Parametric VaR - Multi Exposure Dual Currency

# Parametric Value at Risk (VaR) - Multi Exposure Dual Currency

### Introduction

We looked at the most basic way to calculate value at risk for a dual asset holding of a single currency and introduced correlation into the equation. Yet, so far we only look at assets denoted in a single currency, namely USD. We are going to extend the exercise by having an invesment in a MYR stock such as Maybank and a USD stock such as Apple funded by a borrowing in US dollar.

### The Investment

There are now three elements to consider:

1. MYR denominated asset - Maybank stock
2. USD denominated asset - Apple stock
3. USD denominated borrowing
To simplify our illustration, the following assumptions are made:
1. MYR 500,000 is invested in Maybank at the last traded price of 7.77 which is equivalent to 64,350 shares
2. USD 120,000 was borrowed for six month
3. Investment is made in Apple shares at at the last traded price of 351.59 which is equivalent to 341 shares

### The Data Requirement

The main requirements to calculate value at risk are the exposure and the historical data. We have made assumptions on the exposure and historical closing prices for the following are also required:

1. Maybank share
2. 6-month zero rate
3. Apple share

Table 1: Closing Prices

The sample data are presented in Table 1.

### Returns, Volatilities and Correlation

In our previous articles on value at risk, we calculated volatilities based on returns on the underlying prices. It was a simple process and standardised since the underlying prices are in dollar and cents i.e. in monetary terms. In other words we calculated price volatilities. Our current example however includes additional product i.e. interest rate which prices are already denoted as returns. Calculation of returns for interest rate products, therefore requires different treatment or methods which can be done in two ways decribed below:

1. If we denote $$r_t$$ as the closing zero rate and $$r_{t-1}$$ as the previous closing rate, then $$r_t - r_{t-1}$$ is the return of a one day period. Volatility calculated from these numbers is the interest rate volatility which need to be converted to price volatility using modified duration for the tenor. Alternatively a simpler way is to use PVBP01 to convert rate volatility into price volatility:

2. Alternatively, we can calculate the price returns and volatilities by converting the zero rates to discount factors. If we denote $$df_t$$ as the closing discount factor and $$df_{t-1}$$ as the previous closing factor, then the return is calculated in the same manner as other product i.e $$ln\left(\frac{df_t}{df_{t-1}}\right)$$ where $$df = e^{-rt}$$ and $$t$$ is fraction of a year which is 0.5 in our case. The volatility is then calculated from these numbers.

Various volatilities have been calculated and the result displayed below Table 1. The rate volatility is calculated as discussed in (i) and price volatility calculated as per (ii). Conversion of rate to price volatility requires the PVBP01 mentioned earlier.

\begin{align} \text{PVBP01} &= e^{-(r+0.0001) * t} - e^{-r * t}\\ &=e^{-(0.0018+0.0001) * 0.5} - e^{-0.0018 * 0.5}\\ &=-0.000049953771389\\ \end{align}

The pvbp01 is -0.00004995 and the rate voaltility is 0.0344% or 3.44 basis point. The price volatility therefore is -0.000171828 or -0.0171828% which is not a coincidence in terms of absolute number. The different in the signage of the number is offset by the signage of the correlation number in Table 2.

Table 2: Correlations

Table two is the correlation of the products prices - the 6M rate, Apple stock and Maybank stock. There are two buttons below the table - 'Method 1' and 'Method 2' which will show the correlaton base on (i) and (ii), respectively. Regardless of the method chosen the absolute number is the same. The signage of the correlation of 6M zero rate and the two other stocks are also inverted i.e. (i) has negative signage whereas (ii) has positive signage.

### Mapping The Exposures

We are yet able to calculate the diversified value at risk. We may be able to calculate the individual exposure's undiversified value at risk but we cannot sum them up. Why?

If we calculate the undiversified value at risk for AAPL for instance, the value is denoted in USD, where as if or undiversified value at risk for MMBM is in MYR. If we do need to sum them up, we need to convert one value at risk the other currency eg, USD to MYR or MYR to USD, such that all undiversified value are in single denomination.

For diversifed value at risk, the conversion is done on the exposure itself. Assuming our base currency for value at risk is MYR, our USD borrowing and AAPL stock holding have to be converted to MYR equivalent at the prevailing USD/MYR exchange rate. Let's recall the formula for portfolio value at risk:

\begin{align} \text{VaR}_{portfolio} &= \sqrt{WRW^T} \\ \text{where } W &= \left[\begin{matrix} w_1 & ... & w_n \end{matrix}\right]\\ w_n &= \text{asset}_n \text{ value }\times z \times \sigma_{asset_n}\\ z&= \text{z score for the confidence interval}\\ R&= \text{correlation matrix } \\ &=\left[\begin{matrix} \rho_{1,1} & ... & \rho_{1,n} \\ \vdots & ... & \vdots \\ \rho_{m,1} & ... & \rho_{m,n} \end{matrix} \right]\\ \rho_{m,n} &= \text{ correlation between asset } m \text{ and asset }n \text{, and } m = n\\ W^T &= \text{transpose of matrix } W \end{align}

For a 95% confidence interval, $$w_n$$ is now $$\text{asset}_n \text{ value }\times z \times \sigma_{asset_n}\times \text{USD/MYR fx rate}$$. Let's go through the individual exposure and calculate $$w$$ assuming 1 USD = MYR 4.2000.

##### 6M Borrowing

For interest rate exposure, mapping the cash flow means mapping the present value of the cash flows. For simplicity purposes, we assumed that borrowing was done at the last price of the 6M zero rate, which means the present value of the maturity amount is the borrowed amount. Using (i), for the volatility:

\begin{align} w_1 &= \text{ present value of maturity amount }\times z \times \sigma \times \text{pvbp01}\times \text{ fx rate}\\ &= -120,000 \times 1.645 \times 3.44bps \times -0.00004995 \times 4.200\\ &= 142.46 \end{align}

We denote the maturity amount as negative to indicate a cash outflow for repayment of the maturity amount

##### AAPL Stock

For stock exposure in foreign currency, we just need to modify the original formula by multiplying the old one defined above with the exchange rate:

\begin{align} w_2 &= \text{ AAPL closing price } \times \text{ AAPL holdings}\times z \times \sigma \times \text{ fx rate}\\ &= 351.59 \times 341 \times 1.645 \times 0.025815 \times 4.200\\ &= 21,383.47 \end{align}

##### MBBM Stock

Exposure in shares in the base currency does not require formula adjustment. Hence,

\begin{align} w_3 &= \text{ MBBM closing price } \times \text{ MBBM holdings}\times z \times \sigma \\ &= 7.77 \times 64,350 \times 1.645 \times 0.012919 \\ &= 10,625.87 \end{align}

### The VaR

We can now apply the formula to calulate value at risk for the portfolio.

\begin{align} \text{VaR}_{portfolio} &= \sqrt{ \left[ \begin{matrix} w_1 & w_2 & w_3 \end{matrix} \right] \left[ \begin{matrix} \rho_{1,1} & \rho_{1,2} & \rho_{1,3}\\ \rho_{2,1} & \rho_{2,2} & \rho_{2,3}\\ \rho_{3,1} & \rho_{3,2} & \rho_{3,3} \end{matrix} \right] \left[ \begin{matrix} w_1\\ w_2 \\ w_3 \end{matrix} \right] } \\ &=\sqrt{ \left[ \begin{matrix} 142.46 & 21,383.47 & 10,625.87 \end{matrix} \right] \left[ \begin{matrix} 1.0000 & 0.2098 & 0.0739\\ 0.2098 & 1.0000 & 0.1736\\ 0.0739 & 0.1736 & 1.0000 \end{matrix} \right] \left[ \begin{matrix} 142.46\\ 21,383.47 \\ 10,625.87 \end{matrix} \right] } \\ &=25,506.36 \end{align}

Recall that we also calculate price volatility and correlation based on price volatility. We only need to recalculate $$w_1$$ to recalculate the VaR and use the correlation from Table 2.

\begin{align} w_1 &= \text{ present value of maturity amount }\times z \times \sigma \times \text{ fx rate}\\ &= -120,000 \times 1.645 \times 0.000172 \times 4.200\\ &=-142.60 \end{align}

We obtained -142.60, a slightly different absolute figure due to rounding. The portfolio VaR can then be recalculated.

\begin{align} \text{VaR}_{portfolio} &= \sqrt{ \left[ \begin{matrix} -142.60 & 21,383.47 & 10,625.87 \end{matrix} \right] \left[ \begin{matrix} 1.0000 & -0.2098 & -0.0739\\ -0.2098 & 1.0000 & 0.1736\\ -0.0739 & 0.1736 & 1.0000 \end{matrix} \right] \left[ \begin{matrix} -142.60\\ 21,383.47 \\ 10,625.87 \end{matrix} \right] } \\ &=25,506.39 \end{align}

A minor difference compared to using converted price volatility from rate volatility. Author preference is using (ii) - calculate the discount factors to find the price returns and volatility due to less data processing requirement. The process of getting zero rates require the discount factors and we simply reduce one step in the calculation , a step in having to calculate the dollar duration or pvbp01 and another step in conversion of rate volatility to price volatility.