FRA^ED-fut: actual loan-rate fixed when@@

Suppose I’m IBM, and need to borrow in 3 months’s time. As explained in typical FRA scenario, inspired by CFA Reading 71, I could buy a FRA and agree to pay a pre-agreed rate of 550 bps.  What’s the actual loan rate? As explained in that post,

  • If I borrow on open market, then actual loan rate is the open-market rate on 1 Apr
  • If I borrow from the FRA dealer GS, then loan rate is the pre-agreed 550 bps
  • Either way, I’m indifferent, since in the open-market case, what ever rate I pay is offset by the p/l of the FRA

Instead of FRA, I could go short the eurodollar futures. This contract is always cash-settled, so the actually loan rate is probably the open-market rate, but whatever market rate I pay is offset by the p/l of the futures contract.

Probability of default ^ bond rating

* 1-Y Probability of default (denoted PD) is defined for a single issuer. * Rating (like AA) is defined for one bond among many by the same issuer.

Jon Frye confirmed that an AA rating is not an expression of the firm’s status/viability/strength/health.

I guess the rating does convey something about the LossGivenDefault, another attributes of the bond not the issuer.

BUY a (low) interest rate == Borrow at a lock-in rate

Q: What does “buying at 2% interest rate” mean?

It’s good to get an intuitive and memorable short explanation.

Rule — Buying a 2% interest rate means borrowing at 2%.

Rule — there’s always a repayment period.

Rule — the 2% is a fixed rate not a floating rate. In a way, whenever you buy you buy with a fixed price. You could buy the “floating stream” …. but let’s not digress.

Real, personal, example — I “bought” my first mortgage at 1.18% for first year, locking in a low rate before it went up.

factors affecting bond sensitivity to IR

In this context, we are concerned with the current market value (eg a $9bn bond) and how this holding may devalue due to Rising interest rate for that particular maturity.

* lower (zero) coupon bonds are more sensitive. More of the cash flow occurs in the distant future, therefore subject to more discounting.

* longer bonds are more sensitive. More of the cashflow is “pushed” to the distant future.

* lower yield bonds are more sensitive. On the Price/yield curve, at the left side, the curve is steeper.

(I read the above on a slide by Investment Analytics.)

Note if we hold the bond to maturity, then the dollar value received on maturity is completely deterministic i.e. known in advance, so why worry about “sensitivity”? There are 3 issues with this strategy:

1) if in the interim my bond’s MV drops badly, then this asset offers poor liquidity. I won’t have the flexibility to get contingency cash out of this asset.

1b) Let’s ignore credit risk in the bond itself. If this is a huge position (like $9bn) in the portfolio of a big organization (even for a sovereign fund), a MV drop could threaten the organization’s balance sheet, credit rating and borrowing cost. Put yourself in the shoes of a creditor. Fundamentally, the market and the creditors need to be assured that this borrower could safely liquidity part of this bond asset to meet contingent obligations.

Imagine Citi is a creditor to MTA, and MTA holds a bond. Fundamental risk to the creditor (Citi) — the borrower (MTA)  i.e. the bond holder could become insolvent before bond maturity, when the bond price recovers.

2) over a long horizon like 30Y, that fixed dollar amount may suffer unexpected inflation (devaluation). I feel this issue tends to affect any long-horizon investment.

3) if in the near future interest rises sharply (hurting my MV), that means there are better ways to invest my $9bn.

Gaussian HJM, briefly

… is a subset of HJM models.

An HJM model is Gaussian HJM if vol term is deterministic. Note “vol” term means the coefficient of the dW term. Every Brownian motion must always refer to an implicit measure. In this case, the RN measure.

How about the drift term i.e. the “dt” coefficient? It too has to be deterministic to give us a Gaussian HJM.

Well, Under the RN measure, the drift process is determined completely by the vol process. Both evolve with time, but are considered slow-moving [1] relative to the extremely fast-moving Brownian Motion of “dW”. Extremely because there’s no time-derivative of a BM

[1] I would say “quasi constant”

Language is not yet precise so not ready to publish on recrec…

various discount curves

For each currency

For each Libor tenor i.e. reset frequency like 3M, 6M

There's a yield curve

STIRT traders basically publish these curves via Sprite. Each currency has a bunch of tenor curves + the FX-OIS curve

This is the YC for the AA credit quality. In theory this yield curve is not usable for a different credit quality. For a BB credit quality, the mathematicians would, correctly, assume a yield curve but in reality I doubt there's a separate curve.

In contrast, there is indeed a tenor curve at 1Y, and other tenors too.

Basis swap means interest rate swap between 2 floating “floating streams”.

* swap between 2 currencies

* swap between 2 Libor tenors

* swap between 2 floating indices. These curves below have different credit qualities:

** Libor — AA banks

** OIS — much lower credit risk given the short tenor

** T-bill — US government credit

bonds – relevant across FI models #HJM..

Bonds are no longer the dominant FI instrument, challenged by IRS and futures. However, I feel for Fixed Income models bonds are more important, more useful than any other instrument.

– Bond (unlike swap rates, FRA rates etc) is a tradeable, and obeys many nice martingale pricing rules.

– Zero Bond can be a numeraire.

– For model calibration, zero bond prices are, IMO, more easily observable than swap rates, FRA rates, cap prices, swaption prices etc. I think futures prices could be more “observable” but the maturities available are limited.

– Zero bond’s price is exactly the discount factor, extremely useful in the math. I believe a full stochastic model built with zero bond prices can recover fwd rates, spot rates, short rates, swap rates and all others …

– I believe term structure models could be based on fwd rate or short rate, but they all need to provide an expression for the “zero bond process” i.e. the process that a given bond’s price follows. Note this process must converge to par at maturity.

– Bond as an instrument is closely related to caps and floors and callable bonds.

– Bonds are issued by all types of issuers. Other instruments (like swaps, IR futures) tend to have a smaller scope.

– Bonds are liquid over most regions of the yield curve, except the extreme short end.

arbitrate-free term structure models

Sound byte — if we have 2 unrelated stochastic processes for 2 points on the YC, the 2 calculated prices can induce arbitrage.

Jargon: whenever you see “process”, always think of “distro” under some “measure”.

Jargon: whenever you see “price”, it almost always means a fair “quote” on some contract whose terminal value is yet to be revealed. This price comes directly from the distro by discounting the expected payoff

Jargon: bond prices basically mean loan rates, possibly forward-start. Libor is a loan. FRA is a loan. ED-future represents a loan. A single-period IR swap consists of 2 loans. OIS is based on overnight loans. I notice that from any term structure model, we always want to back out the PROCESS of various bond prices. I guess that's because the actual contracts are invariably written in terms of loans.

Preliminary: IMO, a simple IR option is a call/put on one point of the YC. Example is a call on the 3M Libor. Libor rate changes every day, so we can model it as a stochastic PROCESS. We can also model the evolution of the entire YC with 20 “points” or infinite points.

Earlier formula used to price IR options (including bond options, caps, swaptions) is not from an IR model at all. Black's formula was proposed for options on commodity futures. When adapted to interest rates, Black's formula was kind of OK to price options on one point on the (evolving) yield curve, but problematic when applied to multiple points on the YC.

As a consequence, such a model can

– give a distribution of an (yet unrevealed) discount factor of maturity M1, after calibrating the model params with one set of market data

– give a distribution of an (yet unrevealed) discount factor of maturity M2, after calibrating the model params with another, unrelated set of market data

As a result, the 2 distributions are like the speculations by 2 competing analysts — can be contracting. If a bank uses such a model, and gives quotes based on these 2 distros, the quotes can be self-contradicting, i.e. inducing arbitrage. I would imagine the longer-maturity yield (converted from the discount factor) could turn out to be much lower than short-maturity, or the yield curve could have camel humps.

Following Black's formula, First generation of IR models describe the short rate evolution as a stochastic process, but short rate can't “reproduce” a family photo. In other words, we can't back out the discount factor of every arbitrary maturity.

More precisely, given a target date “t” [1] the model gives the distribution of the (unrevealed) short rate on that target date, but the zero-curve on that target date has infinite points, each being a discount factor from some distant future (like 30Y) to the target date. The short rate distribution is not enough to reproduce the entire YC i.e. the family photo.

The only way I know to give consistent [2] predictions on all points of the YC is a model describing the entire YC's evolution. We have to model the (stochastic) process followed by any arbitrary point on the YC, i.e. any member on the family photo. The model params thus calibrated would be self-consistent.

[1] that's later than the last “revelation” date or valuation date, i.e. any IR rate on the target date is unknown and should have some probability distribution.

[2] arb-free, not self-contradicting

Sound byte — disconnected, unrelated Processes for 2 bonds (eg 29Y vs 30Y) could induce arbitrage. These 2 securities are unlike Toyota vs Facebook stocks. I think the longer maturity bond can be used to arbitrage the shorter maturity. I think it's safe to assume non-negative interest rate. Suppose both have face value $1. Suppose I could buy the long bond at a dirt cheap $0.01 and short-sell the short bond at a high $0.98 and put away the realized profit…..

swaps illustration diagrams — how to read

This write-up covers IRS, x-ccy swap…

These block diagrams are popular and partially useful, but beginners often don't realize:

* initial context — typically a corporation has a periodic liability, or an investor has a periodic income.

** We had better ignore all the other arrows first.

* the motivation — typically to convert the initial single arrow to other arrows. The swap contract adds 2 arrows, one of them cancelling out the pre-existing arrow.

** we had better focus on the 3 arrows and ignore other parts of the diagram.

yield curve , according to Jeff

Jeff’s lecture notes (in 0xpdf) has detailed explanations on

1) EUR OIS YC bootstrapping using specific OIS instruments
2) Libor YC under OIS discounting — so OIS curve + libor curve needed.
3) Libor curve for a non-default tenor, such as 6M or 2M

lots of “root-finding”… but not too hard.

A YC (or a term structure) can be represented as a series of
* spot disc factors
* fwd disc factors
* spot interest rates
* fwd interest rates

Rebanato – good author on fixed income models

recommended by Sian Hwee.

Ronnie said Black model is popular (largely due to simplicity, and historical reason), and many option products are quoted in terms of vols implied from the Black model. 
TA seems to agree that the advanced models (beyond the Black model) are still needed but indeed harder than the earlier lectures before the Black model.

buying (i.e. long) a given interest rate

Tony (FX lecturer) pointed out “buying” any variable means executing at the current “level” and hope the “level” moves up. (Note a mathematician would point out an interest rate is not directly tradeable, but never mind.)

Therefore, buying an interest rate means borrowing (not lending) at a rock bottom rate.
Wrong intuition — “locking in the interest income stream”.
Eg: Say gov bond interest is super low, we would borrow now, and hope for a rise.
Eg: Say swap rate is super low, we would lock it in — pay fixed and lock in the floating income stream, and hope for the swap rate and floating stream both to rise.

par swap rate drop means …@@

For a given tenor (say 1Y)


I think treasury yield rise (or drop) has a simpler interpretation….


I think Libor ED deposit rate drop (or rise) has another simple interpretation …. and has a credit element.


Libor par swap rate drop has a non-trivial interpretation….


OIS swap rate is even more complicated…

FRA^ED-fut, another baby step

I think the differences like convexity adjustment are not “sticky” in my memory, after I tried many times to internalize, so no thin->thick->thin…

Jeff (MSFM) pointed out

* FRA — on expiry date you know the settlement amount. 2 days later that amount physically settles. That’s the accrual period start date!
* ED futures – every day you give or take a bit of the (usually big) settlement amount. On IMM date full amount settles. No 2-day delay.
ED has Neg convexity versus FRA, so whether your realized pnl is P or L, FRA is a bit “better” (for the holder) than ED for the same strike.

bonds let us to lock in a profit today to be realized in N years

After my lecturer touched on this point, I did some research.

For equities, say IBM, if we buy it at $100 and hope to cash out about
5 years from now, we are never confident. At that time, price could
drop below $100 and we may have to wait indefinitely to recover our
capital. That’s the nature of equity investment. Barring another
financial crisis (which i consider unlikely in the next 20 years),
price should recover but I might have bought at the peak, as I did
many times in my experience.

For a bond with a coupon rate 7.5% per year, maturing in 5 years, the
current price could be about $100, which translates to a yield around
7.5%, probably a high yield bond issued by some lesser-known entity
XYZ. If all the coupons are paid out only on maturity without
compounding, then the yield turns to be around 6.5%, as illustrated in
the attached spreadsheet.

The special thing about bond (relative to stocks) is, we kind of lock
in an annualized return of 6.5% at the time we buy it, barring credit

As the attached spreadsheet illustrates, today we pay about $100 to
own the bond, and in 60 months we are sure to receive exactly $137.5
i.e. $7.5 x 5 years coupon payment. This terminal value is not subject
to any market movement. The only uncertainty is credit default. Most
bonds we deal with, even the high yield bonds, are very unlikely to
default. If you buy a bond fund, then you would invest into hundreds
of bonds, so some defaults may be compensated by other bonds’ positive

If you don’t want to worry about defaults at all, then get a
investment grade bond, perhaps at a yield of 4%. You still lock in
that annualized 4% if you hold it till maturity.

The spreadsheet shows that even if there’s a credit crunch some time
before maturity, the bond’s market value (NAV) may drop drastically,
but it is sure to recover. Even if yield goes up in the last year,
barring default, the maturity value is still exactly $137.5. This
guaranteed return is something stocks can’t offer.

There are other factors to muddy the water a bit, but the simple fact
is, barring default, we could effectively lock in a profit today, to
be realized on the bond’s maturity date.

I guess that’s how insurers can guarantee returns over many decades.
They buy very long bonds which offer higher yields.

What do you think?

PRICE is clean or dirty;YIELD is always "dirty"

Traders quote clean price. However, Full price, Dirty price, or Invoice price is the price used for settlement.

YTM is neither clean or dirty. YTM is always “converted” from dirty price. My lecturer said —

There is only one yield to maturity.  It is neither of a “clean” nor “invoice” type.  The formula relating yield to maturity to bond price refers to the invoice price, not the clean price. 

bond "callable" provision is a risk to investor#my take

see also post on price spread between callable and non-callable bonds

To a buyer of a callable bond, the call feature poses a “call risk”.

Q: Why is early repayment a risk to the lender? To me it sounds like an added value.
A: The issuer only exercises this option when advantageous. Advantageous to issuer always means bad luck for the counter-party i.e. bond holders.

Bond is called only in a low-interest period. Imagine you receive 10% pa coupons year after year and suddenly you get the principal back but you can get only 0.0% pa (like Japan or Europe) from now on 😦

When an investor is comparing 2 otherwise identical bonds, the one with the call option is worth less because there’s a risk (“call risk”) that cash-flow will be lower after the call option exercise.

Yield-to-worst is basically the lowest yield that could be realized if the embedded call is exercised. Buyer assumes what could go wrong WILL go wrong — Murphy’s Law. In such an analysis, the worst case to the investor is an early repayment, and reduced cash-flow subsequently.

However, my friend Ross (in the bubble room) in Macquarie said the callable bond they hold could get called and as a result they could realize a windfall profit. He said it’s a good thing but I believe they would forgo the high coupon interest.

bond "callable" provision is a risk to investor#CFA textbook

See also post on why a bond’s callable provision is a risk.

P260 of CFA textbook on Eq and FI has a concise explanation.

– Suppose A issues a riskless non-callable bond, for a $109 price.
– Suppose B issues a callable bond, with identical features otherwise, for $106.8 price. The $2.2 price difference is the value of the embedded call option (to the issuer).

If issuer A uses $2.2 of the sales proceeds to buy a call option (to receive the coupons), then A’s position would match B’s. When rates drop to a certain level, the B issuer would call the bond, ending her obligation to pay those high coupons. A would continue to pay coupons, but his call option would produce a cash flow matching the coupon obligations.

Notice the callable bond is cheaper because the issuer holds a call option — to be exercised if low-interest happens.

Also notice — as expected IR volatility (implied vol) rises, the embedded option is worth more and the price spread between the A vs B bonds will widen.

In fact, since A price isn’t sensitive to vol, B price must drop — intuitive?

The B issuer is (believed to be) more likely to call the bond and refinance at a lower coupon, whereas the A bond will continue to pay the super-high coupon until maturity. So bond A is more valuable to an investor.

bonds ^ swaps – 2 subdomains in FI

3rd and 4th domains would be credit (including muni) and mortgage, but let’s put them aside, despite their very large market sizes.

There’s huge demand for IR swaps and riskfree bonds. To keep things simple (perhaps over-simplifying), we can say
– bonds are income-generation Investments, needed by every investor esp. serious investors including banks, fund managers and pension/insurance.
– swaps (on the long end of YC) and IR futures (on the short end) are needed for risk management (hedging) by large enterprises with interest rate exposures.
Both domains rely on their respective yield curves. The so-called curve instruments are never mixed. Let’s illustrate using USD
– swap curve is built from Libor instruments like Libor swaps, Libor FRA and ED deposits, but not Treasury instruments. In AUD market, there’s no Libor so people use BBSW.
– Treasury curve is built from T and T-futures, not Libor instruments

In Lida’s words, these would be the risky curve (AA curve) vs the riskfree curve (government curve).

Q: how about the OIS curve? I think it’s based on OIS instruments

OIS^Libor — 2 indices

contract maturity –up to 50Y, for both OIS and Libor swaps.

payment frequency – 1Y for long-term OIS swaps. Example – 50 payments for a 50Y OIS swap. If term (ie maturity) is below 1Y, then there’s just one payment, at maturity. See

fixing frequency – 1D for OIS. In contrast, for USD Libor swap, typically 3M.

accrual period — Assuming a 3M payment frequency on the floating leg. The underlying “assumption” is a 3M (unsecured) loan to an AA-rated bank. In contrast, the OIS underlying loan is a 1D (unsecured) loan, typically to a bank. The underlying loan tenor is what I call the “accrual period”. Taking USD as example, the standard “accrual period” is 3M. So if you want a swap with floating accrual period of 6M, then you need to do a tenor basis swap. I feel OIS is simple. The accrual period is always 1D, as the underlying loan is 1D. The 1Y payment frequency is less important.

Both are unsecured loans, but OIS is a closer proxy for the riskfree rate. I guess overnight borrowing by gov (of the named currency) would be a better riskfree rate, but no such thing.

Libor IRS has zero value at inception. The trade execution price is the fixed rate, which equals the “average” of the expected floating payments… Overnight Index swap is a similar contract. The average is geometric, and the reference (i.e. floating) rate is the FF/EONIA/SONIA/… has concise details.

implicits in an ED fut price

At the heart of this price thingy is a __fwd-starting loan__. The price is related to the interest rate on this loan, also known as FRA rate or simply fwd rate. Traders basically guess at (“bet and “trade” are less intuitive) this rate.

Implicit – loan is 3M tenor

Implicit – loan starts 2 days after expiry of the futures contract.

Implicit – this fwd interest rate is always, always, always annualized

bond yield – liquidity premium ^ credit spread

I guess an issuer A’s bond may trade at a Higher yield than an issuer B’s bond, even if A’s credit quality is higher. Paradox?

One reason i can imagine is liquidity preference. Suppose you are an investor. Suppose the 2 bonds have the same coupon and other features. You as well as the market know that bond A is (slightly) less likely to default than B, but you may still be willing to pay a bit more for B, because it’s easier to sell it when you need cash.

Bond A may be an unknown entity, traded on very few markets (including bank’s private distribution networks), so there are far fewer bids of A then B. The lower market access leads to lower buyer competition, lower bid prices when you are forced to sell. 
Therefore, a prudent investor may be willing to pay more for bond B.
Since many investors behave similarly, B gets higher buyer competition, higher valuation and lower yield (spread).

Similarly, McDonald burgers may be smaller but more expensive than the no-name burger next door — liquidity

fwd curve^spot curve: different x-axis

ICAP terminology —
effDate :=  accrual starts. Therefore, fwd-starting contracts have effDate long after execution date.
maturity:= accrual ends.
tenor := accrual period length := maturity – effDate

A Fwd curve consisting of 22 points describes 22 FRA deals with identical accrual length, but 22 different accrual-start dates. (Both Mark Hendricks and NYU agree.)

spot rate r(t)= rate of a loan starting today, with maturity = t
fwd rate f(t) = rate of a FRA starting at t, with a standard and Implicit tenor (say, 6M)

  • spot curve with 33 numbers describes loans with 33 different loan maturities
    • discount curve is similar
  • fwd curve with 22 numbers describe FRA deals with Identical accrual length, but 22 different fwd-start dates.

So the x-axis has different meanings! Relationship between the 2 curves are described by Mark Hendricks + Bruce Tuckman with an intuitive explanation!

I believe the fwd curve (with /infinitesimal/ tenor) is based on the theoretical concept of instantaneous fwd rate (IFR)… but let’s not get bogged down with technicalities.

what departments use the yield curve

In one mkt risk system, the USD (no other currencies!) yc is used to compute FX swap points. That’s the only usage of the yc in that system.

Why some large investment banks have a sizable IT team supporting the yield curve(s) and update it a few times a day? I was told

… that a big user is the IRS desk. IRS contracts last many years. A portfolio may be highly sensitive to the interest rate at some point on the yc. A small shift of the yc may tip the entire portfolio from ITM to OTM. Note ITM/OTM is always for the swap dealer.

I feel if a portfolio is sensitive to the yc, then the trader needs up-to-the-hour yc to help guide his quoting and trading decisions.

use YC slope to predict 5Y bond’s return over the next 12M

Mark’s lecture 4 describes a famous “YC” carry trade strategy using T bonds. To keep things simple, we use zero bonds (coupon bonds same). Given a bond of 5Y maturity, next year’s return is defined as the NAV 12M from now vs the current NAV. In other words, the ratio of next year’s price over today’s price. It’s probably slightly above 1.0 or perhaps below 1.0.

This return factor is observable 365 days later, but we can predict it using the currently observable term spread, i.e. the 5Y yield – the 3M yield seen today. 
Idea is, if the slope is steep, then we expect that return to be high. Steep slope basically means we observe high yield at the 5Y point on today’s yield curve. Suppose it’s a high 8.8%. If we were to hold the 5Y bond to maturity, we would realize approx. (without compounding) 44% return. Instead, we actually plan to sell the bond next year, so we are forecasting this bond price next year, effectively the 4Y point on next year’s yield curve. (Apply P/Y conversion)
We expect that yield to remain around 8.8%, or equivalently, we expect the market yield on this same bond to remain. That would be higher than the riskfree rate (represented by the 1Y yield, say 0.8%).

However, If we are unlucky, the return factor (observable in a year) could come below the riskfree return factor today. (Note both deals cover the same loan period.)
* But then, we could cancel our plan and hold the bond to maturity and realize a total return of 44%. This is somewhat risky, because bond yield could rise further beyond 8.8%, hurting our NAV before maturity.
* Crucially, if the return over the next 12 months turns out to be lower than riskfree rate, then the subsequent 4 years must return more than 8.8% pa, since the return-till-maturity is fixed at 44%.

I have a spreadsheet illustrating that yield shifts in the next year may hurt the then NAV but the total return till maturity is unaffected.

EH (expectation hypothesis) actually says there’s no reason to prefer the 5Y vs the riskfree rate. Contrary to EH, empirical data show that today’s slope is a good predictor of the return over the next 12 months.

alpha or beta@@ illustrated with treasury spread

Mark's slide P7.47 on Liquidity risk posed the question — the widening spread between on-the-run vs off-the-run treasuries invite an arbitrage strategy. The excess return is often considered alpha. Maybe it is actually beta, because this excess return is not free lunch. Instead it is a reward for taking on liquidity risk. Off the run is less liquid when you are forced to sell it before maturity. It's also less valuable as a collateral.

Bottom line – Any excess return that's associated with some risk is beta not alpha.

collateralized 100% financing on a treasury trade

Develop instincts with these concepts and numbers — common knowledge on IR trading desks. P10 of the LTCM case has an example on Treasury trading with repo financing.

Most buy-side shops work hard to get 100% collateralized financing. Goal = avoid locking up own capital. 100% means buying $100m T bond and immediately pledge it for repo and use the borrowed $100m for the bond purchase. If only $99m cash borrowed (1% haircut), then LTCM must commit $1m of own capital, a.k.a. $1m “equity investment”.

P14 explains that many buyers choose overnight and short term repo, but LTCM chose 6-12M term repo, where the repo rate is likely higher.

LTCM managed to keep most of the $6.7b capital in liquid cash, generating about 5% interest income annually. This $350m interest adds almost 50% on top of the average $750m trading profit annually.

4th data source to a yield curve – year-end Turn

for more details.

The year-end turn of the yield curve is defined as the sudden jump in yields during the change of the year. This usually happens at the end of the calendar year, reflecting increased market activity related to
year-end portfolio adjustments and hedging activity….When there is a year turn(s), two discount curves are
constructed: one for turn discount factors and one for the discount factors calculated from the input instruments after adjustments and the discount factor at any time is the multiplication of two.

trading swap spread – LTCM case (with wrong intuitions

See Harvard Biz School case study 9-200-007 on LTCM. I feel this is a good simple scenario to develop analytic instinct/intuition about IRS.

I believe USD swap spread is similar to the TED (which is 3M). A very narrow TED means on the lower side T (i.e. treasury) yield too high and on the upper side fwd Libor too low.

T yield too high means T bonds too cheap. Therefore, LTCM would BUY T bonds.

Expected series of Libor cashflow is too low, so the equivalent fixed leg is also too low. Therefore LTCM would PAY the fixed rate. The par swap rate is the price you lock in today, which buys you the Libor stream, which you believe to be rising.

In the orange case, you as a Libor/orange buyer lock in a price today and you expect the oranges to get bigger soon.

For a 10Y swap, we are saying the forward Libor rates over 3->6M, 6->9M, … 120->123M… are too low and may rise tomorrow. There are many wrong ways to interpret this view.

correct – Since the floating income will rise, we would want to receive those future Libor interests.


correct – We can think of the floating leg as a hen giving eggs periodically. The market now forecasts small eggs, but LTCM feels those eggs will be bigger, and the hen is under valued. So LTCM buys the hen by paying the (low) fixed rate.

trading swap spread – LTCM case, again

Here’s a simpler way to look at it. When the swap spread is too narrow, T yield is too high and swap fixed rate is too low. …. (1)

Key – use a par bond as a simple (but not simplistic) example, so its yield equals its coupon interest rate.

Now we can rephrase (1) as — T bond interest too high and swap fixed rate too low, and they are going to widen. Now it’s obvious we should Buy to receive T interest (too high). And we pay the swap fixed rate (too low), and consequently receive Libor.

When we say “swap rate is too low and is likely to rise in 4 months”, i think we are predicting a “rise” in Libor. Swap rate is like a barometer of the Libor market and the Libor yield curve.

A simple “rise” is a parallel shift of the Libor yield curve. A less trivial “rise” would involve a tilt. Rise doesn’t mean upward sloping though.

It’s rather useful to develop instinct and intuition like this.

difference – discount factor ^ (Libor,fwd,spot…)rates

Discount factor is close to 1.0, but all the rates are annualized and usually between 0.1% ~ 8%.

This simple fact is often lost in the abstract math notations. When I get a long formula with lots of discount factors, forward rates, (forward) Libor rates, floating payments, future fixed payments… I often substitute typical numbers into the formula.

Also, due to annualizing, the rate number for overnight vs long tenors (like 1Y) are similiar, at least the same order of magnitude.

FX vs IR trading desks, briefly

Now I know that in a large sell-side, FX trading is “owned” by 2 desks – the “cash” FX desk and the IR desk. Typically, anything beyond 3 months is owned by the Interest Rate desk (eg STIRT). It seems that these FX instruments have more in common with interest rate products and less in common with FX spot. They are sensitive to interest rates of the 2 currencies.

In one extreme case every fx forward (outright?) deal is executed as a FX spot trade + a FX swap contract. The FX swap is managed by the interest rate desk.

FX vol is a 3rd category, a totally different category.

y use OIS instead of Libor discounting — random notes

Cash-flow discounting (to Present Value) should use a short rate, “instantaneously short”, ideally a risk-free rate, which is theoretical. In reality, there are various candidates —

Candidate: treasury bill rate. The rate is artificially low due to tax benefit leading to over-demand, higher price and lower yield. There are other reasons explained in ….

Candidate: Libor. In recent years, Libor rates are less stable compared to OIS. Libor is also subject to manipulation — the scandals. OIS is actual transaction rate, harder to manipulate.

Q: why OIS wasn’t chosen in the past?
%%A: not as actively traded (and influential) as Libor

selling an existing IR swap@@

I guess technically we can’t sell an IRS as it’s not a product like an orange (or a house, or an option) with an owner. A IRS is a long-term bilateral agreement. Analog? I can’t “sell” my insurance policy to someone else.

A liquid swap market lets us offset our Libor exposure —

Suppose I’m a Payer in Deal 1 with Citi, to receive Libor and pay fixed 4.5%. Five hours (or 5 days or 5 months) later, I could become a Receiver in a JPM deal (Deal 2) to pay Libor and receive fixed 4.6%. Therefore I get rid of my Libor exposure, as long as the reset dates are identical between Deal 1 and Deal 2. But strictly speaking I haven’t Sold an existing swap. Both are long-term commitments that could in theory be unwound (painful) but never “sold” IMO.

By market convention, the counterparty paying the fixed rate is called the “payer” (while receiving the floating rate), and the counterparty receiving the fixed rate is called the “receiver” (while paying the floating rate).

(instantenous) fwd rate

I believe fwd rate refers to an interest rate from a future start date (like next Aug) to a future maturity date (next Nov). We are talking about the market rate to transpire on that start date. That yet-unknown rate could be inferred (in a risk-neutral sense) today, using the live market rates.

The basic calc is documented in my blog …

When the loan tenor becomes overnight (or, theoretically, shorter than a nanosec), we call it the instantaneous fwd rate. This rate, again, can be estimated. Given observation time is today, we can estimate the fwd rate for different “fwd start dates”, denoted tau. We can plot this fwd rate as a function of tau.

FRA/ED-Fut: discount to fwd settlement date

–Example (from Jeff’s lecture notes)–
Assume on 12 Nov you buy (borrow) a 3×9 FRA struck at 5.5% (paying 5.5%) on 1M notional. On 12 Feb, 6M Libor turns out to be 5.74% , compensation due to you =

$1M x (0.0574-0.055) * 180/360 / (1 + 0.0574*180/360) = $1166.52
——–Notation ——-
Libor fixing date = 12 Feb

“accrual end date” (my terminology) = 12 Aug.

settlement could be either before or (occasionally) after the 6M loan tenor. This example uses (more common) fwd settlement.
disc factor from 12 Aug to 12 Feb = 1/ (1 + 0.0574 * 180/360)
Note the “interest due date” is always end of the 6M accrual period. Since we choose fwd settlement, we discount that cashflow to the fixing date.

annualized interest Rate difference = 5.74 %- 5.5%
pro-rated  interest Rate difference = (0.0574-0.055) * 180/360
difference in interest amount (before discounting) = $1M x (0.0574-0.055) * 180/360. This would be the actual settlement amount if we were to settle after the 6M loan period. Since we choose fwd settlement …

discounting it from 12 Aug to 12 Feb = $1166.52
Now we come to the differences between FRA and ED Futures.
1) a simple difference is the accrual basis. ED futures always assumes 90/360 exactly. FRA is act/360.
2) Another simple difference is, ED Futures always uses 3M libor, so our example must be set on Mars where ED futures are 6M-Libor-based.

3) The bigger difference is the discounting to fwd settlement date or fixing date.
– EDF gets away without the PV discounting. It takes Libor rate as upfront interest rate like in Islamic banking. Since Libor turns out to be 5.74% but you “bought” at 5.5%, the difference in interest amount is, under EDF, due immediately, without discounting to present value.
– the payout, or price, is linear with the Libor rate L.
– this is essentially due to daily mark-to-market margin calculation
* FRA takes Libor rate as a traditional loan rate, where interest is due at end of loan period.
** under late settlement, the amount is settled AFTER the 6M, on the proper “interest due date”. (Linear with L)
** under fwd settlement, the amount is settled BEFORE the 6M, but PV-discounted. This leads to a non-linear relationship with libor rate and convexity adjustment.

family photo ^ family video – yield curve

snapshot – The yield curve (yc) is a snapshot.
snapshot – term structure of IR is another name of the yc.
snapshot – discount curve is the same thing

On a given snapshot, we see today’s market prices, yields and rates of various tenors[1]. From this snapshot, we can derive[2] a forward discount factor between any 2 dates. Likewise, we can derive the forward 3M-Libor rate for any target date.

Looking at the formula connecting the various rates, it’s easy to mix the family photo vs the family video.
– family photo is the snapshot
– family video shows the evolution of all major rates (about 10-20) on the family photo.
** an individual video shows the evolution of a particular rate, say the 3M rate. Not a particular bond, since a given bond’s maturity will shrink from 3M to 2M29D in the video.
All the rate relationships are defined on a snapshot, not on a video.

I guess we should never differentiate wrt to “t”, though we do, in a very different context (Black), integrate wrt “t”, the moving variable in the video.

An example of a confusing formula is the forward rate formula. It has “t” all over the place but “t” is really held as a constant. The t in the formula means “on a given family photo dated t”. When studying fixed income (and derivatives) we will encounter many such formula. The photo/video is part of the lingo, so learn it well.

Also, Jeff’s HJM slide P12 shows how the discount bond’s price observed at time t is derived by integrating the inst fwd rates over each day (or each second) on a family photo.

[1] in an idealized, fitted yc, we get a yield for every real-valued tenor between 0 and 30, but in reality, we mostly watch 10 to 20 major tenors.

[2] The derivation is arbitrage free and consistent in a risk-neutral sense.

bond ^ deposit , briefly

Bond and deposit are the 2 basic, basic FI instruments, underlying most interest rate derivatives.

Both pay interest, therefore have accural basis, like act/360 or 30/360

Both have settlement conventions, such as T+2. Note Fed Fund deposit is T+0.

# 1 difference in pricing theories — Maturity value is know for a bond, but in contrast, for some important deposits (money-market deposits) we only know the total market value tomorrow, not beyond. Though many real life fixed-deposits have a long tenor comparable to bonds, the deposits used in pricing theories are “floating” overnight deposits.

# 2 difference — Bond has maturity value exactly $1 and is traded at a discount before maturity, making it an ideal enbodiment of discount factor. A Deposit starts at $1 and grows in value due to interest.

–1) Bonds
eg of bonds — all treasury debts, corp debts, muni debts.

Has secondary market

bonds are the most popular asset for repo.

–2) Deposits is fairly similar to zero bonds.
eg of deposit — Fed Fund deposit, or deposits under other central banks. Unsecured
eg of deposit — Eurodollar deposit, in about 20 major currencies. Unsecured

OIS is based on deposits (Fed Fund deposit)

Libor is based on eurodollar deposits, for a subset (5) of the currencies.

Libor IRS and OIS IRS – all based on deposits.

No secondary market.

I feel deposits tend to be short term (1Y or less)

yield curve -> fwd rate, spot rate …

This is yet another blog post about yield curve, fwd rate, spot rate etc

Let’s say we have a bunch of similar derivative instruments [1] on IBM. Each has an expiry date at each month end. For the Feb instrument, on the expiry date (end of Feb) all uncertainties would vanish and the value of the instrument would be determined/fixed. Therefore it’s practically possible to cash settle on that day. Alternatively the contract may specify a later maturity date (eg 3M from expiry/fixing) for the actual cashflow to occur.

Today, I can record all the current prices of this family of (eg 9) instruments. A minute later I can record their new prices… I keep doing it and get 9 (time-series) streams of live prices.

The “live yield curve” is something similar. The 9 instruments are the 9 deposit maturities we monitor, perhaps {1M, 3M, 6M, 1Y, 2Y, 3Y, 5Y, 10Y, 30Y …} These prices, after converting to yield numbers, actually comprise a 9-point yield curve. From this snapshot yc, we can derive many useful rates, such as (instantaneous) forward rates, spot rates, short rates… all valid at this moment only.

An additional complexity is discounting the cash flow. Whether the cash flow occurs on fixing date or on maturity date, we need to discount to valuation time (moment of observation), using a discounting curve such as the OIS curve.

Every minute, we re-sample live prices, so this 9-point yield curve (and the discount curve) shifts and wiggles by the minute.

[1] Could be bunch of forward contracts, or bunch of binary put options etc

Libor, Eurodollar, OIS, Fed Fund rate … common features

deposit — All are based on the simple instrument of “deposit” — $1 deposited today grows to $1.00x

unsecured — when I deposit my $1 with you, you may go down with my money. Credit risk is low but non-zero.

inter-bank — the deposit (or the lending) is between banks. The lending rate is typically higher when lending to non-banks.

short-term — overnight, 3M etc, up to 12M.

bond investment is safe, in the long run (Hendricks)

If you buy a bond at a $98.5 (1.5% discount from face value), and hold it till maturity, then there’s no uncertainty in how much you will get at the end.

I heard a lot of “common sense” wisdom that bond appreciates and drops with interest rate, and therefore volatile and risky.

(I usually assume default risk is very low for the bonds I consider, at least much lower than the sensitivity to yield.)

However, if indeed a bond loses value due to rate hike, then bond holders always have the “safe” option to hold it till maturity. Its price will eventually rise and end up exactly $100.  Therefore there’s absolutely no uncertainty about the terminal value like there is about options, stocks, or futures contracts.

This is one of the most fundamental features of bond as an asset class. I don’t know another asset having this feature.

##most popular IR drv

According to an IRD veteran friend…

1) Swaps swaptions
2) FRA
3) caps floors. Rate locks aren’t popular

These are widespread enough to qualify as vanilla IR derivatives. All other IR derivatives are exotic.

Both vanilla and exotic are customizable and therefore “structured”.

Exotics are by definition less common, but among them the relatively popular ones are–

* range accruals (often callable)
* Cliquet? popular among Eq derivatives, not among IR derivatives.

yield CURVE ^ yield/price CURVE

There are many curves in bond math, but these 2 curves stand out as by far the 2 most useful.

* the yield curve and twin sister the discount curve, aka the swap curve
* the yield/price graph.

Note duration, convexity, dv01 are defined on the y/p curve.

For a given bond or for a given position, the y/p curve is fundamental. Most bond characteristics are related to or reflected on the y/p curve.

callable bond is a real option #my take

If you are only working in the fixed income space, you may feel you don’t need option knowledge. Well, the ubiquitous callable bond is an embedded option. (Puttables are less ubiquitous.)

To the bond holder (or buyer), A long position in a callable bond is equivalent to
+ a long in a non-callable bond
+ a short position in a call option, which is like a giving out a shopping voucher “get a beer from me for $1”. If exercised, bond holder loses the “asset” — the long position in the regular bond

In other words, You buy a callable when you give counter party the right to “call it away”.
———-To the Issuer (not the dealer), a short position in a callable is equivalent to
+ a short position in a non-callable bond i.e. an obligation to pay fixed coupons + principal
+ a long position in a call option i.e. a right to pay holders a stipulated amount to acquire an “asset” that’s a perfect hedge for (100% cancels out) the short position.
Every call has an underlier asset. For a callable bond, the asset is the piece of paper representing ownership of a bond. The paper also has strips called coupons. When the issuer exercises the call, issuer pays par price to “buy” back the paper. Assuming there was only one holder for this issue, then issuer has no more liability. The bond ceases to exist.

y bond traders need drv, briefly

An IT veteran said — either to reduce or increase exposure. Exposure to many variables, not only interest rate.

As market maker, the “Reduce” is essential.

For both buy-side and sell-side, the “Increase” can be attractive. I feel derivatives provides leverage — up to hundreds of times higher exposure. Lida (MSFM) discussed “exposure…

libor vs gov bond — 2 benchmarks

Most credit instruments, all (secured/unsecured) loans, all IR products, most derivatives based on FX, IR or credit (I'd say virtually all derivatives) need to apply a spread on a reference yield. If the deal has a “maturity date” or “delivery date”, “call date” [1], then we look for that date on the reference yield curve and read a yield number like (222 bps pa) off that curve. That number is the reference yield, a.k.a reference spot rate. You can convert that to a discount factor easily. There are also straightforward and well-defined conversions to/from fwd rates, driven by arbitrage principles.

[1] perhaps among a series such dates.

Question is which reference yield curve to use. Most companies use a single, consolidated curve for each currency. One of the biggest muni security trading desks in the world has just one yield curve for USD, which is typical. Another megabank has a single live Libor curve for the entire bank, updated by the minute.

If you use more than one yield curve built from different data sources, then for any maturity date, you would read 2 yield numbers off them. If sufficiently different, you create arbitrage opportunity and your valuations are inconsistent.

On the short end of the IR space the reference curves are 1) Libor 2) Fed Funding (USD only). Libor is more popular.

On the long end, T-bond dominates the USD market. Many governments issue similar bonds to create a reference riskless rate.

However, the most liquid IR instruments are probably more realistic and reliable as a reflection of market sentiment. ED futures, Bund futures, T-bond inter-dealer market rates are examples.

real world libor curve bootstratp – facts&&figures

* first 3 months use libor deposit rates — O/N, 1w, 2w etc

* 3 month – 3 years use libor futures. For USD, CME lists 3-month futures contracts 10 years out, but liquidity is good for first 2-5 years only.

* 4 years – 50 years use swap rates from tradeweb or bloomberg

Some curves (EUR? GBP? BRL) also use zero-coupon bond prices or year-end adjustments. All 5 types of instruments provide a maturity/rate pair, i.e. one data point on the yield curve space, to be fitted.

Note in this investment bank, US treasury prices are not used. I wonder why. I guess this is strictly a Libor yield curve. Base instrument is ED loans between AAA-rated banks in London, not US treasury bonds. Lida said “risky curve vs riskless curve”

FI — divided into short-term ^ long-term markets

Fixed Income is broad divided into rates + credit markets. Rates business is further divided into short term rates (i.e. money market) and long term rates.

(In terms of pricing, risk…) Since rates are the basis of credit,
^ short term rates are the basis of short term credit — eg repo, short term corporates/munis
^ long term rates are the basis of long term credit

The instruments are obviously different between short vs long term rates. Therefore the /markets/ are distinct, since a short term instrument (like ED futures) is traded only in the short term market.

However, Swap curve covers short/long terms, just like T yield curve.  In contrast, Libor is strictly (unsecured) short term lending.

going long Libor .. means@@

  • If you are long an instrument, you are longing for it to rise. Your delta is positive i.e. in your favor.
  • If you are short an asset, you have a shortage for it and hope it depreciates. Your delta is negative.

… That’s my new “cheat-sheet”. Now let’s compare important derivatives:

  1. I think the meaning of “long eurodollar futures” is non-intuitive. If I long eurodollar futures, I want the expiration-date Libor number to Drop. Suppose today is a week before expiration and today’s Libor is 222 bps, I would long for the rate to Drop to 180 on my expiration date. That would mean my futures’ final price would be 98.2
    1. If you are long the futures, then I guess you are a fixed-rate lender
  2. FRA — you buy the FRA (go long) by agreeing to pay the fixed rate. You want the fixed rate (loan-start-date Libor rate) to rise.
    1. if you buy a FRA then you are a fixed-rate borrower
  3. IRS — fixed-payer is long a swap. As illustrated in earlier posts, the “oranges” to be delivered to you is the stream of floating interest payments

Q: Look at any financial instrument with a fluctuating price. An instrument not necessarily transferable or tradable, but always bettable. What does it mean if I “long” this instrument?
– i pay a fixed price for it, today or predefined dates. No runaway — i must pay.
– i get the right to demand cash flow from this instrument
– I long this instrument because I benefit from appreciation.

This complicated explanation is needed in the Libor and IRS context.

Simplest example — if I long IBM, i buy the stock and stand to gain if it rises.
Simple example — if I long copper, i buy a copper futures contract at $800 ie I pay this price today, and hope the price at contract expiration is higher.
example — if I long 10-year T bond, i buy a futures contract at $101….

In Libor IRS, I’d say every fixed-payer is long Libor. Let me repeat —

* Fixed rate payer is long Libor.
* You BUY a swap if you are long Libor. [1]
* you buy a swap if you are short the bond-market [2]

[2] because you want bond prices to drop and yield to rise along with Libor.

[1] because the fixed rate agreed today is based on current Libor. If I go long on Libor, say, the 90-day eurodollar deposit rate, then I enter IR swap
– I pay a fixed rate of 222 bps/year — ie the “price”, computed at current Libor
– I receive a yet-unknown floating rate, hopefully above 222 bps
– I am long Libor therefore I stand to gain when Libor rises

Fixed Income -more complex due to time dimension

FI is more complex than eq / FX / commodities, largely due to the time dimension.

There’s a maturity in Every fixed income contract.
There’s a maturity in Every derivative contract. For clarity Let’s use the alternative term “expiration”.
There’re 2 distinct maturities in a typical Fixed Income derivative contract.

Do eq/FX/comm spot trades have a time dimension? I don’t know many. FX spot trades have valuation dates, typically T+2, but T+1 for CAD…

Now, every derivative contract has an expiry date. This time-dimension factor interacts with maturity, in a way that’s unique to Fixed Income instruments.

The time dimension is critical
– because prices change over time.
– because volatility changes over time. Roughly half of all derivatives involve vol/optionality.
– because interest rate and credit spread change over time, but more slowly
– because many investors borrow money to trade, at an interest.
– because of time value of money. Money received in the future is worth less than money received today, and the difference is mathematics.
– because interest rate has a term structure
– because der contracts are often held open for longer timeframe than cash positions. Reason?
** der is often used to hedge away risk;
** many contracts can’t be transferred or closed out early.

Yield as relative value comparator

Yield is the most versatile, convenient (therefore most popular and practical) soft market data for relative value comparison. Yield lets you

– compare across currencies
– compare across time horizons (yield curve)
– compare across credit qualities (credit spread)
– compare across disparate coupon frequencies including zero-coupons
– compare across disparate call provisions
– compare government vs private issuers
– compare across vastly different industries
– compare across big (listed) vs small companies and even individual borrowers
– compare across eras like 70’s vs 90’s
– compare with interest rates. In fact, lenders use credit spread and prevailing IR to derive a lending rate on each loan.

Why is yield such a good comparator? Yield is a soft-market-data item derived using many inputs. Yield, in one number, captures the combined effect (on what? Of course valuation) of many factors such as

* different credit qualities
* different probabilities of default
* different embedded options
* different coupon rates
* comparable (but different) maturities

Without capturing all of these differences, it’s unwise to even attempt to compare 2 bonds. You get an obviously biased comparison; or you get an incomplete comparison. No info is better than misleading info.

Yield is so widely adopted that major data sources directly output yield numbers, making yield a “raw” market datum rather than a “soft” market datum.

3rd party evaluation service for structured FI instruments

Products covered – 200,000 structured FI instruments in multiple currencies, including

* regular CMO (biggest group)
* hybrid CMO
* unsecuritized whole loans — In contrast, those securitized loans are sliced up and not “whole”

Twice a day, each bond would get a new evaluation price. Price is always different from previous day because
1) accrued interest
2) benchmark interest rate and yield changes daily,
3) built-in optionality may kick in

Benchmark interest rates in this space are — Libor, government bond yields in different currencies

Upstream Market data includes
* transactions of the day
* indications of interests (like quotes) by both dealers and their clients
= either come in end of the day or come in by email

Methodology – take in market data to CALIBRATE the models. Feed them into Intex engine ..

another large Interest Rate trading desk

Gov desk + IRD desk

IRD trading desk is very close to Gov bond trading desk which includes agency bond.

IRD covers IRS, ED futures, T futures, vanilla derivatives and exotic derivatives .. some of these derivatives take a long time to price.

Core of the entire IRD system is the risk engine. It “Provides real time risk assessment to traders” — a one-sentence explanation by an IT manager in the team.

Gov desk has higher trade volume, mostly flow trades, with some prop trades. IRD vol is lower – about 1000 trades a day, with 8000 positions in one of the top International i-banks.

decreting vs accreting curves in a YC Group — basics

Suppose you hold an asset (say a stock) on borrowed money. Over 1 year,

A) You pay interest on the loan. Interest amount is determined by your credit rating and the prevailing risk-free rate (but more commonly Libor). If your credit is excellent, then this interest amount primarily reflects inflation and the gradual diminishing purchasing power of one unit of USD (or whatever currency).

B) The asset in your portfolio also accrues in market-value in USD, due to dividend or coupon income, or you can lend it out on the repo market to earn a fee.

Both of these sums grow with holding duration. A partially offsets B. Net appreciation of the asset is often positive. Note A depends on the borrower and B depends on the asset.

I believe most professional traders (buy-side or sell-side) do most trades on borrowed money — Leverage. They start with $1m, and through credit relationship, can use $10m — leverage ratio of 10.

Even if you trade using your own $500k, you are still “paying” or forgoing the interest (opportunity cost) you could earn on the $500k.

risk free bond isn’t risk-free

(Thanks to Eric for the enlightenment.)

Risk free treasury is credit-risk free, but not rate-risk free.

If you buy a large quantity of 30-year long bond at $99 and hold it till maturity, and valuation drops steadily from $99 to $98, $97… then your balance sheet will reflect an unrealized loss every year, a large loss if the position is large.

Many people actually looks at your balance sheet. So this large unrealized loss will affect your company's financial health.

If you use this bond as collateral, the depreciation will affect its collateral value. If you borrow money using this collateral and lender demands additional collateral due to depreciation, this risk-free bond could make you bankrupt.

Compared to equities, risk-free bonds won't lose the entire value. You will get back the face value if you hold it to maturity, but your investment has a poor return relative to the prevailing interest rates available on the market. Suppose your coupon is 100bps, but interest rate moves up after your purchase. If you don't liquidate the losing position, you would keep earning the 100bps while other long-term investments would have fetched 200bps returns.

comparing Libor rate and treasury yield@@

(See also TED ie the spread for a 3-month maturity.)

You can compare 12-month USD Libor rate (i.e. Eurodollar deposit interest rate)  with a zero treasury’s yield of the same maturity.

Note Libor covers short maturities, overnight to 12 months, whereas T covers 3 months – 30 years. Even when there’s a match on maturity, usually you can’t directly compare Libor rates with a treasury yields, because treasury generates coupon payments.

+ve carry in treasury market

In many markets, cost of carry is a cost to a holder of a security, because …. frequently it’s bought under borrowed money. A typical short term interest is 200%/yr, around 0.5bps/day.
In the Treasury market (perhaps in agency and muni too), each day you earn accrued interest, which is typically 500bps/yr or about 1.5bps/day. Positive carry.

The earlier a buyer settles the trade, the earlier she starts earning accrued interest[1]. As a result, the earlier a buyer settles, the higher the price to pay.

This reminds me “start investing in 401k early, so you start earning returns” has numeric examples points out that in the comm market cost of carry means storage cost.

unspoken assumptions about a Libor value

By default, Libor refer to the USD lending rate. Other currencies are actually covered by Libor too.

Usually people refer to overnight rate or 3-month rate, but actually Libor covers other maturities.

Libor is inter-bank lending. Borrower is a big bank and has the best credit rating but lower credit than US government. That’s why TED spread is always above 0.

FI desk — how long to hold a FI position

See also post on repo-based automatic financing for treasury trades.

A typical FI trading desk holds cash positions only for a few days, but could hold derivative positions till expiry.

A typical cash bond position ties up much higher capital than equity positions, so even the biggest banks don’t have so much working capital to hold many big FI positions for long. Even if a big FI position is profitable, it might take up too much capital so we can’t do enough other trades. Holding big positions indefinitely — is prop trading.

Margin account allows the desk to trade much larger volumes of securities. Common in options, futures, even FX spot, but I don’t think it’s common for cash bonds.

In this context, most common (and effective) derivative is a fixed-floating IR swap. It doesn’t tie up massive capital.

typical FRA scenario, inspired by CFA Reading 71

Today Jan 1 end user (say a top-rated borrower like IBM) anticipates it needs to borrow $xxxx in 3 months (i.e. 1 Apr), for a term of 6 months. A dealer bank (say GS) quotes 550 bps/year. IBM agrees.

Q: what is the quotation about, specifically?
A: GS offers to lend IBM $xxxx for a 6-month term from 1 Apr to 30 Sep [1]. Rate is 550 bps, regardless of the fluctuating 6-month Libor rate.

[1] more likely to be a fixed 180 days. I feel this way Libor rate is more comparable.

Q3: Locking — IBM effectively locks its borrowing cost at 550. Why?
A3: IBM will inevitably borrow from open market comes 1 Apr. If that becomes 600 bps, IBM pays 600 but _receives_ the difference (from FRA contract). If that becomes 500, IBM pays 500 but _pays_ the difference to GS.

Q3b: why “inevitable”? Why not borrow from the FRA dealer.
%%A: I guess GS may not have enough fund to lend; IBM may borrow from multiple banks; IBM may cancel its project. IBM just wants to lock down the borrowing cost in case it needs to borrow on 1 Apr.
%%A: the contract doesn’t require both parties to physically /close/ the 3-month loan. The contract stipulates cash settlement as in A3 above. Therefore IBM doesn’t have to take the loan or take it from GS.

If Libor becomes 600 bps on 1 Apr, then IBM is lucky to lock its borrowing cost at 550 bps. It receives a sum from GS equal to the difference.

Q5: Why lucky?
A: without the FRA, IBM is likely to end up paying 600 bps 😦
A: withe the FRA, IBM gets to borrow at a pre-agreed lower rate of 550 bps, or IBM could cancel the project and pocket the FRA profit as a windfall.

Q5b: what if it turns to be 500 bps?
A: IBM will end up paying 500 on the physical loan, but on 1 Jan, as a borrower IBM needs predictable borrowing cost, so it can plan long term.

Therefore IBM wants Libor to rise i.e. it is long Libor, but IBM doesn’t get unlimited profit as it probably still needs to borrow at the high Libor rate.

Q8: in a commodity (similarly stock / bond / FX) forward contract, there’s a physical item to be delivered at expiration. What’s the “item” in FRA?
A: At expiration on 1 Apr, IBM receives the right to the 6-month deposit interest at prevailing Libor rate. At the same time it must pay the 550 bps to GS. Net payment is not on 1 Jan, not 30 Sep, but on 1 Apr.
A: The “item” is the 6-month deposit interest at prevailing Libor rate, be it 600 or 500. IBM pays 550 bps (times notionally, and prorated 6-month) to receive this item.

Q8b: how is this similar to a commodity forward?
A: At expiry, IBM receives the commodity, and pays a pre-agreed contract price, not the prevailing price. In FRA, the physical deliverable is … a piece of paper for the interest on a CD, 6-month CD at the prevailing Libor rate at, say, 600 bps / year.

popular FI drv in Asia and in muni trading desks

According to a FI professional in Singapore…
Structured/exotic FI derivatives are much less liquid in Asia than in US.
MBS/ABS? —- People trade US MBS.
Credit Default Swaps — this is main credit product, so popular.
Collateralized Debt Obligations — CDO was very hot before 2008 financial crisis but seems dead after financial crisis
interest rate swaps — this is main rates product, so popular.
swaptions — main rates product, popular.
Citi muni trades a lot of swaptions, caps/floors, treasury futures and also CDS. IRS is the cheapest and most basic derivative, and probably the most widely used.

As of 2012 in US, most common IR derivatives after IRS are 1) Swaption and 2)caps/floors

euro dollar deposit, spot, forward, FRA, IRS

Eurodollar (without “futures”) is a monolithic (non-derivative) CD with a term always from _today_ for 1 day (overnight), or 7 days, 14 days … 3 months, .. up to 12 months.
* Eurodollar deposit rate is a always spot rate, not a forward rate.
* Libor 11am announced rate is always a spot rate, not a forward rate.

In contrast,
– an ED futures price is always a forward rate, not a spot rate.
– the rate in an ED “contract” is always a forward rate

An Eurodollar futures contract is a loan with a 3 month term always starting on an IMM date (never today).

In the interest rate business, “Spot rate” has broader, more generic meanings but here we focus on USD short-term spot rate. The libor eurodollar deposit rate is exactly that. Zeros (STRIPS) are also relevant. Example below is a Zero with semi-annual compounding.

Suppose spot rate == 250 bps/6months for a 2 year term (ie 4 x 6 months), it means on the present market, people are willing to close deals like “Take my $1M today. Repay in 2 years (1.025*1.025*1.025*1.025)*$1m=$1,103,800”

When studying IRS or FRA, always always bear in mind eurodollar is nothing but a simple time deposit but depositor is a lending bank and interest payer is a corporate borrower.

FRA is an contract or agreement referencing “tomorrow’s [2] published libor rate” for eurodollar deposit rates.

[2] In reality, “tomorrow” is more likely months away.

what is a cap (and floor) — a few points

Jargon warning — a bond has an _issuer_ and a _lender_ who receives coupons
Jargon warning — an option in Eq/FI has a _holder_ and a _writer_
Jargon warning — Don't worry about the concepts of “caplet” or “cap”. First understand the basics of call option.

A cap is an instrument combining a floating bond + an option, Specifically, a cap models “paying-capped-floating-coupon” using “holding a call option”. This sounds a mouthful, but in plain English, “I pay you a floating coupon but you write me a call option”

A call option's payoff has infinite upside. A floating coupon is also unlimited. A floating bond issuer has unlimited cost, so he needs an equally unlimited protection.

As a bond issuer, I issue a floating bond but hold a call option (as protection).

exactly y is a bond selling at a discount@@

…because this bond’s coupon rate is (perhaps significantly) lower than current “market rate” i.e. coupon rate of a bond trading at par, such as new issues.

“market rate” might also mean the spot interest rate or prevailing discount factor (i.e. yield) for the same credit quality?

Now, the same bond could appreciate in price in a few years, if market sentiment changes (yield curve drops), and this bond’s coupon rate suddenly looks attractive.

STRIPS: simpler than bonds

To a bond math student, STRIPS (zeros) are always simpler than coupon bonds. A zero-coupon bond, to a bond mathematician, is one single cash inflow at a pre-set time; whereas a coupon bond is a series of periodic payouts, each to be discounted differently to give you the NPV.

To derive the NPV of a coupon bond, you need to add up the NPV of each payout. There’s no clever “silver bullet” to bypass this summation.

2 ways to improve current income in your T-bond porfolio

Suppose your Treasury portfolio (safest investment) is yielding too low.

Technique level 1 — generally increases current income
* high yield bonds
** consider corporate instead of government bonds
* emerging market bonds, often denominated in USD, so if your PnL is in USD, then no fx risk. Soreign risk yes.
* callable bonds

Technique level 2 — often but not always increases current income
* foreign government bond with comparable liquidity and credit rating. Yield sometimes much higher, but with FX risk
* preferred stock of the same issuer
* longer-term bonds of the same issuer

y bond prices in 100 but face value in $1000

Exec summary — Correct way to compute transaction/clearance amount of 5 bonds sold at $99.02 is 99.02% * $1000/bond * 5 bonds

Convention — Bond prices are quoted as a percentage of face value aka principal amount aka par amount which is typically $1000

People often imprecisely talk about a bond price as “2 cents up” at $99.02. Truth is, that price means 99.02% of face value. Somehow, the implicit $100 face-value seems to live on.

In major sell-side dealing rooms of corporate/muni bond, minimum round lot is 5 BONDS with total face value $5000. Investopedia says — In bond trades, a round lot is usually $100,000 worth of bonds. That’s not the reality as of 2011.

y does yield follow libor rates@@

Q: When interest rates (libor or T or others) rises, why does yield rise?

I used to think yield reflects credit quality. I think that’s still correct, but that’s a static “snapshot” view — explaining different yields of 2 bonds at a point in time.

For now, focus on one particular bond. When interests rise, yield does rise but why? Remember yield is a discounting device, so why do traders discount the future payouts more deeply? Here’s my answer.

First, ignore credit risk and look at a $1000 T zero maturing in 12 months. Say we used to discount the payout by 201bp but now interest rate is higher for similar maturities in Libor and Treasury markets. Sellers of this zero would each discount the payout at 222bp. If you stick to 201bp, then you create arbitrage opportuniuty within this particular market alone. Therefore all sellers of this zero all advertize at very similar prices.

Q: What if there are only 3 big sellers in the market and we collude to keep our price high at 201bp? Crucially, to avoid arbitrage, bid yield has to be slightly higher at 202bp.

%%A: arbitrager can BUY 12-month libor (==a zero bond) at a higher yield of 300bp, and hit our bids at 202bp. There’s an arbitrage linkd between them.

%%A: arbitrager can BUY a mix of Treasuries of 6-month and 2-year, both of which have higher yields above 250bp and hit out bids at 202bp. There’s a arbitrage link across the maturities.

Effectively inflation is rising meaning future payouts getting “cheaper”. If you don’t discount future payouts, then people will see you are pricing your cash flow unfairly.

In the end, it’s a matter of valuation. If you still discount your coupon payouts at the old 201bp then you over-value your bond. Consequence is arbitrage. If competing sellers undersell, then you can’t sell. If you monopolize this market and also bid around that 201bp, then your bid will get hit due to arbitrage using similar instruments.

repos intuitively: resembles a pawn shop

Borrower (“seller”) needs quick cash, so she deposits her grandma’s necklace + IBM shares + 30Y T bonds .. with the lender i.e. buyer of the necklace. Unlike pawn shops, the 2 sides agree in advance to return the necklace “tomorrow”.

Main benefit to borrowers — repo rate is cheaper than borrowing from a bank.

haircut – the (money) lender often demands a haircut. Instead of lending $100m cash for a $100m collateral, he only hands out $99m.

requester – is usually the borrower. She needs money so she must compromise and accept the lender’s demand.

trader – is usually the borrower. Often a buy-side, who buys the security and needs money to pay for it. (The repo seller could be considered a trader too.)

Repo maturity is 1 day to 3M. Strictly a money market instrument.

Common collateral for most repos — Government securities are the main collateral for most repos, along with agency securities, mortgage-backed securities, and other money market instruments.

For every repo, someone has a “reverse-repo” position. In every repo deal, there’s a borrower and a lender; there’s a repo position on one side and a reverse-repo position on the other side of the fence.

Is repo part of credit business or rates business? Depends on the underlier. Part of the repo business is credit. Compare an ECN – can trade Treasuries and credit bonds.

UChicago Jeff’s assignment question is the most detailed numerical repo illustration I know of. Another good intro is

commodity^IR^FX interaction illustated

When oil, crops (and gold) appreciate against USD, the objective and rational Treasury bond buyers have to discount the cash “flows” deeper [1,2]. In other words, buyers are willing to pay less for those little papers known as coupons. Sellers reluctantly follow.

Commodity (and gold) prices can also rise due to USD depreciation.

QE2 in the US but not Eurozone ==> USD depreciation against Euro and other currencies =?=> commodities appreciation ==> T yields rise. Each cause-effect in the chain reaction is “one of the factors”.

Now, magnitude of this so-called yield shift depends on how important commodities are to an economy. Is it 3% of the economy or 30% of the economy?

I feel basic commodity prices are a major contributing factor beneath the consumer price index.

[1] “Flows” are the shorthand for the multiple future cash flows of any bond, except zeros.

1+2 bond buy/sell trade flows (offer/bid..)

For a muni bond dealer, there’s just 1 way to sell and 2 ways to buy.

Dealers sell via _advertising_ i.e. publishing offers. Price is fixed. Quantity limit is fixed too. Clients can trade any quantity within the limit.

Similarly, dealers can buy via advertising i.e. publishing fully-disclosed bids. Price is fixed. Quantity limit is fixed too. Clients can trade any quantity within the limit.

Buyers more often initiate a buy via RFQ (bid-wanted). Buyer sends an RFQ (Bid Request) for a given quantity. A dealer (among other dealers) responds with Bid-Answer along with a final price. If he likes a particular bid answer (or “bid”), buyer sends an order.

In all 1+2 cases, price is decided by the dealer.

bond yield — precise definition && example

* yield for a given bond (with a given maturity) is used to discount payout(s) and derive a fair price, aka present value or PV.

* it’s easy to derive price or yield from each other

* yield looks like an annual return, semi-compounded. eg 10% pa, but a coupon rate often looks like $102 pa.

* yield is about the only usable metric to compare bonds across maturities, coupon rates and face values.

* See the posts on discount factor, and how it discounts cash flows. Now, for math-challenged bond traders ;-), we simply assume the discount factor is consistent for every duration. So we discount every income using the same formula —
** cashflow[1] is discounted by 1/(1+10%/2)(1+10%/2) ie twice
** cashflow[1.5] is discounted by 1/(1+10%/2)(1+10%/2)(1+10%/2) ie 3 times
** see link

eurodollar futures — phrasebook

* deposits — first understand the simpler concept of eurodollar deposit
** remember the deposit rate (2% for 3 months) is a forward int rate (not a spot rate) beginning on an IMM date.

* futures — eurodollar futures, very similar to commodity futures

* guaranteed — by exchange
** counterparty — of every trade is the exchange
** margin and daily mark to market settlement, just as in commodities futures

* libor — the reference rate for ED futures
* speculator and hedgers — corporations borrowing money
* rate lock — you can lock in the 3m borrowing cost [1/1/2013 – 4/1/2013] at 2.01%. Similarly, you can lock in 2013 corn price at $1025/tonn.
* IRS pricing
* Libor curve — Note all rates are 3m USD libor rates. We plot the “expected” rates against 40 contract expiration dates.