CarryDecomposition

Time return

Unlike equities, coupon-bearing securities guarantee a certain return to the holder due to the passage of time. If the markets did not move in any way, the holder of a coupon-bearing bond would still receive one or more coupon payments per year. If the holder only owned the bond for part of the year, he would receive part of the regular coupon payment, with the amount proportional to the time he had owned the security. The majority of fixed-income securities pay some sort of coupon, so this is an important source of return in the fixed income markets.

The terms coupon return and time return are often used interchangably, since all can be seen as names for return that does not arise from changes in the market's term structure. However, the two terms are not strictly equivalent. Securities that pay no coupon, such as zero-coupon bonds or bank bills, show return that is purely due to the passage of time, which causes the security's price to approach par. A better alternative is 'yield return' or 'carry'.

There are several ways in which one can measure a security's return due to the passage of time while ignoring changes in the term structure. One of the simplest is to use the security's yield to maturity (or YTM). YTM is the single rate $y$ that, when used to discount a security’s cash flows, gives the current security price $P$:

where $C_i$ is cash flow $i$, and $t_i$ is the interval in years between the present and that cash flow.

YTM has the property that, if unchanged over a short period, it equals the security’s total rate of return. A bond’s return due to yield is therefore given by

where $y$ is the security’s yield to maturity, and $\delta \tau$ is the elapsed time in years.

Current yield and pull to par

Yield return may be further decomposed into a current yield and a pull-to-par yield.

Suppose a bond has just been issued and has a long time to maturity. To a close approximation, its instantaneous return will be given by its coupon divided by the price at which it was bought. This quantity is called the current yield (also flat yield, interest yield, or running yield), and is given by

where $r$ is the current yield return, $c$ is the bond’s coupon, and $p$ is its current clean price (i.e. excluding accrued interest).

However, current yield is only a rough and ready measure of a bond’s return. As the bond approaches maturity, its price will converge towards par, and this will affect the yield. The effect is called pull to par or reduction of maturity. The size of this effect is given by the difference between the yield to maturity and the current yield, and may be regarded as the return due to capital gains or losses between the time the bound is bought and the time it matures. Therefore, a more accurate way to measure non-term structure return is to express it in terms of coupon return (running yield), plus return due to the passage of time (pull-to-par return). The pull-to-par return is given by the difference between the yield-to-maturity return and the current yield return:

FIA offers several options for calculation of yield returns:

• NONE: Suitable for portfolios of zero-coupon or equity-type securities, where there is no yield return;
• AGGREGATED: All returns from accrued interest are assigned to a single category, which measures the return from yield to maturity;
• PULL_TO_PAR: Returns from accrued interest are split into current yield and pull to par yield, with the sum of the two returns equal to the yield to maturity return.

Use the CouponDecomposition flag in the configuration file to indicate which option is required. If this flag is not set, FIA uses the AGGREGATED setting.

The user has the option of supplying a yield to maturity for each record in the weights and returns file. If a yield to maturity is not provided, it is calculated automatically using a numerical iterative routine.