This section describes the mathematical models that underlie asset dedication. If you have no interest in this or you do not have suffi- cient training to follow the technicalities of why asset dedication works, you may want to skip this section and go on the next, which describes the critical path that the Browns’ portfolio must follow after they retire if it is to last until they reach age 100. Chapter 8 will provide a step-by-step description of how to use the web site that supports this book (www.assetdedication.com). Those who pre- fer the do-it-yourself approach can use the web site to find the spe- cific bonds that will make up the income portion of their portfolio.
The web site requires no mathematical training and is self- explanatory. It is really a matter of filling in the blanks and letting the site do the work.
The mathematical algorithms that underlie asset dedication are based on a technique called mathematical programming. Dis- covered in 1947 by a mathematician working for the U.S. Air Force during World War II, it is a tool that has been used by airlines, oil refineries, and shipping companies to save billions of dollars.6 Its applications grow every year, and it has become an established component of all college curriculums that include a course in quan- titative methods.
In the vernacular of mathematical modelers, the problem faced by the Browns is a scheduling problem. They must schedule the maturities of their bonds to match the cash flow needs related to their living expenses over the planning horizon they choose. In finance books, it is referred to as the cash-matchingproblem.
This problem was at one time considered intractable. Recall the quote from Chapter 3 by William Sharpe, 1990 Nobel Prize win- ner in Economic Sciences, in his recent text:
Cash matching is not so easily accomplished. This is because the promised cash outflows may involve an uneven stream of pay- ments for which no zero coupon bonds exist. Indeed, it can be dif- ficult (if not impossible) and expensive to exactly match cash inflows with promised outflows.7
The Distribution Phase: Dedicating Assets to Do Their Job 143
Sharpe is correct when he says that cash matching is difficult, but it is no longer impossible. High-end money managers that con- trol large pension funds with millions or billions of dollars have access to the technology that can deal with this problem. But until now, no one had made it available to individual investors except in a crude, manual way.
If only zero-coupon bonds are used to fund the income portion of the portfolio, higher-level mathematics is not required. Zero- coupon bonds pay no interest until they mature. At that point, they pay all the interest due in one lump sum that is added to the prin- cipal. (Zeros also have some tax disadvantages, which are described shortly). To match an income stream, you simply buy the number of zeros required to match the amount needed each year. Table 7.2 demonstrates the simple mathematics with zeros for Ms. Smith from Chapter 3. Recall that she wanted to withdraw $30,000 plus 4 percent inflation from her portfolio. Her target income stream is shown in the middle column of Table 7.2. Divide the cash flow needed by $1000 and round off. You must round off because zeros, like nearly all bonds, come only in denominations of $1000. The tar- get income stream comes to $168,990 excluding the $30,000 needed for the current year. The face value of the zeros is $169,000. They will cost less than $169,000, of course, reflecting the interest they will earn until they mature plus a lower purchase price if current interest rates are higher than the coupon rates the bonds pay.
The practical problem with this is that even though the inter- est is not received, Ms. Smith must still pay taxes as if it were if the
Table 7.2
Zero-Coupon Bonds to Buy for
$30,000 plus 4 Percent Inflation
Target Zeros to Year Income Buy
1 $31,200 $31,000 2 $32,448 $32,000 3 $33,746 $34,000 4 $35,096 $35,000 5 $36,500 $37,000 Total $168,990 $169,000
funds are in a taxable account. For example, assume that the
$37,000 in zero-coupon bonds purchased to fund the fifth year in Table 7.2 cost $30,000 at today’s interest rates. One year from now, the same zeros are selling for $31,000. As far as the IRS is con- cerned, Ms. Smith earned the equivalent of $1000 on her invest- ment and must pay taxes on that $1000.
But, you may think, “She did not actually receive the money. It is only ‘phantom’ interest. Why should she pay taxes on it?” Too bad—she will have to pay those taxes with money from somewhere else. Zero-coupon bonds hold a somewhat unique position in the tax code, as taxes must be paid regardless of the fact that the bonds did not actually pay interest. Once this phantom interest tax issue is factored in, zero-coupon bonds are not quite the convenient solution that they appear to be. Also, they sometimes pay less interest than coupon bonds. If the money is held in a tax-free account like an IRA, the tax problem does not apply, and zero-coupon bonds can be used if they pay the highest interest. They certainly make the cal- culations easy—you just buy what you need for each year in the future after factoring in inflation.
Coupon bonds, on the other hand, are more common among corporate and municipal bonds.8 They also complicate the mathe- matics. For a 10-year target income stream, the appropriate num- ber of coupon bonds would be used in precisely the same way to build a portfolio that provides the same income stream as the zero- coupon bonds discussed in Table 7.2. But a coupon bond maturing in 10 years actually pays the interest every year between now and then. If $20,000 is invested in a bond with a 5 percent coupon, it will generate $1,000 interest every year for the next 10 years (in fact, probably $500 every 6 months). It therefore supplies $1,000 cash flow each year. The same is true of a similar bond that matures in 9 years, 8 years, 7 years, and so on. The cash flows pro- vided by these interest payments must be factored in when esti- mating how many additional bonds must mature each year in order to match the needed withdrawal exactly. The number to buy for any given year depends on how much interest is already being gener- ated for that year by all the bonds maturing in later years.
Piecing together the right coupon bonds in the right way to generate the right income stream in a precise manner is difficult.
The simultaneity introduced by the interrelationships of bonds with different maturities makes the problem mathematically chal- lenging, especially when the income need is “lumpy,” or irregular. If the Browns want to take a cruise every other year, for example, and
The Distribution Phase: Dedicating Assets to Do Their Job 145
need $10,000 more than normal to pay for it, the extra withdrawals represent lumps in the otherwise smooth payment stream.
Fortunately, mathematical programming can solve this sort of problem. Unfortunately, like many high-level mathematical tech- niques, it is not easily understood. The formulations and solution algorithms are admittedly complex. It is a little like playing the vio- lin: It only looks easy by someone who knows how to do it. Anyone who has had a course that includes mathematical programming can testify to its complexity (unfortunately, I am convinced that some of my MBA students never really catch on). So the actual calculations are best left to a computer, which is exactly what Chapter 8 describes how to do.