Bond Portfolio Strategies, Returns, And Skewness a Note

8/6/2019 Bond Portfolio Strategies, Returns, And Skewness a Note

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSISMarch 1977

BOND PORTFOLIO STRATEGIES, RETURNS, AND SKEWNESS: A NOTE

H, Russell Fogler, William A, Groves, and James G. Richardson*

1, IntroductionThe academic research has produced a series of contributions on optimal

ortfolio strategies (Bradley and Crane (1], Crane (4], Cheng (3], Fisher and(5], Watson (9], Wolf (10]). Several of these studiesBradley and Crane,

"dumbbell" strategy. Such a dumbbell strategy invests only in the shortestd longest maturities, ignoring the intermediate maturities. The logic is

straightforward: liquidity risk is lowest in the shortest maturities and yieldis generally highest in the longest maturities. The risk/return superiority ofch a strategy was empirically verified by Watson, with subsequent confirmation

However, two important issues have not been fully addressed by these stud-First, what is an appropriate objective functionnone of the previous

performance evaluation. A second issue is whether bond portfolio holding-include a guaranteed fixed portion, are more or less sym-

cal than common stock returns; such a condition might present important

Accordingly, the purpose of this note is twofold: (1) to evaluate the

dumbbell strategy; and (2) to exaunine the moments of the return distribu-for bond portfolios. To achieve these purposes, a bond portfolio simula-

movements. The next section will describe the simulation prograun

University of Florida, University of Florida, and Cerny and Ivey Associ-respectively.


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and the results. Section III then discusses the skewness of the returns, aswell as the effect of nonstationarity on the skewness measure,

II, Impact of Objective Function SpecificationWatson's study was a major empirical simulation of the postwar yield cur

history and corresponding bank cash flow patterns. His results indicated ththe dumbbell strategy was superior on a risk/return criterion compared to othstrategies, and he noted that his measure of risk [9, p, 39] and return

...included all coupon instaOsility but only that portion ofthe capital gains and loss instadaility caused by the forcedsale of bonds to meet liquidity needs. Realized capitalgains and losses caused by bond sales implemented solely torestructure the maturities in the portfolio were excludedfrom this measure of risk, [9, p. 38]

Bradley and Crane handled the bond problem much differently. Their forlation had an objective function of terminal (horizon) value maximization suject to four types of constraints: nonnegativity, cash inflows and outflowsinterperiod balancing, and net capital loss. While they did include a constron book (unrealized capital) losses at the horizon time period, constraints each period's capital loss were defined:

in order to reflect bank practice and to represent theportfolio manager's attitude toward risk, constraints wereput on the allowaQ>le net realized capital loss during anyyear, [1, p, 23]

Thus, Watson, Bradley and Crane, and Wolf did not penalize long-termbonds for any risk from price fluctuation if the bonds were not sold. Whilesuch a definition of risk might be considered a reasonable attitude becausebonds have little or no principal risk if held to maturity, a high rate of iflation does create purchasing power risk; it may be argued that loss in primay be caused by inflation rather than merely a cyclical rise in interest raFurther, a price decline has an implicit opportunity cost of not obtaining thighest rate possible. The two implicit problems of purchasing power risk aopportunity costs make a strong case for a holding-period-return criterion.

To assess the results of using a holding-period-return (HPR) criterion,computer program was developed. Twenty maturities were availad^le for theperiods 1-20 years. Yield curves were generated for each year from 1946-197

Because previous studies had dealt only with government securities, inest rates were collected for: (a) 3-month Treasury Bills; (b) Short-Term (year) Governments; (c) Intermediate (7-9 year) Governments; and (d) Long-Ter


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Depending upon which portfolio strategy was being evaluated, bonds were boughtat the beginning of the year and then sold at the end of year prior to calculating the HPR. Transactions costs on purchases and sales were 1/8 percent on one-year maturities, 1/4 percent of maturities from two up to eight years, amd 3/8percent on longer maturities.

Five interest rate patterns and four portfolio strategies were tested. Thselection of interest rate patterns was to provide a set of generating processewhich would replicate all reasonaQjle assumptions. The patterns selected were:(1) cyclical with rising trend1945-1970, (2) cyclical with declining tr end 1970-1945, (3) full trend cycle and cyclical1945-1970-1945; (4) cyclical without trend (1948-1950, repeated eight times for a simulated 24-year history);and (5) rising with humped yield curves1959-1970. Patterns #1 and #5 werethe actual data, while the other three patterns were artificially created fromthe actual data.

The four portfolio strategies were: even-ladder, full-dumbbell, partial-dumbbell, and buy-longest-and-hold (BLAH)-ladder. The even-ladder strategy wasto invest 1/20 of the portfolio's value in each maturity. The full-dumbbellallocated 27 percent evenly spread between the shortest three maturities, andthe remaining 73 percent to the 20-year bond (the role of short maturities in

izing terminal wealth is summarized in Fogler (6, pp. 264-266] and Renshaw(8, pp. 123-149]. The partial dumbbell was similar to the full dumbbell excep

al yield curve. This procedure was selected after extensive testing of oth

The 27 percent was placed in short maturities because it has been shownif you have a risky and riskless asset, you can obtain the highest return

r time (geometric rate of returnand correspondingly, the highest terminal

your wealth in the riskless asset. The riskless asset acts as a sort ofIf your risky asset declines, you can invest more; this allows aof dollar averaging effect, and amounts to a constant ratio plan assuming

stationary return distribution.The approximating formula for determining the percentage in riskless asset

s X = r-i where X_ is the percentage to be invested in the risky asset, i2 the return on the risky asset, i is the return on a riskless ass

such as a one-period Treasury Bill, s is the variance of returnhe risky asset. The 1946-71 annual holding period returns (HPRs) for a 20-

(thus eliminating nonstationarity in the mean, as well aing the resulting returns on a percent of market yield basis). The varianchese returns was approximately 0.52. Using r=8%, and i5%, and s24 .1 6%


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that the 20-year bond was not rolled-over until it had less than fifteen yearto maturity; this is, in each year available cash (interest payments, maturinbonds, and sold bonds minus required reinvestment in the short-term 27 percenwas reinvested in the 20-year bonds which would be held for the next six yearThe BLAH-ladder strategy was similar to the partial dumbbell except long bondwere bought and held until they had less than four years to maturity.

For initial testing, one billion dollars was the starting portfolio, andno further cash inflows or outflows were assumed (while no cash outflows seemreasonaUDle for many types of institutional investors, the full impact of thisassumption will be discussed later on in this Section), Table 1 contains theresults for each strategy and interest rate pattern.

Clearly the full-dumbbell is always inferior to the other three strategconsidering either return or risk/return. The partial-dumbbell strategy wassuperior in its return when rates were either falling (1970-45) or fluctuatincyclically (1948-50); however, the partial-dumbbell was always inferior to bothe even-ladder and the BLAH-ladder on a risk/return basis as measured by thecoefficient of variation.

By a risk/return criterion, either the even-ladder or the BLAH-ladder isalways best. The BLAH-ladder always dominates the even-ladder on both returand risk/return, with the exception of the rising humped yield pattern (1950when the shorter average maturity of the level becomes a distinct advantage.Over a full-trend-and-cycle interest rate pattern (1945-1970-1945), the returadvantage of the BLAH-ladder is a result of buying higher yields on average4holding them. During the shorter period, the advantage is partly capital gduring price declines and partially higher interest. The risk/return superiity of the BLAH-ladder is due to its lower standard deviation; this lower staard deviation results from the property of higher coupon bonds to fluctuate lthan lower coupon bonds (remembering that BLAH is buying the highest couponsyield curves are upward sloping),

regardless of the percentage allocation between short versus long bonds,4This statement can best be documented by reference to Tad>le A-2 in the

Appendix, Table A-2 presents the holding-period returns after both capitalgains and transactions costs were removed. The resultant returns are intereyields, and the BLAH-ladder can be seen to have a slight interest rate advanover the full period 1945-1970-1945,

Initially it was thought that the lower risk might be due to the gradu


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TABLE I

HPRs and Risk of Portfolio Strategies(Transactions Cost Included)

1945-70

1.632.372.072.39

Interest Rate Patterns1970-45 1945-70-45 1948-50

a. Geometric Rates of Return4.484.684.834.76

3.083.573.483.61

2.112.352.472.36

1959-7

2.983.903.423.83

b. Standard Deviations5.065.025.064.61

2.892.022.311.85

5.274.935.274.50c.

1.141.031.06.93

5.325.095.294.56

Coefficients of1.651.381.471.23

2.952.692.982.66

2Variation1.371.131.191.11

5.986.096.026.03

1.901.501.681.51

Holding-Period-Returns (HPRs) are equal to (1.0 plus the geometric rate ofreturn). For ease of analysis, the 1.0 has been subtracted and only thegeometric rate of return is presented. 2.37 means 2.37%. This is thesame concept as bond traders' "total return."

The coefficient of variation is the standard deviation of the returnsdivided by the arithmetic mean of the returns. As such, it gives ameasure of risk/return. This measure seemed most appropriate for abond portfolio, rather than the standard capital market measures(e.g., Treynor, Sharpe, etc.) in which the return is measured as theexcess return above the risk-free rate of return.


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What do the above findings suggest? Basically, they confirm and extendthe Bradley and Crane observation that

the most important difference between Watson's simulationresults and those of the optimization model is that hisefficient portfolios contain no laddering at the long end.This difference probably results from his assumption thatthe long term bond can be sold at the end of each periodand a new one purchased at no cost.... If this unrealisticassumption of no trading cost were removed, his analysiswould in all likelihood show that efficient portfolioswould be laddered on both the long and short end of thematurity range. (1, p. 30]

However, Bradley amd Crane's analysis was limited to approximately threetime periods, versus an infinite horizon solution, because of the computationamagnitude of their stochastic dynaunic prograunming formulation. As they noted,this prevented them from determining "when should the bonds be taken long agai(1, p. 3 0] . The results of our analysis suggest that the answer may well bethat the bonds would never be taken long but rather that a BLAH-ladder might bexpected to evolve, especially for financial institutions with low liquidityrequirements. For institutions with higher liquidity needs, the short-termallocation would be higher, but the remainder of the BLAH-ladder strategy woulbe appropriate.

III. Skewness and Bond Portfolio HPRsDuring the aibove tests, the computer prograun calculated both the geometri

and arithmetic average of the HPRs for each test. These averages were oftenose, much closer than expected given the size of the standard deviatio

skewed, and Table II contains the results of calculating the skewness measureseach portfolio strategy and each interest rate pattern. The results were

Q

skewed in all cases except the 1948-50 pattern.Such skewness raises several questions. Although the impact of skewness

is well known, the exact interpretation of its ex post timeit must be remembered that their model contained two types of constrai

realized losses and an upper limit on unrealized lossThese limits (constraints) forced most strategies to be conservativ

for liquidity, i.e., these extra funds were invested because of the connstraints which emphasized safety from price fluctuations.

On examination of the 1948-50 pattern, the large negative returns occurr


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TABLE II

Skewness of Bond Portfolio HPRs(with Transactions Costs)

1945-70

.941.421.021.58

Interest Rate1970-45

.43

.53

.46

.92

Pattern1945-70-45 1948-50

,60,89,65.93

-.60-.65-.61-.61

1959-70

.891.06.84

1.08

Relative skewness was measured as the third moment about the meandivided by the cube root of the standard deviation. For a synmetricaldistribution, this measure is equal to zero. Larger coefficients indi-cate greater skewness.


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measurement is less clear. For example, Fogler and Radcliffe (7] recentlposi-

HPRs, but negative skewnessHPRs. Further, skewness of annual

To assess whether similar saunple sensitivity existed for bond portfolioit was decided to test the impact of alternative starting points and

intervals. Table III (A) contains the annual HPR means, standardeviations, and skewness measures for the even-ladder and full-dumbbell strate-

qgies for two interest-rate patterns, but for four different starting dates.

re always positive and the fluctuation is nuch less than for stock HPRs. The

Since the above results do indicate some saunple sensitivity, it was ex-

y. However, as Table III (B) illustrates, the skewness is relativelya>le and positive. The even-ladder strategy could not be tested quarterly

enormous storage requirements. Every bond by maturity and datechase (with its aunount and coupon) is stored; for each maturity, therefor

are subclassified by date of purchase. As the semi-amnual and quartesuggest, more frequent rebalancing would have improved the annual returns

s such as 1945-1970-1945, but not during periods of continuallych as 1945-70. Also, since semi-annual and quarterly standardproportionately larger than the annual, the risk/return is higher

r a shorter time horizon.Given the positive skewness of returns regardless of starting point orinterval, it is interesting to notice the postwar period's HPRs194519461947194819491950195119521953

5.0%.9%-1.5%3.4%4.9%

- .4%-2.2%2.6%3.6%

195419551956195719581959196019611962

2.5%- .5%-1.4%7.7%

-5.0%-1.3%12.1%1.5%7.0%

19631964196519661967196819691970

.9%4.2%

.3%5.3%

-1.6%1.7%-4.3%19.2%


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in 1960 and 1970. This skewness is the result of large positive HPRs whichccurred at the end of a liquidity crisis. Negative HPRs were much less ex-treme; price declines occurred more gradually as interest rates rose during

overies. This underlying generating process also explains whysemi-annual and quarterly differencing did not produce negative skewness as inthe case of common stock returns. Finally, this positive skewness may make asubstantial contribution to reducing combined portfolio risk if bond and stockreturns are not highly correlated.

IV. ConclusionsThe empirical results of this note suggest that dumbbell portfolio strate-

were longer. On a risk/return basis, the dumbbell strategies were con-y outperformed by the buy-long-and-hold (BLAH) strategy.

In contrast to the earlier studies, the objective function tested in this

of heavy liquidity requirements explains the different results. However,function and lack of liquidity needs is typical of performanc

portfolios for pension funds. Certainly, this class of institu-

Further, it was found that bond portfolio HPRs were skewed. The measure o

sample sensitivity of common stocks. It is

APPENDIXAs mentioned in the foregoing note, portfolio simulations were duplicated

two additional sets of assumptions: Transactions Costs Set Equal Zeroble A-1) and Holding-Period-Returns Excluding Capital Gains (Losses) and

A-2). These results are provided below.


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TABLE A-1HPRs and Risk of Portfolio Strategies

(No Transactions Costs)

Strategy

Even-LadderFull-DumbbellPartial-DummbbellBLAH-Ladder

1945-70

2.2.2.2.

,41,18,20,45

Interest Rate Patterns1970-45 1945-70-45

a.4.5.4.4.

Geometric Rates73 3.6105 3.6596 5.3083 3.67

1948-50

of Return2.2.2.2.

,38,65.59,42

1959-70

3.3.3.3.

96515690b. Standard Deviations

Even-LadderFull-DumbbellPartial-DumbbellBL.\H-Ladder

5.035.045.124.62

4.925.305.224.50

4.785.335.324.57

2.702.952.952.68

6.125.986.126.07

c. Coefficients of VariationEven-LadderFull-DumbbellPartial-DumbbellBLAH-Ladder

1.992.193.201.81

111.02.02.02.91

1.371.411.421.21

1.111.101.121.09

1.481.631.641.49


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TABLE A-2

HPRs and Risk of Portfolio Strategies

(Unrealized Capital Gains and Losses Ignored, No Transactions Costs)

StrategyInterest Rate Patterns

1946-70 1970-46 1946-70-46 1948-50 1959-70

Even-LadderFull-DumbbellPartial-DumbbellBLAH-Ladder

2.823.313.203.00

a. Geometric4.283.553.634.10

Rates3.483.483.483.52

of Return2.132.022.002.04

4.644.624.584.55

b. Standard DeviationsEven-LadderFull-DumbbellPartial-DumbbellBLAH-Ladder

.901.321.281.10

1.111.381.371.29

1.101.401.401.20

.06

.10

.07

.04

.45

.92

.79

.61c. Coefficients of Variation

Even-LadderFull-DumbbellPartial-DumbbellBLAH-Ladder

.32.40

.40

.37

.26.39

.38

.31

.32.40

.40

.34

.03.05

.04

.02

.10.20

.17

.13

*Note; These standard deviations are much lower than those in the previoustables. This tendency is to be expected because unrealized pricefluctuations are not being considered.


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REFERENCES[1] Bradley, Stephen P., and Dwight B. Crane, "A Dynamic Model for Bond Po

folio Management," Management Science, Vol, 19 (October 1972),[2] , "Management of Commercial Bank Government Security Portfol

An Optimization Approach under Uncertaintv," Journal of Bank Research,Vol. 4 (Spring 1973),

13] Cheng, Pao Lun, "Optimum Bond Portfolio Selection," Management SciencVol. 8 (July 1962), pp. 490-499.[41 Crane, Dwight B. "A Stochastic Prograunming Model for Commercial Bank B

Portfolio Management." Journal of Financial and Quantitative Analysis,Vol, 11, No, 3 (1971), pp, 955-976 .

[5] Fisher, Lawrence, and Roman L, Weil, "Coping with the Risk of InterestRate Fluctuations: Returns to Bondholders from Naive and Optimal Stratgies," Journal of Business (October 1971), pp, 408-431,

[6] Fogler, H, Russell. Analyzing the Stock Market: A Quantitative Approalst ed, Columbus, Ohio: Grid Publishing Company, 1973, pp, 264-266,

[7] Fogler, H, Russell, and Robert C, Radcliffe, "A Note on Measurement ofSkewness," Journal of Financial and Quantitative Analysis (June 1974pp. 485-489,

[8] Renshaw, Edward. "Portfolio Balance Models in Perspective: Some Generzations That Can Be Derived from the Two-Asset Case." Journal of Finanand Quantitative Analysis, Vol, 2 (June 1967), pp, 123-149.

[91 Watson, Ronald D, "Tests of Maturity Structures of Commercial Bank Goment Securities Portfolios: A Simulation Approach," Journal of Bank Rsearch, Vol, 3 (Spring 1972), pp, 34-46,

[101 Wolf, Charles R, "A Model for Selecting Cpmmercial Bank Government SePortfolios," The Review of Economics and Statistics, Vol, 51 (Februar1969), pp, 40-52,


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