Forecasting: Its Purpose and Accuracy

Abstract

The purpose of this monograph is to concisely convey forecasting techniques to applied investment analysis. People forecast when they make an estimate as to the future value of a time series. That is, if I observe that IBM has a stock price of $205.48, as of March 23, 2012, and earned an earnings per share (eps) of $13.06 for fiscal year 2011, then I might wonder at what price IBM would trade for on December 31, 2012, if it achieved the $14.85 eps that 21 analysts, on average, expect it to earn in 2012 (source: MSN, Money, March 23, 2012, 1:30 p.m., AST). The low estimate is $14.18 and the high estimate is $15.28. Ten stock analysts currently recommend IBM as a “Strong Buy,” one as a “Moderate Buy,” and ten analysts recommend “Hold.” Moreover, if IBM achieves its forecasted $16.36 eps average estimate for December 2013, when could be its stock price and should an investor purchase the stock? One sees several possible outcomes; can IBM achieve its forecasted eps figure? How accurate are the analysts’ forecasts? Second, should an investor purchase the stock on the basis of an earnings forecast? Is there a relationship between eps forecasts and stock prices? How accurate is it necessary for analysts to be for investors to make excess returns (stock market profits) trading on the forecasts?

Appendix

Exponential Smoothing

The most simple forecast of a time series can be estimated from an arithmetic mean of the data Davis and Nelson (1937). If one defines f as frequencies, or occurrences of the data, and x as the values of the series, then the arithmetic mean is

$$ A = \frac{{{f_1}{x_1} + {f_2}{x_2} + {f_3}{x_3} + \ldots + {f_t}{x_t}}}{T} $$

(1.22)

where $T = {f_1} + {f_2} + {f_3} + \ldots + { }{f_t}.$

$$ A = \frac{{\Sigma {f_i}{x_i}}}{T}. $$

Alternatively,

$$ \frac{{\Sigma {f_i}({x_i} - x)}}{T} $$

$$ A{ = }x{ +}\frac{{\Sigma {f_i}{(}{x_i} - x)}}{T}. $$

(1.23)

The first moment, mean, is

$$ A{ =}\frac{{\Sigma {f_i}{x_i}}}{T} = \frac{{{m_1}}}{{{m_0}}} $$

$$ {m_0} = \sum {f_i} = T,{ }{m_1} = \sum {f_i}{x_i}. $$

If x = 0, then

$$ {\sigma^2} = \frac{{\Sigma {f_i}x_i^2}}{T} - {A^2} $$

$$ {\sigma^2} = \frac{{{m_2}}}{{{m_{\mathrm{o}}}}} - \frac{{m_1^2}}{{m_{\mathrm{o}}^2}} = ({m_0}{m_2} - m_1^2)m_0^2. $$

(1.24)

Time series models often involve trend, cycle seasonal, and irregular components, Brown (1963). An upward-moving or increasing series over time could be modeled as

$$ {x_t} = a + bt, $$

(1.25)

where a is the mean and b is the trend, or rate at which the series increases over time, t. Brown (1963, p. 61) uses the closing price of IBM common stock as his example of an increasing series. One could use a quadrant term, c. If c is positive, then the series

$$ {x_t} = a + bt + c{t^2} $$

(1.26)

trend is changing toward an increasing trend, whereas a negative c denotes a decreasing rate of trend, from upward to downward.

In an exponential smoothing model, the underlying process is locally constant, x _t  = a, plus random noise, ε _t .

$$ {x_t} = a{\varepsilon_t}. $$

(1.27)

The average value of ε = 0.

A moving average can be estimated over a portion of the data:

$$ {M_t} = \frac{{{x_1} + {x_{t - 1}} + \ldots + {x_{t - N}} + 1}}{N}, $$

(1.28)

where M _t is the actual average of the most recent N observations.

$$ {M_t} = {M_{t - {1}}} + \frac{{{x_t} - {x_{t - N}}}}{N}. $$

(1.29)

An exponential smoothing forecast builds upon the moving average concept.

$$ {s_t}{(}x{)} = \alpha {x_t} + ({1} - \alpha ){ }{s_{t - {1}}}(x), $$

where α = smoothing constant, which is similar to the fraction ${{1} \left/ {T} \right.}$ in a moving average.

$$ \begin{array}{cc}{s_t}{(}x{)} = \alpha {x_t} + ({1} - \alpha )[\alpha {x_{t - {1}}} + ({1} - \alpha ){s_{t - {2}}}{(}x{)}] \hfill \\ = \alpha \Sigma_{ko}^{t - 1}{(1 - \alpha )^k}{x_{t - k}} + {(1 - \alpha )^t}{x_{\mathrm{o}}}, \end{array}$$

(1.30)

where s _t (x) is a linear combination of all past observations. The smoothing constant must be estimated. In a moving average process, the N most recent observations are weighted (equally) by 1/N and the average age of the data is

$$ k = \frac{{0 + 1 + 2 + \ldots + N - 1}}{N} = \frac{{N - 1}}{2}. $$

An N-period moving average is equivalent to an exponential smoothing model having an average age of the data. The one-period forecast for an exponential smoothing model is

$$ {F_{t + {1}}} = {F_t} +\alpha ({y_t}-{F_t}), $$

(1.31)

where α is the constant, 0 < α < 1.

Intuitively, if α is near zero, then the forecast is very close to the previous value’s forecast. Alternatively,

$$ \begin{array}{lll} {F_{t + {1}}} = \alpha {y_t} + ({1} - \alpha ) {F_t} \hfill \\{F_{t + {1}}} = \alpha {y_t} + \alpha ({1} - \alpha ){ }{y_{t - {1}}} + ({1} - \alpha ){2}{F_{t - {1}}}. \hfill \\ \end{array} $$

(1.32)

Makridakis, Wheelwright and Hyndman (1998) express F _t−1 in terms of F _t−2 and, over time,

$$ \begin{array}{llll} {F_{t - {1}}} = \ \alpha {y_t} + \alpha ({1} - \alpha ){ }{y_{t - {1}}} + \alpha {(a - \alpha )^2}{y_{t - {2}}} + a{({1} - \alpha )^3}{y_{t - {3}}} \hfill \cr \qquad \qquad + \alpha {({1} - \alpha )^4}{y_{t - {4}}} + \alpha {({1} - \alpha )^5}{y_{t - {5}}} + \ldots\cr \qquad \qquad + \alpha {({1} - \alpha )^{t - {1}}}{y_t} + {({1} - \alpha )^t}{F_1}. \hfill \\ \end{array} $$

(1.33)

Different values of α produce different mean squared errors. If one sought to minimize the mean absolute percentage error, the adaptive exponential smoothing can be rewritten as

$$ {F_{t + {1}}} = \alpha {y_t} + ({1} - \alpha ){F_t} $$

(1.34)

$$ \alpha t + {1} = \left| {\frac{{{A_t}}}{{{M_t}}}} \right|, $$

${\rm where } \qquad \qquad \qquad \qquad \begin{array}{lll} {A_t} = \beta {E_t} + ({1} - \beta ){ }{A_{t - {1}}} \hfill \\{M_t} = \beta |{E_t}| + ({1} - \beta ){ }{M_{t - {1}}} \hfill \\{E_t} = {y_t} - {F_t}. \hfill \\ \end{array}$

A _t is a smoothed estimate of the forecast error and a weighted average of A _t−1 and the last forecast error, E _t .

One of the great forecasting models is the Holt (1957) model that allowed forecasting of data with trends. Holt’s linear exponential smoothing forecast is

$$ \begin{array}{llll} \quad \ {L_t} = \alpha {y_t} + ({1} - \alpha ){(}{L_{t - {1}}} + {b_{t - {1}}}) \hfill \\ \quad \ {b_t} = \beta ({L_t}-{L_{t - {1}}}) + ({1} - \beta ){b_{t - {1}}} \hfill \\{F_{t + m}} = {L_t} + {b_t}m. \hfill \\ \end{array} $$

(1.35)

L _t is the level of the series at time t, and b _t is the estimate of the slope of the series at time t. The Holt model forecast should be better forecasts than adaptive exponential smoothing models, which lack trends. Makridakis et al. (1998) remind the reader that the Holt model is often referred to as “double exponential smoothing.” If α = β, then the Holt model is equal to Brown’s double exponential smoothing model.

The Hold (1957) and Winters (1960) seasonal model can be written as

$$ {\text{(Level)}}\quad {L_t} = \alpha \frac{{{y_t}}}{{{s_{t - s}}}} + (1 - \alpha )({L_{t - {1}}} + {b_{t - {1}}}) $$

$$ ({{Trend}})\quad {b_t} + \beta ({L_t}-{L_{t - {1}}}) + (a - \beta ){b_{t - {1}}} $$

$$ ({{Seasonal}})\quad {s_t} = \gamma \frac{{{y_t}}}{{{L_t}}} + (a - \gamma ){s_{t - s}} $$

$$ ({{Forecast}})\quad {F_{t + m}} = ({L_t} + {b_t}m){ }{S_{t - s + m}}. $$

Seasonality is the number of months or quarters, L _t is the level of the series, b _t is the trend of the series, and s _t is the seasonal component.