VRM 9: Pricing Conventions, Discounting, and Arbitrage

Governments raise short-term money by issuing Treasury bills. A bill runs for one year or less and two features define it: the face value, also called the principal amount or par value, and the maturity date. The holder collects that face value at the end and nothing before, which makes a bill a pure zero-coupon security. Market makers post a bid quote, at which they buy, and an ask quote, sometimes called the offer quote, at which they sell.

Source: published bill quotes. The day column is derived.

Those quotes are not prices. Each measures the interest a bill earns. Writing Q for the quote, C for the cash price per USD 100 of face value, and n for the calendar days left to maturity, the link runs as follows.

The convention is clearest when n equals 360. Such a bill costs 100 minus Q and repays 100, so Q is the interest over a 360-day stretch measured against face value, and a shorter life scales it in proportion. Two features strike most investors as odd: interest measured over 360 days rather than a calendar year, and measured against face value rather than the money invested, which would instead give Q divided by 100 minus Q.

The conventions were settled when traders worked without computers or calculators, and a 360-day year with interest charged on a round face value kept the arithmetic manageable. Rates in early 2018 already looked low, and two years later, after aggressive Federal Reserve easing, the annual return on a bill was down around 0.13%.

A Treasury bond lasts more than a year from issue, and three features define it: the face value, the coupon rate, and the maturity date. Coupons arrive twice a year at half the annual rate applied to face value, so a 3% coupon rate delivers USD 15 twice a year for every USD 1,000 held. Payment dates run backwards from maturity in six-month steps, usually landing on the 15th of a month or at a month end. Instruments issued with one to ten years of life are often called Treasury notes, though pricing is identical.

Quotes run per USD 100 of face value, and on March 9, 2018 the list stretched from bonds with a week left to bonds with 30 years left. Trades settle one business day later, a convention written as T plus 1 settlement, so a purchase on Friday, March 9, 2018 settles on Monday, March 12, 2018.

Why the quoted price is not the money paid

Add accrued interest to the quoted price and the result is the cash price. Accrued interest covers the stretch from the previous coupon date to the settlement date, and it belongs to the seller, who held the bond over those days.

The two prices carry names. Quoted price is the clean price; the money handed over is the dirty price. Traders quote clean because it behaves better. A clean price moves only with interest rates, while a dirty price moves with rates and with the daily build-up of accrued interest at once, laying a mechanical saw-tooth over the market signal. Picture a Treasury rate of 3% at every maturity, holding there. A bond with a 3% coupon stays at par, so its clean price is flat, while its dirty price climbs through each coupon period and drops when the coupon is paid.

Arbitrage arguments nearly always involve selling something you do not own, so the mechanics come first. When an investor asks a broker to sell short 100 shares trading at USD 50, the broker borrows them from a client who owns them and sells them into the market, and the investor buys them back later. A small fee is often charged for borrowing, and any dividend or interest paid meanwhile goes to the lender.

Follow the same 100 shares. Two months on, the price has fallen to USD 40, and a dividend of USD 2 per share fell due after one month. Closing out then gains USD 8 per share, or USD 800 in total, assuming no borrowing fee: sold at USD 50, bought back at USD 40, with USD 2 of dividends passed on. Ignoring the fee, that short position is the exact mirror of a purchase held over the same two months, which loses USD 800, so holding both leaves nothing at any date.

Shorting carries obligations. The short seller keeps a margin account with the broker, funded with cash or marketable securities, which stops the investor walking away if the share price climbs. Initial margin normally starts at 50% of the share value, added to the sale proceeds, and can be raised when prices rise. A position lives only as long as the asset remains available to borrow. Bonds can be shorted the same way, which is what makes the convergence trades later possible.

The law of one price holds that two portfolios delivering identical future cash flows on identical dates must sell at the same price. Where they do not, a theoretical arbitrage opportunity exists. Arbitrage means buying and selling at the same moment to capture a price difference, whether the instruments are identical or merely equivalent once repackaged.

The argument runs through two traders. Say Portfolio X costs more than Portfolio Y while both throw off the same cash flows on the same dates. A trader holding Portfolio X can sell it, buy Portfolio Y, take in the difference, and still own the same future cash flows. A trader holding neither can sell Portfolio X short and buy Portfolio Y, capturing that difference provided the cost of borrowing X is modest. Both trades push one way, and the gap closes.

Bringing this to bonds needs a common footing, which the discount factor supplies. Four Treasury instruments quoted on March 9, 2018 all matured on May 31, 2018: one bill and three coupon bonds, carrying coupons of 0.875%, 1% and 2.375%. Each has a single payment left, so each implies a discount factor for that date, found by dividing cash price by payment.

For the bill, 83 calendar days separate the two dates. Its bid quote of 1.630 gives 100 minus 1.630 multiplied by 83 and divided by 360, which is 99.6242, and its ask quote of 1.620 gives 99.6265. Because a bill repays exactly 100, those are the bid and ask discount factors with the decimal point moved: 0.996242 and 0.996265. For the 0.875% bond, the cash prices found earlier, USD 100.0577 and USD 100.0733, divide into a final payment of 100.4375 to give 0.996218 and 0.996374.

Source: chapter table of instruments with a common maturity. The last column is derived.

All eight factors sit within a whisker of one another, and the pattern is the one the law of one price predicts. Every ask factor exceeds every bid factor, across instruments as well as within them. Locking in a profit needs some bid factor to top some ask factor, so that an instrument bought at its ask is paid for by selling an equivalent at its bid. No such pair appears.

Consistency across those four instruments does not mean Treasury markets are always free of arbitrage. Promised cash flows drive price, but not alone. Tax treatment matters, since a bond taxed more favourably commands more than an otherwise equivalent bond, and transaction costs matter. Liquidity, a measure of how actively an instrument trades, matters most. Put two bonds side by side promising the same future cash flows, where Bond X changes hands constantly and Bond Y trades a handful of times a year. Bond X will very likely command the higher price, because whoever buys it knows selling it again is straightforward, while a buyer of the illiquid Bond Y cannot be sure what getting out later would take.

The gap is an invitation. An arbitrageur buys the cheap Bond Y and shorts the rich Bond X, taking in the difference straight away. Since the two promise identical cash flows, the receipts and payments cancel over time and the opening difference is the profit. The trade is called a convergence arbitrage, because the prices are expected to converge as the common maturity approaches.

This was, in simplified form, the strategy of the hedge fund Long Term Capital Management, which bought the less liquid portfolio and sold the more liquid one. It worked for several years. In 1998 it did not. Russia defaulted on its debt, triggering a flight to quality, and liquid instruments soared relative to illiquid ones, which is the direction that hurts a convergence position. LTCM was heavily levered, could not meet its margin calls, and went bankrupt.

The positions were not wrong. Held to maturity they would probably have paid off, but the fund could not hold them that long. A successful arbitrage needs two things: the relative valuation has to be right, and the arbitrageur has to survive the short-term price moves along the way.

Treasury bills hand over discount factors directly, but only out to one year. Past that point the market quotes coupon-bearing bonds, and each price blends several unknown factors. Separating them takes a bootstrap, run in maturity order on discount factors rather than interest rates.

Two housekeeping rules keep the arithmetic honest. Use mid-market prices, the average of bid and ask, and use dirty prices, because that is the money actually paid. Then sort by maturity and start at the front. The shortest bond is easy: if only one payment remains, its dirty price divided by that payment is the factor for its date. Every bond after it has known factors for all payments but the last, so the last drops out of a single equation.

Source: chapter tables of coupon-bearing bonds and their factors. The coupon column is derived.

Those dirty prices come from the quotes in the usual way. On the first bond the bid and ask clean prices of USD 100.8906 and USD 100.9063 average to USD 100.8985, and accrued interest of USD 0.2762 lifts that to a dirty price of USD 101.1747.

A discount function says what a bond ought to cost. A replicating portfolio is the trade that enforces it: can one bond’s cash flows be rebuilt out of other bonds paying on the same dates, and does the package then cost what the bond costs?

Take a second bond maturing on February 15, 2020, carrying an 8.5% coupon and quoted at a bid of USD 111.8906 and an ask of USD 111.9063. Its mid-market clean price is USD 111.8985, and accrued interest lifts the dirty price to USD 112.4855. Per USD 100 of face value it pays 4.25 on each of the three earlier dates in the previous table and 104.25 at maturity, so the four bonds already bootstrapped serve as building blocks. Replication runs backwards, because only the longest bond reaches the final date, and each shorter bond covers what the longer positions leave uncovered.

Source: chapter replicating portfolio table, reordered. The share column is derived.

Replication costs USD 112.26232 against a mid-market price of USD 112.4855, so the bond looks expensive by USD 0.22318 per USD 100 of face value. The indicated trade is to short the 8.5% bond and buy the package, worth roughly USD 22,318 on USD 10 million of face value. The calculation runs on mid-market prices, and in practice the bid-ask spread across five instruments would quite likely swallow the gap.

Replication builds a synthetic zero-coupon exposure out of coupon bonds. The Treasury market also sells the real thing. STRIPS stands for Separate Trading of Registered Interest and Principal of Securities, and a STRIP is what emerges when a coupon bond is broken into individual payments, each trading afterwards as a standalone zero-coupon security.

Investment dealers do the taking apart. A dealer delivers a coupon-bearing bond to the Treasury and receives its principal component and coupon components back as separate registered securities. Those from the coupon payments go by TINTs, INTs or C-STRIPS, the one from the principal payment by TPs, Ps or P-STRIPS.

Work an example. A Treasury bond maturing on May 15, 2030 pays a 6.25% coupon, so USD 1,000,000 of face value pays USD 31,250 each May 15 and November 15 and returns USD 1,000,000 at the end. Stripped on March 9, 2018, it becomes 25 C-STRIPS, one for every remaining half-yearly coupon out to May 2030, plus one P-STRIP for the principal.

Because a STRIP is one payment on one date, its price is a discount factor with the decimal point shifted, and reading a discount function off STRIP prices takes no bootstrap at all. On March 9, 2018 a P-STRIP maturing in November 2047 was bid at 38.945 and asked at 39.059, so averaging 0.38945 and 0.39059 gives a mid-market factor of 0.39002.

The two families do not always agree, and where they diverge the law of one price is violated on paper. On the same date a C-STRIP maturing in May 2021 could be bought at 92.426 while a P-STRIP maturing that day could be sold short at 92.499, both promising 100. Buying the first and shorting the second banks USD 0.073 per USD 100 of face value and leaves no net cash flow at maturity, though transaction costs would very probably eliminate the gain.

Every accrual in this lesson quietly assumed a rule for measuring elapsed time, and that rule is the day-count convention. Interest over a reference period is usually known, typically the gap between two coupons, and interest over a shorter holding period inside it has to be worked out. A convention is written as X over Y: X measures the holding period, Y the reference period.

Three conventions cover most of what a risk manager meets: actual/actual (in period), 30/360, and actual/360.

Actual/actual (in period)

U.S. Treasury bonds use actual/actual, the convention already applied in the accrued interest formula. Both counts are real calendar days, so the fraction of the next coupon earned is days held divided by actual days in the coupon period.

30/360

Corporate and municipal bonds issued in the United States use 30/360, which treats every month as 30 days and every year as 360. Take coupons of 10% per year due each March 5 and September 5, and ask what accrues by June 10. The convention allows 30 days each to March, April and May and then five more, making 95 days. Annual interest on USD 1,000 of face value is USD 100, so the amount earned is 95 divided by 360, multiplied by 100, or 26.39. Two odd consequences follow: three days of interest accrue between February 28 and March 1, in leap years and ordinary years alike, and nothing accrues on the 31st of any month.

Actual/360

Money market instruments in the United States, the Treasury bills from earlier included, use actual/360. Quoted interest applies to a 360-day period while real days in the holding period drive what is earned, with interest struck against face value rather than the price paid. Conventions are national: in Canada, Australia and New Zealand, money market quotes run on an actual/365 basis instead.