Normal and Lognormal Shortfall-risk

NORMAL
AND
LOGNORMAL
SHORTFALL-RISK
PETER ALBRECHT
ABSTRACT
Shortfall-risk
- the probability
that a specified miminum return level will
not be exceeded is an important
measure of risk that is more consistent with the
investors’ perception of risk than the traditional
measure of risk, the variance
of returns.
The present paper treats the calculation and the control of the shortfall-risk
in the case of normal and lognormal returns over one and over many periods.
1. INTRODUCTION
Shortfall-risk (downside-risk) - the probability that a special minimum return level (target return, benchmark return) will not be exceeded
- is a measure of risk which has recently attracted considerable interest in modern investment management, cf. e.g. Leibowitz/Henriksson
(1989), Leibowitz/Kogelman
(1991a, 1991b), Leibowitz/Krasker
(1988),
Leibowitz/Langetieg
(1990) and Sortino/Van Der Meer (1991). In contrast to the traditional measure of risk in portfolio theory, the variance
of returns, shortfall-risk is more consistent with the investors’ intuitive
perception of risk in that it focusses more on the real economical risk
of an investor, whereas the variance or returns is rather a measure of
the volatility of financial assets. “Upward” volatility results in investment chances, only “downward” volatility will result in investment risks,
so that asymmetric measures of risk, cf. Harlow (1991), are attractive
alternatives to the variance. Shortfall-risk is the most elementary asymmetrical risk measure.
This paper concentrates on the calculation and the control of shortfall-risk in the case of normal and lognormal returns over one and over
many periods.
Peter Albrecht
418
Especially for life insurance companies the control of the shortfall
risk of investment returns over short and over long horizons should
develop into an important tool in investment and risk management.
2. ONE-PERIOD SHORTFALL-RISK
Let R denote the one-period return of an asset. The expected value
resp. the variance of the return will be denoted by
P = WR),
ill
o2 = Var(R)
We will assume that R is normally distributed
PI
or alternatively
m and v2, i.e.
PI
R - NbL,02)
that 1 + R is lognormally
distributed
with parameters
ln( 1 + R) - N(m, v2).
To ensure that [l] still is valid, we have to choose the parameters m
and v2 as follows (the relevant properties of the lognormal distribution
used in this paper are given in the appendix):
&ln[l+(&)‘]
PI
?7l -= ln(1 + p) - $3.
To quantify the shortfall-risk we have to specify a desired minimum
return M, the shortfall-risk then is given by
[51
SR(M)
= P(R 5 M).
A graphical illustration of SR(M) is given in figure 1.
To control the shortfall-risk SR(M) we have to specify a (small)
probability E and to set up the shortfall-constraint
PI
P(RIM)<E.
Normal
and
Lognormal
Shortfall-risk
419
Let F, denote the (1 - e)-quantile of the distribution F of R, i.e.
> Fe) = E, then the probability constraint [6] is equivalent to the
“deterministic” constraint
P(R
A4 2 Fl-,
.
Let N, denote the (1 - e)-quantile of the standard normal distribution, then the (1 - z)-quantiles NE(p) O) of a normal distribution resp.
LN,(m,v)
of a lognormal distribution are given by
LN,(m,
IJ)= exp[m+ NEvl
As we have Nl-, = -NE for a standard normal distribution,
relevant shortfall-constraint
for a normal distribution is given by
the
In a (a, p)-diagram the shortfall-constraint
[6] can be visualized as
follows. The straight line p = M + N,a divides the (c, p)-plane into two
seperate sectors. The sector above the line (including the line itself)
contains all (0, ,u)-positions with controlled shortfall-risk. Figures 2 and
3 give two examples for the corresponding lines in the case of normal
distribution.
Peter Albrecht
420
Fig.
2 Shortfall-Risk
in the normal
case:
M = 0.06; E = 0.25, 0.2,O.l.
Fig.
3 Shortfall-Risk
in the
case:
E = 0.2; M = 0.03 0.06, 0.09.
normal
The case of the lognormal distribution
we have from [7] in connection with [8]
WI
is much more complex. First
m > In(l + M) + N,w ,
but we are interested in the possibility of visualization and therefore we
need a constraint involving 0 and /J. From the appendix we obtain the
Normal
and Lognormal
adequate transformation
421
Shortfalhisk
as follows. Let GE(v) be defined by
GE(v)=
1 - exp[-l\r,v - 1/2v2]
’
[exp(v2) - 1]112
Then [lo] is equivalent to [12]
/I L M + Ge(v)u.
P21
As G,(v) = GE(/b,o) it becomes clear that [12] is only an implicit
inequality.
The following two figures are the counterpart
3 for the lognormal case.
0.7
I
I
1
I
I
0.0s
I
0.1
I
to the figLtres 2 and
I
I
I
0.6 -
0.) -
0.4 -
0.3 -
0.2 -
0 0
I
0.1s
I
0.2
I
0.25
I
0.3
I
0.35
0.4
v
-
9o%-Qullnlil
...‘.’ 8wrQurnli
-Fig.
75Y,Quultil
4 Shortfall-risk
in the
loguxrnal
case:
A/I = 0.06; E = 0.25, 0.2, 0.1.
422
Peter Albrecht
0.
I
I
I
I
I
I
I
I
0.05
I
0.1
I
a.15
I
03
I
0.25
I
0.3
I
0.35
0.
0.
0.
0.
0.4
- wo.09
...... Ivl=o.o6
-Fig.
M4.03
5 Shortfall-risk
in the lognormal
case:
E =
0.2; M = 0.03,0.06, 0.09.
Finally Figure 6 contains a comparison of the normal and the lognormal situation in a special case.
Normal
0.7
Lognormal
and
423
Shortfall-risk
I
I
I
I
I
I
I
I
0.05
I
0.1
I
0.15
I
0.2
I
0.25
I
0.3
I
0.35
0.6 -
0.5 -
0.4 -
0'
0
0.4
.‘.“. LOGNORMAL
- NORMAL
Fig. (i Tiorn~al case vs. lognormal
M = 0.07, & = 0.10.
case:
3. MULTI-PERIOD
SHORTFALL-RISK
In the multi-period case there are two possibilities to annualize a
series of successive, say T, one-period return RI,. . . , RT.
First the arithmetically
P31
The geometrically
annualized return RA(T) is given by
Ra(T)=;(R~+...+&).
annualized return Rc(T)
is given by
M
The geometrical annualization clearly is the correct one, but the
arithmetical annualization is often applied in practice because of its
simplicity.
Peter Albmcht
424
In the following we will make different distributional assumptions
which have their reason in the corresponding properties of the normal
resp. lognormal distribution.
When working with RA(7’) we will assume
Rt N N(p, a2), which gives
1151
i.e E(R*)
= p and I
= a/e.
The dispersion of the (arithmetically)
annualized return is mono
tonically decreasing with hicreasing time horizon.
When working with Rc;(T), we will assunie
which gives
1 +&(T)
WI
17~(1
-t R,) N N(m,v2),
- LN
We obtain:
E(Rc)=exp(m+kv”)
Var(R,)
= (1 + PL)~[exp (g)
-l=p
-11 =(1+~1)2[~~-1]
.
Again we have the property of a monotonically decreasing (geometrically) annualized return.
Figure 7 gives an illustration of this effect in the lognormal case.
Fig.
7 Ooe-sigma-intro-val
of the geometrically
a~~~lualizctl
return.
Normal
and Lognormcll
Shortjall-lisk
425
Controlling the shortfall-risk of the arithmetically
RA(T) results in the shortfall-constraint
annualized return
P(RA(T) I M) I E,
PI
which is equivalent to the constraint
Compared to the corresponding one-period constraint [9] this shows
that the time horizon is taken into consideration in dividing the second
term on the right side of inequality [9] by 0.
Figure 8 illustrates this
effect graphically.
Fig.
8 Multi-period
shortfall
risk
(arithmetically
annualized
return):
M = 0.06; E = 0.10; T = 1, 5, 15.
In case of a geometrically annualized return &(T)
the shortfall-risk results in the shortfall-constraint
PO1
P(RG(T)I M) I E,
which is equivalent to the (implicit)
WI
the control of
constraint
p > M + G&/J?;)a/J?;.
Figure 9 is the counterpart
to figure 8 in the lognormal case.
426
Peter Albrecht
0.6
I
I
I
I
I
I
I
I
0.03
I
0.1
I
0.11
I
0.2
I
0.23
I
0.3
I
0.31
0.3 -
0.4 -
0.3 -
0.2 -
0
0
T=l
‘.._ T=S
-- T=lS
Fig.
9 Multi-period
0.4
0
shortfall-risk
(geometrically
annualized
return):
A4 = 0.06; E = 0.10; T = 1, 5, 15.
The shortfall-constraints
[IS] req. [20] only allow the shortfall risk
to be controlled over the total horizon of time. Only the annualized
return is controlled, the desired minimum return will be realized with
high “on the average” probability . To be able to control the shortfallrisk in addition over sub-intervals of time one could proceed as follows.
Specify the periods of time for which control of the return is desired, e.g.
1 < tr < . . . < tN 2 T, the desired minimum returns &fr, . . . ,ndN and
the desired control levels ~1, . . , &N. An appropriate control criterion in
the case of arithmetically annualized returns could be
P(h(tN)
5
MN)
5 EN
.
. i.e. a collection of controls of the type [18]. The case of geometrically
annualized returns can be treated in complete analogy. The following
427
Norn~cd and Lognormnl %ortfall-risk
figure illustrates the following constraints:
(1) P&4(3)
2 0.03) IO.10
(2) P(R,&O) 5 0.05) 5 0.10
(3) P(RA(15) 5 0.07) 5 0.10.
c
0.16-
0.02
l
1.2swn
0.140.0s.
1.2swAO
0.120.07
0.1
l
1.2swns
-
0.020
0
O.OS
,
0.1
1
1
0.1s
I
0.2
1
0.m
.
.
Fig. 10 Shortfall-constraints over different the horizons.
4. IMPLICATIONS FOR PORTFOLIO SELECTION
In portfolio selection one has to identify the optimal portfolio composition for a given investor with given preferences. In the framework of
the Markowitz approach, the preferences only depend on E(R) and (T(R)
and the optimal portfolio has to be selected from the efficient frontier.
This selection of the optimal portfolio on the efficient frontier can
be easily performed on the basis of the multi-period shortfall-constraints
as follows. Solve the optimization problem
E[RA(T)] = E(R) -+ max
subject to the constraints
Because of [19] in case of the normal distribution 1231 is equivalent to a linear programming problem. It is obvious that the resulting
Peter Albrecht
428
portfolio must be efficient in the Markowitz sense, so one could as well
first identify the efficient frontier and then choose that portfolio on the
efficient frontier fulfilling the constraints [22] and possessing the highest
expected value. Figure 11 illustrates this graphically.
‘,
0.03 . 1.29wn
0.10.00
O.lO-
0
1
0
Fig.
5.
l
1.2!3wrlo
0.07.1.2sod-l9
I
O.O¶
11 Portfolio
APPENDIX:
1
0.L
selection
PROPERTIES
A random variable
normally distributed:
X(>
I
0.15
under
0~
I
0.1
THE
-
0.25
l
shortfall-constraints.
LOGNORMAL
0) is lognormally
DISTRIBUTION
distributed,
if 1nX is
x - LN(p, cT2) 4=+ 1nX N N([L, 02)
+=+ lnXo-
p N N(0, 1) *
The density function of t,he lognormal distribution
fx(x)
=
1
-exp[(l
&FG
{ 0
nx - PL)~
2a2
]
xc>0
x50.
The first two moments of the lognormal distribution
E(X) = exp
Var(X)
is given by
are given by
P+ i*’
(
1
= exp(2p + a2) [ exp(a2) - 1] = E(X)2 [ exp(a2) - 11 .
Normal and Lognormal
Shortfall-risk
429
These moments uniquely determine the parameters of the lognormal
distribution:
= 2lnE(X)
For n stochastically
independent
.
lognormal distributions
xi - LN(b”i,u,“)
*
+ Var(X)]
- ~ln[E(X)”
we have
i=l,...,?l
,
fiXi-LN
i=l
in case of identical distributions
therefore
fi Xi - LN(np,
TZCT~).
i=l
For the geometric mean we obtain therefore
Let LN,(p, a) d enote the (1 -&)-quantile of a LN(p, a2)-distribution
and N, the (1 - t-)-quantile of the standard normal distribution, then
we have
LN,(p,
From Johnson/Katz
LN1-,
(1970, p. 117) we have
- E(S)
4X)
Because of Nr-,
0) = exdp + NE4 .
= -N,
=
1
exp [ N1--EO - l2” 2 -1
[ exp(cT2) - l] 1’2
and introducing
1 - exp
’
the function
- N,a - ;u2]
G(a) =
[ exp(cr2) - l] 1’2
430
Peter
Albrecht
this is equivalent to
U-VI-,
= E(X) - G(a) CT(X) =
= E(X) - G@(X),
as the parameter
a(X))a(X),
CTis itself a function of E(X) and CT(X).
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(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
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Cambridge,
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Asset
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S. KOTZ,
Distributions
in Statistics,
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Portfolio
optimization
with shortfall
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A confidence-limit
approach to managing downside
risk,
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S. KOGELMAN,
Assert
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constraints,
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S. KOGELMAN,
Return
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to the risk/nzturn
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The persistence of risk:
Stocks
versus
bonds over the lon,g term,
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Shortfall
r-i&s and th.e asset allocation
decision,
in: Fabozzi, F.J. (Ed.): Managing Institutional
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SORTINO,
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