Foreign Exchange Risk: Simple Formulation (FRM Part 1)

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Foreign Exchange Risk: Simple Formulation (FRM Part 1)

1. Context

Here is a quick and simple formulation for gaining insights into foreign exchange risk and hedging it. The reading discusses hedging using examples of lending and borrowing by a Model Bank operating with a straightforward balance sheet. The formulation below will help understand the reading from a simple math perspective. The details of the reading are given below:

AreaFinancial Markets and Products
ReadingForeign Exchange Risk
Reference Anthony Saunders and Marcia Millon Cornett, Ch13. Foreign Exchange Risk. In Financial Institutions Management: A Risk Management Approach, 8th Edition, McGraw-Hill, New York, 2014.

2. Trade and Terminology

As in the reading, we analyze the foreign exchange risk of a simple borrowing and lending transaction. The bank intends to make money via the interest rate spread between it’s borrowing rate and lending rate. Provided these interest rate differentials persist, it can choose to earn the net spread either in it’s local domestic currency (DC), or foreign currency (FC) or both. Here we analyze the risk of lending in foreign currency, which is potentially a currency that offers a high rate of interest. We play around with three cases below – the base case where it doesn’t perform any hedging, the case where it performs a on-balance sheet hedge and finally, the case where it performs an off-balance sheet hedge (i.e. via transactions like forwards, options). Here is a list of symbols and their meaning:

$NA(DC)$Notional Amount in Domestic Currency
$r_{FC}^{A}$Interest rate offered by Foreign Currency assets or deposits.
$r_{DC}^{L}$Interest to be paid on Domestic Currency liabilities or deposits.
$X_0\left( DC/FC \right)$Current exchange rate between DC and FC in DC per unit of FC.
$X_T\left( DC/FC \right)$Final exchange rate between DC and FC in DC per unit of FC.
$PnL(DC)$Profit or Loss of the trade expressed in DC.
$F(DC/FC)$Forward exchange rate for same expiry as of loan.

3. Base Case: No Hedging

In this case, the bank sources it’s funds in DC, converts them to FC at the current exchange rate ($X_0\left( DC/FC \right)$), lends the proceeds at $r_{FC}^{A}$. The balance sheet of the bank is balanced with both assets and liabilities as $NA(DC)$. Finally, at maturity, the bank converts back the receipts in FC (which have accumulated interest at $r_{FC}^{A}$) to DC to pay back it’s depositors their principal and interest (being charged at $r_{DC}^{L}$). Following the above steps, it’s final PnL in DC will be:

$$\begin{align} & PnL\left( DC \right)=\frac{NA\left( DC \right)}{{{X}_{0}}\left( DC/FC \right)}\left( 1+r_{FC}^{A} \right)\cdot {{X}_{T}}\left( DC/FC \right)-NA\left( DC \right)\cdot \left( 1+r_{DC}^{L} \right) \\ & =\left[ \frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)} \right]\cdot \left( 1+r_{FC}^{A} \right)\cdot NA\left( DC \right)-NA\left( DC \right)\left( 1+r_{DC}^{L} \right) \\ & =NA\left( DC \right)\left[ \frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)}\left( 1+r_{FC}^{A} \right)-\left( 1+r_{DC}^{L} \right) \right] \\ & =NA\left( DC \right)\left[ 1\cdot \left( \frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)}-1 \right)+\left( \frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)}r_{FC}^{A}-r_{DC}^{L} \right) \right] \\ \end{align}$$

The risk in this PnL comes from the variability of $X_T\left( DC/FC \right)$, which as written above affects both the principal component (the first term) and the interest component (the second term). An adverse movement in $X_T\left( DC/FC \right)$ can very easily wipe out any gains from the interest rate differential. Following the defnition in the reading, the net foreign exchange exposure of the bank as of today ($t=0$) is $\frac{NA(DC)}{X_0(DC/FC)}$ and it grows to $\frac{NA(DC)}{X_0(DC/FC)}(1+r^A_{FC})$ by maturity.

4. Case I: On-Balance Sheet Hedging

In this case the bank makes it’s current net foreign currency exposure zero by switching it’s borrowing (funding) from DC to FC (for which, it has to now pay an interest of $r^L_{FC}$). It’s balance sheet is still balanced as the assets and liabilities are still equal (both equal $NA(DC)$ in DC). Following this strategy, it’s final PnL will be given by:

$$\begin{align} & PnL\left( DC \right)=NA\left( DC \right)\frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)}(1+r_{FC}^{A})-\frac{NA\left( DC \right)}{{{X}_{0}}\left( DC/FC \right)}\left( 1+r_{FC}^{A} \right){{X}_{T}}\left( DC/FC \right) \\ & =NA\left( DC \right)\cdot \left( r_{FC}^{A}-r_{FC}^{L} \right)\cdot \frac{{{X}_{T}}\left( DC/FC \right)}{{{X}_{0}}\left( DC/FC \right)} \\ \end{align}$$

This step of switching the funding currency helped in reducing the current net exposure, which in turn reduced the foreign exchange risk. The transaction isn’t totally risk free though, since such on-balance sheet hedging does not make the final (future) foreign currency exposure zero due to difference in the rate at which the FC assets and liabilities grow ($r^A_{FC}$ vs $r^L_{FC}$). The interest rate differential that the bank earns is still subject to foreign currency risk. Since this final exposure is much smaller than before, the on-balance sheet hedge does help.

5. Case II: Off-Balance Sheet Hedging

In this final case, the bank reduces it’s foreign currency exposure to perfect zero via trading in derivatives (e.g. forwards) that are off-balance sheet in nature. Since in this simple case, the bank knows with certainity the timing and amount of foreign currency cash flows it can perfectly lock-in the exchange rate it will receive at maturity via currency forwards. The final PnL is shown below and since it has absolutely no dependence on $X_T(DC/FC)$, the bank has perfectly hedged it’s foreign exchange risk. This scenario rests on the practically implausible assumption that both the timing and amount of foreign currency cash flow is known with certainity. $$\begin{align} & PnL\left( DC \right)=\frac{NA\left( DC \right)\cdot \left( 1+r_{FC}^{A} \right)\cdot F\left( DC/FC \right)}{{{X}_{O}}\left( DC/FC \right)}-NA\left( DC \right)\left( 1+r_{DC}^{L} \right) \\ & =NA\left( DC \right)\left[ \frac{F\left( DC/FC \right)}{{{X}_{O}}\left( DC/FC \right)}\left( 1+r_{FC}^{A} \right)-\left( 1+r_{FC}^{L} \right) \right] \\ \end{align}$$