This memorandum presents some basic equalities and inequalities about rates of return in discrete time, without auto-correlation. The arithmetic and geometric means are discussed. Estimation of the expected payout and the median payout is discussed, including maximum likelihood estimation.

Arie ten Cate

English

Research Papers in Economics

CPB Memorandum
CPB Netherlands Bureau for Economic Policy Analysis
Unit(s)/Project : IT
Author(s) : Arie ten Cate
Number : 223
Date : April 14, 2009
Arithmetic and geometric mean rates of return in discrete time
This memorandum presents some basic equalities and inequalities about rates of return in
discrete time, without auto-correlation. The arithmetic and geometric means are discussed.
Estimation of the expected payout and the median payout is discussed, including maximum
likelihood estimation.Contents
1 Introduction 1
2 The general model 2
3 Example 3
4 The lognormal model 3
5 The expected geometric mean return 4
6 The arithmetic and geometric returns 5
7 Maximum likelihood estimation of m and E[ST] 5
8 The median 5
9 When to consider geometric? 6
9.1 Unbiased estimation of E[ST] 6
9.2 The median 6
10 Summary of results 7
References 7
1 Introduction
This memorandum1 presents some basic equalities and inequalities about rates of return in
discrete time, without auto-correlation.
Modelling stochastic rates of return in discrete time might be simpler than using the concept
of continuous time Brownian motion. On the other hand, many results below are only
1 The author thanks Pascal Janssen (PGGM) and Gijsbert Zwart and Ed Westerhout (CPB) for their help.
1approximations, using assumptions such as the one-period return and variance being both much
smaller than one. In both approaches numerical results must be computed with small time steps.
First a general model is discussed, requiring only an expected return which is constant over
time. Starting in section 4, the case of lognormally distributed returns is discussed, including
maximum likelihood estimation and the median payout.
Although most if not all results are not new, this overview might be useful. Your author
would have saved time if it were available when he needed it. Comments are invited.
2 The general model
Let St be the value of stocks at time t, with:
St = (1+rt)St−1 (2.1)
for all t = 1,...,T. Let S0 be a given positive number. The rt are stochastic; they are
independently distributed. The form of the distribution is not speciﬁed. Assuming limited
liability implies for all t:
Pr(rt < −1) = 0 (2.2)
The St are independent of the previous series St−1,...,S1. Also, the rt have the same
expectation E[rt], denoted by m:
E[rt] = m (2.3)
for all t. Since the rt and the St−1 are independently distributed, we have for some T:
E[ST] = E[(1+rT)ST−1] = E[1+rT]E[ST−1] = (1+m)E[ST−1] (2.4)
Repeating this all the way down to S0 gives:
E[ST] = (1+m)TS0 (2.5)
or
(E[ST/S0])
1/T −1 = m (2.6)
The arithmetic mean is
b mA ≡
1
T å
t
rt =
1
T å
t

St
St−1
−1

(2.7)
Of course this is an unbiased estimator of m, due to equation (2.3) above:
E[b mA] = m (2.8)
2See section 7 below for the maximum likelihood estimation of m under the assumption of
lognormal returns.
Let 1+ b mG be the geometric mean of the 1+rt:
1+ b mG ≡

Õ
t
1+rt
1/T
=

ST
S0
1/T
(2.9)
Hence ST/S0 is equal to one plus the geometric mean, to the power T. However, its expected
value is equal to one plus the expected value of the arithmetic mean, to the power T.
A geometric mean of non-negative numbers is smaller than (or equal to) the arithmetic mean.
(For example, with rt = ±1/2 the arithmetic mean of 1+rt is 1 and the geometric mean is
√
0.75 = 0.87.) Hence we have b mG ≤ b mA for every realization of the series 1+rt, and hence,
with (2.8):
E[b mG] < m (2.10)
Nonzero variance of the rt is assumed here, giving strict inequality in (2.10).
3 Example
There are several small numerical examples in the literature which illustrate the previous section.
For instance the well-written Exhibit 10.6 in the section “Geometric versus arithmetic average”
of McKinsey & Company et al. (2000).
A very simple example is the case of T = 2 with rt = ±1/2, like the example before equation
(2.10). With equal probabilities this gives m = 0. With S0 = 1, the four possible values of ST are
0.25, 0.75 (twice) and 2.25. They have equal probability and hence E[ST] = 1, which agrees
with equation (2.5) above.
The geometric mean return b mG, deﬁned in equation (2.9) above, can take on the four values
√
0.25−1,
√
0.75−1 (twice) and
√
2.25−1, giving E[b mG] = −0.07 which is smaller than
m = 0. This agrees with equation (2.10) above.
Note that after one high and one low rt (in either order) we get a
p
S2/S0 equal to the
geometric mean computed in the example before equation (2.10), smaller than 1+m. This is
because the other two possibilities are omitted: twice high and twice low.
4 The lognormal model
Additionally to the assumptions in section 2, it is assumed that all rt are identically distributed.
Moreover, let the distribution of the 1+rt be lognormal:
log(1+rt) ∼ N
 
µ,σ2
(4.1)
3for all t. The limited-liability restriction (2.2) above now becomes Pr(rt ≤ −1) = 0.
Also:
log(ST/S0) ∼ N
 
Tµ,Tσ2
(4.2)
log(1+ b mG) ∼ N

µ,
1
T
σ2

(4.3)
It follows from the formulas for the expectation and variance of a lognormal variable that:
E[1+rt] = exp

µ +
1
2
σ2

(4.4)
E[1+ b mG] = exp

µ +
1
2T
σ2

(4.5)
Var[rt] = (1+m)2 
exp
 
σ2
−1

≈ (1+m)2σ2 ≈ σ2 (4.6)
where the two ≈ signs are associated with small σ2 and small m, respectively.
5 The expected geometric mean return
An approximation of the difference between the two sides of inequality (2.10) is derived, with
lognormal 1+rt. With (4.5) above, we have
E[1+ b mG] = exp

µ +
1
2T
σ2

= exp

µ +
1
2
σ2−
1
2
σ2+
1
2T
σ2

= (1+m)exp

−
1
2
σ2+
1
2T
σ2

(5.1)
Hence with large T we have:
E[1+ b mG]
1+m
≈ exp

−
1
2
σ2

< 1 (5.2)
Then we have also:
E[b mG] ≈ (1+m)exp

−
1
2
σ2

−1
≈ (1+m)

1−
1
2
σ2

−1 = m−
1
2
σ2−m
1
2
σ2
≈ m−
1
2
σ2
= E[b mA]−
1
2
σ2 (5.3)
where the last two ≈ signs are associated with small σ2 and small m, respectively. Compare with
(2.10).
46 The arithmetic and geometric returns
The result of the previous section suggest the following two concepts, used in communication
between practitioners:
arithmetic rate of return ≡ m (6.1)
geometric rate of return ≡ m−
1
2
Var[rt] ≈ m−
1
2
σ2 (6.2)
Equations (4.6) and (5.3) are used for (6.2).
7 Maximum likelihood estimation of m and E[ST]
In section 2 above, the unbiased estimation of m = E[rt] was discussed. The maximum
likelihood estimate of the same is, using equation (4.4) above:
b mML = exp

b µ +
1
2
b σ2

−1 (7.1)
where b µ and b σ are the maximum likelihood estimate of µ and σ, respectively. These are the
sample mean and standard deviation of the normally distributed log(1+rt), respectively. (See
any econometrics or statistics textbook.) Then, with equation (2.9) above, we have:
b µ = log(1+ b mG) (7.2)
and hence with equation (7.1):
1+ b mG
1+ b mML
=
exp(b µ)
exp
 
b µ + 1
2 b σ2 = exp

−
1
2
b σ2

< 1 (7.3)
Compare with equation (5.2) above. For large T the standard error of b σ2 goes to zero and we
have b σ2 ≈ σ2; compare Campbell et al. (1997), equation (9.3.31)2.
The maximum likelihood estimate of E[ST] is (1+ b mML)TS0, using (2.5). In practice of
course the b mML is computed over a historical time range before t = 0.
8 The median
The median of a variable is equal to the exp of the median of the log, since both functions are
monotonous. With lognormal 1+rt, log(1+ b mG) is normally distributed; see (4.3) above. Hence
2 Our T is the n of Campbell et al. (1997) and our rt is their Rt. We follow them in making no distinction between b σ2 and
c σ2, because the two are the same, of course. The σ above their equation (9.3.26) must be σ2.
5the median of log(1+ b mG) is equal to its expectation. Then
Median

ST/S0

= exp
 
Median[log(ST/S0)]

= exp

Median

å
t
log(1+rt)

= exp
 
Median[T log(1+ b mG)]

= exp
 
T Median[log(1+ b mG)]

= exp
 
T E[log(1+ b mG)]

= exp
 
Tµ

(8.1)
Then
Median

ST/S0

E

ST/S0
 =
exp
 
Tµ

exp
 
T
 
µ + 1
2σ2 = ρT (8.2)
with
ρ ≡ exp

−
1
2
σ2

< 1 (8.3)
Also, with T  σ2/µ we have, using (8.1) and (4.5):
Median
h
(ST/S0)
1/T
i
=
 
Median[ST/S0]
1/T
= exp(µ) ≈ E[1+ b mG] (8.4)
9 When to consider geometric?
9.1 Unbiased estimation of E[ST]
The conclusion below equation (2.9) might suggest the use of the arithmetic mean for the
estmation of the expected multi-period payout.
However, substituting the arithmetic mean for the m in (2.5) gives a biased estimate of the
expected payout, due to the non-linearity of the power raising. Jacquier et al. (2003) ﬁnd that
using a weighted average of the geometric and arithmetic means gives an unbiased estimate of
the expected multi-period payout.
The alternative is the maximum likelihood estimator of E[ST], in section 7. (Note that at
Campbell et al. (1997), p.367, maximum likelihood estimation of option prices is the only
estimation method discussed, without bothering about ﬁnite sample bias.)
Finally, note that in simulations where parameter values are assumed, this discussion about
estimation is not relevant.
9.2 The median
One might be interested in the median payout rather than the expected payout. Focussing only
on a high value the latter may be “courting with ruin” (Samuelson, 1971) if the median payout
6tends to zero for large T. As can be seen from equation (8.4), for the case of lognormal returns
this implies a negative expected geometric mean – or a negative geometric return, from (6.2).
This is also relevant in simulations where parameter values are assumed.
10 Summary of results
With lognormal returns we have:
arithmetic rate of return ≡ E[rt] ≡ m = E[b mA] ≈ E[b mML] ≈
 
E

ST/S0
1/T
−1 (10.1)
and
geometric rate of return ≡ m−
1
2
Var[rt] ≈ m−
1
2
σ2
≈ E[b mG] = E
h 
ST/S0
1/Ti
−1
≈
 
Median

ST/S0
1/T
−1
= Median
h 
ST/S0
1/Ti
−1 (10.2)
References
Campbell, J., A. Lo and A.C. MacKinlay, 1997, The econometrics of ﬁnancial markets,
Princeton University Press, Princeton.
Jacquier, E., A. Kane and A. Marcus, 2003, Geometric or arithmetic mean: A reconsideration,
Financial Analysts Journal, vol. 59, pp. 46–53.
McKinsey & Company, T. Copeland, T. Koller and J. Murrin, 2000, Valuation; Measuring and
managing the value of companies, Wiley, New York, 3rd ed.
Samuelson, P., 1971, The “fallacy” of maximizing the geometric mean in long sequences of
investing or gambling, in Proceedings of the National Academy of Sciences, vol. 68, pp.
2493–2496, www.pnas.org.
7

Geometric or arithmetic mean: A reconsideration,

The “fallacy” of maximizing the geometric mean in long sequences of investing or gambling,

The econometrics of ﬁnancial markets,

Valuation; Measuring and managing the value of companies,

Arithmetic and geometric mean rates of return in discrete time

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Arithmetic and geometric mean rates of return in discrete time

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