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Lehman Brothers - The Short-End of the Curve | Swap (Finance) | Central Banks
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Lehman Brothers | The Short-End of the Curve
10 July 2007 1
The Short-End of the Curve
Traditional Swap Curve
construction does not treat the very front of the curve
consistently. The use of Libor deposits prevents the accurate pricing of FRAs and short-
term interest rate swaps, due to a lack of consideration of market expectations of Central
Bank Monetary Policy. Furthermore, risk management based upon deposit rates is not
useful, due to the inability of most trading desks to trade them. This paper provides an
introduction to an approach that uses Overnight Indexed Swaps and Money Market basis
swaps to infer market policy expectations and construct the front of the curve with a one
day resolution. Furthermore, it demonstrates the usefulness of risk managing with
respect to changes in the policy rate.
Funding risk is the fundamental factor when considering position exposures in various
currencies. It is possible to break this down into several primary elements:
Day-to-day cash liquidity issues that cause variance around the central
bank’s policy rate
Cross-currency funding demand (e.g. – the carry trade)
Term liquidity premium
Clearly, the most significant exposure is to changes in the Central Bank Policy rate, and
thus, a good a risk management and pricing system should have some understanding of
Traditionally, the very short-end of the curve has been constructed as a kludge of deposit
rates and interest rate futures, but this introduces consistency problems over the rates
used. For example, including a 1M deposit rate in a 3M Forward curve ignores the lower
credit risk of lending for 1M vs 3M. Secondly, if cash deposits are used, the rates
implied are often quite different to the Libor fixings due to commercial banks
dominating the market, but if the Libor fixings are used, then the forward curve sub 3M
is stationary. This may not seem like too much of a problem initially, but consider a
surprise interest rate move
after the fixing has occurred; clearly the forwards in the
curve will be very bizarre at the very front of the curve as the first future is ~25bp
different to the (overlapping) 3M Libor fix. All of these issues have the effect of mis-
pricing very short-term interest rates, most obviously when comparing market traded
(especially the 1x4 and 2x5 FRAs). A more difficult question is where FRAs
starting the day after a policy meeting should be priced and how to hedge them. The
existing methodology does not provide any information about the 1D forwards priced
into the curve in this respect.
Furthermore, risk management based upon deposit rates does not make sense as very few
market participants may actually trade the deposits themselves (how many Swap traders
have ever traded a deposit?). Moreover, how much “real” risk is involved in a 1M
deposit position with $100k DV01 when there is no Central Bank meeting in this period?
The current VaR is around $200k, but given that the only factors affecting this position
The Swap Curve, Lehman Brothers Fixed Income Research, Fei Zhou, 2002
The author thanks Zhengyun Hu, Francis Butterworth, Alan Hookham and Jean-Baptiste Home for helpful
comments and suggestions. Chris Pulman works on the FX Trading Desk.
Or indeed any move in rates post-fixing
In the event of a 25bp rate move
+44 20 710 32240
chris.pulman@lehman.com
10 July 2007 2
are liquidity based funding issues, surely the risk is far lower? Conversely, what if there
is a policy meeting in this period, surely the risk is much higher?
A further issue is how to represent the risk between different instruments that are very
similar. For example, it is well-known that FX Forwards are deposit-like instruments, but
how should they trade in relation to FRAs, or Overnight Indexed Swaps (OIS)? How
should one account for the term spread of 1D funded assets vs 3M funded assets, or
indeed Cross-Currency funded assets?
Clearly, existing methods of risk-management and pricing are insufficient in both their
consistency and accuracy at the very front of the curve, and a new approach that fully
accounts for the above effects needs to be considered.
2. OVERNIGHT INDEXED SWAPS (OIS)
OIS are a form of (generally) short-term interest rate swap in which the floating leg fixes
to a daily interest rate, making them useful for gaining exposure to monetary policy
changes. The OIS market grew out of the French T4M swap market in the 1990s, and has
gradually expanded into many currencies over the past few years to be extremely liquid
in the majors. These swaps are generally quoted in monthly tenors out to 1Y, or even
beyond in some currencies, as well as weekly out to 1M. Recently, an inter-bank market
has developed for swaps that are dated between Central Bank meetings, although these
are generally only quoted via voice brokers.
An OIS is a swap which has a single payment at the end of the swap (usually on the day
after the maturity of the swap), representing the difference in interest on the two legs of
the swap. The fixed leg can be considered much like a synthetic deposit, and is quoted in
the market as a yield that is applied over the tenure of the swap. The floating leg, on the
other hand, has a daily fix, usually to the weighted-average of overnight cash deposits
traded that day. The interest on the floating leg is then compounded up (apart from Fed
Fund swaps, where it is averaged) and at the end of the swap, the net difference in
interest is paid to the successful counterparty. As the interest is paid in a different period
to that which it applies, there is a convexity issue; however, this can be shown to be
Figure 1. Overnight Indexed Swap Cash Flows
R is the swap rate, and ri are the daily overnight fixes
The fact that these swaps have exposure to overnight interest rates makes them
extremely useful in determining what the market is pricing in terms of the expectation of
the path of central bank policy rates. Specifically, if we one is able to find some way of
stripping out the short-term liquidity effects that are priced into these swap rates then one
Overnight Indexed Swaps and Floored Compounded Instrument in HJM One-Factor Model, Bank for International
Settlements, Marc Henrard, 2004
10 July 2007 3
is left with an instrument that describes the market expectations of central bank policy.
Indeed, these swaps provide a resolution that interest rate futures do not
, being available
at monthly tenors out to a year. Thus, noting that there is hardly ever more than one
central bank policy meeting per month
, it is possible to attribute each Swap tenor as
representing a single meeting – e.g. the 1M OIS represents the 1
meeting, the 2M the
How can this be done, quantitatively? As a first approximation, assume that there are no
liquidity effects, so that they can be ignored. Then, note that the overnight rate should
only change on the day of the policy meeting. Thus, knowing the current policy rate, r
compound up the interest for the number of days until the meeting, and then compound
up at the new rate, r
, for the remaining number of days, and solve for r
. The below
equation has the NPV of the fixed leg of the swap on the left-hand side, and the NPV of
the floating leg on the RHS:
is the 1M OIS rate, T
is the maturity date of the swap, t
is the start date of the
swap, T is the payment date on both legs of the swap (usual one business day after T
but this varies by currency), q
is the overnight rate on day l and n
for which that rate compounds (e.g.- usually equal to one unless day l is followed by a
weekend, when it will be three, or a holiday), D is the day-count basis and dF
discount factor to date T.
If, as discussed above, the q
are allowed only to change after the meeting date, t
is associated with the m-th period OIS, then q
for 0 < l <t
≤ l<T
Finally, cancelling the dF
This can then be solved for r
. For the purposes of maintaining a suitable abstraction in
this article, the binomial approximation is used so that:
Then, re-labelling t
, for the meeting date in the first month, the equation reduces
With the exception of Fed Funds Futures in the US and Cash Rate Futures in Australia and New Zealand
If there is more than one meeting per month, then additional inputs, such as forward-starting swaps, that cover
unique periods are needed, otherwise the problem is not generally soluble.
t t n and that ( )
10 July 2007 4
One can do this recursively, using the swap rates out to say 6M, to find the expected
policy rates, r
, for each meeting. Actually, this is somewhat an over-simplification, as a
treatment of cash liquidity (see §5) must be considered when solving for these rates,
which is modelled as a daily funding spread s
, so the real equations to solve are:
,,, (A)
However, the end result is almost the same: information that represents the 1D Central
Bank Forward Curve, which is constrained to change level only on Monetary Policy
meetings (fig 2). Once this has been determined, one can add back in the liquidity
adjustments, s
, to produce a 1D OIS Forward Curve that contains the required 1 day
resolution in the forwards.
Figure 2. Reserve Bank of Australia (RBA) 1D Forward Curve
Apr-07 May-07 Jun-07 Jul-07 Aug-07 Sep-07 Oct-07
3. MONEY-MARKET & CROSS-CURRENCY BASIS SWAPS
It’s all very well building a step-curve for pricing OIS, but how can this information be
used to better construct the front of the Libor curve? Money-Market and Cross-Currency
basis swaps have been discussed in detail elsewhere
, so for brevity they shall just be
briefly commented upon here. These are swaps where two floating rate streams are
exchanged, for example 3M Libor vs. 6M Libor. There will be a basis-spread on one of
the legs of the swap that generally represents investor preference of one index over the
other and depends upon things such as the credit risk of lending for one term vs. another
(e.g. lending for 6M is riskier than for 3M. So generally, one would expect to
pay/receive 6M vs. 3M + spread. Similarly, one could enter a basis swap in which one
paid the 1D OIS fix compounded up quarterly and received the 3M Libor fix + a spread,
or almost equivalent, paid 1D OIS + spread vs. 3M Libor. One would expect the latter
spread to be positive, as lending for 1 day is much less risky than for 3M, and also, one
would expect interest-rate expectations to be priced into the next 3M which would also
Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency Basis Swaps, Lehman Brothers Fixed
Income Liquid Markets Research, Bruce Tuckman & Pedro Porfirio, 2003.
10 July 2007 5
affect that rate – in an upward sloping curve this would be more-positive, and in a
downward sloping curve this would be less-positive (and possibly negative, depending
on the amount and rate of easing priced in).
Figure 3. OIS – Libor Basis Swap
Lj is the 3m Libor payment on day j, ri is the1d OIS fixing, and x is the OIS-Libor basis spread. The payments on the
1d leg of the swap are compounded up and paid quarterly.
Given that there are usually FRA prices for 1x4, 2x5 etc, and OIS out to 1Y with
monthly resolution, one can bootstrap the FRAs and the OIS separately and price the 1D-
3M basis swaps to maturities 1M – 1Y
. This can be done once a day as a calibration, or
left free-floating. These basis spreads can then be applied to the step-shape OIS forward
curve by recursively solving the following equation, to create a 1D OIS Index curve and
a 3m Libor Index curve, which represents the NPV of the each leg of the OIS-Libor basis
dF L dF
1 ,,, (B)
The product is over the 1d fixings within the j-th 3m period (i.e. - compounding over the
3m period), L
is the j-th 3m Libor fixing, ∆
is the day count fraction for the Libor
coupons and each element of the sums is the 3-monthly payment on the basis swap. The
result is a new curve that represents the 3M Libor forwards, the 3M Libor Index Curve,
with a 1D resolution out to say 6M. The remaining portion of the curve can be
constructed from the 2
future out to the 8
future, and then with swap rates as
per normal swap-curve construction.
For completeness, it is possible to extend this methodology to build the Funding Curve
(often referred to as the ‘Cross-Currency Basis Curve’, as it is constructed from Cross-
Currency Basis Swaps). The discount factors in the above equation can be found by
including the Cross-Currency Basis spread, either directly (for 1Y+, these are market
traded), or by calibration from FX Forwards. A Cross-Currency Basis Swap is a swap in
which a 3m Libor stream in one currency is exchanged for one in a second currency plus
a spread, y, with an exchange of principal at the beginning and the end, and is used in
pricing FX Forwards
Note that the 1M, 2M & 3M basis swaps will just be single stub payments.
See Tuckman & Porfirio.
10 July 2007 6
Figure 4. Cross-Currency Basis Swap
Where 1 unit of notional in foreign currency is exchanged for S dollars, Li is 3m USD Libor and L~i is 3m foreign
The discount factors can be found by recursively solving the following equation, which
represents the NPV of the Cross-Currency basis swap legs:
SdF dF L S F d F d y L S + ∆ + = + ∆ + +
( ,,, (C)
are the foreign currency discount factors,
discount factor to maturity,
dF is the dollar discount factor to maturity, S is the number
of US dollars per foreign unit of notional, and
is the day count fraction for the
foreign Libor coupons. The USD discount factors and Libor forwards are found by
constructing the Swap curve as discussed earlier, with the step-shape at the very front,
and traditionally, from the 2
future out, similarly for the foreign Libor forwards. It is
purely the foreign discount factors that are different. Finally, for sub 1Y discount factors,
either the 1Y Basis swap can be used (providing the short-term liquidity model is good
enough) or the Basis swap spreads can be implied by noting that the FX Forwards can be
are the discount factors to spot in USD and in the foreign currency
respectively. This equation may be substituted into equation (C).
To summarise, by including the solution of equations (A), (B) and (C) in a curve-
building algorithm, one is able to construct a fully consistent set of Index and Funding
curves that are capable of pricing Swaps, FRAs, OIS, Cross-Currency Basis Swaps, OIS-
3m Libor Basis Swaps
and FX Forwards
. Significantly, they will also have proper
knowledge of the 1D forwards at the very short-end of the curve (see fig 5).
Further Index curves can be constructed by supplying basis spreads to the floating term – e.g. 3m-6m basis
FX Forwards are priced by taking discount factors from the Funding Curve.
10 July 2007 7
Figure 5. Australian Index & Funding Curves
Apr-07 Jul-07 Oct-07 Jan-08 Apr-08 Jul-08 Oct-08
RBA 1D Index 3m Bank Bill Index (1d Forwards)
Funding (1d Forwards) 3m Bank Bill (3m Forwards)
4. CENTRAL BANK RISK REPRESENTATION
Now that it has been shown how to construct the curve in terms of policy adjustments, it
is instructive to represent risk in terms of these adjustments. Continuing in the above
frame-work, ignoring the liquidity effect, one has a set of equations that represent the
curve. The usual method for estimating delta risk is to calculate the analytic Jacobian of
the price of the assets used in the curve
.For ease, below, the Jacobian of the swap rates
with respect to the meeting rates has been calculated and then multiplied by the
(diagonal) Jacobian of discount factors to the Swap tenors with respect to the Swap rates:
Using the earlier approximation and generalising to the j-month OIS and j-th overnight
rate, the curve equations are:
The products are over each of the overnight rates, with their interest factors raised to the
power of the number of days for which they are valid (i.e. – to the previous meeting date,
rather than to the previous OIS maturity date). Then, separate into two products,
associated with the compounding up to a meeting date and that from the meeting date up
to the OIS maturity date:
Where the first product is set so that the compounding starts from the initial policy rate,
, and then also define T
as the start date of the OIS, as noted earlier.
A detailed description of this approach is beyond the scope of this paper, but “Fixed-Income Securities:
Valuation, Risk Management and Portfolio Strategies” by Martellini, Priaulet & Priaulet contains an excellent
10 July 2007 8
Differentiate with respect to the k
meeting rate, r
1 ln δ δ
δ is the Kronecker delta-function
. Re-arranging terms gives the Jacobian:
1 δ δ
If the approximation
that the discount factors can be given by the below equations is
Then multiplying the diagonal matrix:
by the Jacobian above will give the Jacobian of discount factors with respect to the
δ δ J
If the cash flows can be represented, by some interpolation method, as an equivalent set
of cash flows on the swap maturity dates, then a reasonably good estimate of the CB01 –
“PV change as a result of a 1bp move in the expectations of the Central Bank meeting
- can be provided.
The Kronecker delta-function is defined by
As described in Section 3, the discount factors should actually be from the Funding Curve and the numerical
examples are calculated using the Funding Curve.
One could, of course, construct a set of curves, with consecutive meeting inputs perturbed, although this is
10 July 2007 9
For completeness, note that given CB01 values for each meeting , inverting the Jacobian
and multiplying by the vector, c, representing the CB01 for each meeting will return the
notional, n, of each OIS that is required to hedge that given risk to Central Bank
4.1 Worked Example
A demonstration of the usefulness of this approach is now appropriate. In Australia, the
OIS fixes to the RBA target rate, so there is no liquidity effect to take into account.
Figure 5 shows the RBA policy expectations as of April 10
corresponding expectations for rates (the current RBA policy rate is 6.25%) are:
Figure 6. RBA Policy Expectations
Date Expected Policy Rate Easing/Hiking
02-May-07 6.36 +11
06-Jun-07 6.43 +7
04-Jul-07 6.43 0
08-Aug-07 6.47 +4
15-Sep-07 6.49 +2
03-Oct-07 6.51 +2
The corresponding Jacobian of discount factors with respect to the meeting dates is:
54 . 1 0 0 0 0 0
19 . 5 04 . 1 0 0 0 0
97 . 9 97 . 9 31 . 1 0 0 0
29 . 9 29 . 9 29 . 9 86 . 1 0 0
51 . 7 51 . 7 51 . 7 51 . 7 61 . 1 0
49 . 9 49 . 9 49 . 9 49 . 9 49 . 9 44 . 2
The rows can be thought of as representing the monthly discount factors/swap rates, and
the columns representing the meetings.
Suppose that one has the view that the RBA will hike rates on 2
May 2007. He/she
could enter a trade that would monetise the 14bp shown above if correct, and loose 11bp
if incorrect. Consider risking $100k – i.e. AUD -11,134 per bp of CB01, taking 0.8165
for AUD/USD spot. As this meeting falls within the 1M OIS, only the left-corner
element of the Jacobian is needed (or invert it and multiply by the vector c, which has
=-11,134, and c
= 0 for i > 1) which is -2.44 per 1m AUD. Thus, in order to generate
the required level of risk, one should pay on AUD 4.563bn of the 1M OIS in order to get
this CB01 (-11,134/-2.44 *AUD 1m). Compare this with the DV01 of AUD -38,323
$31,291), which implies a risk value that is 244% too high.
4.2 Risk-Bucketing Comparison
A comparison of the bucketing of risk in both the new and traditional method would be
useful. Figure 7 shows the market data used in constructing the curves. For the Central
Calculated as the basis point difference between in the expected policy rate vs. the previous expected policy
For a single cash flow, the DV01 is given by
− = × ×
10 July 2007 10
Bank approach, meetings have been used up to the 2
, and then the usual bucketing
structure thereafter.
Figure 7. Australian Market Data
Traditional Step-based
Bucket Price Bucket Step-based
1M Deposit 6.3950 02-May-07 RBA 6.36
2M Deposit 6.4483 06-Jun-07 RBA 6.43
3M Deposit 6.4967 04-Jul-07 RBA 6.43
YBAM7 Future 9342.5 08-Aug-07 RBA 6.47
YBAU7 Future 9341.5 YBAU7 Future 9341.5
YBAZ7 Future 9334.5 YBAZ7 Future 9334.5
YBAH8 Future 9332.5 YBAH8 Future 9332.5
YBAM8 Future 9337.0 YBAM8 Future 9337.0
YBAU8 Future 9340.0 YBAU8 Future 9340.0
YBAZ8 Future 9340.5 YBAZ8 Future 9340.5
YBAH9 Future 9341.0 YBAH9 Future 9341.0
3Y Swap 6.6100 3Y Swap 6.6100
4Y Swap 6.6275 4Y Swap 6.6275
5Y Swap 6.5875 5Y Swap 6.5875
7Y Swap 6.5075 7Y Swap 6.5075
10Y Swap 6.4255 10Y Swap 6.4255
In the below examples, the PV01s and CB01s are defined as the change in value for 1bp
increase in the interest rate. The cash flow dates have been chosen randomly to illustrate
the risk-bucketing in the different regions of the curve.
Deposit Region
Figure 8. Comparison of risk-bucketing in the Deposit Region
Bucket Risk Bucket Risk
1M Deposit 394 02-May-07 RBA -9,402
2M Deposit -10,016 06-Jun-07 RBA -5,379
3M Deposit -10,491 04-Jul-07 RBA 0
YBAM7 Future 377 08-Aug-07 RBA 0
YBAU7 Future 0 YBAU7 Future 0
YBAZ7 Future 0 YBAZ7 Future 0
YBAH8 Future 0 YBAH8 Future 0
YBAM8 Future 0 YBAM8 Future 0
YBAU8 Future 0 YBAU8 Future 0
YBAZ8 Future 0 YBAZ8 Future 0
YBAH9 Future 0 YBAH9 Future 0
3Y Swap 0 3Y Swap 0
4Y Swap 0 4Y Swap 0
5Y Swap 0 5Y Swap 0
7Y Swap 0 7Y Swap 0
10Y Swap 0 10Y Swap 0
Total Risk -19,736 Total Risk -14,781
A single cash flow of AUD 1bn, receivable on 26
10 July 2007 11
Figure 9. Comparison of risk-bucketing in the Future Region
1M Deposit -54 02-May-07 RBA -897
2M Deposit -1,712 06-Jun-07 RBA -719
3M Deposit 314 04-Jul-07 RBA -897
YBAM7 Future -2,525 08-Aug-07 RBA -1,024
YBAU7 Future -2,382 YBAU7 Future -2,309
YBAZ7 Future -2,131 YBAZ7 Future -2,131
YBAH8 Future 41 YBAH8 Future 0
Total Risk -8,449 Total Risk -7,977
A single cash flow of AUD 100m, receivable on 10
Swaps Region
Figure 10. Comparison of risk-bucketing in the Swaps
1M Deposit 4 02-May-07 RBA 107
2M Deposit 131 06-Jun-07 RBA 86
3M Deposit -28 04-Jul-07 RBA 107
YBAM7 Future 280 08-Aug-07 RBA 123
YBAU7 Future 217 YBAU7 Future 229
YBAZ7 Future 197 YBAZ7 Future 212
YBAH8 Future 156 YBAH8 Future 168
YBAM8 Future 139 YBAM8 Future 151
YBAU8 Future 98 YBAU8 Future 108
YBAZ8 Future 82 YBAZ8 Future 91
YBAH9 Future 92 YBAH9 Future 52
3Y Swap 1,580 3Y Swap 1,393
4Y Swap 7,538 4Y Swap 2,000
5Y Swap -39,859 5Y Swap -30,005
7Y Swap -8,037 7Y Swap -11,587
10Y Swap 741 10Y Swap 0
Total Risk -37,410 Total Risk -36,765
A single cash flow of AUD 100m receivable on 10
It can be seen that the total risk for the Futures Region and the Swaps Region is very
close in both methods. Indeed, the bucketing from around the forth future out is similar.
10 July 2007 12
However, in the Deposit Region, the traditional method over-estimates the risk
4.3 OIS – Libor Basis Risk
The next thing that needs to be monitored is the OIS-Libor basis risk. As there is not a
generally traded basis swap at the short-end it is a risk to be monitored in terms of its
volatility. Alternatively, one could trade OIS vs. FRAs or Swaps to synthetically create
the Basis Swap.
The main driver of these basis spreads is the difference in credit quality in lending for 1d
vs. 3m – there is a significantly higher risk of lending for 3m than for 1d. Across most
G10 currencies, this is thought to be 10-15bp, as is assumed when one is trying to assess
what is priced in to 3m Libor-based futures
. However, the spread between the Central
Bank target rate and where 3m Libor is setting is also determined by what Monetary
Policy is priced into the next 3m, which causes it to change depending whether there is
cutting, easing or hiking priced into the curve. Therefore, it is not particularly useful to
try and work out what is priced in from interest rate futures based upon this assumption,
as one may deduce pricings that are somewhat different to the real expectations.
Furthermore, it is inappropriate to apply this spread to the constructed OIS curve. If a
spread curve is to be properly constructed, then 3m Libor has to be compared with a 1d
rate that is compounded up over 3m so that the expectations over the next 3m are
properly included in the calculation of the spread. Conveniently, the 1d payments in an
OIS-Libor basis swap are compounded up and paid quarterly vs. 3m Libor. Then, the 3m
basis swap can almost be thought of as the difference between today’s 3m Libor fix and
the market quoted 3m OIS
. As noted in §1, if there is a surprise move in interest rates
by the Central Bank, the market traded instrument will move to reflect this, and in this
case, the 3m basis swap can move dramatically (see fig. 11). Indeed this spread will
move throughout the day anyway, as market news can cause expectations of policy to
change too. A stark example of this was in January 2007 when a large downward
surprise on Australian CPI caused the market to completely price out the chance of an
immediate RBA hike – see the spike on the chart.
Figure 11. Recent History of the 3m OIS-Libor Basis Swap Spread
Apr-06 Jun-06 Aug-06 Sep-06 Nov-06 Jan-07 Feb-07
A large downward surprise in CPI caused the pricing out of an immediate RBA hike (red oval)
Introducing Reposcope: A tool for monitoring UK and euro-area monetary policy expectations, Liquid Markets
Research Quarterly, Alexei Jiltsov and Adam Purzitsky, 2006
Of course, the spread is also compounded so this is a small approximation.
10 July 2007 13
Generally, however, for longer-dated basis swaps, such as the 6M or the 12M, this effect
is more muted, as it will only apply to the first payment on the swap. Indeed, the 12M
Basis Swap spread (fig. 12) is pretty stable, with mean 8.82bp and standard deviation
1.46bp over the past year. It only moves significantly around policy meetings, but this is
not too much of an issue if one has hedged their CB01, as this move in the spread is
perfectly offset with the CB01 position.
Figure 12. Recent History of the 12m OIS-Libor Basis Swap vs. RBA Policy
The red ovals show when there was an RBA Monetary Policy meeting that wasn’t fully priced for a hike going in,
and the RBA subsequently hiked. The blue ovals show when it wasn’t fully priced for a hike, and then
the tightening wasn’t delivered
The spread move highlights the inadequacy of the traditional method of constructing the
short-end, as it uses instruments that are stationary, when the real curve is moving – the
spread move captures that movement. To wit, consider the 1m forward-starting 3m OIS-
Libor basis swap spread, which can be effectively thought of as the difference between
the 1x4 Libor FRA and the 1x4 OIS
. Both of these instruments should have the market
expectations of policy built into them, and because they have not had a fixing yet, they
will not be distorted by the fixing effect.
5. FUNDING AND LIQUIDITY ISSUES
Once the risk with respect to Central Bank policy moves has been removed, positions
still need to be funded. While this is less of an issue for Libor swap books, OIS, FX
Forwards and Bonds still have either a cash position to be rolled or a fixing to funding
instruments (in the case of OIS). There are, of course, subtle differences between the
financing of these positions, in that the OIS fixes to a weighted average funding rate,
while the FX Forwards and Bonds (via repo) are funded wherever the trader is able to
It is also worth further distinguishing between Libor financing of swaps and real
financing of physical cash positions. Libor is a daily fixing that is determined by where
several banks think that they “ought” to lend to AA-rated banking counterparties. Thus,
it is not a real lending rate, and is often manipulated by constituent banks’ swap fixing
requirements. As it is used as the primary fixing rate for 1M, 3M & 6M Libor swaps, the
other fixings may bear no resemblance to real market funding rates at all. Furthermore, it
Again, remembering the spread is also compounded.
10 July 2007 14
is an un-secured lending rate, which makes it further different to FX Forward implied
and cash deposit rates. (figure 13).
Figure 13. Comparison of 3m Deposits
3m Bank Bill 3m FX Forward Implied (Close) 3m Cash Deposit (Close)
After stripping out the central bank and basis spread risk, the remaining exposure is due
to overnight funding, which is driven by many liquidity factors. For example, figure 14
shows the daily EONIA
refinancing rate spread. The mean spread between the
two rates is about 7bp, and this is stable, apart from around certain dates when it moves
markedly. One can imagine that certain events such as corporate tax days, year-ends and
month-ends would yield a higher demand for cash, and thus push the funding rate up. Of
course, there are a great many drivers of this funding spread, and the ECB has produced
a very sophisticated model of the EONIA - ECB rate spread
, but below is a qualitative
discussion of a few of these factors.
It is also worthwhile commenting on the non-existence of the so called “arbitrage” between deposits and FX
Forwards. This arises from the risk of lending to one counterparty and borrowing from another – if the counterparty
holding one’s funds defaults, one loses the entire principal, while still having to pay back the currency that has been
borrowed. On an FX Forward, however, the entire transaction is with a single counterparty, so if they default, one
can withhold payment of “borrowed” currency, and hence is just left with an FX spot position arising from the initial
exchange of principal. Thus the difference between FX Forward implied rates and cash deposits can be thought of
as representing the difference between the risk of default on the notional and that of an FX position – i.e. this
“arbitrage” is really selling the risk of default.
A Comprehensive Model of the Euro Overnight Rate, ECB Working Paper No. 207, Flemming Reinhardt, 2003
10 July 2007 15
Figure 14. EONIA – ECB Refinancing Rate Spread
Oct-05 Jan-06 Apr-06 Jul-06 Sep-06 Dec-06 Mar-07
Mean spread 7.09bp, standard deviation 7.06bp
5.1 Example Factors
All companies are required to pay their tax, and thus, free cash-flow held in bank
accounts will be withdrawn in order to pay these bills. This creates a deficit in the
banking system, as banks borrow to ensure that they have enough cash for this with-
drawl. This pushes up borrowing costs.
Various accounting regulations require firms not to lend out their balance sheet when
year-end accounts are produced. Again, this drives up financing costs.
As for above, but to less an extent. Often, the effect is even larger at quarter-ends.
On a micro-economical level, the consumer generally withdraws money from his/her
bank account on a Friday evening in order to have cash to spend over the weekend. As a
weekend roll is generally for 3 days rather than 1 day, it is also slightly more risky and
will also have this premium built in. Again, an upward move in funding costs.
On a Monday, spare cash left over from the weekend is often deposited back into current
accounts, as well as many companies paying in wages on a Monday. This exerts
downward pressure on funding, as banks struggle to lend out their cash.
Central Bank reserve periods
Most Central Banks, in order to minimise systemic risk in the banking system, require
banks to post collateral with them to a certain average value per month. This can create
interesting effects around central bank policy meetings. As an example, the Federal
Reserve requires banks to deposit an average amount of cash over a monthly period,
dependent upon their liabilities. Over the course of the recent hiking cycle, these bank
treasury desks have sort to deposit a large amount for just a few days prior to the Federal
Open Market Committee (FOMC) meetings. This ensured that the reserve requirement
was met, on average, for the next month, by borrowing capital at the lower funds rate
10 July 2007 16
and therefore avoiding the higher cost of financing post-FOMC hike. This put upward
pressure on the overnight deposit rate in the days leading up to the meeting.
Correspondingly, in the few days after the meeting, the very large amount of maturing
repo and deposits caused a net cash surplus in the banking system, which caused rates to
fall significantly (figure 15). The reverse of this would be expected in a cutting cycle.
Maturing Repo
Term repo executed with the Central Bank is part of the Central Bank’s efforts to ensure
that money market funds are trading close to its target rate. Clearly, a large amount of
maturing repo will flood the market with cash on that maturity date, and will thereby put
downward pressure on the funding rate. This is very evident if the maturity date of
occurs shortly after a hike in the central bank rate. An example of this is the ECB bi-
weekly open market operation which generally matures on the Tuesday following an
ECB Governing Council Meeting, and the average cash rate on that day over the past
couple of years has been ~30bp below the re-financing rate.
Yield-curve effects
There are also effects deriving from the slope of the curve – for example, if the Fed are
hiking rates, and the expectation is that they will continue to do so, there may be a large
portion of the market short Treasuries – and this will put upward pressure on repo
Catastrophes/One-off events
The one thing that everybody wants in the event of a catastrophe is very quick access to
cash. For example, there was a serious worry that the Millennium bug might cause
computer systems globally to fail, and this pushed up short term borrowing costs over
the year-end turn by 45bp. Nobody wanted to have to borrow cash back if they were
short, as no-one would want to lend cash if their computer systems were down. In the
face of possible systemic risk to the banking system, the Federal Reserve announced that
it would supply all of the liquidity that would be needed. Subsequently, the year-turn
premium collapsed to zero. However, it has become customary, ever since, for traders to
price in a year-turn premium, even though this is rarely realised. In fact, the overnight
deposit has regularly collapsed on the actual day of funding.
Figure 15. US Overnight Fed Funds Effective vs. Target
Aug-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-06 Apr-06
Fed Funds Ef f ective Fed Fund Target
Note that the Fed Funds effective rate creeps up shortly before the policy meetings when the Fed is hiking
Measures of Asset Swap Spreads and their Corresponding Trades, Lehman Brothers Fixed Income Research,
Bruce Tuckman, 2003
10 July 2007 17
5.2 Isolating the funding spread
It is instructive to construct a quantitative model of spread of the funding rate to policy
rate will trade and risk manage with respect to this:
s r f ε + + =
is the funding rate on day l, r
applicable on day l, and ε
Ideally, ε
would be small and mean-reverting over a period of 1M, and any model of the
funding spread should have this goal in mind. The adjustments may be found either by
averaging or some regression or weighting method. Figure 12 shows a trivial model of
Base Rate Spread that includes a month-end and premium and a
daily average premium/discount based upon recent spread fixes. One can make the
model as complex as required to reduce the residual error, or even keep it very simple by
simply using the mean spread.
Once a spread model has been constructed, it can be used to extract the market policy
expectations as described by the previously discussed equation:
Thus, provided the residual error term is mean reverting, or at least normally distributed,
then, funding spread risk can be monitored using a standard VaR measure based upon
the historical volatility of the residual error term ε
Figure 16. Simple Model of the SONIA-MPC Base Rate Spread
MPC 1d Forwards SONIA 1d Forwards
A further point that is worth noting is that the only instrument with which one can hedge
these funding differences is the Cross-Currency Basis Swap, which has an initial
exchange of principal at the beginning and at the end. The swap is effectively the
exchange of two floating rate notes (FRN) and so has no duration exposure, except after
a fixing, when it effectively becomes a 3M FX Forward, plus a 3M forward-starting
Cross-Currency Basis Swap. The near exchange of principal clearly has to be funded,
and this is usually done by executing a 3M FX Forward. Alternatively, one might wish to
10 July 2007 18
lock in the OIS fixing each day, and this could be performed by executing an OIS-3M
Libor basis swap in each currency – which effectively turns the Cross-Currency Basis
Swap into two floating rate notes that pay the 1D OIS fixing. If the initial exchange is
rolled daily, via an overnight FX Forward, the residual is the spread between when the
trader is able to fund in the market, and where the market average was.
Clearly, there is little point in actually doing this, due to transaction costs, and generally,
the funding exposure can be reasonably well hedged by executing 1W OIS (provided
there is no meeting in the next week), and assuming that one is able to fund close to that.
Alternatively, one could just execute Cross-Currency Basis Swaps to cover the funding
risk, rolling quarterly via an FX Forward, and take on the risk of the Cross-Currency
Spread moving (this is reasonably stationary, except for high-yield vs. Yen crosses, as a
result of the carry trade via Uridashi Issuance
5.3 Funding and Index Curves
It is natural to differentiate between Funding and Index instruments. DV01 and CB01 are
primarily defined and hedged by Index instruments:
Libor Deposits
Funding instruments, capture the (small) residuals
Money-Market Basis Swaps
The inclusion of OIS as an Index instrument, despite it containing information about
funding, is that it still has an interest rate fix, like Libor swaps.
6. EXTRACTING MARKET PROBABILITIES
Finally, it is natural to try and extract the probabilities of certain outcomes occurring and
compare them to personal views for trade decision-making. The market in options on
Fed Funds futures has allowed market participants to transparently see what the market-
based probability
of Fed actions is, and thus take positions based purely on these.
However, for other markets, futures
and options upon the policy rate do not exist
despite efforts to try and link options on Libor futures
, the lack of resolution (3m
futures generally contain three meetings, so there is an infinite number of solutions to the
policy moves in that period
) and the problems in applying the policy-Libor spread (see
§4.3).
Uridashi issuance is the issuance of high yield debt, in New Zealand, for example, sold to Japanese investors.
The financing of this cross-currency issuance causes banks to be paying JPY Libor and receiving NZD Bank Bills
(the New Zealand equivalent of Libor) which causes them to hedge this basis risk by paying NZD Bills and
receiving JPY Libor on the basis swap, causing the spread to widen.
The Fediscope, Bruce Tuckman and Dana Calistru, Lehman Brothers Fixed Income Research, 2003
Although extraction of expectations from OIS is equivalent.
Actually, futures have recently been introduced for Australia and New Zealand, but they are not very liquid as of
See Jiltsov and Purzitsky
Analogous to the three-body problem in Physics.
10 July 2007 19
Given the rapid expansion of the OIS market, it seems natural that a swaption market
, allowing the similar direct trading of the probabilities of central
bank action. However, until then, the approach, which is well known from the trading of
Fed Funds futures, is to apply some assumptions to the outcomes of meetings and then
use the expected policy rates to infer estimates of the probabilities. Essentially, these
assumptions sell tail-risk for the purpose of maintaining simplicity – when analysing
what is priced into a future, one assumes that only two outcomes can occur – e.g. if the
future implies a higher rate than the current rate, then there is a chance of a 25bp hike
and a chance of unchanged, but the assumption is that there is no probability of cuts.
That inference is merely an estimate of the probability, however, as there is still a real
probability of a rate cut, or a 50bp hike etc. The problem is that the future does not
capture this probability – it merely provides the expectation. Options on Fed Funds
futures are required to fully describe the probability distribution of market expectations
of Fed Policy
One approach, similar to that mentioned above, for estimating the probabilities is to
assume that if the expected rate outcome is greater than the current policy rate then there
is zero probability of a rate cut at that meeting. Then, say that the difference in rates, ∆,
(column 3 in figure 6) divided by 25bp is the probability of a 25bp hike, as for the Fed
Funds futures above. For example, at the first RBA meeting in figure 5, there is 11bp
priced into the expectation, so there is an 11/25 = 44% chance of a 25bp hike, and a
100% - 44% = 56% chance of no move. Conversely, if the expected rate outcome is
lower than the policy rate, then assume there is zero probability of a hike, and then apply
the same logic. If the difference in rates is greater than 25bp, then there is a real
probability of a 50bp move. In this case, attribute (∆-25bp)/25bp*100% as the
probability of a 50bp move and 100% minus this as the probability of a 25bp hike. This
then assumes a zero probability of a rate cut or an unchanged rates outcome.
Furthermore, it assumes that moves of greater than 50bp do not occur (50bp moves,
themselves, are pretty rare these days). Each meeting is then treated in isolation.
Obviously, this approach is far from perfect, but it does provide at least some
A more sophisticated method could include an application of the Reposcope
to calibrate the probabilities at the Libor future maturity dates and then some suitable
form of interpolation (although, what form of interpolation, remains to be answered).
However, the above approach, when applied to the RBA, gives the following probability
Figure 17. Estimated Probability Distribution for the RBA
Date -50bp -25bp 0bp +25bp +50bp Expectation bps
02-May-07 0% 0% 56% 44% 0% 6.36 +11
06-Jun-07 0% 0% 72% 28% 0% 6.43 +7
04-Jul-07 0% 0% 100% 0% 0% 6.43 0
08-Aug-07 0% 0% 84% 16% 0% 6.47 +4
15-Sep-07 0% 0% 92% 8% 0% 6.49 +2
03-Oct-07 0% 0% 92% 8% 0% 6.51 +2
Marc Henrard has discussed options on OIS in his article.
See Fediscope, Tuckman
10 July 2007 20
This paper presents a new approach to the construction of the short-end of the Swap
Curve which avoids the inconsistencies of the traditional approach. Through the use of
Overnight Indexed Swaps and Money Market Basis Swaps, market expectations of
Central Bank policy can be included that allow the accurate pricing of short-term interest
rate instruments such as FRAs. This method is extended, through the inclusion of Cross-
Currency Basis Swaps to allow the pricing of FX Forwards. Furthermore, this framework
allows risk-management in terms of the outcomes of Monetary Policy meetings, whilst
maintaining consistency across different instruments, through the use of basis swaps.
Finally, integration of this approach with the usual instruments used in Swap Curve
construction (Futures & Swaps) from around the 6m point on the curve out, produces a
fully consistent framework that allows the accurate pricing and simple risk-management
of the entire curve
The Swap Curve, Lehman Brothers Fixed Income Research, Fei Zhou,
Overnight Indexed Swaps and Floored Compounded Instruments in HJM
One-Factor Model, Bank for International Settlements, 2004
Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency
Basis Swaps, Lehman Brothers Fixed Income Liquid Markets Research,
Bruce Tuckman and Pedro Porfirio 2003
Strategies, Martellini, Priaulet and Priaulet, 2003
Introducing Reposcope: A tool for monitoring UK and euro-area monetary
policy expectations, Lehman Brothers Liquid Markets Research Quarterly,
Alexei Jiltsov and Adam Purzitsky, 2006
A Comprehensive Model of the Euro Overnight Rate, ECB Working Paper
No. 247, Flemming Reinhardt, 2003
Measures of Asset Swap Spreads and their Corresponding Trades, Lehman
Brothers Fixed Income Research, Bruce Tuckman, 2003
The Fediscope, Lehman Brothers Fixed Income Research, Bruce Tuckman
and Dana Calistru, 2003
are liquidity based funding issues, surely the risk is far lower? Conversely, what if there is a policy meeting in this period, surely the risk is much higher? A further issue is how to represent the risk between different instruments that are very similar. For example, it is well-known that FX Forwards are deposit-like instruments, but how should they trade in relation to FRAs, or Overnight Indexed Swaps (OIS)? How should one account for the term spread of 1D funded assets vs 3M funded assets, or indeed Cross-Currency funded assets? Clearly, existing methods of risk-management and pricing are insufficient in both their consistency and accuracy at the very front of the curve, and a new approach that fully accounts for the above effects needs to be considered. 2. OVERNIGHT INDEXED SWAPS (OIS) OIS are a form of (generally) short-term interest rate swap in which the floating leg fixes to a daily interest rate, making them useful for gaining exposure to monetary policy changes. The OIS market grew out of the French T4M swap market in the 1990s, and has gradually expanded into many currencies over the past few years to be extremely liquid in the majors. These swaps are generally quoted in monthly tenors out to 1Y, or even beyond in some currencies, as well as weekly out to 1M. Recently, an inter-bank market has developed for swaps that are dated between Central Bank meetings, although these are generally only quoted via voice brokers. An OIS is a swap which has a single payment at the end of the swap (usually on the day after the maturity of the swap), representing the difference in interest on the two legs of the swap. The fixed leg can be considered much like a synthetic deposit, and is quoted in the market as a yield that is applied over the tenure of the swap. The floating leg, on the other hand, has a daily fix, usually to the weighted-average of overnight cash deposits traded that day. The interest on the floating leg is then compounded up (apart from Fed Fund swaps, where it is averaged) and at the end of the swap, the net difference in interest is paid to the successful counterparty. As the interest is paid in a different period to that which it applies, there is a convexity issue; however, this can be shown to be negligible6.
The fact that these swaps have exposure to overnight interest rates makes them extremely useful in determining what the market is pricing in terms of the expectation of the path of central bank policy rates. Specifically, if we one is able to find some way of stripping out the short-term liquidity effects that are priced into these swap rates then one
Overnight Indexed Swaps and Floored Compounded Instrument in HJM One-Factor Model, Bank for International Settlements, Marc Henrard, 2004
How can this be done. t0 is the start date of the swap. as discussed above. D is the day-count basis and dFT is the discount factor to date T. The below equation has the NPV of the fixed leg of the swap on the left-hand side. t m − t 0 −1 T1 − t m −1 8 9 7 Note ∏ (n ) = t l l =0 m − t0 and that ∏ (n ) = T − t l 1 l =0 m 10 July 2007 3 . that cover unique periods are needed. the 2M the 2nd.usually equal to one unless day l is followed by a weekend. it is possible to attribute each Swap tenor as representing a single meeting – e. Then. otherwise the problem is not generally soluble. the above equation becomes: t − t −1 T − t −1  R1 (T1 − t0 )  m 0  nl r0  1 m  nl r1  1+ = ∏ 1 +    ∏ 1 +  D D  l =0  D    l =0  This can then be solved for r1. If. then additional inputs. Thus. compound up the interest for the number of days until the meeting. note that the overnight rate should only change on the day of the policy meeting. T is the payment date on both legs of the swap (usual one business day after T1. or a holiday). being available at monthly tenors out to a year. and solve for r1. re-labelling tm as t1. noting that there is hardly ever more than one central bank policy meeting per month8. when it will be three. such as forward-starting swaps. assume that there are no liquidity effects. the ql are allowed only to change after the meeting date. so that they can be ignored. r0.g. the 1M OIS represents the 1st meeting. the binomial approximation is used so that: r   nl r0   1 +  ≈ 1 + 0  D   D  nl Then. ql is the overnight rate on day l and nl is the number of days for which that rate compounds (e.Lehman Brothers | The Short-End of the Curve is left with an instrument that describes the market expectations of central bank policy. and so on. Indeed. then ql=r0 for 0 < l <tm and ql=r1 for tm ≤ l<T1. quantitatively? As a first approximation. these swaps provide a resolution that interest rate futures do not7. but this varies by currency).. Thus. for the meeting date in the first month. r1. knowing the current policy rate. Finally. tm. and the NPV of the floating leg on the RHS:  T1 −t0 −1  nl ql  R1 (T1 − t 0 )  1 + dFT =  ∏ 1 +  D D    l =0    dFT   Where R1 is the 1M OIS rate. where tm is associated with the m-th period OIS. T1 is the maturity date of the swap. for the remaining number of days. the equation reduces to9:  R1 (T1 − t 0 )   r0   = 1 +  1 + D   D  This is easily solved: (t1 −t0 ) r   1 + 1   D (T1 −t1 ) With the exception of Fed Funds Futures in the US and Cash Rate Futures in Australia and New Zealand If there is more than one meeting per month. cancelling the dFT. For the purposes of maintaining a suitable abstraction in this article.g. and then compound up at the new rate.
Once this has been determined. (A) However. paid 1D OIS + spread vs. to produce a 1D OIS Forward Curve that contains the required 1 day resolution in the forwards. this is somewhat an over-simplification. Similarly. but how can this information be used to better construct the front of the Libor curve? Money-Market and Cross-Currency basis swaps have been discussed in detail elsewhere10. one can add back in the liquidity adjustments.5 6. and also. which is modelled as a daily funding spread sl. for example 3M Libor vs.Lehman Brothers | The Short-End of the Curve  1          R1 (T1 − t 0 )   T1 −t1      1 + D   r1 = D  − 1 (t −t )    r0  1 0     1 + D          One can do this recursively. So generally.25 6 A pr-07 M ay-07 Jun-07 Jul-07 A ug-07 Sep-07 Oct-07 3. to find the expected policy rates. Money Market Basis Swaps. another (e. Bruce Tuckman & Pedro Porfirio. 3M + spread. One would expect the latter spread to be positive.g. one would expect to pay/receive 6M vs. Actually. si. the end result is almost the same: information that represents the 1D Central Bank Forward Curve. one could enter a basis swap in which one paid the 1D OIS fix compounded up quarterly and received the 3M Libor fix + a spread. and Cross-Currency Basis Swaps. There will be a basis-spread on one of the legs of the swap that generally represents investor preference of one index over the other and depends upon things such as the credit risk of lending for one term vs. using the swap rates out to say 6M.. which is constrained to change level only on Monetary Policy meetings (fig 2). These are swaps where two floating rate streams are exchanged. 6M Libor. 2003.75 6. so the real equations to solve are: t −t T −t  R1 (T1 − t0 )  m 0 nl (r0 + sl )  1 m  nl (r1 + sl )  1 +  = ∏ 1 +  ∏ 1 +  D D D   l =0   l =0   . MONEY-MARKET & CROSS-CURRENCY BASIS SWAPS It’s all very well building a step-curve for pricing OIS. for each meeting. 10 10 July 2007 4 . as a treatment of cash liquidity (see §5) must be considered when solving for these rates.. so for brevity they shall just be briefly commented upon here. Figure 2. or almost equivalent. Reserve Bank of Australia (RBA) 1D Forward Curve 6. Lehman Brothers Fixed Income Liquid Markets Research. one would expect interest-rate expectations to be priced into the next 3M which would also Interest Rate Parity. as lending for 1 day is much less risky than for 3M. lending for 6M is riskier than for 3M. ri. 3M Libor.
∆j is the day count fraction for the Libor coupons and each element of the sums is the 3-monthly payment on the basis swap. These basis spreads can then be applied to the step-shape OIS forward curve by recursively solving the following equation. 2M & 3M basis swaps will just be single stub payments. A Cross-Currency Basis Swap is a swap in which a 3m Libor stream in one currency is exchanged for one in a second currency plus a spread. Lj is the j-th 3m Libor fixing. The result is a new curve that represents the 3M Libor forwards. or by calibration from FX Forwards.. ri is the1d OIS fixing.Lehman Brothers | The Short-End of the Curve affect that rate – in an upward sloping curve this would be more-positive.e. OIS – Libor Basis Swap Lj ri x ………… ………… ………… Lj is the 3m Libor payment on day j. and OIS out to 1Y with monthly resolution. one can bootstrap the FRAs and the OIS separately and price the 1D3M basis swaps to maturities 1M – 1Y11. and in a downward sloping curve this would be less-positive (and possibly negative. 10 July 2007 5 . with an exchange of principal at the beginning and the end. For completeness. 2x5 etc. to create a 1D OIS Index curve and a 3m Libor Index curve. either directly (for 1Y+. Given that there are usually FRA prices for 1x4. y.. the 3M Libor Index Curve. This can be done once a day as a calibration. (B) The product is over the 1d fixings within the j-th 3m period (i. The discount factors in the above equation can be found by including the Cross-Currency Basis spread. The payments on the 1d leg of the swap are compounded up and paid quarterly. with a 1D resolution out to say 6M. Figure 3. as it is constructed from CrossCurrency Basis Swaps). which represents the NPV of the each leg of the OIS-Libor basis swap: T  T j +1 −T j −1  ni (ri + x)   ∑  ∏ 1 + D  − 1dF j = ∑ L j ∆ j dF j  i= j    j  j  T . it is possible to extend this methodology to build the Funding Curve (often referred to as the ‘Cross-Currency Basis Curve’. and x is the OIS-Libor basis spread. depending on the amount and rate of easing priced in). and then with swap rates as per normal swap-curve construction.compounding over the 3m period). 11 12 Note that the 1M. and is used in pricing FX Forwards12. . these are market traded). The remaining portion of the curve can be constructed from the 2nd future out to the 8th or 12th future. See Tuckman & Porfirio. or left free-floating.
Significantly.. 3m-6m basis spreads. similarly for the foreign Libor forwards. by including the solution of equations (A).Lehman Brothers | The Short-End of the Curve Figure 4. Cross-Currency Basis Swap 1 S SLi L˜i y S 1 Where 1 unit of notional in foreign currency is exchanged for S dollars. 14 FX Forwards are priced by taking discount factors from the Funding Curve. OIS. (B) and (C) in a curvebuilding algorithm. OIS3m Libor Basis Swaps13 and FX Forwards14. It is purely the foreign discount factors that are different. dFT is the foreign currency discount factor to maturity. Li is 3m USD Libor and L~i is 3m foreign Libor The discount factors can be found by recursively solving the following equation. ~ Further Index curves can be constructed by supplying basis spreads to the floating term – e. To summarise. 13 10 July 2007 6 . S is the number of US dollars per foreign unit of notional.. either the 1Y Basis swap can be used (providing the short-term liquidity model is good enough) or the Basis swap spreads can be implied by noting that the FX Forwards can be described by the following equation: ~ ~ ~ ~  dF dF  FT = S  T ~S   dF dF  S   T Where dFS and dFS are the discount factors to spot in USD and in the foreign currency respectively. This equation may be substituted into equation (C). and traditionally. from the 2nd future out. they will also have proper knowledge of the 1D forwards at the very short-end of the curve (see fig 5). for sub 1Y discount factors.g. (C) Where dF j are the foreign currency discount factors. which represents the NPV of the Cross-Currency basis swap legs: T T ~ ~ ~ ~ S + ∑ ( L j + y )∆ j dF j + dFT = 1 + S ∑ L j ∆ j dF j + SdFT j j . The USD discount factors and Libor forwards are found by constructing the Swap curve as discussed earlier. with the step-shape at the very front. Finally. and ∆ j is the day count fraction for the foreign Libor coupons. one is able to construct a fully consistent set of Index and Funding curves that are capable of pricing Swaps. dFT is the dollar discount factor to maturity. FRAs. Cross-Currency Basis Swaps.
75 6. Risk Management and Portfolio Strategies” by Martellini. ignoring the liquidity effect.e. The usual method for estimating delta risk is to calculate the analytic Jacobian of the price of the assets used in the curve15. Australian Index & Funding Curves 6. as noted earlier. separate into two products. the Jacobian of the swap rates with respect to the meeting rates has been calculated and then multiplied by the (diagonal) Jacobian of discount factors to the Swap tenors with respect to the Swap rates: Using the earlier approximation and generalising to the j-month OIS and j-th overnight rate. Continuing in the above frame-work.25 6 Apr-07 Jul-07 Oct-07 Jan-08 Apr-08 Jul-08 Oct-08 RBA 1D Index Funding (1d Forw ards) 3m Bank Bill Index (1d Forw ards) 3m Bank Bill (3m Forw ards) 4. rather than to the previous OIS maturity date). 15 10 July 2007 7 . associated with the compounding up to a meeting date and that from the meeting date up to the OIS maturity date: (t i −Ti −1 ) j (T − t ) j  R j (T j − t0 )  ri  i i  ri −1   1 +  = ∏ 1 +  ∏ 1 + D    D D  i =1    i =1  Where the first product is set so that the compounding starts from the initial policy rate. CENTRAL BANK RISK REPRESENTATION Now that it has been shown how to construct the curve in terms of policy adjustments.5 6. Taking the logarithm of both sides gives: A detailed description of this approach is beyond the scope of this paper. one has a set of equations that represent the curve. – to the previous meeting date.For ease. with their interest factors raised to the power of the number of days for which they are valid (i. Priaulet & Priaulet contains an excellent exposition. Then. r0. below. the curve equations are: (t −t ) j  R j (T j − t 0 )  ri  i i −1  1 +  = ∏ 1 +    D   i =0  D  The products are over each of the overnight rates. but “Fixed-Income Securities: Valuation.Lehman Brothers | The Short-End of the Curve Figure 5. it is instructive to represent risk in terms of these adjustments. and then also define T0 = t0 as the start date of the OIS.
k =   ∂rk  T j − t0  D  i =1       ri −1  1 +  D  (ti − Ti −1 ) + δ i.k is the Kronecker delta-function16. then a reasonably good estimate of the CB01 – “PV change as a result of a 1bp move in the expectations of the Central Bank meeting outcome”18 .Lehman Brothers | The Short-End of the Curve j j  R (T − t )  r   r   ln1 + j j 0  = ∑ (ti − Ti −1 )ln1 + i −1  + ∑ (Ti − ti )ln1 + i    D D  i =1   D   i =1 Differentiate with respect to the kth meeting rate.k  (T j − t0 )1 + R j (T j − t0 )  i =1      D    j  ri −1  1 +  D  (ti − Ti −1 ) + δ i. of course. construct a set of curves. the discount factors should actually be from the Funding Curve and the numerical examples are calculated using the Funding Curve.k  (Ti − ti )   ri    1 +    D  If the cash flows can be represented.k  ri −1   i =1  1 +   D      (Ti − ti )  r   1 + i    D δ i. rk:  j    R j (T j − t0 )   ∂  ln1 +   = ∑  δ i −1. with consecutive meeting inputs perturbed. 18 One could. by some interpolation method. k  ∂rk   D   i =1      Where    j  (ti − Ti −1 )  +  δ ∑ i. 16 The Kronecker delta-function is defined by δi. Re-arranging terms gives the Jacobian:   D  R j (T j − t0 )  j  ∂R j 1 + ∑ δ i −1.can be provided.k =  1 0 i=k i≠k As described in Section 3. as an equivalent set of cash flows on the swap maturity dates. 17 10 July 2007 8 . although this is computationally expensive.k  (Ti − ti )   ri    1 +    D  If the approximation17 that the discount factors can be given by the below equations is made: dFi = 1  Ri (Ti − t 0 )  1 +  D   Then multiplying the diagonal matrix: (Ti − t 0 ) ∂ (dFi ) =− 2 ∂Ri  Ri (Ti − t 0 )  D 1 +  D   by the Jacobian above will give the Jacobian of discount factors with respect to the meeting dates: J= ∂ (dF j ) ∂rk   1 =− ∑ δ i −1.
which has c1=-11.0001× Notional = − 2 ∂R  Ri (Ti − t 0 )  D1 +  D   10 July 2007 9 . 4. Thus.e.04 0 − 9.49 − 7. note that given CB01 values for each meeting . and the columns representing the meetings.134/-2.44 per 1m AUD.51 − 9. inverting the Jacobian and multiplying by the vector.36 6.86  0 0  0  0 0 0   0 0 0  − 9.43 6.49   − 7.97 − 1.43 6. the previous expected policy rate. Figure 7 shows the market data used in constructing the curves.Lehman Brothers | The Short-End of the Curve For completeness.0001× Notional × 0.291).44 *AUD 1m).19   − 1.134. in order to generate the required level of risk. Suppose that one has the view that the RBA will hike rates on 2nd May 2007.32320 ($31. 20 For a single cash flow. He/she could enter a trade that would monetise the 14bp shown above if correct.49 6.29 − 1.134 per bp of CB01. As this meeting falls within the 1M OIS. representing the CB01 for each meeting will return the notional. taking 0.563bn of the 1M OIS in order to get this CB01 (-11.49  − 1. n.49 − 9.31 0 0 − 9. RBA Policy Expectations Date 02-May-07 06-Jun-07 04-Jul-07 08-Aug-07 15-Sep-07 03-Oct-07 Expected Policy Rate 6.51  0  0 0 − 1.29   − 9. so there is no liquidity effect to take into account. and the corresponding expectations for rates (the current RBA policy rate is 6.29 − 9. the OIS fixes to the RBA target rate.44 − 9. For the Central Calculated as the basis point difference between in the expected policy rate vs. of each OIS that is required to hedge that given risk to Central Bank meetings: n = J −1 c 4.49 − 7. In Australia.51 − 9.97  − 5.47 6.51 Easing/Hiking19 +11 +7 0 +4 +2 +2 The corresponding Jacobian of discount factors with respect to the meeting dates is:  − 2. Figure 5 shows the RBA policy expectations as of April 10th 2007. only the left-corner element of the Jacobian is needed (or invert it and multiply by the vector c.25%) are: Figure 6. Consider risking $100k – i. and ci = 0 for i > 1) which is -2. Compare this with the DV01 of AUD -38. the DV01 is given by 19 (Ti − t 0 ) ∂ (dF ) × 0. AUD -11.8165 for AUD/USD spot. and loose 11bp if incorrect.51  − 9.1 Worked Example A demonstration of the usefulness of this approach is now appropriate. c.2 Risk-Bucketing Comparison A comparison of the bucketing of risk in both the new and traditional method would be useful. one should pay on AUD 4. which implies a risk value that is 244% too high.54   The rows can be thought of as representing the monthly discount factors/swap rates.61 − 7.
6100 6.5 9341.5875 6.5875 6.5 9332.6275 6.379 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -14.402 -5.Lehman Brothers | The Short-End of the Curve Bank approach.5 9332.0 9340.6275 6.5 9334.016 -10. Figure 7.5 9334.5 9337.736 th Step-based Bucket 02-May-07 RBA 06-Jun-07 RBA 04-Jul-07 RBA 08-Aug-07 RBA YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Risk -9.5075 6. meetings have been used up to the 2nd.43 6.781 A single cash flow of AUD 1bn.47 9341. and then the usual bucketing structure thereafter. the PV01s and CB01s are defined as the change in value for 1bp increase in the interest rate.0 9340.5075 6.4255 Bucket 02-May-07 RBA 06-Jun-07 RBA 04-Jul-07 RBA 08-Aug-07 RBA YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Step-based Step-based 6.3950 6. Deposit Region Figure 8.491 377 0 0 0 0 0 0 0 0 0 0 0 0 -19. The cash flow dates have been chosen randomly to illustrate the risk-bucketing in the different regions of the curve. receivable on 26 June 2007 10 July 2007 10 .4483 6.43 6.0 6. Comparison of risk-bucketing in the Deposit Region Traditional Bucket 1M Deposit 2M Deposit 3M Deposit YBAM7 Future YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Risk 394 -10.0 9340.5 9341.5 9341. Australian Market Data Traditional Bucket 1M Deposit 2M Deposit 3M Deposit YBAM7 Future YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Price 6.36 6.4967 9342.0 6.5 9337.0 9340.4255 In the below examples.6100 6.
Lehman Brothers | The Short-End of the Curve Futures Region Figure 9.131 0 0 0 0 0 0 0 0 0 0 -7.131 41 0 0 0 0 0 0 0 0 0 -8.449 th Step-based Bucket 02-May-07 RBA 06-Jun-07 RBA 04-Jul-07 RBA 08-Aug-07 RBA YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Risk -897 -719 -897 -1.765 A single cash flow of AUD 100m receivable on 10th September 2012 It can be seen that the total risk for the Futures Region and the Swaps Region is very close in both methods.000 -30.037 741 -37.859 -8. the bucketing from around the forth future out is similar.538 -39.977 A single cash flow of AUD 100m.005 -11.587 0 -36. 10 July 2007 11 .382 -2.410 Bucket 02-May-07 RBA 06-Jun-07 RBA 04-Jul-07 RBA 08-Aug-07 RBA YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Step-based Risk 107 86 107 123 229 212 168 151 108 91 52 1.712 314 -2. Comparison of risk-bucketing in the Swaps Traditional Bucket 1M Deposit 2M Deposit 3M Deposit YBAM7 Future YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Risk 4 131 -28 280 217 197 156 139 98 82 92 1. Indeed.024 -2.580 7.393 2.525 -2. Comparison of risk-bucketing in the Future Region Traditional Bucket 1M Deposit 2M Deposit 3M Deposit YBAM7 Future YBAU7 Future YBAZ7 Future YBAH8 Future YBAM8 Future YBAU8 Future YBAZ8 Future YBAH9 Future 3Y Swap 4Y Swap 5Y Swap 7Y Swap 10Y Swap Total Risk Risk -54 -1. receivable on 10 March 2008 Swaps Region Figure 10.309 -2.
the market traded instrument will move to reflect this. which causes it to change depending whether there is cutting. Alexei Jiltsov and Adam Purzitsky. FRAs or Swaps to synthetically create the Basis Swap. 11). and in this case. the traditional method over-estimates the risk significantly. As there is not a generally traded basis swap at the short-end it is a risk to be monitored in terms of its volatility. Figure 11. the 3m basis swap can move dramatically (see fig.3 OIS – Libor Basis Risk The next thing that needs to be monitored is the OIS-Libor basis risk. one could trade OIS vs. 4. 2006 22 Of course. If a spread curve is to be properly constructed. As noted in §1. However. A stark example of this was in January 2007 when a large downward surprise on Australian CPI caused the market to completely price out the chance of an immediate RBA hike – see the spike on the chart. Alternatively. this is thought to be 10-15bp. the 1d payments in an OIS-Libor basis swap are compounded up and paid quarterly vs. Liquid Markets Research Quarterly. the 3m basis swap can almost be thought of as the difference between today’s 3m Libor fix and the market quoted 3m OIS22. it is inappropriate to apply this spread to the constructed OIS curve. Indeed this spread will move throughout the day anyway. Furthermore. Recent History of the 3m OIS-Libor Basis Swap Spread 18 16 14 12 10 8 6 4 2 0 Apr-06 Jun-06 Aug-06 Sep-06 Nov-06 Jan-07 Feb-07 A large downward surprise in CPI caused the pricing out of an immediate RBA hike (red oval) Introducing Reposcope: A tool for monitoring UK and euro-area monetary policy expectations. Across most G10 currencies. 3m – there is a significantly higher risk of lending for 3m than for 1d. in the Deposit Region. 21 10 July 2007 12 . The main driver of these basis spreads is the difference in credit quality in lending for 1d vs. it is not particularly useful to try and work out what is priced in from interest rate futures based upon this assumption. 3m Libor.Lehman Brothers | The Short-End of the Curve However. easing or hiking priced into the curve. Therefore. then 3m Libor has to be compared with a 1d rate that is compounded up over 3m so that the expectations over the next 3m are properly included in the calculation of the spread. Conveniently. as market news can cause expectations of policy to change too. as is assumed when one is trying to assess what is priced in to 3m Libor-based futures21. if there is a surprise move in interest rates by the Central Bank. as one may deduce pricings that are somewhat different to the real expectations. the spread is also compounded so this is a small approximation. Then. the spread between the Central Bank target rate and where 3m Libor is setting is also determined by what Monetary Policy is priced into the next 3m.
of course. The blue ovals show when it wasn’t fully priced for a hike. which can be effectively thought of as the difference between the 1x4 Libor FRA and the 1x4 OIS23. the other fixings may bear no resemblance to real market funding rates at all. FUNDING AND LIQUIDITY ISSUES Once the risk with respect to Central Bank policy moves has been removed. While this is less of an issue for Libor swap books. and because they have not had a fixing yet. as it uses instruments that are stationary. subtle differences between the financing of these positions. consider the 1m forward-starting 3m OISLibor basis swap spread. with mean 8. Figure 12. but this is not too much of an issue if one has hedged their CB01. as it will only apply to the first payment on the swap. Recent History of the 12m OIS-Libor Basis Swap vs. It only moves significantly around policy meetings. and then the tightening wasn’t delivered The spread move highlights the inadequacy of the traditional method of constructing the short-end. 3M & 6M Libor swaps. while the FX Forwards and Bonds (via repo) are funded wherever the trader is able to during the day. FX Forwards and Bonds still have either a cash position to be rolled or a fixing to funding instruments (in the case of OIS). Indeed. when the real curve is moving – the spread move captures that movement. Furthermore. 12) is pretty stable. As it is used as the primary fixing rate for 1M. There are.46bp over the past year. remembering the spread is also compounded. they will not be distorted by the fixing effect. as this move in the spread is perfectly offset with the CB01 position. It is also worth further distinguishing between Libor financing of swaps and real financing of physical cash positions. the 12M Basis Swap spread (fig. RBA Policy Meetings 14 12 10 8 6 4 2 0 Apr-06 Jun-06 Aug-06 Sep-06 Nov-06 Jan-07 Feb-07 The red ovals show when there was an RBA Monetary Policy meeting that wasn’t fully priced for a hike going in. Libor is a daily fixing that is determined by where several banks think that they “ought” to lend to AA-rated banking counterparties. it is not a real lending rate. for longer-dated basis swaps. Thus.82bp and standard deviation 1. and the RBA subsequently hiked. 10 July 2007 13 . and is often manipulated by constituent banks’ swap fixing requirements. positions still need to be funded. 5. OIS. it 23 Again. in that the OIS fixes to a weighted average funding rate.Lehman Brothers | The Short-End of the Curve Generally. such as the 6M or the 12M. Both of these instruments should have the market expectations of policy built into them. To wit. however. this effect is more muted.
5 6. one can withhold payment of “borrowed” currency. Flemming Reinhardt. 2003 24 10 July 2007 14 . The mean spread between the two rates is about 7bp.75 6. one loses the entire principal. 25 Euro Overnight Index Average 26 European Central Bank 27 A Comprehensive Model of the Euro Overnight Rate. Of course. and hence is just left with an FX spot position arising from the initial exchange of principal. (figure 13). 207. but below is a qualitative discussion of a few of these factors. so if they default. and thus push the funding rate up.ECB rate spread27. For example. the entire transaction is with a single counterparty. figure 14 shows the daily EONIA25 .25 6 5. apart from around certain dates when it moves markedly. there are a great many drivers of this funding spread. while still having to pay back the currency that has been borrowed. which is driven by many liquidity factors. On an FX Forward. however. which makes it further different to FX Forward implied rates24 and cash deposit rates. and this is stable. It is also worthwhile commenting on the non-existence of the so called “arbitrage” between deposits and FX Forwards. One can imagine that certain events such as corporate tax days. the remaining exposure is due to overnight funding. ECB Working Paper No. This arises from the risk of lending to one counterparty and borrowing from another – if the counterparty holding one’s funds defaults. Comparison of 3m Deposits 6.5 Apr-06 Jun-06 3m Bank Bill Aug-06 Sep-06 Nov-06 Jan-07 Feb-07 3m FX Forw ard Implied (Close) 3m Cash Deposit (Close) After stripping out the central bank and basis spread risk.75 5. year-ends and month-ends would yield a higher demand for cash.e. this “arbitrage” is really selling the risk of default.Lehman Brothers | The Short-End of the Curve is an un-secured lending rate. Thus the difference between FX Forward implied rates and cash deposits can be thought of as representing the difference between the risk of default on the notional and that of an FX position – i. and the ECB has produced a very sophisticated model of the EONIA . Figure 13.ECB26 refinancing rate spread.
1 Example Factors Corporate Tax Day All companies are required to pay their tax. require banks to post collateral with them to a certain average value per month. standard deviation 7. the Federal Reserve requires banks to deposit an average amount of cash over a monthly period. Central Bank reserve periods Most Central Banks. it is also slightly more risky and will also have this premium built in. Over the course of the recent hiking cycle. As an example. EONIA – ECB Refinancing Rate Spread 30 20 10 Oct-05 -10 -20 -30 -40 -50 -60 -70 Mean spread 7. This exerts downward pressure on funding. Month-end As for above.06bp Jan-06 Apr-06 Jul-06 Sep-06 Dec-06 Mar-07 5. an upward move in funding costs. these bank treasury desks have sort to deposit a large amount for just a few days prior to the Federal Open Market Committee (FOMC) meetings. Often. on average. Financial year-end Various accounting regulations require firms not to lend out their balance sheet when year-end accounts are produced. the effect is even larger at quarter-ends. This can create interesting effects around central bank policy meetings. this drives up financing costs. as banks struggle to lend out their cash. as well as many companies paying in wages on a Monday. This creates a deficit in the banking system. Again. but to less an extent. free cash-flow held in bank accounts will be withdrawn in order to pay these bills. for the next month. dependent upon their liabilities. This ensured that the reserve requirement was met. Mondays On a Monday. the consumer generally withdraws money from his/her bank account on a Friday evening in order to have cash to spend over the weekend.09bp. spare cash left over from the weekend is often deposited back into current accounts. Again. As a weekend roll is generally for 3 days rather than 1 day. in order to minimise systemic risk in the banking system.Lehman Brothers | The Short-End of the Curve Figure 14. Weekends On a micro-economical level. and thus. as banks borrow to ensure that they have enough cash for this withdrawl. This pushes up borrowing costs. by borrowing capital at the lower funds rate 10 July 2007 15 .
An example of this is the ECB biweekly open market operation which generally matures on the Tuesday following an ECB Governing Council Meeting. However. This put upward pressure on the overnight deposit rate in the days leading up to the meeting. Figure 15. and the expectation is that they will continue to do so. Lehman Brothers Fixed Income Research.Lehman Brothers | The Short-End of the Curve and therefore avoiding the higher cost of financing post-FOMC hike. which caused rates to fall significantly (figure 15). the overnight deposit has regularly collapsed on the actual day of funding. it has become customary. as no-one would want to lend cash if their computer systems were down. for traders to price in a year-turn premium. there was a serious worry that the Millennium bug might cause computer systems globally to fail. and this pushed up short term borrowing costs over the year-end turn by 45bp. 2003 28 10 July 2007 16 . even though this is rarely realised. in the few days after the meeting. there may be a large portion of the market short Treasuries – and this will put upward pressure on repo28. the Federal Reserve announced that it would supply all of the liquidity that would be needed. In fact. and will thereby put downward pressure on the funding rate. Catastrophes/One-off events The one thing that everybody wants in the event of a catastrophe is very quick access to cash. Yield-curve effects There are also effects deriving from the slope of the curve – for example. In the face of possible systemic risk to the banking system. US Overnight Fed Funds Effective vs. Maturing Repo Term repo executed with the Central Bank is part of the Central Bank’s efforts to ensure that money market funds are trading close to its target rate. Subsequently. Correspondingly. Clearly. The reverse of this would be expected in a cutting cycle. Bruce Tuckman. the year-turn premium collapsed to zero. This is very evident if the maturity date of occurs shortly after a hike in the central bank rate. ever since. a large amount of maturing repo will flood the market with cash on that maturity date. the very large amount of maturing repo and deposits caused a net cash surplus in the banking system. and the average cash rate on that day over the past couple of years has been ~30bp below the re-financing rate. For example. if the Fed are hiking rates. Nobody wanted to have to borrow cash back if they were short. Target 5 4 3 2 1 Aug-04 Oct-04 Jan-05 Apr-05 Jul-05 Oct-05 Jan-06 Apr-06 Fed Funds Effective Fed Fund Target Note that the Fed Funds effective rate creeps up shortly before the policy meetings when the Fed is hiking Measures of Asset Swap Spreads and their Corresponding Trades.
or even keep it very simple by simply using the mean spread. and this is usually done by executing a 3M FX Forward. which has an initial exchange of principal at the beginning and at the end. plus a 3M forward-starting Cross-Currency Basis Swap. or at least normally distributed. and any model of the funding spread should have this goal in mind. funding spread risk can be monitored using a standard VaR measure based upon the historical volatility of the residual error term εl. then. Alternatively.25 5 Apr-07 May-07 Jun-07 Jul-07 Aug-07 Sep-07 Oct-07 MPC 1d Forw ards SONIA 1d Forw ards A further point that is worth noting is that the only instrument with which one can hedge these funding differences is the Cross-Currency Basis Swap. it can be used to extract the market policy expectations as described by the previously discussed equation: t −t T −t  R1 (T1 − t0 )  m 0 nl (r0 + sl )  1 m  nl (r1 + sl )  1 +  = ∏ 1 +  ∏ 1 +  D D D   l =0   l =0   Thus.Lehman Brothers | The Short-End of the Curve 5. Ideally. Once a spread model has been constructed.5 5. One can make the model as complex as required to reduce the residual error. The adjustments may be found either by averaging or some regression or weighting method. when it effectively becomes a 3M FX Forward. εl would be small and mean-reverting over a period of 1M. provided the residual error term is mean reverting. Simple Model of the SONIA-MPC Base Rate Spread 6 5. and εl is the residual error term. rl is the ri applicable on day l. one might wish to 29 30 Sterling Overnight Index Average Bank of England Monetary Policy Committee 10 July 2007 17 . Figure 12 shows a trivial model of the SONIA29-MPC30 Base Rate Spread that includes a month-end and premium and a daily average premium/discount based upon recent spread fixes. The near exchange of principal clearly has to be funded. Figure 16. except after a fixing. The swap is effectively the exchange of two floating rate notes (FRN) and so has no duration exposure.75 5.2 Isolating the funding spread It is instructive to construct a quantitative model of spread of the funding rate to policy rate will trade and risk manage with respect to this: f l = rl + sl + ε l Where fl is the funding rate on day l.
capture the (small) residuals Cross-Currency Basis Swaps Money-Market Basis Swaps FX Forwards Cash Deposits The inclusion of OIS as an Index instrument. the lack of resolution (3m futures generally contain three meetings. via an overnight FX Forward. futures have recently been introduced for Australia and New Zealand. there is little point in actually doing this. 32 The Fediscope. and despite efforts to try and link options on Libor futures35. and this could be performed by executing an OIS-3M Libor basis swap in each currency – which effectively turns the Cross-Currency Basis Swap into two floating rate notes that pay the 1D OIS fixing. and assuming that one is able to fund close to that. sold to Japanese investors. in New Zealand. one could just execute Cross-Currency Basis Swaps to cover the funding risk. it is natural to try and extract the probabilities of certain outcomes occurring and compare them to personal views for trade decision-making. is that it still has an interest rate fix.Lehman Brothers | The Short-End of the Curve lock in the OIS fixing each day. 5. 35 See Jiltsov and Purzitsky 36 Analogous to the three-body problem in Physics. for example. and where the market average was. 31 10 July 2007 18 . as a result of the carry trade via Uridashi Issuance31). If the initial exchange is rolled daily. However. Lehman Brothers Fixed Income Research. despite it containing information about funding. and thus take positions based purely on these. rolling quarterly via an FX Forward. the residual is the spread between when the trader is able to fund in the market. The financing of this cross-currency issuance causes banks to be paying JPY Libor and receiving NZD Bank Bills (the New Zealand equivalent of Libor) which causes them to hedge this basis risk by paying NZD Bills and receiving JPY Libor on the basis swap. except for high-yield vs. DV01 and CB01 are primarily defined and hedged by Index instruments: Libor Deposits FRAs Futures Libor Swaps OIS Funding instruments. like Libor swaps. and generally. 2003 33 Although extraction of expectations from OIS is equivalent. due to transaction costs. EXTRACTING MARKET PROBABILITIES Finally. Yen crosses. The market in options on Fed Funds futures has allowed market participants to transparently see what the marketbased probability32 of Fed actions is. Alternatively. but they are not very liquid as of yet. Clearly. 6. for other markets.3 Funding and Index Curves It is natural to differentiate between Funding and Index instruments. and take on the risk of the Cross-Currency Spread moving (this is reasonably stationary. 34 Actually. the funding exposure can be reasonably well hedged by executing 1W OIS (provided there is no meeting in the next week). Bruce Tuckman and Dana Calistru.3). futures33 and options upon the policy rate do not exist34. so there is an infinite number of solutions to the policy moves in that period36) and the problems in applying the policy-Libor spread (see §4. Uridashi issuance is the issuance of high yield debt. causing the spread to widen.
A more sophisticated method could include an application of the Reposcope39 approach to calibrate the probabilities at the Libor future maturity dates and then some suitable form of interpolation (although. however.43 6. themselves. The problem is that the future does not capture this probability – it merely provides the expectation. This then assumes a zero probability of a rate cut or an unchanged rates outcome. For example. Estimated Probability Distribution for the RBA Date 02-May-07 06-Jun-07 04-Jul-07 08-Aug-07 15-Sep-07 03-Oct-07 -50bp 0% 0% 0% 0% 0% 0% -25bp 0% 0% 0% 0% 0% 0% 0bp 56% 72% 100% 84% 92% 92% +25bp 44% 28% 0% 16% 8% 8% +50bp 0% 0% 0% 0% 0% 0% Expectation 6. Furthermore. However.Lehman Brothers | The Short-End of the Curve Given the rapid expansion of the OIS market. so there is an 11/25 = 44% chance of a 25bp hike.36 6. but it does provide at least some transparency. when applied to the RBA. then there is a chance of a 25bp hike and a chance of unchanged. are pretty rare these days). (column 3 in figure 6) divided by 25bp is the probability of a 25bp hike. as there is still a real probability of a rate cut. the approach. if the expected rate outcome is lower than the policy rate. then assume there is zero probability of a hike. until then. Tuckman See Jiltsov and Purzitsky 10 July 2007 19 . at the first RBA meeting in figure 5. Essentially. it assumes that moves of greater than 50bp do not occur (50bp moves. attribute (∆-25bp)/25bp*100% as the probability of a 50bp move and 100% minus this as the probability of a 25bp hike. In this case. Each meeting is then treated in isolation. Conversely.49 6. for estimating the probabilities is to assume that if the expected rate outcome is greater than the current policy rate then there is zero probability of a rate cut at that meeting. remains to be answered).47 6. ∆. but the assumption is that there is no probability of cuts. One approach. there is 11bp priced into the expectation. Obviously.44% = 56% chance of no move. and a 100% . it seems natural that a swaption market will soon develop37. the above approach. or a 50bp hike etc. say that the difference in rates. Then. if the future implies a higher rate than the current rate. is to apply some assumptions to the outcomes of meetings and then use the expected policy rates to infer estimates of the probabilities. and then apply the same logic.g.51 bps +11 +7 0 +4 +2 +2 37 38 39 Marc Henrard has discussed options on OIS in his article.43 6. as for the Fed Funds futures above. these assumptions sell tail-risk for the purpose of maintaining simplicity – when analysing what is priced into a future. See Fediscope. similar to that mentioned above. allowing the similar direct trading of the probabilities of central bank action. If the difference in rates is greater than 25bp. one assumes that only two outcomes can occur – e. That inference is merely an estimate of the probability. which is well known from the trading of Fed Funds futures. then there is a real probability of a 50bp move. However. Options on Fed Funds futures are required to fully describe the probability distribution of market expectations of Fed Policy38. what form of interpolation. gives the following probability distribution: Figure 17. this approach is far from perfect.
integration of this approach with the usual instruments used in Swap Curve construction (Futures & Swaps) from around the 6m point on the curve out. Flemming Reinhardt. 247. Priaulet and Priaulet. this framework allows risk-management in terms of the outcomes of Monetary Policy meetings. REFERENCES The Swap Curve. 2003 10 July 2007 20 . Finally. Bruce Tuckman and Dana Calistru. Money Market Basis Swaps. Furthermore. through the inclusion of CrossCurrency Basis Swaps to allow the pricing of FX Forwards. Bruce Tuckman and Pedro Porfirio 2003 Fixed-Income Securities: Valuation. CONCLUSION This paper presents a new approach to the construction of the short-end of the Swap Curve which avoids the inconsistencies of the traditional approach. Through the use of Overnight Indexed Swaps and Money Market Basis Swaps. and Cross-Currency Basis Swaps. Lehman Brothers Fixed Income Research. Alexei Jiltsov and Adam Purzitsky. Fei Zhou.Lehman Brothers | The Short-End of the Curve 7. 2002 Overnight Indexed Swaps and Floored Compounded Instruments in HJM One-Factor Model. Bank for International Settlements. Lehman Brothers Fixed Income Research. market expectations of Central Bank policy can be included that allow the accurate pricing of short-term interest rate instruments such as FRAs. through the use of basis swaps. Bruce Tuckman. whilst maintaining consistency across different instruments. Lehman Brothers Fixed Income Liquid Markets Research. 2003 Measures of Asset Swap Spreads and their Corresponding Trades. Risk Management and Portfolio Strategies. ECB Working Paper No. produces a fully consistent framework that allows the accurate pricing and simple risk-management of the entire curve 8. 2003 Introducing Reposcope: A tool for monitoring UK and euro-area monetary policy expectations. Martellini. Lehman Brothers Liquid Markets Research Quarterly. 2003 The Fediscope. This method is extended. 2006 A Comprehensive Model of the Euro Overnight Rate. Lehman Brothers Fixed Income Research. 2004 Interest Rate Parity.
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