Currency Swaps: Valuation Approaches (FRM Part 1)
1. Context
Here, we look at currency swaps – their mechanics and valuation approaches both as positions in bonds and using currency forwards. We also establish the equivalence of the two approaches. The details of the reading in which this topic appears are given below:
| Area | Financial Markets and Products |
| Reading | Swaps |
| Reference | Hull, John, Ch10. Mechanics of Options Markets. In Options, Futures, and Other Derivatives, 9th Edition, Pearson, New York, 2014. |
2. Symbols
We use the following symbols in the sections that follow:
| $NA$ | Notional Amount |
| $\tau$ | Settlement period, swap pays every $\tau$ years |
| $r_{fixed,1}$ | Fixed rate in currency 1 |
| $r_{fixed,2}$ | Fixed rate in currency 2 |
| $X_0$ | Current exchange rate, in currency 1 per unit of currency 2 |
| $X(t_i)$ | Future exchange rate, on date $t_i$, unknown as of today |
| $X_F(t_i)$ | Forward exchange rate, for $t_i$ expiry |
| $CF_{1,i}$ | Cash flow in currency 1 (interest and/or principal), for future settlement date $t_i$ |
| $CF_{2,i}$ | Cash flow in currency 2 (interest and/or principal), for future settlement date $t_i$ |
| $DF_{1,i}$ | Discount factor for currency 1, for future date $t_i$, arrived at using zero rate $r_{1,i}$ |
| $DF_{2,i}$ | Discount factor for currency 2, for future date $t_i$, arrived at using zero rate $r_{2,i}$ |
| $V_1$ | Current price of a bond that pays a currency 1 coupon of $r_{fixed,1}$ every $\tau$ years |
| $V_2$ | Current price of a bond that pays a currency 2 coupon of $r_{fixed,2}$ every $\tau$ years |
3. Currency Swaps: Mechanics
A cross currency swap or simply currency swap exchanges interest payments in different currencies, coupled with principal exchanges both at initiation and expiry. The principals in the two currencies are relate to each other via the current (spot) exchange rate $?_0$ $(??_1 = ??_2\cdot ?_0)$. Both currency legs pay fixed rates ($?_{?????,1}$ and $?_{?????,2}$) every $\tau$ yrs.
4. Valuing as Bond Positions
We will find the value of the currency swap from perspective of party that receives currency 1 flows and pays currency 2 flows. The value we arrive at will be in currency 1. If one were to aggregate cash flows in currency 1, and then separately aggregate cash flows that are in currency 2, we arrive at positions in two bonds (in currency 1 and currency 2). If the current values of the two bonds are $V_1$ and $V_2$ respectively, the value of swap will then be: $$V_{swap}=V_1 – X_0 \cdot V_2$$ Essentially, in this approach, we have first aggregated cash flows (with proper discounting) separately in the two currencies to arrive at the respective bond prices and then performed a single currency translation of one bond price (one in currency 2) into our valuation currency (currency 1) using current exchange rate $X_0$.
5. Valuing using Currency Forwards
The second approach to value currency swaps is to aggregate cash flows (not separately in the two currencies as we did above), but on each settlement date. Since the cash flows are in different currencies and hence cannot be simply aggregated, we make use of the relevant exchange rate on every settlement date ($X(t_i)$) to arrive at the net cash flow in our valuation currency (currency 1). Since this exchange rate is unknown as of today, we use the rates implied from currency forwards i.e. $X_F(t_i)$. The value of swap is given by: $$V_{swap}= \sum _{i=1}^n DF_{1,i} \left [CF_{1,i}- X_F(t_i)\cdot CF_{2,i}\right]$$ So, in this step, we have performed a currency translation at every settlement date using the forward exchange rate to get net cash flows in currency 1 and then performed the discounting using currency 1 discount curve.
6. Equivalence of Approaches
We can convince ourselves that the two approaches above give the same swap valuation by the simple arguments below. As a building block, we would need the following result from our study of currency forwards: $$\begin{align} X_F(t_i)&=X_0 e^{\left(r_{1,i}-r_{2,i}\right) \left( t_i –t_0\right)} \\ &=X_0\:\frac{e^{-r_{2,i}(t_i – t_0)}}{e^{-r_{1,i}(t_i – t_0)}} \\ &=X_0 \: \frac{DF_{2,i}}{DF_{1,i}} \\ \end{align}$$ We can now proceed as follows. First, we express $V_1$ and $V_2$ in terms of discounted values of coupons and principals. Then, we aggregate the cash flows together (date-by-date) and finally, evoke the definition of the currency forward exchange rate above. $$\begin{align} V_{swap} &= V_1 – X_0 \cdot V_2 \\ &=\left [\sum\limits_{i=1}^{n}\left ( r _{fixed, 1}\cdot \tau\cdot NA_1 \right)\cdot DF_{1,i}+NA _1 \cdot DF_{1,n}\right] \\ &\:\:-X_0 \left [\sum\limits_{i=1}^{n}\left ( r _{fixed, 2}\cdot \tau\cdot NA_2 \right)\cdot DF_{2,i}+NA _2 \cdot DF_{2,n}\right] \\ &=\sum_{i=1}^{n}CF_{1,i}\cdot DF_{1,i}-X_0\cdot CF_{2,i}\cdot DF_{2,i}\\ &=\sum_{i=1}^n DF_{1,i} \left [CF_{1,i}-\left (\frac{X_0\cdot DF_{2,i}}{DF_{1,i}}\right) CF_{2,i}\right]\\ &= \sum _{i=1}^n DF_{1,i} \left [CF_{1,i}- X_F(t_i)\cdot CF_{2,i}\right]\\ \end{align}$$ Therefore, starting from the valuation formula of first approach, we have arrived at the formula of second approach and proved that the two approaches are essentially the same.