then the new portfolio value would be $V_{k,l+1}=V_l \left(c_{l,1}\left(1+R_{k+1}\right) + c_{k,2} \left(1+e^{r_k}\right)\right)$, and the profit or loss would be $X_k=V_{k,l+1}-V_l$, you can rank those $X_k$'s, and look for the desired VaR's confidence level.