Investments, tenth edition

  Forecasts for the Long Haul 

 We use arithmetic averages to forecast future rates of return because they are unbiased 

estimates of expected rates over equivalent holding periods. But the arithmetic average of 

short-term returns can be misleading when used to forecast long-term cumulative returns. 

This is because sampling errors in the estimate of expected return will have asymmetric 

impact when compounded over long periods. Positive sampling variation will compound 

to greater upward errors than negative variation. 

 Jacquier, Kane, and Marcus show that an unbiased forecast of total return over long 

horizons requires compounding at a weighted average of the arithmetic and geometric 

historical averages.  

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   The proper weight applied to the geometric average equals the ratio 

of the length of the forecast horizon to the length of the estimation period. For example, if 

we wish to forecast the cumulative return for a 25-year horizon from an 80-year history, an 

unbiased estimate would be to compound at a rate of   

Geometric average 3

25

80

1 Arithmetic average 3

(80 2 25)

80

    

 This correction would take about .6% off the historical arithmetic average risk premium on 

large stocks and about 2% off the arithmetic average premium of small stocks. A forecast 

for the next 80 years would require compounding at only the geometric average, and for 

longer horizons at an even lower number. The forecast horizons that are relevant for cur-

rent investors would depend on their life expectancies.     

  

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 For an introduction to this approach, see Yacine Aït-Sahalia and Jean Jacod, “Analyzing the Spectrum of Asset 

Returns: Jump and Volatility Components in High Frequency Data,”  Journal of Economic Literature  50 (2012), 

pp. 1007–50. 

  

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 Eric Jacquier, Alex Kane, and Alan J. Marcus, “Geometric or Arithmetic Means: A Reconsideration,”  Financial 

Analysts Journal,  November/December 2003. 

     1.   The economy’s equilibrium level of real interest rates depends on the willingness of households 

to save, as reflected in the supply curve of funds, and on the expected profitability of business 

investment in plant, equipment, and inventories, as reflected in the demand curve for funds. It 

depends also on government fiscal and monetary policy.  

    2.   The nominal rate of interest is the equilibrium real rate plus the expected rate of inflation. In gen-

eral, we can directly observe only nominal interest rates; from them, we must infer expected real 

rates, using inflation forecasts.  

    3.   The equilibrium expected rate of return on any security is the sum of the equilibrium real rate of 

interest, the expected rate of inflation, and a security-specific risk premium.  

    4.   Investors face a trade-off between risk and expected return. Historical data confirm our intui-

tion that assets with low degrees of risk provide lower returns on average than do those of 

higher risk.  

   SUMMARY 

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