12

Other security-specific sources of return

12.1 PAYDOWN

Amortising securities are structured so that their principal is repaid over the lifetime of the security, rather than as a single lump sum at maturity. Typically, the ratio of interest to principal is designed to change over time so that the borrower can make equal payments throughout the security’s lifetime.

Some types of amortising bonds, such as mortgage-backed securities (MBSs), passthroughs and collateralised mortgage obligations (CMOs), have the additional feature that the borrower can pay the principal back to the lender faster than scheduled. When early repayments of principal are made, such securities can generate paydown return in addition to return from carry, curve and credit shifts:

• If the security is trading at a premium (price > $100 per $100 face value), any paydown of principal generates a loss.
• If the security is trading at a discount (price < $100 per $100 face value), any paydown of principal generates a profit.

To see why, suppose that a security is purchased at $90 per $100 face value. The owner is guaranteed to make a profit, irrespective of what happens in the marketplace. All they have to do is to hold it to maturity, wait for the price to converge to its par value, and bank a capital gain. This pull to par effect is described in Chapter 9.

However, this assumption holds only if the face value of the investment stays unchanged at $100, which is precisely what sinking securities do not do. Suppose that the security had half of its face value returned to the purchaser just after it had been bought at $90. The effect of this paydown is to halve the rate of return, because only 5% of the nominal face value will be generated by pull to par, not 10% as previously.

Paydown return is therefore closely related to pull-to-par return, but should be displayed separately as it is not a function of yield. In general, paydown return r_p is given by

where:

• P is the security’s market price;
• δf is the change in the amount paid down over the calculation interval. f starts at 0.0 when the security is issued and increases over the security’s lifetime to a value of 1.0 when all capital is repaid. A non-amortising security’s f value is 0.0 for its entire lifetime.

Example

The following example is from Fabozzi (2012, p. 1675):

• An MBS has a market price of 105.
• Prepayment of capital over the next month will be 2.4%.

Return due to paydown is then

The security was trading at a premium, so its paydown return was negative over the interval.

In general, paydown return should be calculated and displayed for all sinking securities, in addition to carry and market returns, when they allow early repayment of principal. Vanilla bonds such as US Treasuries do not show any paydown return, because their principal remains unchanged throughout their lifetime.

12.2 CONVEXITY

Equation (8.6) shows that the relationship between yield and price is roughly linear, with nonlinearity depending on the security’s convexity.

12.2.2 Trading convexity

In some cases it can be useful to separate a bond’s convexity return from its duration return. Suppose that a trader owns a bond with the following characteristics (see Table 12.1):

Table 12.1 Barbell strategy before restructuring

The trader believes there will be significant volatility in the marketplace, but does not know which way the markets will move. Their strategy is to sell the bond and buy two other bonds as follows (see Table 12.2):

Table 12.2 Barbell strategy after restructuring

The purpose of this restructure is to construct a new portfolio with the same modified duration as the original, but with much higher convexity. This has increased the trader’s portfolio’s second-order sensitivity to yield changes. As long as there is a substantial change in yields, the new portfolio will outperform the old one, and this will occur whether the market moves up or down.

The strategy is based on two assumptions:

• That the market will move in a parallel manner. If non-parallel movements occur, the trader is only hedged against parallel curve movements, and the more complex cash flow structure of their new portfolio may generate losses if, for instance, the yield curve twists or flattens.
• That there will be any movement in the curve. In general, the yield of the new portfolio will be lower than that of the old. If there is no market movement, no returns will be made by the convexity position, but the lower yield will result in lower carry return than the older portfolio would otherwise have generated.

This is known as a barbell strategy, as it moves the main cashflows of a portfolio outwards so that they lie at opposite ends of the term structure, with the graph of exposures against maturity resembling a barbell.

To assess whether this strategy worked, one needs to know the portfolio’s return from convexity, as distinct from modified duration. A first-principles approach to pricing will not generate a value for modified duration, in which case it will be necessary to calculate this quantity separately as follows.

Suppose that the current YTM is given by y. Given a pricing function p = p(y, t), its numerical derivative may be substituted into equations (8.4) and (8.5) to give the effective duration and convexity:¹

where δ << 1.

The modified duration and convexity returns are then given by the second and third terms in equation (8.6), respectively.

12.3 ROLLDOWN

Consider a security that has a single cash flow one year in the future. The yield curve is steeply sloped at the 1-year point, but flattens out at longer maturities.

Suppose further that market conditions do not change for a month, so that the level and shape of the yield curve remains unchanged over this interval.

At the end of the month, this one-year bond will have become an 11-month bond, and the yield used to price this security will now be read from the 11-month point rather than the 12-month point on the yield curve.

Since the yield curve is downwards sloping, the 11-month yield will be lower. Since the yield of the security is lower, the price will be higher, and a positive return will have been generated.

This effect is sometimes called riding the yield curve, as it is most effective when a security’s cash flows are positioned at maturities where the curve is most steeply sloped.

Note that this return has not been generated by movements in the market, since we explicitly assumed that market conditions were unchanged. Nor has it been generated by elapsed time because the return is generated entirely by a change in yield. Rolldown is distinct from either source of return, and should be measured separately.

Everything else being equal, it can be worth positioning one’s securities so that the bulk of their cash flows are positioned at the most steeply sloped parts of the curve. However, rolldown return is seldom substantial, and will be overwhelmed by even quite small changes in the level or shape of the curve.

A bond’s change in yield due to rolldown is given by

where Y^t is the yield curve at time t, Y^t(m₁) its value at maturity m₁ and Y^t(m₂) its value at maturity m₂ where m₂ < m₁.

Return generated by rolldown may then be calculated in the same way as carry:

Pull-to-par return is generated even if the yield curve is flat, as it is driven by differences between the price of the bond and its eventual par price. In contrast, rolldown return is driven entirely by the shape of the yield curve.

12.4 LIQUIDITY RETURN

A security’s price may be affected by liquidity, which is a measure of the ease with which it may be bought or sold without affecting its price. For instance, a US bond that is on-the-run (a reference issue) will be traded in significantly larger volumes than one that is off-the-run, despite being identical in all other respects. The off-the-run bond may trade at a lower price, or a higher yield, because of perceived difficulties in unwinding the position.

Issue size can also affect liquidity, and hence the bond’s yield. For instance, a corporate issue that has identical cash flows and creditworthiness to a highly liquid sovereign issue can trade at a substantially higher yield, because of perceived difficulties in unwinding a position in that security. The effect may be especially pronounced if the amount on issue of the corporate bond is small.

Liquidity considerations can generate returns in two ways:

• By generating higher yields. A security that is structurally identical to a liquid bond but has a smaller amount on issue will probably trade at a higher yield, because of perceived difficulties in buying or selling. This higher yield will generate a higher carry return to compensate.
• By generating additional changes in yield. A bond that was on-the-run a year ago but is now off-the-run will trade at a slightly higher yield, because of its lower price. This increase in yield will generate a negative return, although its magnitude may be so small as to make no appreciable difference to the bond’s overall return over this interval.