VaR Mapping: Forward Contracts (FRM Part 2)
1. Context
In this swatch we look at the topic of VaR mapping, as applied to a currency forward, which we map on to exposures on spot exchange rate, domestic currency bond and foreign currency bond. The details of the related reading are given below:
| Area | Market Risk Measurement and Management |
| Reading | VaR Mapping |
| Reference | Philippe Jorion, Ch11. VaR Mapping In Value-at-Risk: The New Benchmark for Managing Financial Risk, New York, McGraw-Hill, 2007. |
2. Symbols
We will use the following symbols in the sections below:
| $S_t$ | Current / Spot price of underlying at time $t$ |
| $F_{t,T}$ | Forward price at time $t$ for expiry $T$ |
| $T-t$ | Residual time to maturity of forward |
| $r$ | Continuously compounded risk free interest rate, assumed to be a variable that changes as per market forces |
| $q$ | Continuously compounded rate of income from underlying asset (dividend yield or risk free rate in foreign currency), assumed to be a variable |
| $r_{FC}$ | Continuously compounded risk free rate in foreign currency, income $q$ applicable to a currency forward, assumed to be a variable |
| $V$ | The current value or accumulated profit (or loss)) of a trade since inception. |
| $P_{t,T}$ | Price of ZCB of expiry $T$ at time $t$ |
3. VaR Mapping: The Principle
VaR mapping refers to the exercise of decomposing or mapping a trade onto primitive risk factors for easier and more tenable risk analysis, calculation and aggregation. The final output of the mapping procedure is a set of exposures (exposure refers to dollar amount of positions) on the chosen risk factors. The mapping exercise should ensure that both the value and risk of the original trade is preserved before and after the mapping. In other words, the value and risk of a trade / position should (approximately, if not exactly) match the aggregate value and risk of all positions in primitive securities (risk factors). This notion will become clearer when we apply it to forward contracts below.
4. Forward: Pricing and Current Value
Pricing and valuation of a forward contract was covered in detail in FRM Part 1 curriculum. As a recap, the fair exercise price of a forward with expiry $T$ at initiation ($t=0$) is given by: $$F_{0,T}=S_t e^{(r-q)(T-t)}$$ where, $r$ is the risk free rate and $q$ is the continuous income: dividend yield in case of equity (index) forwards or foreign currency risk free rate in case of currency forwards. For a party with a long position in this forward, the value of the trade at time $t$ will be (again, covered in Part 1): $$\begin{align} V_{t,T}&=\left (F_{t,T}-F_{0,T}\right)e^{-r(T-t)} \\ &=S_{t}e^{-q(T-t)}-F_{0,T}e^{-r(T-t)} \end{align}$$ The above relation satisfies our consistency check that value of a forward starts at zero value to both long and short counterparties (try putting $t=0$ and check. You might need to use the relation of fair exercise price here).
5. Zero Coupon Bond: A Useful Building Block
The price of a zero coupon bond (ZCB) at time $t$ and expiry $T$ is given by: $$P_{t,T}=e^{-r(T-t)}$$ which also demonstrates the inverse relationship between prices and yields that we know of. The risk in investing in this ZCB is that it’s price fluctuates as interest rates (or yields) fluctuate. With a fluction of $dr$, the price impact on bond price $dP_{t,T}$ will be: $$\begin{align} dP_{t,T}&=-(T-t)e^{-r(T-t)}dr \\ \Rightarrow \left (\frac{dP_{t,T}}{P_{t,T}}\right)&=-(T-t)dr \end{align} $$ We will use the above relation to transform the risk of any security to risk of fluctuations in interest rates ($dr$) into risk of fluctuations in zero coupon bond prices $dP_{t,T}$. This tranformation will help us as variable $P_{t,T}$ represents the price of a tradeable security. Owing to the minus sign above, just remember when you’re performing this transformation, the sign of final exposure on $P_{t,T}$ will have an opposite sign (and a different magnitude) than the exposure on $r$.
6. Getting It Right Intuitively
Before we dig into any math, it is useful and intuitive to get the mapping instruments and direction of exposures right and then, leave it to the math to get us the exposures. To do so, we start with the relation: $$V_{t,T}=S_{t}e^{-q(T-t)}-F_{0,T}e^{-r(T-t)}$$ On a day to day basis, our accumulated profit (loss) is sensitive to fluctuations in $S_t$, $r$ and $q$. The last two may fluctuate a lot less compared to $S_t$, but they still do. We can quickly lay down the directions:
| Variable | Impact | Position |
|---|---|---|
| Spot($S_t$) | $S_t \uparrow \Rightarrow V_t \uparrow$ | Long |
| Rate($r$) | $r \uparrow \Rightarrow V_t \uparrow$ | Long |
| Income($q$) | $q \uparrow \Rightarrow V_t \downarrow$ | Short |
We now convert the above positions from rates and yields to prices of tradeable securities (zero coupon bonds). This is doable for a currency forward as we have zero coupon bonds available (hopefully for same maturity as the forward) for both domestic currency ($P_{t,T}$) and foreign currency ($P^{FC}_{t,T}$). For equity forwards, interest rate risk can be expressed in terms of zero coupon bonds as above, but one may not be able to transform dividend yield risk into an analogous liquid tradeable security. Continuing with example of currency forward, and flipping the directions (and signs) for positions on rates, we have:
| Variable | Impact | Position |
|---|---|---|
| Spot($S_t$) | $S_t \uparrow \Rightarrow V_t \uparrow$ | Long |
| Bonds($P_{t,T}$) | $r \uparrow \Rightarrow P_{t,T} \downarrow \\ \Rightarrow V_t \uparrow$ | Short |
| Bonds($P^{FC}_{t,T}$) | $r_{FC} \uparrow \Rightarrow P^{FC}_{t,T} \downarrow \\ \Rightarrow V_t \downarrow$ | Long |
Hence, a currency forward can be mapped on to positions in spot currency position (long), bonds in domestic currency (short) and bonds in foreign currency (long) . We will validate these results and get the actual exposure amounts using some math below. An intuitive feel of the positions (instruments and direction) might suffice as far as the exam is concerned.
7. Getting It Right Mathematically
Again, we start with relation for current value of currency forward: $$V_{t,T}=S_{t}e^{-r_{FC}(T-t)}-F_{0,T}e^{-r(T-t)}$$ Using a bit of calculus, we can write the change in value of a currency forward (when any or all of $S_t$, $r$ and $r_{FC}$ change, as: $$\begin{align} dV_{t,T}&=\frac {\partial V_{t,T}}{\partial S_{t}} \: dS_{t}+ \frac{\partial V_{t,T}}{\partial r_{FC}}\: dr_{FC}+\frac {\partial V_{t,T}}{\partial r}\: dr \\ &=\left( e^{-r_{FC}(T-t)} \right) dS_{t}-S_{t}(T-t)e^{-r_{FC}(T-t)} dr_{FC}+F_{0,T}(T-t)e^{-r(T-t)}dr\\ \end{align} $$ Now, transforming risk on $dr$ and $dr_{FC}$ into $dP_{t,T}$ and $dP^{FC}_{t,T}$ respectively, we have: $$dV_{t,T}=\left [e^{-r_{FC}(T-t)}S_t \right] \frac {dS_t}{S_t}+ \left[S_t e^{r_{FC}(T-t)}\right] \frac {dP^{FC}_{t,T}}{P_{t,T}^{FC}}-\left [F_{0,T}e^{-r(T-t)}\right] \frac {dP_{t,T}}{P_{t,T}}$$ From the above treatment, we obtain the terms in the square brackets which denote in the dollar exposures in respective securities. Again, we get the same conclusion that a currency forward is mapped on to a long position in spot currency (note the + sign of exposure), short position in local currency bond (note the – sign of exposure) and long position in foreign currency bond (note the + sign of exposure).