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Financial institutions are currently in the process of updating
their derivative pricing systems to support Risk Free Rates (RFRs).
With the end of IBOR fast approaching (end of 2021) and the amount
of work to be done looming over the development teams, some
institutions may wonder if approximating RFR payoffs with IBOR
payoffs is suitable.
We investigate the quantitative impact for a plain vanilla
interest rate swap and European swaption by introducing the three
ways of approximating:
1. LEGACY: Use a RFR projection curve in place
of the IBOR curve in the legacy IBOR payoff formula
2. PROXY: Use a proxy IBOR projection curve
(built to reproduce the value of forward compounded rates) in the
legacy IBOR payoff formula
3. OIS: Use RFR projection curve in legacy OIS
compounding instruments
Summary:
It is found that a proxy curve can work quite well for
approximating the present value of a RFR payoff with a IBOR payoff.
However, the approximations deteriorate significantly for forward
valuations required for PFE and xVA simulations. Due to the fixing
of some of the daily rates contributing to the partially compounded
first coupon after valuation, the RFR instruments could not be
reproduced by using the legacy IBOR forward rate projection.
An alternative approximation of using an OIS payoff produces
better results. However, the OIS legs may incorrectly capture the
RFR compounding conventions (look-back, lock-out, backward shift)
and may not be available in legacy system for pricing products like
notional resetting cross currency swaps, caps and swaptions. In
this case, significant approximation errors exist. Given such a
legacy system, we found that the maximum future exposure difference
from the values with RFR is up to around 200bp by notional in the
PFE context and up to around 50 bp in the xVA context. It is good
practice to keep monitoring the risk measure errors on PFE/xVA
until these banks will fully support the RFR instruments in their
risk system.
Set Up:
We consider sample trades of the following product types;
interest rate swaps and European swaption. All trades are receiver
swaps or receiver swaptions, and the RFRs are daily compounded and
backward-shifted. The European swaptions are physically settled.
All notionals are set to 1000 and the fixed rates are 1%, and the
trade maturities or swap lengths (for swaptions) are 20 years and
coupon frequency is semi-annual. The valuation date is February 10
2021. In our setup we assume that banks cannot not use the OIS leg
in the swaption case.
Quantitative Results (swaps):
The following table presents the difference from true RFR payoff
for three proxy methods. LEGACY method is where the RFR projection
curves is used in place of the IBOR projection curve in the IBOR
payoff. PROXY approach is where we bootstrap a IBOR projection
curve to match with the daily compounded risk-free rates and use
that in the IBOR payoff formulas. The OIS approach is where we use
the RFR projection curve in existing OIS instruments.
The present value (MTM_T0Diff) matches very well for the PROXY,
and less so for the other two methods. This is attributed to the
backward shift methodology not being reproduced in those two
methods.
The expected positive exposure (EPE) and tails of the
distribution at future exposure dates show significantly larger
error. We find up to 200 bps error at the worst future exposure
date. This quantitative impact would depend on the calibrated model
volatility for the yield curve. Our model uses about 20% model
volatility and a shifted log-normal model with 100bp shift.
Further insight into the reason for the larger differences at
future dates can be seen by plotting the difference in the exposure
profiles. In the following figure we plot the future MTM
differences at multiple percentile points of the exposure
distribution: 1%, 5%, 50%, 95% and 99%, as well as the EPE. The
figure contains 6 sub panels. The 3 top panels show the future MTM
differences in percentile points over time up to the trade maturity
for the LEGACY, PROXY and OIS cases. The 3 bottom panels show the
same values in the first 2 years.
The OIS proxy performs well, with the difference being
attributed to the backward shift convention used for RFR
compounding. For the LIBOR and PROXY curve approach we see a
periodic spike in the error. The periodic frequency corresponds to
the semi-annual coupon frequency. We investigate the causes of this
error by drilling into the valuation at two consecutive exposure
dates 2022/2/15 and 2022/2/25, as highlighted in the red box of the
lower middle panel. Note that this IRS trade has the payment dates
at 2/20 and 8/20 each year prior to the holiday adjustment and
lag.
The below figure plots the coupon rate differences (for a single
Monte Carlo path) between RFR and PROXY. The blue curve is for
February 15 valuation and the orange for Februay 22. Note that a
coupon period pays/starts on February 20, part way between these
two exposure dates.
The first thing to note is that for the forward starting coupons
the projected coupon rates match very well. This is because the
payoff formula for geometric compounding reduces to the IBOR payoff
formula. However, this is not the case for the partially compounded
coupons; the coupon paying February 20 for the blue curve and
August 20 for the orange curve. The second thing to note is the
error is much larger when the exposure date is near the end of the
partially compounded coupon. The exposure date February 15 sits
near the end of the coupon period ending February 20, in fact 118
out of the 120 daily compounded rates are already fixed as of this
exposure dates. These fixed rates are determined by simulation up
to each compounding date. Due to the volatility of rates these can
differ significantly from the proxy term Libor calculated at the
start of the coupon period. In contrast to this, the exposure date
on February 25 sits near the beginning of the coupon period ending
August 20. In this case only 5 out of 118 dates in the period are
fixed. Hence most of the daily rates are projected with the curve
as of the valuation date February 25. The first coupon rate
difference causes the large future MTM error. And the first coupon
rate difference is caused by projection versus simulation of the
RFR rates to their fixing dates.
Using the OIS leg, the fixing of the daily compounded rates is
accurately captured. This fact explains why the maximum MTM
differences for OIS are much smaller than the other two cases, i.e.
LEGACY and PROXY. The relatively small difference with the OIS leg
is caused by the RFR averaging type difference.
Quantitative Results (swaptions):
We repeat the same analysis for swaption trades, i.e. European
swaptions with 10 years option expiry by 20 years underlying swap.
However, we don't include the OIS proxy as it is assumed that OIS
swaptions are not available in the legacy library.
Similar results are found as in the swap case. Present value is
well approximated by the proxy curve, however forward valuations
show significant errors.
The below figure plots the difference in the exposure profile
for the swaption. Before exercise the proxies behave quite well,
but after exercise we observe the same periodic spike in error.
This error is attributed to the error in the partially compounded
coupon rates of the swap that the swaption has exercised into.
Recap:
We have seen that it is successful to introduce the proxy RFR
curve for computing the present value for swaps and swaptions.
However, the proxy behaves poorly when used for future exposure
calculations. Banks with the OIS swap legs can approximate PFE with
the RFR swaps very well. However, if the OIS leg is not available
for swaptions they have to use the legacy IBOR based swaptions
instead, and in this case significant errors are found. Those banks
should pay attention on the large PFE errors (200 bps in our
example). In the xVA context, the calculation is based on the EPE
profile. The EPE differences from RFR with the legacy leg are
around 50 bp and 40 bp, respectively, in both swaps and swaptions.
It is good practice to keep monitoring the pricing errors on
PFE/xVA until these banks will fully support the RFR instruments in
their risk system.
