A Model of Bond Fund Investing

William E. Shambora

Ohio University

Chulho Jung

Ohio University

Abstract

In this paper we develop a game-theoretic model which can explain seemingly irrational or naïve behavior of bond fund investors. The stylized facts suggest that bond fund investors purchase bond funds when the interest rate is low (i.e., bond prices are high) and sell bond funds when the interest rate is high (i.e., bond prices are low). Consistently buying high and selling low could be considered irrational behavior and potentially brings capital losses to investors. We explain this behavior using a game-theoretic model. Naïve or slow investment decisions by a majority of investors are shown to lead to apparently self-defeating investment behavior in the bond funds market.

Corresponding author: Economics Dept; 335 Bentley Annex; Athens, OH 45701; (740) 593-1845; shambora@ohio.edu.

A Model of Bond Fund InvestingAbstract

In this paper we develop a game-theoretic model which can explain seemingly irrational or naïve behavior of bond fund investors. The stylized facts suggest that bond fund investors purchase bond funds when the interest rate is low (i.e., bond prices are high) and sell bond funds when the interest rate is high (i.e., bond prices are low). Consistently buying high and selling low could be considered irrational behavior and potentially brings capital losses to investors. We explain this behavior using a game-theoretic model. Naïve or slow investment decisions by a majority of investors are shown to lead to apparently self-defeating investment behavior in the bond funds market.

1. Introduction

There is a common perception that mutual fund investors, particularly bond fund

investors, make naïve investment decisions. They are thought to buy when bond interest rates

are relatively low and sell when rates are relatively high, thus minimizing their total returns. If

true, this could create problems for bond fund portfolio managers as they try to balance current

marketing considerations with optimizing long-run returns. It is the purpose of this paper to

model the motivation and results of this behavior.

In an article for the Wall St. Journal Damato (2001) describes the results of a telephone

survey which showed that only 31% of investors know that bond prices decline when interest

rates increase. If investors ignore or misunderstand the effects of changing interest rates on

portfolio values, it is quite possible that they poorly time their bond fund purchases and sales.

Naïve investors seeking high relative current yields instead of high overall total return may buy

long term bond funds when long rates are high relative to short rates. A steep yield curve is

often thought to precede an advancing economy and increasing interest rates. Such timing

presents bond fund managers with a dilemma. Should they emphasize the long end of the

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maturity spectrum to lure investors with high current yields or should they shorten maturities to

minimize the potential erosion of principle? If they opt to attract customers with high current

yield, the overall total return will suffer.

In this paper we model behavior where many bond fund investors do not act in their

own best interest. Their behavior even appears to be irrational in the sense that their naïve and

slow action might lead to potential capital losses. They purchase bond funds when bond prices

are high and sell when bond prices are low. We develop a game-theoretic model which can

explain the irrational behavior of bond fund investors.

Background and qualitative information is presented in Section 2. Section 3 introduces a

game-theoretic model that explains mutual fund investor behavior. Section 4 concludes.

2. Background

2.1 Mutual Funds and Bond Funds

Most mutual funds fall into one or more of three major asset classes: Fixed income,

equity, and money market. The Investment Company Institute [ICI] reports on four general fund

categories: Bond funds, equity funds, money market funds and hybrid funds. Hybrid funds

represent a small portion, about 5%, of the total value of mutual funds. Bond mutual funds,

representing the fixed income category, use pooled investor money to invest in diversified

portfolios of bonds. Stock funds invest in equities and money market funds invest in short-term,

near-cash, instruments such as certificates of deposit and Treasury bills. Investors can generally

change their allocation among these types of funds without cost within a family of funds.

Presumably, investors allocate among the fund types based on their assessment of the risk and

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return associated with each type. Shares of mutual funds are created when investors add money

to a fund and destroyed when investors withdraw money from a fund. In other words supply

always adjusts to demand. When confronted with net inflows or outflows, fund managers must

make adjustments to the fund’s investment portfolio.

ICI (1998) reports that 80% of bond fund holders also hold stock funds. Bond fund

holders are about the same age, but slightly better educated and wealthier than mutual fund

holders in general. Investment in all types of mutual funds has increased dramatically during the

past two decades. For the period from 1990 through 2002, assets in bond funds increased from

$291 billion to $1.1 trillion, stock funds increased from $239 billion to $2.7 trillion, and money

market funds increased from $498 billion to $2.3 trillion. To put this in perspective, while

mutual fund assets increased nearly six-fold, nominal GDP and corporate profits only doubled

during this period. At the mid-point of 2003, the total bond mutual fund holdings of $1.5 trillion

accounted for about 11.6% of the total corporate, U.S. government, agency, and municipal

bonds. Equity mutual fund holdings of $2.5 trillion accounted for about 18.9% of the total

equity value of $13.3 trillion1. Although the overall trend has been for increased investment in

mutual funds, fund inflows and outflows can change dramatically. The flows to any class of

fund can be influenced by investors’ ability or willingness to invest in mutual funds in general or

changes in asset allocation among funds.

2.2 Bond Fund Flows and Returns

1 Data from FRB Flow of Funds as of September 19, 2003. Data descriptions can be found at http://www.federalreserve.gov/releases/Z1/.

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A rational bond investor would buy long-term bonds when interest rates are high to

capture capital gains when interest rates decrease. Figure 1 shows the relationship between yields

on Aaa rated corporate bonds and net flows into bond mutual funds. The series look almost like

mirror images, suggesting an inverse relationship between the two. Such a relationship would

imply that investors buy bond funds near interest rate lows and sell bond funds near interest rate

highs. This behavior would not be in the investors’ best interests and would tend to reduce their

overall return on these investments.

(Figure 1 about here)

An informal analysis of flows around local interest rate peaks and troughs from 1984

through 2002 is quite revealing. The cumulative flow within three months before and three

months after each of six peaks sums to more than $84 billion of net outflow from bond funds.

The cumulative flow within three months of each of five troughs sums to over $117 billion of net

inflow to bond funds. This suggests that bond fund investors may indeed take action opposite to

their best interest.

There have been several key studies that examine the relationship between mutual fund

flows and returns. Warther (1995) looks at flows in stock, bond, and gold funds. The author

finds correlation between flows into stock funds with stock returns, bond funds with bond

returns, and gold funds with gold returns. Positive correlation exists between flows and

subsequent returns in weekly data, but negative correlations in monthly data. The author

suggests that flows and returns move together. Remolona, Kleiman, and Gruenstein (1997) find

very weak correlation between short-term returns and flows. Flows into funds with more

conservative investment objectives appear to be more sensitive to return. Edelen and Warner

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(2001) examine daily flows for a year and a half for stock funds only. They find a positive

concurrent relationship between flows and returns. One-day lagged flow is also positively

correlated suggesting a possible positive feedback effect. Fant (1999) finds equity fund returns

to be positively concurrently correlated with exchanges from other funds into the funds and

negatively correlated with exchanges out of the funds. He finds no evidence of a relationship

between returns and new sales or redemptions. Edwards and Zhang (1998) find that returns on

equity and bond funds positively impact flows, but flows do not impact returns.

Goetzmann, Massa, and Rouwenhorst (2003) study behavioral factors affecting daily

flows. The authors find positive correlation between daily flows and contemporaneous fund

returns for equity funds, but not for bond funds. They find negative correlation between equity

and bond fund flows. Using factor analysis principal components approach, the authors find the

polarity between stock and bond fund flows to be the first component, explaining more than 30%

of the variation. The authors attribute this to investor sentiment regarding the equity premium.

Sirri and Tufano (1998) find that investors base their decisions on equity fund purchases on prior

performance. They also uncover evidence that fund buyers are fee sensitive and respond to fund

volatility. Zheng (1999) examines investors’ fund selection ability. That paper finds that

investors can pick equity fund winners, largely due to persistence of returns. It finds abnormal

inflows to funds that become winners.

3. A Model of Mutual Fund Buyer Behavior

3.1 Basic Model

Given the behavioral characteristics implied by the articles and stylized facts discussed in

the previous sections, we attempt to model that behavior in this section. Such a model should

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prove useful in predicting investor behavior. We assume that only bonds and stocks are available

to investors. We do not consider other types of financial assets for reasons of simplicity. We

examine four cases under three different states of interest rates. We then examine the

implications of the model under gradually changing interest rates.

Let us assume that there are two types of bond fund investors classified into “Wise” and

“Naïve.” Naive is composed of uninformed and naïve investors. Those who are in this group do

not understand the relationship between bond prices and interest rates. They do not understand

the basics of economic theory. Wise is composed of sophisticated and well-educated investors.

Those who are in this group not only understand economic theory, but also know how the stock

and bond markets work.

We also assume that there are three states of the world: high interest rate state; medium

interest state; and low interest rate state. In the high interest rate state, the economy is at the

peak of its business cycle. The stock market is also at its peak. In this situation, Naive keep

their money in the stock market since Naive are ignorant of the ramifications of high interest

rates. They do not understand that high interest rates mean low bond prices. Thus, they stay in

the stock market since the stock market is doing very well. For example, when the stock market

was at its peak in the middle of 2000 many people who did not have a good grasp of economics

stayed in the stock market.

Wise, however, comprised of well-educated and smart investors, understand that high

interest rates mean low bond prices, and that the economy will be in the contraction stage of the

business cycles in the near future. Thus, they move their money from the stock market into the

bond market and keep their money in the bond market until the contraction stage is over.

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Figure 3 shows the situation explained above. The game matrix shows choices for Naive

and Wise. The stock market is denoted by “S” and “B” denotes the bond market. The numbers

in parentheses show the level of satisfaction the investors can get from staying in either stock or

bond markets. The satisfaction level is binary where 1 is satisfied and 0 is unsatisfied. Staying

in the stock market is the dominant strategy for Naive, and staying in the bond market is the

dominant strategy for Wise. Thus, (S,B) = (1,1) is the dominant strategy equilibrium (also a

Nash equilibrium) for the mutual fund investment society. If, for example, 90% of the investors

are Naive and 10% are Wise, then only 10% of the investors are in the bond market, which

explains why bond fund flow could be low when the interest rate is high. Instead of using

specific proportions (90% and 10%) for each type, we can use (1 - w) and w to find more general

results as will be explained later.

(Figure 3 about here)

Let us next assume that the economy is in the middle of the contraction stage of the

business cycle and thus in the medium interest rate state. In this case, the economy is worsening

and the stock market is declining. Wise know that the economy is not at the bottom yet and they

correctly expect the interest rate to go down further in the future. Thus, Wise keep their money

in the bond market. However, Naive are not sure whether the economy will get better or worse

in the near future. Thus, as can be seen in Figure 4, Naive are indifferent between stock and bond

markets.

(Figure 4 about here)

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As previously mentioned, the numbers in parentheses show the level of satisfaction the

investors can get from investing in either stock or bond markets. Staying in the bond market is

the dominant strategy for Wise, however, Naive is indifferent between the two strategies. Thus,

(S,B) and (B,B) are the two Nash equilibria for the mutual fund investment society. If 90% of

the investors are Naive and 10% are Wise, then a half of Naive and all of Wise will stay in the

bond market, which means 55% of the investors would be in the bond market. It explains why

bond fund flow is at its medium level when the interest rate is at the medium level. It is also

reasonable to say that as the state of the economy changes from high interest rate state to

medium interest rate state (or from its peak to contraction stage), bond mutual fund flow

increases.

Let us assume that the economy is at the bottom (trough) of the business cycles. In this

case, the economy is in the low interest rate state. Naive finally realize that staying in the stock

market is not a good idea and they move all their money to the bond market without realizing

what will happen to the economy in the future. Wise understand that the economy and the

interest rate are at the bottom, and that bond prices are at their peak. Wise move all their money

to the stock market. As shown in Figure 5, staying in the stock market is the dominant strategy

for Naive, and staying in the bond market is the dominant strategy for Wise. Thus, (B,S) = (1,1)

is the dominant strategy equilibrium in this case for the investment society. If 90% of the

investors are Naive and 10% are Wise, then 90% of the investors are in the bond market, which

explains why bond fund flow is high when the interest rate is low.

(Figure 5 about here)

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Next we assume that the economy is in the middle of the expansion stage of the business

cycle and thus in the medium interest rate state. In this state, both the economy and the stock

market improve. Wise know that the economy is not at the peak yet and they correctly expect

the interest rate to go up further in the future. Thus, Wise keep their money in the stock market.

However, Naïve are not sure whether the economy will get better or worse in the future. Thus, as

can be seen in Figure 6, Naive are indifferent between stock and bond markets.

(Figure 6 about here)

Staying in the stock market is the dominant strategy for Wise; however Naive are

indifferent between the two strategies. Thus, (S,S) and (B,S) are the two Nash equilibria for the

mutual fund investment society. If 90% of the investors are Naive and 10% are Wise, then one

half of Naive will stay in the bond market, which means 45% of the investors are in the bond

market. This explains why bond fund flow is at its medium level when the interest rate is at the

medium level when the economy is in the process of expansion. It is also reasonable to say that

as the state of the economy changes from low interest rate state to medium interest rate state (or

from its trough to expansion stage), bond mutual fund flow decreases.

3.2 Various Specific Cases

Using the above model with gradual interest rate changes, we can examine more realistic

and interesting results. Let us assume that interest rates move in the range of zero and one. Thus,

0 r 1, where r = interest rate. Also assume that the proportion of the mutual fund

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investors who are Wise is w. Thus, 0 w 1, where w = proportion of mutual fund investors

who are Wise.

We assume that as the interest rate falls the proportion of those who are Naive who move

from the stock market to the bond market changes gradually and in inverse proportion to the

change in the interest rate. Thus, the proportion of Naive who are in the bond market equals (1 –

r). As r falls from 1, the proportion of Naive in the bond market increases gradually.

We can find the proportion of mutual fund investors who are in the bond market for each

value of r. When the economy moves down from its peak, (1 - r)(1 - w) + w will be in the bond

market for 0 < r 1. When the economy moves up from its trough, (1 - r)(1 - w) will be in the

bond market for 0 r < 1. When w = 0, i.e., every investor is Naive, the above function is

continuous at r = 0 or 1. The graph of the above two functions when w = 0 is shown in Figure 7.

In order to see the bond mutual fund flow more clearly, the vertical axis measures fund flow and

interest rates on the same scale. The horizontal axis measures time. As can be seen in Figure 7,

the two variables show a mirror image similar to that of Figure 1 in Section 2. If w increases

from zero to one, then the two lines of the bond fund flow become flatter. Figure 8 shows the

case where w = 0.1 as mentioned in the examples associated with Figures 3 through 6. Figure 9

shows the case when w = 0.5. If w = 1, then the two curves become like the ones depicted in

Figure 10.

When w is relatively small, we see the mirror image like that of the actual data.

However, as w increases, the mirror image disappears and the discontinuous shift in the two lines

at the peak and trough of the economy becomes increasingly larger. Since Wise show a

complete shift in their strategy instead of a gradual one shown by Naive, the discontinuity of the

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two lines becomes larger and larger. When w =1, we see two horizontal lines discontinuous at r

= 0 and r = 1. If the proportion of Wise investors is big, then we would see more drastic changes

in the bond mutual fund flow. This means that the proportion of investors who are Wise is

probably small since the actual movement in bond mutual fund flow is rather gradual.

The model also shows that poor timing and slow action of bond investors lead to

irrational behavior in the bond market and creates the mirror image shown in Figure 1. Their

poor timing and slow action is due to their inability to understand what is happening to the

market.

4. Conclusion

In this paper we suggested that bond fund investors appear to behave irrationally in the

market. As shown in Figure 1, bond fund investors purchase bonds when bond prices are high

and sell bonds when bond prices are low. In Section 3, in order to explain the stylized facts

about investor behavior, we developed a game-theoretic model. The model explains that poor

timing and slow action of bond investors lead to apparently irrational behavior in the bond

market and creates the mirror image of flows and interest rates shown in Figure 1. The poor

timing and slow action is due to their inability to understand what is happening to the market.

The lesson to mutual fund investors is simple. A disciplined, random approach to asset

allocation and market timing would be more rewarding for many naïve fund investors. The

famous dollar cost averaging system and a rigid asset allocation mix would serve many of these

folks better than trying to time the market by chasing what appears to be today’s “best bet”.

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REFERENCES

Blake, Christopher R., Edwin Elton J., Martin J. Gruber, 1993, The Performance of Bond Mutual Funds, Journal of Business 66, 371-403.

Box, George E. P., and Gwilym M. Jenkins, 1976, Time Series Analysis: Forecasting and Control, San Francisco: Holden-Day.

Carhart, Mark, 1997, On the persistence of mutual fund performance, Journal of Finance, 52, 57-82.

Cornell, Bradford, Kevin Green, 1991, The Investment Performance of Low-grade Bond Funds, The Journal of Finance 46, 29-48.

Damato, Karen, 2001, Investors turn to bonds without grasp of basics, The Wall Street Journal. November 9, 2001.

Edelen, Roger M., 1998, Investor flows and the assessed performance of open-end mutual funds, Journal of Financial Economics 53, 439-466.

Edelen, Roger M., Jerold B. Warner, 2001, Aggregate price effects of institutional trading: a study of mutual fund flow and market returns, Journal of Financial Economics 59, 195-220.

Edwards, Franklin R., Xin Zhang, 1998, Mutual Funds and Stock and Bond Market Stability, Journal of Financial Services Research 13:3, 257-282.

Elton, Edwin J., Martin J. Gruber, Christopher R. Blake, 1995, Fundamental Economic Variables, Expected Returns, and Bond Fund Performance, The Journal of Finance 50, 1229-1256.

Enders, Walter, 2004, Applied Econometric Time Series, John Wiley and Sons, New York.

Fant, Franklin L., 1999, Investment behavior of mutual fund shareholders: The evidence from aggregate fund flows, Journal of Financial Markets 2, 391-402.

Goetzmann, William N., Massimo Massa, K. Geert Rouwenhorst, 2000, Behavioral Factors in Mutual Fund Flows, Yale International Center for Finance, Yale School of Management.

Investment Company Institute, 1998, A guide to Bond Mutual Funds, Washington, DC.

Investment Company Institute, 2003, Mutual Fund Yearbook, Washington, DC.

Lehmann, Bruce N., David M. Modest, 1987, Mutual Fund Performance Evaluation: A Comparison of Benchmarks and Benchmark Comparisons, The Journal of Finance 42, 233-265.

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Remolona, Eli M., Paul Kleiman, Debbie Gruenstein, 1997, Market Returns and Mutual Fund Flows, FRBNY Economic Policy Review, July 1997, 33-52.

Sirri, Erik R., Peter Tufano, 1998, Costly Search and Mutual Fund Flows, The Journal of Finance 53, 1589-1622.

Warther, Vincent A., 1995, Aggregate Mutual fund flows and security returns, Journal of Financial Economics, 39, 205-235.

Wei, William W. S., 1990, Time Series Analysis, Addison-Wesley, Redwood City, California.

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Figures (1 – 10)

Figure 1. Bond fund flow and bond yields

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Figure 2. Term Premium and Bond Fund Flow*

*Data are from the series described in section 3. In (a) both the bond flow and yield spread data are split at their respective median values while in (b) both are split at their mean values. The term premium is the yield spread between the ten-year Treasury and the three-month Treasury bill. Numbers in the cells represent the number of observations in each of the four categories. Total number of observations =228.

(a) Split at Median FlowLow High

Yield Spread:10 Yr vs. T-bill

Low 71 43High 42 72

(b) Split at Mean FlowLow High

Yield Spread:10 Yr vs. T-bill

Low 82 38High 45 63

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Figure 3. Economy at peak of business cycle

Figure 4. Contraction of business cycle

Figure 5. Business cycle bottom

Figure 6. Business cycle expansion

WiseS B

Naïve S (1,0) (1,1)

B (0,0) (0,1)

WiseS B

Naïve S (1,0) (1,1)

B (1,0) (1,1)

WiseS B

Naïve S (0,1) (0,0)

B (1,1) (1,0)

WiseS B

Naïve S (1,1) (1,0)

B (1,1) (1,0)

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Figure 7. Wise = 0.0% (w = 0.0)

Interest Rate and Bond Fund Flows

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

Interest Rate Flow

Figure 8. Wise = 10% (w = 0.1)

Interest Rate and Bond Fund Flows

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

Interest Rate Flow

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Figure 9. Wise = 50% (w = 0.5)

Interest Rate and Bond Fund Flows

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

Interest Rate Flow

Figure 10. Wise = 100% (w = 1.0)

Interest Rate and Bond Fund Flows

0

0.2

0.4

0.6

0.8

1

1.2

Time

Interest Rate Flow

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