Cash flows duration

Most market participants are familiar with the concept of duration and its effect on the returns from fixed income instruments. However, the difference in the calculations of duration of a bond and a portfolio is worth noting. Most index and analytics providers (including Barclays Capital) calculate index and portfolio duration as the market value weighted duration of individual bonds that constitute the index or portfolio. The other measure, known as cash flow duration, calculates the duration based on the cash flow of the entire portfolio, using the internal rate of return (IRR) of those cash flows and then measuring the sensitivity of portfolio value to change in that IRR. 

A prerequisite to a corrodable material being used is that it is known to have a useful and reasonably predictable life. Planned, or unplanned downtime costs money and the intervals between planned replacements must be of reasonable duration. In practice, the replacement interval is usually conservative at first and then as experience accumulates, the intervals between planned replacements will usually extend. The main reason for choosing a planned maintenance policy is that on a discounted cash flow (DCF) calculation over the life of the plant, the cost of regular replacements including maintenance labour and downtime is less than the extra initial capital cost of a more durable material. 

The payback period method is an approximate measure of preference. First, it does not consider the timing of cash flows prior to payback, ignoring the time value of money. It weighs cash flows 10 years from now the same as cash flows occurring today. Second, it ignores the duration of the cash flows. Cash flows after the payback period, such as major overhauls, are not included in the calculation. These weaknesses in the payback period method render it less desirable than the other measures of merit presented in this section. 

Consider a uniform continuous cash flow, A, which begins at time m and continues for an uncertain duration t. Assume that m and t are statistically independent random variables with known probability functions f m) and f t). It may be shown that... 

To illustrate, consider a uniform cash flow of 1000 per year beginning at some uncertain time m and continuing for a duration of t years. The delay to initiation is uniformly distributed between 6 months and 1 year. The project duration is gamma distributed with mean of 3 years and standard deviation of 1 year the parameters of the gamma distribution yielding these statistics are a = 3 and b = 9. The nominal interest rate is 10% compounded continuously. It is assumed that the initiation time and project dmation are independent random variables. Our problem is to determine the equivalent present value of these cash flows. (This problem is taken from Park and Sharp-Bette (1990, p. 411)... 

From market observation we know that index-linked bonds can experience considerable volatility in prices, similar to conventional bonds, and therefore, there is an element of volatility in the real yield return of these bonds. Traditional economic theory states that the level of real interest rates is cmistant however, in practice they do vary over time. In addition, there are liquidity and supply and demand factors that affect the market prices of index-linked bonds. In this chapter, we present analytical techniques that can be applied to index-linked bonds, the duration and volatility of index-linked bonds and the concept of the real interest rate term structure. Moreover, we show the valuation of inflation-linked bonds with different cash flow structures and embedded options. 

In the traditional approach the duration value is calculated using nominal cash flows, discounted at the nominal yield. A more common approach is to assume a constant average rate of inflation, and adjust cash flows using this inflation rate. The real yield is then used to discount the assumed future cash... 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

A repeat of the above procedure, with the additional step, after the shift in the yield curve, of recalculating the bond cash flows based on a new inflation forecast. This produces a duration measure that is a function of the level of nominal yields. This measure is in effect an inflation duration, or the sensitivity to changes in market inflation expectations, which is a different measure to the real yield duration ... 

To obtain the price of an inflation-linked bond, it is necessary to determine the value of coupon payments and principal repayment. Inflation-linked bonds can be structured with a different cash flow indexation. As noted above, duration, tax treatment and reinvestment risk, are the main factors that affect the instrument design. For instance, index-aimuity bmids that give to the investor a fixed annuity payment and a variable element to compensate the inflation have the shortest duration and the highest reinvestment risk of aU inflation-linked bonds. Conversely, inflation-linked zero-coupon bonds have the highest duration of all inflation-linked bonds and do not have reinvestment risk. In addition, also the tax treatment affects the cash flow structure. In some bond markets, the inflation adjustment on the principal is treated as current income for tax purpose, while in other markets it is not. 

Portfolio managers must also take account of a further relationship between default risk and interest-rate risk. That is, if two corporate bonds have the same duration but one bond has a higher default probability, it essentially has a shorter duration because there is a greater chance that it will experience premature cash flows, in the event of default. 

There are valuation models that can be used to value bonds with embedded options. These models take into account how changes in yield will affect the expected cash flows. Thus, when V and V+ are the values produced from these valuation models, the resulting duration takes into account both the discounting at different interest rates and how the expected cash flows may change. When duration is calculated in this manner, it is referred to as effective duration or option-adjusted duration or OAS duration. Below we explain how effective duration is calculated based on the lattice model and the Monte Carlo model. 

Duration measures the sensitivity of a bond s price to a given change in yield. The traditional formulation is derived under the assumption that the reference yield curve is flat and moves in parallel shifts. Simply put, all bond yields are the same regardless of when the cash flows are delivered across time and changes in yields are perfectly correlated. Several recent attempts have been made to address this inadequacy and develop interest rate risk measures that allow for more realistic changes in the yield curve s shape. ... 

The third and final step is to calculate the percentage change in the bond s portfolio value when each key rate and neighboring spot rates are changed. There will be as many key rate durations as there are preselected key rates. Let s illustrate this process by calculating the key rate duration for a coupon bond. Our hypothetical 6% coupon bond has a maturity value of 100 and matures in five years. The bond delivers coupon payments semiannually. Valuation is accomplished by discounting each cash flow using the appropriate spot rate. The bond s current value is 107.32 and the process is illustrated in Exhibit 4.27. The initial hypothetical (and short) spot curve is contained in column (3). The present values of each of the bond s cash flows is presented in the last column. 

The prices used in equation (4.4) to calculate convexity can be obtained by either assuming that when the yield changes the expected cash flows either do not change or they do change. In the former case, the resulting convexity is referred to as modified convexity. (Actually, in the industry, convexity is not qualified by the adjective modified. ) In contrast, effective convexity assumes that the cash flows do change when yields change. This is the same distinction made for duration. 

A maximum of 10% of the cover pool can be commercial property loans and substitution assets cannot exceed 20%. To limit cash flow mismatching risk, the Irish bonds exhibit tight matching requirements. For example, the nominal value of the cover assets must at all times exceed the value of the corresponding securities. The aggregate interest from the assets must also exceed that of the covered bond and the currency of the cover assets must be similar to the related bonds. In addition to this, the duration of the cover assets must be greater than the duration of the bonds. 

The appropriateness of either model (cash-flow-based versus market value-based) will depend on the asset manager s trading style as well as the particulars of the asset class the asset s market liquidity, duration profile, and credit spread volatility. In terms of mechanics, cash flow arbitrage CDOs are no different than balance sheet CDOs, (again, the only difference being their intended pnrpose and asset sourcing strategy). Consequently, one should see the section on Balance Sheet CDOs for further details. Now, we shift the discussion to market value CDOs. 

The former approach, even though an approximation, has been deemed desirable by EFFAS in the face of the enormity of the task of calculating duration based on the individual cash flows of the constituent bonds. In the trivial case where all the bonds have exactly same yield, the average duration is the same as the true duration of the portfolio or the index. That is. 

Investors can also use interest rate swaps for a similar purpose. These contracts exchange fixed-rate cash flows for floating-rate cash flows based on LIBOR/EURIBOR. Investors on the paying (fixed) leg of the swap reduce the duration of their portfolio, while those on the receiving (fixed) leg increase the duration of the portfolio. Since interest rate swaps are extremely liquid contracts, they are an efficient way of expressing a short-term view on interest rates. 

Portfolio convexity depends on the distribution of cash flows in the portfolio. A portfolio with an even distribution of cash flows has higher convexity than one where cash flows are concentrated in a particular maturity bucket, assuming equal duration and no optionality. By extension, considering a bond to be a portfolio of cash flows, the obvious conclusion is that bonds with higher coupons have higher convexity than bonds with low or zero coupons. 

The average time until receipt of a bond s cash flows, weighted according to the present values of these cash flows, measured in years, is known as duration or Macaulays duration, referring to the man who introduced the concept in 1938—see Macaulay (1999) in References. Macaulay introduced duration as an alternative for the length of time remaining before a bond reached maturity. 

Duration varies with maturity, coupon, and yield. Broadly, it increases with maturity. A bonds duration is generally shorter than its maturity. This is because the cash flows received in the early years of the bond s life have the greatest present values and therefore are given the greatest weight. That shortens the avetj e time in which cash flows are received. A zero-coupon bond s cash flows are all received at redemption, so there is no present-value weighting. Therefore, a zero-coupon bond s duration is equal to its term to maturity. 

Duration increases as coupon and yield decrease. The lower the coupon, the greater the relative weight of the cash flows received on the maturity date, and this causes duration to rise. T ong the non—plain vanilla types of bonds are some whose coupon rate varies according to an index, usually the consumer price index. Index-linked bonds generally have much lower coupons than vanilla bonds with similar maturities. This is true because they are inflation-protected, causing the real yield required to be lower than the nominal yield, but their durations tend to be higher. 

Yield s relationship to duration is a function of its role in discounting future cash flows. As yield increases, the present values of all future cash flows fall, but those of the more distant cash flows fall relatively more. This has the effect of increasing the relative weight of the earlier cash flows and hence of reducing duration. 

Effective duration recognizes that yield changes may effect the future cash flow of a bond and so its price. For bonds with embedded options the difference between traditional duration and effective duration can be significant. The effective duration of a callable bond, for example, is sometimes half its traditional duration. As noted in chapter l4, for mortgage-backed securities, the difference is sometimes greater still. 

Just as standard duration is not appropriate for bonds with embedded options, neither is traditional convexity. This is because traditional convexity, like traditional duration, fails to take into account the impact on a bond s future cash flows of a change in market interest rates. As discussed in chapter l4, the approximate convexity of any bond may be derived, following in Fabozzi (1997), using equation (11.14). 

There are five basic methods of linking the cash flows from a bond to an inflation index interest indexation, capital indexation, zero-coupon indexation, annuity indexation, and current pay. Which method is chosen depends on the requirements of the issuers and of the investors they wish to attract. The principal factors considered in making this choice, according to Deacon and Derry (1998), are duration, reinvestment risk, and tax treatment. 

As explained in chapter 2, a bond s modified or Macaulay s duration is the average time to receipt of its cash flows, weighted according to their present values. To compute a mortgage-backed bond s duration, it is necessary to project its cash flows using an assumed prepayment rate. These projections, together with the bond price and the periodic interest rate, derived from the yield, may then be used to arrive at the bond s periodic duration, which is divided by twelve (or four, in the case of a bond that pays quarterly) to arrive at its duration in years. 

Given the nature of a mortgage-backed bond s cash flows, its exact yield cannot be calculated. Market participants, however, commonly compare an MBS s cash flow yield to the redemption yield of a government bond with a similar duration or a term to maturity similar to the MBS s average life. The usual convention is to quote the spread over the government bond. 

The modified duration of a bond measures its price sensitivity to a change in yield. It is essentially a snapshot of one point in time. It assumes that no change in expected cash flows will result from a change in market interest rates and is thus inappropriate as a measure of the interest rate risk borne by a mortgage-backed bond, whose cash flows are affected by rate changes because of the prepayment effect. 

Effective duration is essentially approximate duration where P and P are obtained using a valuation model—such as a static cash flow model, a binomial model, or a simulation model—that incorporates the eflFect of a change in interest rates on the expected cash flows. The values of P and T, depend on the assumed prepayment rate. Generally analysts assume a higher prepayment rate when the interest rate is at the lower level of the two rates—interest and prepayment. 

The left side of equation (16.6) is the bond s market price broken into clean price plus accrued interest, as it was in (16.4). In fact, it can be shown that rmu is identical to the initial yield in (16.4), mt. The value of % that results in a stable future bond value is the Macaulay duration. At this point, assuming the existence of only one, parallel yield shift, a change in yield will not impact the future value of the bond. The bond s cash flows are immunized, and the instrument can be used to match a liability existing on that date. 

Liquidity diflferences often produce yield differences among bonds with similar durations. Institutional investors prefer to hold the benchmark bond—the current 2-, 5-, 10-, or 30-year issue— which both increases its liquidity and depresses its yield. The converse is also true because more-liquid bonds are easier to convert into cash if necessary, demand is higher for them, and their yields are thus lower. The effect of liquidity on yield can be observed by comparing the market price of a six-month bond with its theoretical value, derived by discounting its cash flow at the current 6-month T-bill rate. The market price— which is equal to the present value of its cash flow discounted at its yield— is lower than the theoretical value, reflecting the fact that the T-bill yield is lower than the bond yield, even though the two securities cash flows fall on the same day. The reason is liquidity the T-bill is more readily realizable into cash at any time. 

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