6: FX Swaps to FX Implied to Par CCS

Mini Chapter Six

FX Swaps to FX Implied to Par CCS

Connecting the Dots. We have discussed both FX swaps and cross-currency swaps and now understand that both products are a reflection of the interest rate differential between the two currencies, with just different cash-flow schedules. I’ll share an illustration below that takes market-traded USDINR FX points, to first calculate the implied zero INR rate (or what’s popularly known as the FX implied INR yield) and use those zero rates to calculate the par CCS yields via bootstrapping.

INR FX implied yield

INR FX implied yield is effectively the zero rate implied by USDINR FX points and the USD zero rate. In the Quantitative easing era zero rates for several years were literally close to zero, such that FX implied yield was almost interchangeably used with the cost of generating local currency funding; dollar liquidity glut made USD funding available at almost 0 spread over risk free rate. But as USD rates went up and steep spreads on USD risk free rates were baked into funding costs the investment carry dynamics completely changed (more on this in the financing section).

• Let’s apply the interest rate parity cash flows to an FX swap transaction (refer to the Cash FX section in Time Value of money) to obtain the INR Zero rate

• USDINR Spot FX rate = 76.515, Outright forward = spot + points

• Initial exchange of USD with INR notional would earn the respective zero rates on those notionals for the tenor of the swap. Therefore for a 3 month tenor,

\[ \{ 1 \times {( 1 + 1.35\%)}^{0.25} \} \times 77.24 = 76.515 \times {( 1 + y\%)}^{0.25} \]

,where y is the annualised implied INR zero rate.

• Future value of $1 in INR terms =

\[ \{ 1 \times {( 1 + 1.35\%)}^{0.25} \} \times 72.50 = 77.50 \]

• Thus, implied INR zero rate =

\[ [\dfrac{ 77.50}{ 76.515} ]{}^{(\dfrac{1}{ 0.25} )} - 1 = 5.25\% \]

Table 3 – Conversion of FX points to FX implied Zero Rates

Source: Pandemonium.

Par CCS yields (semi-annual in this case as per convention) - would follow bootstrapping on zero INR yields as below:

Table 4 – Bootstrapping Zero rates to Par rates

*Zero rates and CCS yields have been rounded up to the second decimal

Source: Pandemonium.

• Recall that cash-flows are discounted using zero rates, so let’s calculate discount factors for every tenor as:

\[ \dfrac{1}{ {( 1 + R_{z,t} )}^t } \]

,where R_z,t is the zero rate for tenor t.

So,

\[ 3m \text{discount factor} = \dfrac{1}{ {( 1 + 5.25\%)}^{0.25} } = 0.9873 \]

• Since coupon exchange on a par INR ND CCS swap is semi-annual, there are no interim cash-flows for less than 1 year tenors. But that still doesn’t mean that up to 6 month tenors on par CCS yields are the same as FX implied yields because – the former is a semi-annual coupon (C) that’s discounted by an annually compounded zero rate scaled by the time period of the swap. For the sake of simplicity, consider this notation for a 6 month par CCS yield assuming $1 as the par value today,

\[ 1 = \dfrac{ ( 1 + \dfrac{C}{2} )}{ {( 1 + R_{z,6m} )}^{0.5} } \]

which clearly shows:

\[ C_t \ne R_{z,t}, \]

• Assuming a present value of CCS cash-flows for a 9m period at par, the CCS semi yield of C being the coupon, the first 3 months would be a short stub that would have a 3 month cash flow followed by a regular 6 monthly payment thereafter:

\[ 1 = ( \dfrac{ C }{ 4 } ) \times 0.9873 + ( \dfrac{ C }{2} ) \times 0.9557 + 1 \times 0.9557 \]

The above solves for C=6.11%

Connecting the Dots. Let’s crack down on some frequently used interest rates lingo to define the frequency of their payments/compounding so all terms used in this section are absolutely clear:

• Interest rate paid semi-annually – Is a per annum (annual) rate payable every six months. As an example from Table 4, the 2 year par CCS yield of 7% is an annual rate of 7% per annum paid as 3.5% every 6 months.

• Annualised rate accommodates compounding as per its payment frequency assuming reinvestment at the same rate. Taking the same example, the 2y par CCS yield paid semi-annually is equal to an ‘annualised’ yield of 7.1225% compounded every 6 months i.e. a dollar invested at 7% per annum paid semi-annually would earn an annualised rate of 7.1225% if compounded semi-annually.

MTM risk of a basis swap

Given that this is an exchange of notional in two different currencies on floating rates there is negligible rates duration in a basis swap. Risk on it largely comes from the FX movement in course of the tenor of the swap and only pending final exchange of notional.

Trivia. Pre-CSA basis swaps in Japan would reset the notional on every coupon date. Basis swaps – since floating – are less sensitive to movements in rates due to short duration tenors of the floating legs. However mark-to-market is sensitive to FX rates as the trade’s notional exposure to the FX rate. This also implies a higher counterparty credit exposure. The JPY-USD basis market tackled this issue by resetting the foreign (JPY) notional on each coupon reset date. For example if a basis swap was entered for 5 years at a spot rate of 100 JPY per USD and the FX rate at the end of the 1st floating rate period moved to 120, the difference of 20 JPY would be additionally lent by the JPY lender to the JPY borrower thus changing the JPY notional lent/ borrowed on the swap for the next period. Notice then, that not only are the respective currency coupons floating, the foreign/local currency notional are also floating. This matching of JPY lent or borrowed was to be done every interest reset period to exchange the mark-to-market on FX movements, which is exactly what happens under a CSA (albeit daily) thus reducing the credit exposure on these swaps. These were known as basis swaps with resetting notional.

Table 5 – Comparison between Long Term FX Swap and CCS

Source: Pandemonium.

Fixed-Fixed Cross Currency Swap

For the sake of completion of all payment frequency types of CCS consider an exchange of different currency notionals at fixed coupons. Salient Features below:

• Sensitivity to interest rates and full duration risk in both currencies
• FX risk is the same as any cross currency swap
• As a dealer hedges this instrument the risk can be broken down to a fixed/float cross currency swap and a fixed interest rate swap against the floating benchmark on the CCS. For eg. a USDKRW fixed/fixed cross currency swap is composed of a USD SOFR float/KRW fixed rate cross currency swap and a USD SOFR fixed interest rate swap

I do hope I’ve been able to keep you engaged and intrigued so far by following a building blocks approach to learning. Extending it further let’s combine ALL that we have discussed so far to build a bigger block of Cash and Financing products.

Look forward to meeting you there.