LIBOR vs. OIS:
The Derivatives Discounting
Dilemma
John Hull
PRMIA
May 2012
Copyright © 2012 John Hull. All Rights Reserved
1
Agenda
OIS and LIBOR
CVA and DVA
The Main Result
Potential Sources of Confusion
FVA and DVA
See John Hull and Alan White: “LIBOR vs OIS: The Derivatives
Discounting Dilemma”, www.rotman.utoronto.ca/~hull
Copyright © 2012 John Hull. All Rights Reserved
2
Risk-neutral Valuation
Project market variables in a risk-neutral
world and discount expected payoff at the
risk-free rate
Risk-free rate defines the expected growth
rates of market variables in a risk-neutral
world and is used for discounting
We focus on the rate that should be used for
discounting
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3
The Risk-Free Rate
Many academics like to assume that the
Treasury rate is the risk-free rate
Pre-crisis practitioners assumed that a “riskfree” zero curve can be calculated from
LIBOR rates, Eurodollar futures, and swap
rates
Post-crisis most banks have started to use
OIS rates for discounting collateralized
transactions and LIBOR/swap rates for
discounting non-collateralized transactions
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Why the Change?
Banks became increasing reluctant to lend to
each other during the crisis.
The TED spread was very high during the
crisis reaching 450 basis points in October
2008
The LIBOR-OIS spread was also very high
during the crisis and reached a record 364
basis points in October 2008
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5
LIBOR Is Not Risk-Free
The crisis emphasizes that the LIBOR/swap
curve is not risk-free
LIBOR rates are the unsecured short-term
borrowing rates of a AA-rated financial
institution
Swap rates are “continually refreshed” shortterm rates. They correspond to the risk in a
series of unsecured short-term loans to AArated financial institutions.
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6
Our Research Conclusions
OIS is a better proxy for the risk-free rate than
LIBOR
It should be used as the discount rate for both
collateralized and non-collateralized portfolios
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7
OIS and the Effective Fed Funds
Rate
The effective fed funds rate is the average of
unsecured overnight borrowing rates
(arranged by brokers using the Fedwire
system) between financial institutions
3 month OIS rate is the rate swapped for the
geometric average of effective fed funds rates
Overnight rates and index swaps are defined
similarly in other countries (eg, EONIA,
SONIA)
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OIS Zero Curve
Can be bootstrapped similarly to LIBOR zero
curve
Maturities of overnight indexed swaps not as
long as LIBOR swaps
Natural approach is to assume that spread
between LIBOR/swap zero rates and OIS
zero rates at the long end is the same as it is
for the longest maturity
OIS rates are very close to risk-free
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9
CVA
For a portfolio of derivatives between dealer
and counterparty, CVA is the cost to the
dealer of a possible default by the
counterparty
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The CVA Calculation
Time 0
t2
t1
t3
t4
………………
tn=T
Default probability
q1
q2
q3
q4
………………
qn
PV of net exposure
v1
v2
v3
v4
………………
vn
n
CVA  (1  R) qi vi
where R is the recovery rate
i 1
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CVA Calculation continued
The default probabilities (i.e., the qi) are
calculated from credit spreads
The PVs of the net exposures (i.e., the vi) is
calculated using Monte Carlo simulation.
Random paths are chosen for all the market
variables underlying the derivatives and the net
exposure is calculated at the mid point of each
time interval. (These are the “default times”)
The vi is the present value of the average net
exposure at the ith default time
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Calculation of Net Exposure
If no collateralization, the net exposure at a default time
is the maximum of the value of the derivatives and zero
If collateral is posted, we assume that a certain number
of days elapse between the counterparty failing to post
collateral and the position being unwound
This is referred to as the “cure period” or “margin period
at risk”
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13
DVA (more controversial than CVA)
DVA is an estimate of the cost to the
counterparty of a default by the dealer
Same formulas apply except that vi is
counterparty’s exposure to dealer, qi is
dealer’s probability of default, etc.
Accounting standards have pushed banks in
the direction of quantifying DVA
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3rd Quarter Increases in Credit
Spreads of US Banks in 2011
Wells Fargo
63 bps
JPMorgan
81 bps
Citigroup
179 bps
Bank of America
266 bps
Morgan Stanley
329 bps
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Use of CVA and DVA in
Valuation
Value of Derivatives Portfolio with Counterparty
equals
No-default Value + DVA − CVA
Seems correct intuitively
Adjustment for double default possibility. See Brigo
and Morini (2011)
Consistent with a modification of Black-ScholesMerton hedging arguments to incorporate credit risk
developed by Burgard and Kjaer (2011)
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Use of CVA and DVA in
Valuation continued
No-default Value + DVA − CVA
This is true for collateralized and noncollateralized portfolios
If we increase the discount rate for noncollateralized portfolios there is a danger that
we double count for credit risk
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Can LIBOR Discounting Work for NonCollateralized Portfolios?
We show that LIBOR discounting gives the correct answer if
CVA is calculated as the excess of the actual expected loss
to the dealer from a counterparty default over the expected
loss if the counterparty’s borrowing rates are given by the
LIBOR/swap curve
DVA is calculated as the excess of the actual expected loss
to the counterparty from dealer defaults over the expected
loss if the dealer’s borrowing rates are given by the
LIBOR/swap curve
(Using the LIBOR/swap rate instead of OIS rate as a benchmark
when calculating credit spreads may give a reasonable
approximation to the correct answer)
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For Non-Collateralized Portfolios, Can We Use the
Discount Rate to Adjust for Credit Risk?
If portfolio will always have a positive value to the
dealer, it can be correctly valued by discounting at
the counterparty’s borrowing cost
If the portfolio will always have a negative value to
the dealer, it can be correctly valued by discounting
at the dealer’s borrowing cost
If the counterparty and dealer are equally
creditworthy, any portfolio can be valued by
discounting at the common borrowing cost of the two
sides
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Using LIBOR /Swap Rates for Discounting NonCollateralized Portfolios Can Cause Confusion
because…
Interest rates are also used to determine expected
returns on assets in a risk-neutral world as well as for
discounting. The interest rate used for the first
purpose should always be the (OIS) risk-free rate
Two different methodologies for calculating CVA and
DVA are necessary
The discount rate for DVA and CVA calculations
should be the OIS rate even if LIBOR has been used
as the discount rate for the main valuation
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20
FVA and DVA
Some banks calculate:
DVA for their borrowing (as well as for their
derivatives)
FVA to reflect that they cannot fund at the risk-free
(OIS) rate
These two should in theory cancel each other
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21
Conclusions
Crisis has taught us the importance of finding
a better proxy for the risk-free rate
OIS rate appears to be the best proxy for the
risk-free rate
The OIS rate should be used as the discount
rate for all derivatives portfolios, not just those
that are collateralized
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Just Out…..
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