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Chapter 40 Formula investing.


Formula investing is a strategy that seeks to limit the role of emotions in investing by adhering to a strict set of rules. Typically, formula investment strategies involve making fixed periodic investments through "dollar cost averaging" or "dividend reinvestment plans." These two strategies can effectively lower the average cost of both equity and fixed-income securities in fluctuating markets. Other formula investing techniques, such as "bond laddering" and "bond barbell strategies," are designed to minimize risk of interest rate fluctuations. (1)


Dollar cost averaging is a simple strategy practiced by many investors who may not even realize it. The strategy is to simply make regular, periodic investments in a security without regard to price. 401(k) investors who automatically invest a certain amount each paycheck are practicing dollar cost averaging. The premise behind dollar cost averaging is to take advantage of market fluctuations to buy more shares when prices are low and fewer when prices are high.

Contrast an investment strategy where an investor purchases the same number of shares of a mutual fund each month with a dollar cost averaging strategy where an investor purchases the same dollar amount of a mutual fund each month. An investor has approximately $1,000 of discretionary income each month and would like to purchase shares of a mutual fund that is currently selling for $10 per share. Consider the following two strategies:

* Strategy A--Purchase 100 shares per month; and

* Strategy B--Purchase $1,000 per month.

The price for each month of the year and the results of each strategy are presented in Figure 40.1. Under Strategy A, the investor has 1,200 shares at the end of the year with an average cost of $10.583 ($12,700 cost / 1,200 shares). The value of the 1,200 shares is $13,200.00 at the final price of $11.00/share ($11 x 1,200 shares), and there is an unrealized gain of $500.00 ($13,200 value $12,700 cost). Under Strategy B, the investor has 1,141.914 shares with an average cost of $10.509 ($12,000 cost / 1,141.914 shares). The value of the shares is $12,561.05 ($11 x 1,141.914 shares) and there is an unrealized gain of $561.05 ($12,561.05 value - $12,000 cost). By purchasing more shares when the price is low and fewer shares when the price is high, the investor lowered his average cost basis with Strategy B. Consequently, the investor finished with greater investment gains (as measured by the unrealized gains of $561.05 vs. $500.00) on the amounts actually invested. Furthermore, the investor maintained a predictable investment outlay--he was not struck with the need to find additional funds for investment in months when the share price was higher. Dollar cost averaging worked well in this example because prices were fluctuating up and down. What if prices were always trending up?

Figure 40.2 shows the impact of the two strategies in a continuous uptrend. In this case, Strategy A has an average cost of $10.55 ($12,660 cost / 1,200 shares), a value of $13,320 ($11 x 1,200 shares), and an unrealized gain of $660 ($13,320 value - $12,660 cost). Strategy B has an average cost of $10.54 ($12,000 / 1,138.661 shares), a value of $12.639.14 ($11 x 1,138.661 shares), and an unrealized gain of 639.14 ($12.639.14 value - $12,000 cost). Note that the dollar cost averaging Strategy B resulted in the lower average cost, but not the maximum gain in this case.

Dollar cost averaging strategy can also be compared to an initial lump sum purchase. For example, assume that Andrea, a 30-year old with limited investments, earns a $12,000 bonus received at the end of January. She has a choice of investing the entire amount in an S&P 500 index fund at one time or spreading the investment evenly over the next 12 months. Each share in the fund currently sells for $9.12. If she invests the entire lump sum at once, she can purchase just over 1,315.8 shares. She is not sure whether the market will rise or fall over the next 12 months and decides to spread the investment over the course of the year.

Here are the month-end values for the fund purchases Andrea makes:
     Month           Fund        Shares

January               $9.12        109.6
February              $9.16        109.2
March                 $8.15        122.7
April                 $8.86        112.9
May                   $9.36        106.8
June                  $8.80        113.6
July                  $8.56        116.8
August                $8.41        118.9
September             $8.48        117.9
October               $9.16        109.2
November              $9.64        103.7
December              $9.75        102.6
Total:                           1,343.9

At the end of the year, she has about 1,344 shares, 28 more than she would have purchased had she bought them all up front (1,344 - 1,316). At the final price of $9.75/share, her investment has grown to $13,103 ($9.75 x 1,344 shares) instead of $12,829 ($9.75 x 1,316 shares) - an additional $274. The reason is that, as the market fell early in the year, she was able to buy a larger number of shares each month (at a lower cost). For example, when the market was at 815 in March she bought nearly 123 shares in the fund. When the market briefly rallied to 936 in May, she purchased only 107 shares at the higher price. Because she was able to take advantage of market fluctuations, she ended the year with more than 2% of additional value than she would have had by investing as a lump sum up-front.

Dollar cost averaging works best when markets are declining or fluctuating, as the investor is able to buy more shares when prices fall (and purchases fewer shares when prices are higher). In this sense, it is a contrarian strategy. Dollar cost averaging does not work in a steadily rising market, as the investor would be better off buying as much as possible at the lower initial price than to pay more each month for a smaller number of shares. Since markets do tend to rise over the long term, it would usually not make sense for Andrea to spread her investment over years rather than months. However, over a span of months, it is quite possible that Andrea will have the opportunity to make purchases at lower prices.

Even though Andrea has a long time horizon and should buy as much in the early years as possible, she must currently make a decision about investing a specific amount of available cash. By spreading the investment over 12 months, she is able to benefit from market fluctuations (since the market did occasionally decline) and at least ensure that she does not (accidentally) invest all of her money at a market peak. It also allows her to establish an investing discipline.

The consequences of investing at a market peak instead of dollar cost averaging can be quite substantial. For example, assume the returns from the above example occurred in reverse order, such that the market started at 9.75 and finished at 9.12. The results of this scenario are as follows:
    Month         Fund       Shares

January            $9.75       102.6
February           $9.64       103.7
March              $9.16       109.2
April              $8.48       117.9
May                $8.41       118.9
June               $8.56       116.8
July               $8.80       113.6
August             $9.36       106.8
September          $8.86       112.9
October            $8.15       122.7
November           $9.16       109.2
December           $9.12       109.6
Total:                       1,343.9

At the end of the year, Andrea still has about 1,344 shares through a dollar cost averaging program. However, in this case the total shares in the end are worth only $12,256 (1,343.9 shares x $9.12/share). But if Andrea had made a lump-sum investment of her $12,000, she would have only been able to purchase about 1,231 shares at the high price of $9.75/share--at the end of the term, this would have only been worth $11,225 (1,230.8 shares x $9.12/share)! In the first case of a fluctuating but rising market, Andrea's dollar cost averaging strategy allowed her to end the year with an extra 2% of additional value--in the latter case, she enjoys an extra $1,031, or over 9% of additional value!

For many investors, dollar cost averaging is as much a matter of necessity as of choice. For those that do not have a lump sum available to invest, dollar cost averaging becomes the de facto strategy simply because they make purchases as cash becomes available to invest.


Dividend reinvestment plans (DRIPs) allow shareholders to take dividends in the form of additional shares rather than cash. Many companies with dividend reinvestment plans allow shareholders to purchase shares commission-free (or with a minimal transaction fee) through their dividend reinvestment plan, and others offer shareholders discounted share prices in return for reinvesting dividends.

At their simplest, dividend reinvestment plans are a form of dollar cost averaging. Instead of accepting a cash dividend, the dividend is reinvested at the then-prevailing price. For the investor, the most basic advantage relates to cash planning. Why should an investor accept dividends if he doesn't need the income for many years? By reducing costly transaction fees and commissions, the investor is able to put more of his investment dollars to work over the long run and earn more. For plans that allow discounted purchases or other perquisites for shareholders, the advantages can be even greater.


Just as with equity investments, bonds are subject to risk. Likewise, bond investors share the desire to maximize their return for a given level of risk.

For risk adverse investors, there are ways to reduce the risk of investing in bonds. For example, an investor who does not want to face any default risk could limit investments to U.S. government bonds. If a somewhat greater return is sought, investing in a diversified portfolio of corporate bonds might be appropriate.

Diversification reduces the negative impact of any single bond defaulting by reducing the amount invested in any particular bond. Risk can also be reduced (although not eliminated) when investing in individual bonds through strategies such as immunization (see below).

As with any investment, bond investments can be actively managed or left alone. A passive approach usually entails selecting a portfolio of bonds to match an investor's preferences toward factors such as time horizon, tax status, and risk, and simply holding the portfolio until maturity. A passive approach often assumes that markets are relatively efficient and that superior returns cannot be obtained by timing the market or continuously looking for mispriced assets. However, care should still be taken in selecting the bonds to include in the portfolio. As with all investments, individual assets and the portfolio as a whole should be examined for appropriateness. The passive approach can also involve techniques such as immunization (see below) and laddering (see below) to reduce portfolio risk over time.

An active approach, on the other hand, generally assumes that markets are less than efficient and that superior (to the passive approach) returns can be generated through market timing and asset selection. Active management can involve analyzing particular bonds and sectors or forecasting anything from interest rates to inflation to economic outlook - looking for mispriced securities and seeking to avoid areas that (in the investor's opinion) may decline in value. This will usually result in more active trading or swapping of bonds within the portfolio, and may require substantial additional investments of time for research and analysis. Passive and active strategies are not always mutually exclusive; some techniques have elements of both and can be useful to all investors.


One fixed-income risk reducing strategy is known as immunization. Immunization involves selecting a bond or portfolio of bonds such that interest rate movements result in bond price fluctuations (interest rate risk) and cash flows for reinvestment (reinvestment risk) that offset one another. It was noted in earlier chapters that these two risks move in opposite directions. If interest rates rise, interest rate risk causes the value of the bond to drop, but the investor earns more on any coupon payments that are reinvested. On the other hand, if rates decline, the value of the bond rises, but less is earned from reinvestment.

Macaulay duration measures the point in time when these two risks precisely offset each other.2 If an investor purchases a bond and sells it when it reaches its duration (as calculated upon purchase), any loss in bond value caused by an increase in interest rates should be offset by the increase in the reinvestment earnings of interest payments. Similarly, if interest rates decline, the loss in reinvested earnings should be offset by an increase in the value of the bonds. By matching the duration of the bond to the investment horizon, interest rate and reinvestment risk can be offset.

Immunization can be accomplished easily using zero coupon bonds, since their maturity is their duration. No computations are necessary. Additionally, zero coupon bonds naturally have no reinvestment risk related to interest payments, since there are no interest payments. If an investor purchases a zero coupon bond and holds it to maturity, the investor will receive his expected yield to maturity (provided there is no default).

Ladder Portfolio

As noted earlier, reinvestment risk is a concern when receiving periodic payments such as interest. Reinvestment risk related to interest payments can be eliminated or reduced by using zero coupon bonds or an immunization strategy (see above). Reinvestment risk also occurs when bonds mature earlier than the investor needs the proceeds - for example, when no bonds are available that match the investment horizon, or when other bonds seem to present a better investment opportunity based upon other factors (e.g., various forecasts or a discovery based upon analysis and research).

One option might be to purchase very long-term bonds (using immunization or duration, see above) to secure the long-term expected return. The danger in this strategy is that long-term bonds are more sensitive to changes in interest rates than shorter-term bonds. If it turns out that the investor must use the money for an emergency or chooses to retire early, the investor may be stuck at an inopportune time with depressed bond prices in the face of a rising interest rate environment.

Another option is to create a laddered portfolio. In a laddered portfolio, an investor invests in an assortment of bonds with staggered maturities. For example, the investor can structure a portfolio where 10% of the bonds mature each year. The bonds would not all mature in the same interest rate environment. If rates rise, the value of the portfolio may fall, the investor does not need to sell bonds that haven't matured. If cash is needed, the maturing bonds offer a ready source. If cash is not needed, the investor can reinvest the proceeds of the maturing bonds at the new (higher) interest rate.

If interest rates fall, the investor will reinvest the proceeds at a lower rate, but only for 10% of the portfolio (and the longer maturity bonds would rise in value.) With a laddered portfolio, the proceeds of maturing bonds would be reinvested in new bonds with a maturity later than those currently in the portfolio.

A bond ladder reduces interest rate risk by staggering maturities among several bonds (each of which represents a rung on the ladder). For a long-term investor, this ends up being similar to a dollar cost averaging strategy. Shorter maturities cushion interest rate (i.e., bond price) risks, while the fact that only a portion of the bonds mature in a given period reduces reinvestment risk. If rates should rise, the maturing bonds can be used to buy bonds offering a higher yield.

Perhaps a greater reason for using a bond ladder is that it enables the investor to match cash flows with planned expenditures. If the investor knows tuition payments will be due each semester for four years, beginning in 15 years, he can buy bonds maturing at exactly the right times to make the tuition payments. Alternatively, a retired investor can plan a laddered bond portfolio to include regularly maturing bonds that provide the cash necessary for annual living expenses.

Bond ladders also work for unplanned expenses. For example, if the investor loses his job, maturing bonds can be used to supplement lost income. Figure 40.3 presents an example of a bond ladder using treasury bonds of various maturities.

Barbell Strategy

A barbell strategy offers another means of balancing the risks in a bond portfolio. While a ladder portfolio may stagger holdings across all maturities, the barbell places heavy weights on very long and very short maturities, with no position in intermediate-term securities. Because it requires only long and short maturities, one advantage of the barbell is that it can be established with fewer bonds, reducing complexity and transaction costs. A disadvantage is that the investor may be unable to match maturities with cash needs as effectively.

A barbell allows the investor to create a fairly effective hedge against both interest rate and reinvestment risk, simply and at low cost. If rates go up, the short portion of the barbell can be reinvested at the higher rate and help offset losses in the longer maturity. If rates decline, the long maturity gains make up for the lower interest rate available for reinvestment of the short maturity portion. Figure 40.4 provides an example of a barbell.

A barbell strategy can also be appropriate based upon the current shape of the yield curve and expected changes in interest rates. A barbell strategy can also result in a portfolio with the same duration as a bullet strategy (a single bond maturity), but with higher convexity. Convexity measures the curvature of the relationship between bond yields and prices. (3) Positive convexity is always a good thing for the bond investor, regardless of whether interest rates rise or fall. If interest rates fall, then convexity will augment the rise in the price of the bond. Interestingly, if interest rates rise, convexity will dampen the decline in the price.


1. Frank J. Fabozzi, Ed., The Handbook of Fixed-Income Securities, 6th Edition (New York, NY: McGraw Hill, 2000).

2. Annette Thau, The Bond Book, 2nd Edition (New York, NY: McGraw Hill, 2001).

3. Marilyn Cohen and Nick Watson, The Bond Bible (New York, NY: New York Institute of Finance, 2000).


Question--An investor is convinced that the stock market will fall in the current year, but rise over the long term. She has a lump sum available to invest. How can she take advantage of her beliefs about the market?

Answer--She can dollar cost average by investing equal amounts over the course of the coming year. If she is wrong and the market rises, she will have taken advantage of lower prices early. If she is correct about the market falling, she will buy more shares at the

lower prices later in the year. In this manner, she is able to mitigate the risk of being wrong while still enjoying some benefit if she is correct.

Question--Bob Boomer started his family late and was able to amass a large portfolio due to his low expenses. However, he now faces the prospect of 3 children going to college over a 10-year period, immediately followed by his own retirement. Of the strategies discussed in this chapter, which would be most appropriate for the fixed-income portion of his portfolio?

Answer--Boomer could structure a bond ladder that matches his needs in terms of his future cash inflows and outflows.


(1.) See LeClair, "Ladders and Barbells", Bob LeClair's Finance and Markets Newsletter, 10.11.2003 at

(2.) For further discussion of Macaulay duration, see Chapter 34, "Measuring Yield."

(3.) For further discussion of convexity, see Chapter 34, "Measuring Yield."
Figure 40.1

             Mutual Fund      Strategy A--      Strategy A--
Month        Share Price         Shares          Total Cost

1              $10.00              100            $1,000.00
2              $11.00              100            $1,100.00
3              $11.00              100            $1,100.00
4              $10.50              100            $1,050.00
5              $10.00              100            $1,000.00
6               $9.50              100              $950.00
7               $9.00              100              $900.00
8              $10.00              100            $1,000.00
9              $11.00              100            $1,100.00
10             $12.00              100            $1,200.00
11             $12.00              100            $1,200.00
12             $11.00              100            $1,100.00
Total                             1,200             $12,700

            Strategy B--      Strategy B--
Month          Shares          Total Cost

1             100.000           $1,000.00
2              90.909           $1,000.00
3              90.909           $1,000.00
4              95.238           $1,000.00
5             100.000           $1,000.00
6             105.263           $1,000.00
7             111.111           $1,000.00
8             100.000           $1,000.00
9              90.909           $1,000.00
10             83.333           $1,000.00
11             83.333           $1,000.00
12             90.909           $1,000.00
Total       1,141.914             $12,000

Figure 40.2

           Mutual Fund      Strategy A--      Strategy A--
Month      Share Price         Shares          Total Cost

1            $10.00              100            $1,000.00
2            $10.10              100            $1,010.00
3            $10.20              100            $1,020.00
4            $10.30              100            $1,030.00
5            $10.40              100            $1,040.00
6            $10.50              100            $1,050.00
7            $10.60              100            $1,060.00
8            $10.70              100            $1,070.00
9            $10.80              100            $1,080.00
10           $10.90              100            $1,090.00
11           $11.00              100            $1,100.00
12           $11.10              100            $1,110.00
Total                          1,200              $12,660

          Strategy B--      Strategy B--
Month        Shares          Total Cost

1           100.000           $1,000.00
2            99.010           $1,000.00
3            98.039           $1,000.00
4            97.087           $1,000.00
5            96.154           $1,000.00
6            95.238           $1,000.00
7            94.340           $1,000.00
8            93.458           $1,000.00
9            92.593           $1,000.00
10           91.743           $1,000.00
11           90.909           $1,000.00
12           90.090           $1,000.00
Total     1,138.661             $12,000

Figure 40.3

Maturity   Yield   Invested              Purpose

3 month    1.0%    $10,000            Emergency fund
2 year     1.8%    $10,000      Hedge against rising rates
3 year     2.3%    $10,000          Higher yield hedge
5 year     3.3%    $10,000                Yield
10 year    4.4%    $10,000                Yield
15 year    4.5%    $20,000       Planned tuition payment
16 year    4.6%    $22,000       Planned tuition payment
17 year    4.6%    $24,000       Planned tuition payment
18 year    4.7%    $26,000       Planned tuition payment
20 year    4.8%    $10,000                Yield
25 year    5.0%    $20,000        Yield, hedge against
                                      falling rates
30 year    5.2%    $30,000     Yield, retirement and hedge
                                  against falling rates

Figure 40.4

  Maturity     Yield   Amount Invested

3 month        1.0%        $50,000
30 year        5.2%        $50,000

  Maturity                        Purpose

3 month        Emergency fund and hedge against rising rates
30 year        Long-term needs and profit from falling rates
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Title Annotation:Techniques of Investment Planning
Publication:Tools & Techniques of Investment Planning, 2nd ed.
Geographic Code:1U2NY
Date:Jan 1, 2006
Previous Article:Chapter 39 Investment strategies.
Next Article:Chapter 41 Hedging and option strategies.

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