Carry return
9.2 Carry-based investment strategies
9.9 Yield for non-bond securities
9.10 Using yield to maturity in attribution reports
9.1 INTRODUCTION
What is the return of a portfolio when the markets are closed?
For equity markets, the answer is straightforward. The reference prices at which the markets last closed should be used for revaluation. Since prices do not change and no dividends are announced, the portfolio’s return will always be zero.
Matters are slightly more complex in the fixed income markets. Here, most securities offer regular, known cash flows to their owners, even if underlying interest rates fluctuate wildly. A bond’s value will therefore increase over time as the next coupon draws closer, and this accrued interest is generated whether or not the markets are open. In other words, bonds are designed to generate a trickle of return every day of the week, including weekends and holidays.
The measurement of this carry return,1 as distinct to return due to changes in other market conditions, is the subject of the present chapter.
9.2 CARRY-BASED INVESTMENT STRATEGIES
A common investment strategy is to structure a portfolio so that it generates higher carry return than its benchmark, while remaining hedged against other fixed income risks. The carry generated by a security is proportional to its yield, so this outperformance arises from investing in securities with higher yields than their counterparts in the benchmark. Such portfolios are often labelled high-yield.
A hedged high-yield portfolio should outperform its benchmark regardless of how underlying interest rates change. However, this outperformance will come with greater exposure to credit events and the risk of more frequent adverse events such as missed coupons, default, early redemption of callable bonds and market illiquidity, all of which can drive the portfolio’s return down. A trader who manages such a portfolio must be skilled at assessing the risks involved in investing in these higher-yielding securities, and deciding where the market is wrong.
An attribution report will confirm that a high-yield strategy has actually worked, and that the portfolio is genuinely generating more yield-driven return than the benchmark. In addition, it will isolate market sectors where the strategy may not be working. Equally importantly, the report will confirm that the manager is not making or losing return because of other sources of risk, such as movements in the yield curve or changes in the credit spread.
9.3 TYPES OF YIELD
Several measures for a yield are used in the markets, according to the investment horizon and future intentions of the security’s owner.
Some of these yield measurements are only applicable to coupon-paying securities, but the concept of yield applies to all fixed income instruments, including those that pay no coupons, and derivative securities.
9.3.1 Nominal yield
Consider a bond with a face (or par) value of $100 and a 5% annual coupon that matures in five years.
If this bond was purchased at par when it was first issued, and held to maturity when it is again priced at par, the annual rate of return made by holding the bond will be $5/$100 = 5%. This is called the nominal (or flat) yield, and is equal to the annual income of the bond, divided by its par value.
In practice, you would only use this measure of yield if you never expected to trade the bond. Even if this were the case, you might still need to mark-to-market the investment, since the market price of the bond is likely to vary away from par over its lifetime.
9.3.2 Simple yield
Suppose the same bond only has a year to run until maturity. For each $100 invested, the owner will receive a coupon payment of $5 and a profit of $5 due to the increase of the bond’s price from $95 now to $100 at maturity. The return over the twelve months to maturity is therefore $10/$95 = 10.52%. This return is called the simple yield.
9.3.3 Running yield
The current, or running, yield is the annual coupon of the bond divided by its current clean price, which is the market price excluding accrued interest. If the clean price of this $100 par value bond is $95, then its running yield is $5/$95 = 5.36%. This is the bond’s yield under the assumption that it can later be sold for the same price.
Running yield is most useful when one needs to compare the relative returns of various investment options over a short horizon compared to the lifetime of the investments.
9.3.4 Yield to maturity
The most widely used measure of yield is redemption yield, or yield to maturity (YTM). YTM is the single yield y at which future cash flows must be discounted to equal the current market price of the bond P:
where Ci and ti are the value and time from the present (in years) of cash flow i.
This equation may be solved for y either numerically or by trial and error.
Yield to maturity measures the return generated by the bond if the security is purchased now and held to maturity. It represents a single-valued proxy for the bond’s cash flows and the current levels of the yield curve.
Yield to maturity differs from running yield in one crucial respect. To use a phrase from game theory, fixed income securities are subject to ‘the shadow of the future’ (Dawkins, 1976), in that almost all have a known maturity date at which they will be worth par, irrespective of their current value. The effect of this known price point in the future can and does affect the current price. Yield to maturity takes these resulting capital gains or losses into account, while running yield does not. This pull-to-par effect is covered in more detail in Section 9.8.1.
9.4 CALCULATING CARRY RETURN
There exist at least two ways to calculate carry return:
- The first is to reprice the security from first principles at different times, under the assumption that the security’s yield remains unchanged, so that the only cause of a change in the price is the passage of time. The return implied by the change in price is the carry.
While conceptually straightforward, this method is not recommended. To calculate carry return using direct pricing over a period when a coupon is paid requires that you know the exact dates on which payment occurred, and this can be difficult because of weekends, bank holidays and country-specific settlement conventions. - The second, far easier method is to use the security’s yield to maturity. The carry return of the bond over an interval is simply given by
where y is the bond’s yield to maturity, and δt is the elapsed time over the interval.
It is just as accurate to calculate carry return using yield to maturity as it is to use a first principles approach, and much less work, especially if you already have market values of YTM available.
Example
Over a week, a bond with a YTM of 5.5% generates carry return of
9.5 PROS AND CONS OF YTM
Despite being complex to calculate, yield to maturity is probably the most useful of all the yield measures shown here, as it makes calculation of carry return straightforward.
However, YTM does have some shortfalls. An important underlying assumption is that the coupons can be reinvested at the initial YTM of the bond, which may not be the case.
9.6 DECOMPOSING CARRY
Carry return can be decomposed in at least two ways:
- Into pull-to-par and running yield returns.
- Into risk-free carry and credit carry, by decomposing the YTM of a security into a portion due to risk-free yield, and a credit spread component.
9.7 WHICH YIELD TO USE?
Which is the most useful yield measure for attribution? It depends on the time-scale over which you hold the bond:
- If you buy the bond at issue and hold it to maturity, you will be unaffected by any fluctuations in the market price of the bond. In this case, the nominal yield may be appropriate for measuring the return on your investment.
- If you are trading bonds over the short term, then the running yield may be most useful, since this is a day-to-day measure of yield and takes no account of the pull-to-parity effect.
- If you are buying a bond now to hold to maturity, the yield to maturity will probably give you the clearest picture of the bond’s value relative to others.
In practice, YTM is most widely used in the marketplace, since it provides a fairly realistic measure of the relative returns of securities that may have completely different characteristics.
9.8 DECOMPOSING CARRY RETURN
Carry return may be decomposed in several ways that reflect different investment strategies.
9.8.1 Pull to par and running yield
Yield to maturity is a more general measure of return than running yield because it includes both the interest payments the bondholder receives, and the capital gain (or loss) made when the bond matures. If a security’s price is substantially different from par as it approaches maturity, this may have a noticeable effect on the security’s return because its value has to converge to par. Even if interest rates are substantially higher than when the bond was issued, so that it is priced at a discount, the price will still be pulled upwards by this effect, which will drive the bond’s yield downwards as maturity approaches.2
This effect is known as pull to par (or reduction of maturity), and can have a pronounced effect on the returns of bonds that are nearing maturity.3
Consider the 5% bond described earlier. The income portion of the yield to maturity is the running yield of the bond, as calculated above.
One way to measure the capital gain portion is to take the difference between the current clean price and the par value of the bond (100 − 95 = 5), divided by the number of years to maturity (5/5 = 1), divided by the current clean price of the bond (1/0.95 = 1.05%). We can therefore expect an extra 105 basis points from pull-to-par effects over the next five years.
Most attribution calculations are run over a shorter interval, in which case one can use the expression
The YTM and the running yield will typically be available for reporting, so it is straightforward to calculate the portion of the bond’s yield due to pull to par, and hence the resulting carry return.
9.8.2 Risk-free carry and credit carry
Everything else being equal, a bond with a lower credit rating will trade at a higher YTM than a treasury equivalent. This reflects the higher risk taken by the bond owner, for which higher returns are required.
For portfolios with large number of high-yielding securities (equivalently, those with lower credit ratings), it may be useful to decompose yield return into risk-free return and risk return.
To take a relatively extreme example, in 2012 Greek government bonds were yielding 16% with a credit rating of CCC, indicating that the market had factored in a 50% probability of default. German bonds with the same maturity and coupon had a yield to maturity of around 3% and a credit rating of AAA, reflecting the market’s consensus that these bonds carried a virtually zero percent chance of default. The risk-free yield of the Greek bond was 3%, and the risk-driven yield was 16% − 3% = 13%. These yields may then be used to calculate the risk-free carry and the credit carry for the bond.
In general,
9.8.3 Risk-free carry, sector carry and security-specific carry
A corporate security’s yield may be a combination of yield due to its underlying risk-free curve, yield due to its sector curve and security-specific yield according to its position relative to its sector curve.
An attribution analysis will measure the return generated by changes in all these quantities, but their absolute levels also generate carry return. These levels may be determined using the techniques described in Chapter 11. In general,
9.8.4 Carry due to optionality
A bond with an embedded call allows the issuer to withdraw the bond at certain predefined times and return the principal, plus any accrued interest, to the purchaser. For the bondholder, this means that their funds will have to be reinvested elsewhere.
The bond issuer usually only does this if (1) they no longer need the funds, or (2) prevailing rates are lower than when the bond was issued, so that they can cancel the bond and raise more money at a lower rate. This is seldom a welcome event for the bondholder. Since interest rates have fallen, the funds now have to be reinvested at lower rate.
This reinvestment risk adds extra uncertainty for the bondholder. To compensate for the possibility of lower future income, a bond with a call option trades at a higher yield than a similar security with no optionality. The extra carry return generated by this higher yield may be displayed in a separate category in an attribution report.
Conversely, a bond with an embedded put allows the bondholder to exchange the bond for cash at any time and to reinvest it at a higher rate if market conditions allow. In this case the bond issuer pays a lower yield, because the reinvestment risk is taken by the bond issuer.
Depending on the bond’s option features, alternative measures of yield may be appropriate for an attribution report, such as yield-to-call and yield-to-worst.
The topic of return generated by optionality on bonds is covered in more detail in Chapter 21.
9.8.5 Inflation carry
The return on all security types discussed so far can be regarded as the sum of their real return and some compensation for the reduction in value from inflation. An inflation-linked bond removes the uncertainly due to inflation by indexing its value to some price index, to ensure that its real value does not fall.
The YTM of an inflation-linked bond is the current inflation rate, plus the real yield to maturity. The carry of an inflation-linked bond therefore requires both these quantities. For more information, refer to Chapter 17.
9.8.6 Other sources of carry
Even when all other effects are removed, bonds with the same maturity, coupon and credit rate can still show different yields. Reasons include:
- Liquidity. A bond from an obscure issuer, or one that is not heavily traded, may trade at a higher yield than one that is more liquid but otherwise identical, to compensate the buyer for the difficulties in selling the bond at a later date.
- Cheap/dear effects, or whether the bond is on or off the run; that is, a heavily traded reference issue.
- Tax effects – whether a bond is bid by the marketplace because of preferential tax treatment.
- Repo effects – whether the security is bid in the repo market.
- Convexity effects – whether a bond is more or less desirable because of high levels of convexity, leading to changes in its price and hence its yield. Higher convexity will lead to measurable outperformance when the term structure moves, but affects yield to a much lower degree.
- Market noise – change in yield that cannot be allocated to any other effect.
I am not aware of any sensible ways to split these effects apart without extensive further analysis. Fortunately, they tend to be minor.
9.9 YIELD FOR NON-BOND SECURITIES
Many fixed income securities pay coupons over a fixed term, but there are some exceptions. For instance, bank bills generate no running yield return, while perpetual bonds produce no pull-to-par return. The treatment of carry for particular security types is covered in Part 4.
9.10 USING YIELD TO MATURITY IN ATTRIBUTION REPORTS
The market convention is to use yield to maturity for reporting time return on attribution reports. Even if you do not actually need YTM to calculate carry return, it is still preferable to cast the attribution report in terms of YTM and to use equation (9.2) for reporting purposes. This follows market convention and is straightforward to explain to traders and investors.
1 Carry return is also referred to as yield return, time return or coupon return. A security or portfolio that has a high yield will generate a high carry return, so the two terms are often used interchangeably in attribution analyses.
2 Bonds priced at a premium will be subject to the opposite effect.
3 The exception is for perpetual bonds, or consols, which have no set maturity date, and some types of constant maturity futures on bonds.