16

Money market securities

16.1 INTRODUCTION

This chapter covers attribution techniques for securities with maturities of less than a year, the prices of which are affected by movements in short-term interest rates. These include cash, discount securities, accrual instruments, floating rate notes, repos and forwards.

16.2 MONEY MARKET YIELD CURVES

When running attribution on money market instruments, the reader should be aware that standard curve data may be quite inaccurate for this purpose, simply because there can be fine detail in this part of the curve that is lost if one takes a broad-brush view over all maturities. For instance, the Nelson-Siegel function is designed to model gross features of the curve over maturities out to 50 years using a small number of parameters. For this reason, the same function may not model money market–specific curves accurately unless specifically fitted to the shorter part of the curve.

For attribution on a money market portfolio, where investment decisions have been made in terms of expected curve movements at low maturities, it may be preferable to use specific high-resolution curves in preference to those used for longer-dated portfolios, even though they refer to the same market.

Fortunately, money market yield curves are particularly simple to construct. As the majority of instruments traded at these maturities pay no coupons, their market yields are identical to their zero coupon yields, so a zero coupon curve can be constructed from the yields of a representative set of securities trading at various maturities.

If the yield curve is sufficiently accurate, short-term rates such as the base rate and the Fed Funds rate may be read from the appropriate curve using interpolation. Use of a single curve as the central source of reference rates for all securities can be a time-saving short cut, compared to the potential effort required to supply and update them separately.

16.3 MONEY MARKET CURVE DECOMPOSITION

Typically, it is not necessary to look at money market yield curves in terms of shift, twist and curvature unless the manager is running the portfolio with a focus on these types of curve movement. More usually, a money market portfolio will focus on credit, so it is these returns that should be highlighted in the attribution report.

16.4 CASH

In principle, cash is a simple asset type to model. In practice, tracking down exactly how much cash is held in a portfolio often presents accounting headaches and can require a disproportionate amount of effort by the performance analyst.

Assuming continuous compounding, the return r_cash generated by a cash holding is given by

where

• y is the deposit yield;
• t is the elapsed time as a fraction of a year.

By definition, cash carries no interest rate or credit risk, so its return arises entirely from carry. Cash always has zero modified duration and convexity.

In practice, interest rates used in the calculation of interest return may be:

• zero (for settlements);
• the overnight or Fed Funds rate, if funds are borrowed or lent between large banks;
• the tom/next (tomorrow/next) rates for trades executed tomorrow, delivery on next business day;
• the official bank rate, or repo rate, the interest rate charged by a central bank for overnight lending;
• the inverse repo rate, the rate at which banks deposit funds with a central bank.

Although the price of cash is unaffected by interest rate changes, its presence will affect the interest rate sensitivity, and hence the modified duration of a portfolio. For instance, cash drag occurs when part of the capital in a managed fund has not been invested in revenue-generating assets. The market value of the assets remains unchanged, but the interest rate sensitivity of the portfolio is lessened.

Example: cash accrual

Suppose the AUD base rate is 3.5%. The continuously compounded return for an AUD cash deposit over six months is

16.5 BANK BILLS AND DISCOUNT SECURITIES

Discount securities include zero coupon bonds, bank bills, letters of credit and promissory notes. While the label may vary, the underlying structure of these securities is identical.

A discount security generates a single cash flow at maturity. The present value of the security is therefore the discounted value of this future cash flow at the present time.

The duration of a discount security is therefore exactly the same as its time to maturity.

Although discount securities do not pay coupons, they do generate carry return. This can sometimes cause confusion, for if a security pays no coupon, how can the passage of time generate return? The reason is that carry return is the sum of two returns: running yield and pull to par. Although running yield is zero for a security that does not pay coupons, the pull to par return need not be. As the maturity date of the security comes closer, the market price must converge to the face value, and this generates return even if interest rates remain unchanged.

The price p of a discount security is given by

where

• F is its face value;
• y is the yield to maturity;
• t is the time to maturity, in years.

Since there is only one cash flow to consider, only one point on the yield curve matters when pricing or performing attribution, and this makes attribution on discount securities particularly simple. For instance, suppose that the YTM of a bank bill is 3.6% at the start of a month and 3.9% at the end; 0.2% of this movement is due to parallel shifts in the curve, and 0.1% is due to other curve movements.

A discount security is sufficiently simple to price that we can perform attribution from first principles. Consider a bill with a maturity date of 31 December 2012 and a face value of 100. On 1 January 2012, its yield to maturity is 4.00% and it has 365 days left until maturity. On 8 January 2012, its yield has dropped to 3.95% and it has 358 days left to run.

The price p₀ at the start of period is

and the price p₁ at the end of the period is

The overall return of the security over this interval is given by

Using this pricing formula, the return of the security can be broken down into market effects and time effects. The return due to changes in yield can be derived by pricing the security at the same date but with different yields:

Using the same p₀ and p₁ as above, the return due to the passage of time is given by

and the return due to changes in yield by

The total return is 0.000481 + 0.000744 = 0.001225, as before.

Compare this to the results given by the perturbational attribution equation r = yδt − MDδy, where the first term approximates the return due to the passage of time, and the second the return due to changes in yield. The yield at the start of the interval is 4.00%, the duration is just 1.000 (the time to maturity in years) and the modified duration is given by

Using these values, the return due to the passage of time is given by

r_t = 7/365 × 4.00% = 0.000767

and the return due to changes in yield by

r_y = −0.961538 × (3.95% − 4.00%) = 0.000481

Both values are close to those derived from first principles.

16.6 ACCRUAL SECURITIES

Accrual securities are very similar to discount securities in terms of return and risk, but are structured slightly differently. Instead of selling at a discount and maturing at face value, they are issued at face value and mature at face value plus interest. Their return is therefore given by the expression presented earlier for discount securities.

16.7 FLOATING RATE NOTES

A floating rate note, or FRN, is structurally identical to a bond, but with the important difference that its coupon varies according to market conditions. FRNs are among the most complex type of money market security.

The coupon on an FRN is set to a money market reference rate such as 3-month LIBOR or the Fed Funds rate, plus a fixed increment called the reset margin. The coupon is reset at specific intervals, typically at the start of a coupon payment interval. The coupon reset interval is usually quarterly, but can also be semi-annual or monthly.

Floating rate notes tend to be long-dated and are therefore closer to coupon-paying bonds in structure. However, they are usually regarded as money market securities as they have no exposure to longer-dated interest rates.

The reason is that an FRN can be regarded as a zero coupon bond with a face value equal to the sum of its principal and the next coupon payment (for a more detailed pricing treatment, see below). Suppose that an FRN has a face value of 100, its coupon has just been paid and the reset rate is r. The next coupon is payable in six months. Then the value of the future cash flow is 100 × (1 + r/2), but this will be discounted at the same rate, so the FRN’s current market price remains unchanged at 100. In other words, its price is not affected by the reset rate.

Any decrease in value for an FRN due to increasing interest rates is thus compensated for by rising coupon payments. Unlike a bond, for which prices fall when interest rates rise, FRN prices are generally unaffected by changes in interest rates. They can therefore form a useful alternative investment to bonds in a rising interest rate environment.

If credit conditions are unchanged, the price of the FRN will revert to par on each coupon reset date. However, this seldom happens in reality. Usually, the creditworthiness of the FRN has changed, so the price will be a little above or below par on the reset date.

16.8 INTEREST RATE AND CREDIT RISK

Because the FRN’s price is largely determined by the level of the next coupon, its modified duration is always close to zero. FRNs are therefore relatively immune to interest rate risk. This is why FRNs are regarded as money market instruments, despite their long maturities.

Unless you are focusing on the shortest part of the yield curve where maturities are a year or less, interest rate returns generated by FRNs may safely be ignored for attribution. In practice, the main source of risk and return for an FRN is credit risk. FRNs are usually issued by companies with below investment grade ratings¹ and have average credit ratings around B, so their returns are largely driven by widening or tightening of credit conditions.

16.9 FRN TYPES

Many types of FRN exist, including perpetual, variable rate, structured, reverse, capped, floored, collared, step-up recovery, range, corridor and leveraged. They are mostly differentiated on the basis of how their coupons are calculated:

• A stepped FRN has its reset margin changed at different times during the security’s lifetime.
• A capped FRN has a maximum coupon payment value.
• A floored FRN has a lowest coupon payment.
• A reverse FRN’s coupon payment moves in the opposite direction to the reference rate. For instance, its coupon might be set as reset margin minus LIBOR.
• A callable FRN has an embedded option, allowing the issuer to repurchase the FRN at times of low interest rates and reissue the debt as fixed rate bonds.
• A variable rate note is an FRN for which the reset margin itself is variable.

16.10 YIELDS AND DISCOUNT MARGINS

Fixed coupon bonds can be compared using their yield to maturity, which allows their respective return if held to maturity to be compared, independently of structural features such as coupons and tenor.

Since the FRN’s cash flows cannot be calculated in advance, YTM is not normally used to assess FRNs. Instead they are compared using a measure called the discount margin, which measures spread relative to the reference rate, and hence expected return. This spread is the sum of the reset margin, defined when the FRN is issued, and the credit spread that compensates the FRN’s holder for its credit risk. Discount margins apply only to FRNs, and the yield of the FRN is the sum of the reference rate and the discount margin.

To understand the discount margin, consider the return made by the owner of the security due to the pull-to-par effect, over and above the return made by coupon payments. If the FRN is bought at less than its maturity value, extra return will be made over the security’s lifetime as its price converges to par at maturity. The discount margin is this return, plus the reset margin; it can be thought of as the option-adjusted spread for an FRN.

The simple margin (SM) of an FRN measures its effective spread, which is a simple approximation to the discount margin. It is given by

where

• P is the FRN’s clean price;
• M is the FRN’s time to maturity, in years;
• r is the reset margin.

The FRN’s discount margin is measured in a similar way, but involves discounting future cash flows by the current reference rate plus margin. The FRN’s price is then calculated and compared to the market value, and the margin is repeatedly adjusted until a value is found that equates the market price to the sum of the discounted cash flows. Since discount margins are usually supplied for attribution, I do not describe this process in further detail; see Fabozzi and Mann (2000) for more information.

16.11 FRN DURATIONS

Unlike a bond, an FRN has two duration measures, reflecting its exposure to both interest rates and discount margins:

• Interest rate duration measures the FRN’s sensitivity to changes in interest rates, as discussed earlier. The interest rate duration is closely approximated by the time to the next coupon, which is usually a fraction of a year. FRNs therefore have relatively low modified durations.
• Spread duration measures the FRN’s sensitivity to changes in the discount margin. Spread duration is typically much larger than interest rate duration, and is measured by calculating the modified duration of the FRN in the same way as that of a vanilla bond, using all its cash flows. An FRN’s returns are typically driven by changes in the discount margin, reflecting the fact that spread durations are much higher than interest rate durations for this asset class.

16.12 DECOMPOSING THE RETURN OF AN FRN

The price P of an FRN per $100 face value is given by a rather complex expression that involves various reference rates, reset and discount margins, the terms and conditions of the security, and the current date. The details are given in Appendix B.

While it is possible to use this expression to run attribution on an FRN by repricing at each valuation date and examining the effect of changing each parameter, this requires a significant amount of data. Fortunately, there is a much simpler way to treat FRNs in an attribution framework. To a high degree of accuracy, the return r of an FRN is given by

or, splitting the yield into reference return and discount margin

where

• y is the yield of the FRN;
• δt is the elapsed time, as a fraction of a year;
• MD_s is the spread duration;
• δDM is the change in the discount margin;
• R_T and DM are as defined above.

Note that I have not included return due to changes in the interest rates, since this is usually negligible. In practice, (16.5) replicates the returns of an FRN very accurately. For attribution purposes, it is much easier to calculate the FRN’s spread duration than it is to reprice the FRN. Even if you are using a first-principles approach elsewhere in your attribution analysis, it is preferable to use (16.5) for FRNs.

16.13 YIELD CURVE ATTRIBUTION

Since the FRN’s return depends on short-term interest rates and discount margin, the shape of the yield curve at longer maturities has little effect on the security’s returns. For this reason, it is seldom necessary to decompose returns into parallel and non-parallel shifts when running attribution on FRNs; a single measure of yield shift is often all that is required. In the context of a portfolio containing other types of security, it is perfectly permissible to show an FRN’s returns as the sum of carry return and a security-specific (credit) return, and to assume curve change return is zero.

16.14 ATTRIBUTION WITH COMPLETE DATA

Suppose that the spread duration, the reference rate and the discount margin are all available on a daily basis. The FRN’s yield will then be the sum of the reference rate and the discount margin, and attribution can proceed as above.

16.15 ATTRIBUTION WITH INCOMPLETE DATA

Suppose that the only daily market data available apart from the return of the FRN is a risk-free yield curve. In this case we can still run attribution, but with slightly less detail than in the previous case.

First, note that the reference rates used to price FRNs are very closely related to particular maturities on the yield curve. For instance, 3-month LIBOR rate and 3-month CD rates are closely correlated, as are the Federal Funds rate and the 6-month treasury bill yield. A yield curve that is accurate at the short end will supply reference rates that are easily accurate enough for attribution.

Secondly, the yield of the FRN is the sum of its reference rate and its discount margin. A yield curve will supply the reference rate, which is read from the appropriate maturity on the yield curve, but not the discount margin, which is a function of the day-to-day creditworthiness of the security. Then we can write the reference return of the FRN as the first term in equation (16.5), and assign the rest of the return to discount margin return.

16.16 TREATMENT OF FRNs IN COMMERCIAL SYSTEMS

The main difficulty with running attribution on FRNs is that they do not fit into a conventional bond attribution framework. Their dependency on changes in the discount margin rather than yield requires an extra term in the attribution equation, which is often unavailable in commercial systems.

Should this be the case, the suggested treatment is to set up an FRN as follows:

• Set the yield to be the reference rate for the FRN, plus the margin. For instance, in the above example this will be LIBOR plus 20 basis points. Reference rates are not usually volatile, so it may be sufficient to reset the rate each time a coupon is paid.
• Set the modified duration to zero. This ensures that there will be no market return, and all remaining return from the FRN will be assigned to the residual bucket.
• Relabel the residual bucket as ‘Discount margin’.

This approach will give exactly the same results as for the full treatment, except that the yield return will be understated because it will not show the full carry return due to the extra yield from the discount margin. However, one might argue that this is actually more accurate, because it aggregates all returns arising from the discount margin (carry and duration return) into one figure.

16.17 FRNs AND SECURITISATION

FRNs often appear in securitisation tranches (see Chapter 22). While the details of the securitisation may be complex, attribution on a tranche that has been structured as an FRN is identical to attribution on a stand-alone FRN.

16.18 CURRENCY FORWARDS

A currency forward locks in the price at which a given amount of currency can be bought or sold at a given date in the future. Assuming continuous compounding, the cost F of a forward is given by

where

• S is the amount to hedge, in base currency;
• r_d is the domestic (base) interest rate;
• r_f is the foreign (local) interest rate;
• t is the period over which the forward contract is active.

A forward can be regarded as two continuously compounded bank bills, one long the base currency and the other short the local currency, with the cash flows from settlement arising at the maturation of the contract.

Forwards are usually used for hedging against adverse currency movements for multi-currency portfolios and for hedging equity and fixed income benchmarks. Their use in attribution is covered in more detail in the section on Karnosky-Singer attribution in Chapter 4.

16.19 REPURCHASE AGREEMENTS (REPOs)

A repo agreement (or repo) is a short-term sale of a bond, together with an agreed repurchase in the future, in order to raise short-term cash. The difference between the sale and repurchase prices can be interpreted as a rate of return, called the repo rate. A repo therefore behaves like a secured deposit rather than a set of transactions.

A reverse repo is the opposite of a repo. If bank A engages in a repo agreement with bank B, then bank B has entered into a reverse repo with bank A.

Term repo agreements can be made for a fixed term of between one day and a year. Conversely, an open repo agreement does not have a set maturity date on which the position is unwound.

The various types of repo are particularly simple to treat in an attribution framework. Since the returns made by investing in this asset class are similar to a cash deposit at the repo rate, all repo return can be treated as simple interest. As a result, repos should never show any residual in an attribution report.

It may be useful to split out repos from other asset classes in an attribution report, to emphasise that the return they generate is entirely different to other sources of return, such as carry and risk-free curve movements.

16.20 MONEY MARKET BENCHMARKS

Money market benchmarks tend to be very straightforward in design. For instance, the UBSW Bank Bill Index is simply formed of 13 bank bills that mature on successive weeks. As the shortest-dated bills mature, new longer-dated bills enter the index.