international investment risk analysis ... budgeting models r. e. jensen trinity university, 715...
TRANSCRIPT
Mathl Modelling, Vol. 9, No. 3-5, pp. 265-284, 1987 Printed in Great Britain. All rights reserved
0270-0255187 $3.00 + 0.00 Copyright 0 1987 Pergamon Journals Ltd
INTERNATIONAL INVESTMENT RISK ANALYSIS: EXTENSIONS FOR MULTINATIONAL CORPORATION
CAPITAL BUDGETING MODELS
R. E. JENSEN
Trinity University, 715 Stadium Drive, San Antonio, TX 78284, U.S.A.
Abstract-Multinational corporation managers often rely upon net present value models using differing foreign discount rates that contain implicit pjx(t) relative risk premiums equal to the difference between Country j and Country k discount rates at Time t. Recent surveys show that these risk premiums are usually subjective and rarely justified in an analytical context.
The major purpose of this paper is to propose a means whereby p,,(t) relative risk premiums may be derived from investor judgments regarding rjjl(t) perceived multiple country foreign investment risk differentials. The rg(t) responses are based upon the amount of a Country j return that would be required in Year t for the respondent to be indifferent per dollar of return from Country k having different types and levels of investment risk.
This paper proposes means whereby the rjt(t) risk differential equivalencies may be elicited as dynamic time functions and aggregated as geometric means for multiple respondents. Inconsistent r,*(t) responses may be adjusted for consistency either by eigenvector scalings or least-squares scalings. This paper illustrates an optima1 least-squares scaling procedure.
The analysis proposed in this paper facilitates multiple-criterion analysis via Analytic Hierarchy Process (AHP) analysis. An AHP formulation is illustrated in the paper.
1. BACKGROUND
In a recent survey of 90 prominent multinational corporations (MNCs), Newman and Czechowicz [I] found that those MNCs using discounted cash-flow (DCF) criteria for investment decisions frequently set higher (premium) discount (target, hurdle, return) rates on foreign investments to allow for higher risk visd-vis domestic investments. They assert [ 1, pp. 134-135-J:
“Countries differ significantly with respect to political risk. Thus, the premium should be commensurate with the risk of each country.”
Implementation, however, is very difficult. Newman and Czechowicz [i,p. 1331 report that the discount rate adjustments for risk tend to be quite arbitrary (emphasis added):
“Many MNCs arbitrarily use a higher discount rate for their foreign projects than that used for domestic projects. The premium or increase in the discount rate is usually
arrived at subjectively and is rarely justz$ed. The only reason usually given is that higher risks are associated with foreign operations. Increasing the discount rate seems to be a sensible as well as convenient method used in MNCs’ capital budgeting processes.”
Increasing the required rate of return for risky investments (vis-d-vis low-risk investments) has considerable intuitive appeal to investors. However, very little guidance is available on how to determine such risk premiums. Academic literature in finance tends to be critical of the practice of adding risk premiums to discount rates because of the confounding of this risk over time in compound interest formulas, e.g. see the harsh criticisms in Ref. [2].
In theory, a better approach is to reformulate deterministic DCF models into stochastic models. Difficulties of estimation, however, have blocked progress along such lines. Bey [3] formulated a stochastic DFC model, but it is restricted to highly limited assumptions that cash flows follow a triangular distribution and economic states are a stationary first-order Markov process. A more general stochastic model introduced in Ref. [4] allows for expert subjective judgments regarding distributions of cash flows, discount rates and inflation rates. This approach, however, requires extensive and complicated judgmental input and remains impractical in practice. Even more general approaches not based upon DCFs have been proposed (notably arbitrage pricing theory) for risky investment capital budgeting analysis, e.g. see Copeland and Weston [2, Chap. 111, Constantinides [S], Ross [6] and Gehr [7]. However, in spite of theoretical advantages, unrealistic assumptions
265
266 R. E. JENSEN , A
of perfect capital markets and computational impracticality of estimating the state contingent prices of comparison alternatives and risk-free rates remain nightmares that are even more exaggerated when considered for international capital investments. MNCs need much mo‘re practical approaches that also have more realistic underlying assumptions.
2. PURPOSES OF THIS STUDY
Given the reported inclinations of MNC managements to use DCF models and to increase discount (target, hurdle, return) rates as foreign risk perceptions increase, the main purpose of this study is to provide a new approach to relating risk premiums to perceived risk. This approach relies upon expert judgments about risk differential indifference points for alternative projects to derive a foreign investment project relative risk premium. It is an attempt to reduce the impractical stochastic DCF model proposed in Ref. [4] to a more practical level that requires less judgmental input.
The approach proposed in this paper requires paired comparisons of differential risk equivalencies by one or more experts, e.g. by the management of an MNC contemplating various investment proposals. For each pair of countries the question posed might be worded something like: “In Year t, what is the average amount the Country j project would have to return per dollar of Country k’s investment return to justify the Country j risk differential?” Responses are then coded into rjk(t) ratios of risk differential equivalencies. It will be assumed that rkj(t) = l/rjk(t) such that, if n countries are compared pairwise, the reciprocal assumption reduces the number of required responses to be elicited to n(n - 1)/2 for each Year t. Means of reducing the response elicitation task over multiple time periods will also be proposed in this paper. For any Year t, the rij(t) elicited responses comprise a risk differential response matrix
rl 1(t). . . r& CNOI = !
[ 1 ml@). . r,,(t)
that is square, nonnegative and reciprocal (hence nonsymmetric). Because MNC managers often like to view foreign investment risk in terms of discount (target,
hurdle, return) rate premiums, formulas will be derived in this paper that translate rij(t) risk differential responses into relative risk premiums, termed pit(t) values, that can be added to cost of capital rates to allow for foreign risk differentials.
It is stressed that even though pj,Jt) relative risk premiums in some sense might be viewed as single-index values of between-country risk, they are derived from subjective expert judgment of rjk(t) paired comparison risk differentials. It is assumed here chat experts will consider multivariate aspects of foreign risk prior to making a paired comparison judgment used in deriving each r,,(t) risk dtflerential ratio. In this sense, the approach proposed in this paper is a multivariate approach to risk assessment where the aggregation (composition) of perceived multiple risk factors (variates) is left to the subjective judgment of the individual respondents. Initially, it will be assumed that rjk(t) responses are derived from a single expert. Means of combining multiple expert judgments are discussed later in the paper.
3. DRAWBACKS OF A SINGLE PUBLISHED RISK INDEX
Rather than set an arbitrarily higher discount rateforforeign investments, managers in some MNCs attempt to use variable discount rate premiums derived from a single published risk index. A case study on the International Machine Corporation (IMC) utilizing such an index for an investment project in Mexico is contained in Ref. [l, pp. 127-1371.t The projected cash flows in the IMC case
7 The IMC case was prepared for the Business International Corporation by V. B. Bavishi and H. A. Shawky. It is reported in Ref. [l].
Risk analysis for multinational corporations 267
are reproduced in Table 1. Purportedly, the difference between forecasted Mexican (20%) and U.S. (10%) inflation rates is adjusted for in the projected cash flows. It appears remaining U.S. inflation at 10% over the lo-year project life is allowed for in the 12% cost of capital rate. The Mexico and U.S. inflation rates seem too low at the present time, but this aspect will not be addressed in the present paper. The Mexican investment risk premium of 4.0% is derived from a single published “political risk” index that is described as follows:
One service provides an index ranking countries by degree of political risk. The index ranges from 0 to lOt-the higher the number, the lower the risk. The number for Mexico is 75. It must be noted that this index (as with all others) is in relative term-hence, investing in Mexico (75) is riskier than investing in Germany (86), as reflected in the relative magnitude of the numbers.
An approach to systematically incorporating the information provided by a country risk index is suggested below for the case of IMC’s Mexican investment.
The added political risk in investing in Mexico rather than investing in the U.S.A. is 25 points (on a lOO-point scale). Hence, we should expect commensurate compensation for the added risk and thus adjust the discount rate in such a manner as to incorporate the magnitude of the above relative riskiness indicated by the index numbers.
Applying this to IMC’s Mexican investment, we have
IMC worldwide weighted average cost of capital = 12%
Mexico’s risk level = 75.
To invest in Mexico, IMC must adjust its required rate of return (12%) to account for the added political risk. Required return for Mexican project:
0.12 =-= 0.75
16%.
The 16% rate for discounting the Mexican project’s cash flow in Table 1 yields:
NPV = -0.7(0.862) - 0.8(0.743) - l.O(O.641) + 2.2(0.552)
+ 2.7(0.476) + 3.3(0.410) + 4.1(0.354) + 5.1(0.305)
+ 6.2(0.263) + 20.2(0.227) - 6.0
= -0.6 - 0.6 - 0.6 + 1.2 + 1.3 + 1.4 + 1.5 + 1.6 + 1.6 + 4.6 - 6.0;
NPV = $5.4m (more accurately this is $5.2mt).
The NPV (net present value) is positive after adjusting for political risk [l, p. 1341.
If such a single risk index is at hand, the above approach is very simple to apply and seemingly easy to comprehend. However, this is deceptively simple and hides some tremendous compli- cations that are not immediately obvious. A major complication is that the calculation of a 4% = 16% - 12% risk premium from a single index combines many different kinds of risk, e.g. risks of sudden expropriation, creeping expropriation, currency restrictions, export barriers, devaluation of currency, price controls, repatriation limits, labor restrictions, production quotas, adverse tax impositions, civil disorder, terrorism and war. A longer list of such factors rated by average perceived importance appears in Ref. [ 1, p. 183.
Use of a single published political risk index does not take into account the fact that two countries having similar single risk index values may have vastly different underlying kinds of risk. For example, suppose hypothetically that Iraq and Israel both have an identical published risk index score, but that beneath this apparent identical risk composite Israel has a greater currency.
t The NPV = U.4m reported in Ref. [l, p. 1341 contains apparent rounding error. A more exact figure of NPV = $5.232m is derived in Case C of Table 4.
“I 9:3/5-H
Tab
le
I. M
exic
an
inve
stm
ent
proj
ect
cash
fl
ows
m t
he
IMC
ca
se (
sour
ce:
Ref
. [l
, p.
12
91)
Yea
rsie
xwct
ed
exch
anee
ra
te
Cas
h-fl
ow
Item
s
0 I
2 3
4 5
6 7
8 9
IO
P22:
$I”
P24.
2:$1
P2
6.6:
Sl
P29
3.$
1 P3
2.2:
$1
P35.
4:$1
P3
9.0:
$1
P42.
9:$1
P4
7.2:
$1
P5
I .9-
s 1 P
57
I:$1
I.
Initi
al
inve
stm
ent,
in $
mill
ions
(6
.0)
2.
Lic
ensm
g fe
es,
(a)
in m
illio
ns
of p
esos
14
I8
24
32
43
56
74
98
12
9 17
0 (b
) in
$
mill
ions
0.
6 0.
7 0.
8 10
1.
2 1.
4 1.
7 2.
1 2.
5 3.
0
(c)
Les
s U
.S.
taxe
s at
42%
, in
$ n
ullio
n (0
.3)
(0.3
) (0
.3)
(0.4
) (0
.5)
(0.6
) (0
7)
(0.9
) (1
.1)
(1.3
) (d
) N
et
afte
r ta
x,
in $
mill
ions
0.
3 0.
4 0.
5 0.
6 0.
7 0.
8 1.
0 1.
2 1.
4 1.
7
3.
Div
iden
ds
rece
ived
, (a
) in
mill
ions
of
pes
os
0 0
0 50
71
97
13
4 I8
3 24
7 33
4 (b
) in
$ m
illio
ns
1.6
2.0
2.5
3.1
3.9
4.x
5.9
4.
Aft
er-t
ax
effe
cts
of l
ost
expo
rts
in $
mul
lions
(1
.0)
(1.2
) (1
.5)
5. T
erm
inal
pa
ymen
t, in
$ m
llhon
s 12
.6b
6. T
otal
ca
sh
flow
s I
+ 2
d +
3b
+ 4
t
5. i
n $
mill
ions
(6
.0)
(0.7
) (0
.8)
(1.0
) 2.
2 2.
7 3.
3 4.
1 5.
1 62
20
.2
7.
Dis
coun
t fa
ctor
at
12
%’
1.0
0.89
3 0.
797
0.71
2 0.
636
0 56
7 0
507
0.45
2 0.
404
0.36
1 0.
322
8.
Pre
sent
va
lue.
m
$ m
illio
ns
(6.0
) (0
.6)
(0.6
) (0
.7)
1.4
1.5
1.7
1.9
?.I
2.2
6.5
9.
Cum
ulat
ive
pres
ent
valu
e,
in
$ m
illio
ns
(6.0
) (6
.6)
(7.2
) (7
.9)
(6.5
) (5
.0)
(3 3
) I .4
0.
7 2.
9 9.
4
“For
th
e lif
e of
the
pr
ojec
t, in
flat
mn
in
Mex
tco
is e
xpec
ted
to
aver
age
20%
an
nual
ly:
in
the
U.S
.A
It 1
5 ex
pect
ed
to
be
10%
. T
he
IO-p
erce
ntag
e-po
int
diff
eren
ce
repr
esen
ts
the
expe
cted
de
chne
in
th
e va
lue
of t
he p
eso
I.L
.&L
.;.\ t
he
dolla
r.
Thu
s,
proj
ect
cash
R
ow I
S ad
just
ed
for
expe
cted
ch
ange
s II
I ex
chan
ge
rate
s.
“IM
C’s
sh
are
of t
he
Mex
ican
au
bsld
iary
’s
net
wor
th
= 6
0%
(beg
mni
ng
equt
y +
net
m
com
e ye
ars
I to
10
dwde
nd
dlst
rlbu
ted
year
s 4
thro
ugh
IO)
= 6
0%
(220
+ 2
835
- 18
60)
= P
7.17
m
= $
12.6
11~
‘The
di
scou
nt
rate
of
12
%
is t
he
wor
ldw
ide
wag
hted
av
erag
e ca
t of
cap
ital.
.I
- 1^
-.
_
. _
_
_
_
- _
- -.
._
--~
-.
-.
-
_
__
-
-..
._-.
.
Risk analysis for multinational corporations 269
devaluation risk and a lower risk of expropriation. An MNC that is more sensitive to expropriation losses may, therefore, want to place a higher risk premium on Iraq, whereas another MNC that is safely diversified? such that Iraq’s expropriation risk is small may be much more concerned about Israel’s hyperinflation. The use of a single published political risk index as illustrated in the IMC case places the same risk premium on both countries if they have the same risk index level.
It follows, therefore, that MNCs are better advised to derive investment project risk premiums by examining their unique dependence upon multiple types of risk that vary from country to country rather than upon a single published risk index that may compositionally offset variations in individual underlying risk factors. At a minimum, the MNC should understand what risk factors lie behind the single risk index and how these factors are weighted when composed into a single published index.
4. DRAWBACKS OF ADDING A RISK PREMIUM TO THE COST OF CAPITAL RATE
In the previous IMC case, a 4% risk premium was derived for the Mexican investment project and added to the 12% IMC cost of capital rate to yield a risk-adjusted discount rate of 16% that, in turn, yielded a $5.2m NPV for the Table 1 cash flows. This is misleadingly conservative, however, due to risk premium compounding bias that arises from simply adding the 4% risk premium to the 12% cost of capital rate. Assuming for simplicity that 12% is the U.S. reinvestment rate and 4% is the intended risk premium for Mexico, the 4% risk premium in a given year is compounded when discounted to earlier years. Furthermore, its compounding increases with each compounding period (every year under annual compounding assumptions) and the net effect is difficult to visualize over multiple period aggregations.
For example, in the preceding IMC case, the cash flows and NPVs under 12% and 16% discounting are as follows (using Table 1 data):
(1) (2) (3) (4) (5) t Cash flow 12% NPV 16% NPV @k(4)
0 $ - 6.0 $-6.0 S-6.0 $0.0
1 -0.7 - 0.6 -0.6 0.0
2 -0.8 -0.6 -0.6 0.0
3 - 1.0 -0.7 -0.6 -0.1
4 2.2 1.4 1.2 0.2
5 2.7 1.5 1.3 0.2
6 3.3 1.7 1.4 0.3
7 4.1 1.9 1.5 0.4
8 5.1 2.1 1.6 0.5
9 6.2 2.2 1.6 0.6
10 20.2 6.5 4.6 1.9
Total $35.3m $9.4m $5.2m $4.2m
“The NPV = $5.4m reported in Ref. [l, p. 1341 contains apparent rounding error. A more exact figure of NPV = $5.232m is derived in Case C of Table 4.
In other words, what might appear to an MNC manager as a 0.04/0.16 = 25% risk premium component in the Mexican project discount rate actually constitutes a 45% reduction in NPV from $9.4m to $5.2m. This much higher reduction in NPV arises largely due to the risk premium being compounded over time.
t For example, an MNC that produces components for its products (e.g. automobiles) in various countries often reduces expropriation risk both by making itself less dependent upon one country and by making the country depend upon the MNC for buying the components produced.
270 R. E. JENSEN
5. DRAWBACKS OF USING DIFFERENT REINVESTMENT DISCOUNT RATES FOR DIFFERENT COUNTRIES
Instead of setting a target discount rate of return in an NPV DCF approach, it is possible to equate the streams of cash inflows to outflows over time and derive each project’s internal rate of return (IRR). For example, the IRR for the Mexican project in the IMC case in Table 1 is the 24.491% shown later on as Case D in Table 4. In theory, investment alternatives with the highest IRRs might then be chosen since high IRRs have higher cushions for risk. This may be highly misleading, however, due to implicit reinvestment assumptions that, if such assumptions are not realized, may greatly alter the planned IRR from the realized IRR. In other words, to attain a very high planned IRR, all interim cash flows released from the project must be reinvested at the project’s IRR which usually varies from project to project. Hence, ultimately realized reinvestment rates are seldom a given project’s realized IRR. For example, all cash inflows in Table 1 must be reinvested at 24.491% to attain the projected IRR even though the domestic (U.S.A.) opportunity cost of capital is assumed to be only 12.0%. Many finance and economics textbooks downplay the use of IRR criteria in discounted cash flow analysis. For example, Copeland and Weston [8, p. 361 stated (emphasis added):
“The IRR rule errs in several ways. First, it does not obey the value additivity principle, and consequently managers who use the IRR cannot consider projects independently of each other. Second, the IRR rule assumes that funds invested in projects have opportunity costs equal to the IRR for the project. This implicit reinvestment rate assumption violates the requirement that cash flows be discounted at the market-determined opportunity cost of capital. Finally, the IRR rule can lead to multiple rates of return whenever the sign of cash flows changes more than once. However, we saw that this problem can be avoided by the simple expedient of assuming that all cash inflows are loaned to the firm by the project at the market opportunity cost and that the rate of return on negative cash flows invested in the project is the IRR.
The NPV rule avoids all the problems which the IRR is heir to. It obeys the value additivity principle, it correctly discounts at the opportunity cost of funds, and most important, it is precisely the same thing as maximizing the shareholders’ wealth.”
In discounted cash flow analysis, an NPV rule rather than IRR is usually the preferred approach. Hence, a discount rate must be set in advance prior to discounting future cash flows. Management then faces the dilemma of whether to include a risk premium in that discount rate and, if so, what risk premium. In setting such a risk premium, however, they implicitly assume differential reinvestment rates.
Clearly, the use of different discount rates (for different countries having unequal risk premiums) on cash flows returned to MNC headquarters suffers from the varying reinvestment rate drawbacks of IRR calculations. For example, why should a dividend sent from a Mexican subsidiary to the U.S. be assumed to be reinvested at 16% while the same dividend received from a U.S. subsidiary is assumed to be reinvested at 12x? This problem is best overcome by attempting to use a discount rate equal
to the estimated actual reinvested rate. If this is not practical, then an average cost of capital rate for the entire MNC makes more sense than varying rates that may differ widely from ultimate actual reinvestment rates. In this paper reinvestment rates will be set at the estimated opportunity reinvestment rate at MNC headquarters, which in this paper is arbitrarily assumed to be located in the U.S.A.
6. SOLVING FOR RELATIVE RISK PREMIUMS
A method will now be proposed which might be deemed a compromise between preset risk premium adjusted NPV and a stochastic model that is impractical to apply in practice. It overcomes the IRR limitations of reinvestment at the IRR, i.e. instead reinvestment discount rates will be set at the same implicit opportunity reinvestment rate. Accordingly, the addition of arbitrary risk premium rates to discount rates resulting in obscure risk compounding is also avoided. The discount rate risk premium will be derived as “relative risk premium”, termed pjk(t) following other
Risk analysis for multinational corporations 271
types of input judgments. Let pjk(t) depict the relative risk premium for Year t cash flows from a Country j investment relative to a Country k investment. Other notation used in this paper is defined in Table 2. For convenience, assume the MNC’s home office is in the U.S.A. Also, suppose k = 1 is the U.S. indicator, iI the firm’s assumed reinvestment (discount, target, hurdle, return) rate utilized for U.S. domestic investments, pjl(t) is the risk premium added to iI for an investment in Country j, and rjl(t) is the investor’s perceived risk differential response per dollar cash flow from a given Country j investment project and investment in the U.S.A. at the iI discount rate. For example, a question posed to the IMC management in Table 1 might be: “In Year 10, what is the average amount the Mexican project would have to return per dollar of a U.S. investment return to justify the Mexican investment risk differential?’ If the reply was that a $4.00 Mexican project return is the risk indifference point per $1.00 return from a U.S. project, then r,,(lO) = S4.OO/Sl.O0 = 4.00, provided j = 2, depicts the Mexico project and k = 1 depicts the U.S.A. It will be assumed rkj(t) = l/rjk(t) reciprocal relationships hold such that r,,(lO) = 0.25 in the above example.
The relative risk premium, Pjk(t), is that amount which when added to i&t) equates the risk- adjusted discount rates of Countries j and k. For example, for Years t = 1 to 10 suppose the MNC opportunity reinvestment is estimated to be iI = 0.12 (12%) for each t. A forecasted X,(10) = $20.2m dividend sent from Mexico to the U.S.A. has a risk-adjusted equivalent in Year 10 of
X,(lO)r,,(lO) = ($20.2m)(0.25)
= $5.05m.
In other words, if the MNC manager subjectively evaluates all types of differential investment risk between Mexico and the U.S.A. and concludes that, 10 years into the future, the Mexican investment
MNC = Multinational corporation.
Table 2. Definition of symbols
j, k E Country designators. In this paper four countries are compared where k = I for the U.S.A., k = 2 for Mexico, k = 3 for Korea and k = 4 for Canada.
X,(I) = Cash dividend from an investment project in Country k in Year t.
IRR = Internal rate of return of all X,(t) cash flows over the life of the project.
NPV = Net present value of all X,(t) cash flows over the life of the project.
it(t) = Year I (target, hurdle, interest, return) rate used to discount X,(t) back to a present value in the NPV calculation for Country k.
r,&) - Risk differential equivalency (of Year f cash flows) that depicts the risk indifference point per dollar of return from Country j relative to k. For example, if rl,( 10) = 4.00 this implies that in Year IO an investor considers a $4.00 return from Country 2 to be risk equivalent to a $1.00 return from Country 1. If pp(t) and it(t) values are specified, then rjdt) = [I + i&J + p,,(t)]‘[I + i&)1 -‘. However, unless noted otherwise it is assumed that r,(t) values are obtained from response judgments elictted for purposes of deriving pjk(t) relative risk premiums.
[R(t)] = Response matrix comprised of r,(t) risk difTerential responses in Year t.
1 = Maximum eigenvalue of [R(t)].
C.E. = Consistency ratio that indicates the degree of response consistency in [R(t)].
x E Summation over all j values. I
log = Natural (Napierian) logarithm to base e
e E lim (1 + I/r)’ 12.7182818284 I-m
exp(y) - ey
w?(t) = Least-squares weight that minimizes aggregate squared error between wi/wr consistent ratios and r,,(t) inconsistent response judgments. The w:(r)/w:(t) ratios are least-squares consistency surrogates for r,*(t) response judgments.
[w*(t)] = Matrix comprised of w~(t)/w:(r) values for Year I consistency surrogates of rjt(r) responses.
[l7’*(t)] = Matrix comprised of the geometric means of w:(t)/w:(t) values for multiple experts.
pit(f) = i,(t) - it(t) = Relative risk premium added to the discount rate of Country j to make its returns risk equivalent to Country k
returns. When rj&) and ir(f) values are specified in advance, p &) = - 1 - i,(r) + exp[ {logr,(t) + flog [l + i&)])/tj = [(r,)“’ - I] [l + it(t)].
AHP s The Analytic Hierarchy Process
u,” - Performance rating of a Country j investment project on criterion Variate v.
ujus = Performance rating of a Country j investment project on a hierarchical subcriterion Variate s under Variate D.
p. = Priority rating of investment criterion Variate 0.
p.. - Priority rating of investment subcriterion Variate s under Variate v.
272 R. E. JENSEN
would have to return $4.00 for every U.S. return of $1.00 to be risk equivalent, then the forecasted S20.2m dividend in Year 10 from Mexico has a risk-adjusted equivalent to a $5.05m dividend from U.S. investments.
An implicit relative risk premium, pjk(t), to be added to the ik(t) Country k discount rate may then be derived from the following equation:
xj(t) -=
xj(t)rkj(t)
Cl + W + Pj/dt)l’ Cl + b(t)1
xj(t)
= Ijk(t)[l + &(t)]”
After some algebraic manipulation, the following formula may be derived using the notation in Table 2: _
Pjk(t) = - 1 - ik(t) + exp [{log rjk(t)
= - 1 - jr(t) + exp [{log rjk(t)
= {[rjk(t)]“’ - I}[1 + &Jr)].
For the above illustration this relative risk premium is
~~~(10) = - 1 - 0.12 + exp {[log4.00
+ t 1% Cl + w)lq
+ t log Cl + ~k@)l)b]
+ lOlog(1 + 0.12)]/10}
= - 1 - 0.12 + expC(1.38629 + 1.33333)/10]
= - 1.12 + exp(0.25196)
= -1.12 + 1.28654
= 0.16654.
Alternatively,
p21(10) = [(4.00)“‘” - l](l + 0.12)
= 0.16654.
Thus, a 16.654% risk premium is added to the 12.000% U.S. discount rate to derive a Mexican discount rate of 28.654% that is consistent with a r,,(lO) = 4.00 judgmental response that a $4.00 return on average from the Mexican project in Year 10 is needed to be risk equivalent to a $1.00 U.S. return. The equivalent NPVs are:
$5.05m $20.2m $20.2m
(1 + 0.12)‘0 = (1 + 0.12 + 0.16654)” = 4.00(1 + 0.12)”
= ,,,
In other words, the risk-adjusted NPV of a $20.2m dividend from Mexico in Year 10 reduces to S1.6260m.
Risk analysis for multinational corporations 213
The above illustration is contained in the t = 10 line of Case A in Table 3. In Case A it is assumed that the r,,(t) values are derived from MNC management judgment of risk differentials. It is also assumed that various kinds of risks are taken into account by management, e.g. risks of sudden expropriation, creeping expropriation, currency restrictions, export barriers, devaluations of currency, price controls, repatriation limits etc. MNC management performs a subjective evaluation of all such risks and provides the rzl(t) ratios shown in Case A of Table 3. The iI = 0.12 cost of capital rate is also specified by MNC management. Given r,,(t) and iI values, the pzl(t) relative risk premiums may be derived as shown in Table 3. The NPVs of the Table 1 cash flows aggregate to -$1.9742m in Case A. This implies that in Case A, the MNC management places such a high premium on Mexican investment risk that the Mexican project does not meet minimum target rates that include the pzl(t) risk premium values.
In Case B of Table 3, it is assumed that MNC management provides somewhat lower r,,(t) risk differential judgments when comparing Mexico investment risk with U.S. investment risk. In this case, p2i(t) relative risk premiums added to the iI = 0.12 discount rate yield an aggregate NPV = $0. This implies that the Mexican project exactly meets the target rates adjusted for subjectively perceived r,,(t) risk differentials in Case B of Table 3. It did not, however, do so in Case A where rzl(t) elicited risk differential equivalencies were greater.
7. SOLVING FOR RISK DIFFERENTIAL EQUIVALENCIES
Unless noted otherwise in this paper, it will be assumed that rjk(t) risk differential equivalencies are response judgments of one or more experts comparing pairwise the investment risks in foreign countries. In this section, however, it will be shown how rjk(t) values may be derived if, for some reason, the pj,Jt) relative risk premiums are instead specijed in advance. For example, in the IMC case in Table 1, it was assumed by the analysts that pzl(t) = 0.04 relative risk premiums were to be added to a iI = 0.12U.S. discount rate for each of t = 1,. ., 10 years in the Mexican project. In this case, the equation
xj(t) xj(t) Cl + it(t) + Pjfctt)l’ = rjdt)Cl + it(t
Table 3. Comparisons of alternative risk differential equivalency perceptions of MNC managers: derivation of p2,(f) from r,,(t) and i,(t)
Risk differential responses Relative Discount rates Table 1 YeaI
Mexico (2) to U.S.A. (I) = r2,(f) risk premium’ U.S.A. Mexico cash flow
Risk-adjusted discounted cash flow
&t(f) + i,(r) = i,(r) + D,,(l) x*(t) xM/Ct + id0 + Gaul’ . . _ .
t=o %l.OO/sl.oo = 1.0000 0.000% + 12.000% = 12.ooo% - %6.Om -6.0/(1 + 0.12C00)” = -$6.OOOm 1 %1.30/51.00 = 1.3OOlI 33.600% + 12.ooo% = 45.600% - %0.7m -0.7/(1 + 0.45600)’ = -50.4808m 2 %1.60/$1.00 = 1.6000 29.670% + 12.000% = 41.670% - S0.8m -O.S/(l + 0.41670)2 = -$0.3986m
4;
%l.90/s1.00 = 1.9wO 26.719% + 12.Oc@% = 38.719% -Sl.om ~ l.O/(l + 0.38719)” = -W3746m %2.20/51.00 = 2.2000 24.403% + 12.Oco% = 36.403% S2.2m 2.2/(1 + 0.36403)4 = $0.6355m
26 %2.5O/Sl.O0 = 2.5000 22.526% + 12.000% = 34.526% $2.7m 2.7/(1 + 0.34526)5 = SO.6128m
G78
$2.80/$1.00 = 2.8KIO 20.967% + l2.o00% = 32.967% 63.3m 3.3/(1 + 0.32967)6 = S0.597lm %3.10/$1.00 = 3.lOeO 19.647% + l2.o00% = 31.647% S4.lm 4.1/(1 + 0.31617)’ = S0.5983m $3.40/ojrs1.00 = 3.4coO 18.513% + 12.000% = 30.513% S5.lm 5.1/l + 0.30513)’ = S0.6058m
9 $3.70/$1.00 = 3.7000 17.524% + l2.o00% = 29.524% S6.2m 6.2/(1 + 0.29524)9 = $0.6043m 10 s4.OO/sl.OO = 4.oooo 16.654% + 12.000% = 28.654% S20.2m 20.2/(1 + 0.28654)‘O = $1.6260111
Time f = 0 NPV of Table I cash flows = NPV = [-61.9742mI Case A
f=O %1.00/$1.00 = l.oooo 0.000% + l2.Oco% = l2.o00% - S6.Om I sl.oz/sl.OO = 1.0200 2 %1.05/$1.00 = 1.0500
ca{
%l.l0/$1.00 = l.looo $1.30/$1.00 = 1.3000
z6 $1.60/$1.00 = 1.6000
67
$1.70/61.00 = 1.7ooo $2.00/$1.00 = 2.txOO
8 52.30/$1.00 = 2.3000 9 $2.60/$1.00 = 2.6000
10 $3.00/$1.00 = 3.0000
2.240%
2.766% 3.615% 7.592%
11.039% 10.356% 11.658% 12.289% 12.545% 13.006%
+ 12.000% = + 12.000% = + l2.c00% = + l2.c00% = + 12.000% = + 12.000% = + 12.000% = + 12.000% = + 12.000% = + 12.000% =
14.240% 14.766% 15.615% 19.592% 23.039% 22.356% 23.658% 24.289% 24.545% 25.006%
- S0.7m - 60.8m -$l.Om
S2.2m 62.7m $3.3m S4.lm 95.lm S6.2m
%20.2m
-6.0111 + 0.120001° = -S6.OOOOm
-0.7jil + 0.1424$ = -O.S/(l + 0.14766)’ = - l.O/(l + 0.15615)” =
2.2/(1 + 0.19592)’ = 2.7/(1 + 0.23039)’ = 3.3/(1 + 0.22356)6 = 4.1/(1 + 0.23658)’ = 5.1/(1 + 0.24289)’ = 6.2/(1 + 0.24545)9 =
20.2/(1 + 0.25006)“’ =
Time t = ONPV of Table 1 cash flows = NPV = 1 S0.0000 1 Case B
-SO.l627m - f0.6074m - so.647 1 In
$l.O755m S0.9575m %0.9835m $0.9273m %0.8956m S0.8599m $2.1679m
‘Pjdf) = - 1 - MO + exp [{logrj~(t) + clog [I + it(r)]}/t]l = {[r,(t)]” - I}[1 + idt)].
274 R. E. JENSEN
may be used to algebraically derive
r ct)
Jk = Cl + ik(c) + PjkCt)lf
[l + ik(t)]’ ’
For example, for t = 10, the risk differential equivalency in the Mexican project becomes
r 21
(lo) = (1 + 0.12 + 0.04)‘O
(1 + 0.12)‘0 = 1.4204.
In other words, a p2i(t) = 0.04 relative risk premium implies that the MNC management views a $1.4204 dividend in Year 10 from Mexico as being risk equivalent to a $1.00 dividend from a U.S. investment.
It may be informative to point this out to MNC managers who used the ~~~(10) = 0.04 risk premium perhaps somewhat arbitrarily. The question posed might be: “Do you really feel that the $1.4204 Mexican return is truly risk equivalent to a $1.00 U.S. return ten years into the future?’ If they answer no, then the assumed 16% Mexican discount rate with its 4% risk premium is not appropriate. The r,,(t) risk differentials for all pjdt) = 0.04 risk premiums assumed in the IMC case (reproduced in Table 1) are derived in Case C in Table 4. For example, in Year 5 it is implicitly assumed that a $1.1919 dividend from Mexico is risk equivalent to a $1.00 U.S. project dividend ifp,,(5) = 0.04. However, in Case D of Table 4, where ~~~(5) = 0.12491 is specified, the corresponding r,,(5) = 1.6967 implies a $1.6967 Mexican dividend is risk equivalent to a $1.00 U.S. dividend. Other r,,(t) values corresponding to a pzl(t) = 0.12491 relative risk premium are shown in Case D of Table 4.
Recall that pzl(t) + i,(lO) = 0.24491 in Case D is also the IRR of the Mexican project. This implicitly assumes reinvestment of risk-adjusted Mexican dividends at the ii(t) = 0.12 domestic cost
Table 4. Comwaisons of alternative relative risk premiums: derivation of r,,(r) from o,,(t) and i,(t)
Year Risk differential equivalencies’
Mexico (2) to U.S.A. (1) = r,,(f)
l=O $l.OcclO/sl.OO = la300 I S1.0357/S1.00 = 1.0357 2 $1.0727/$1.00 = 1.0727
“I $1.1110/$1.00= $.1.1507/s1.00 = = 2 $1.1919/$1.00 1.1110 1.1507 1.1919 6
6;
$1.2342/$1.00 = 1.2342 $1.2783/$1.00 = 1.2783 $1.3239/%1.00 = 1.3239
9 $1.3711/$1.00 = 1.3711 10 $1.4204/$1.00 = 1.4204
Relative risk premium
P2l(f) +
O.ooo% + 4.000% + 4.ooo% + 4.ooo% + 4.OKl% + 4.Oca% + 4.000% + 4.ooo% + 4.tIcuwa + 4.ooo% + 4.Oca% +
Discount rates Table 1
U.S.A. Mexico cash flow
i,(t) = i](t) + &t) X,(t)
12.wO% = 12.000% - $6.0m 12.ooo% = 16.000% - $0.7m 12.ooo% = 16.WO% - SO.Bm 12.coO% = 16.ooO% -$l.Om
12.coO% = 16.000% $2.2m 12.cmxwo = 16.000% S2.7m 12.000% = 16.000% $3.3m 12.ooo% = 16.000% S4.lm 12.ooo% = 16.000% SS.llIl 12.000% = 16.000% $6.2m 12aIO% = 16.OW/o $20.2m
Risk-adjusted discounted cash Row
x,(t)/Cl + i,(t) + P,,W
-6.0/(1 + 0.12000)” = -66.OOOOm -0.7/(1 + 0.16000)’ = -$06034m -O.S/(l + 0.16000)2 = -$0.5945m - l.O/(l + 0.16OLXI)’ = -E0.6407m
2.2/(1 + 0.16000)4 = 61.2150m 2.7/(1 + 0.16000) = $1.2855m 3.3/(1 + 0.16000)6 = S1.3545m 4.1/(1 + 0.1600@ = $1.4507m 5.1/(1 + 0.1600@* = S1.5556m 6.2/(1 + 0.16000)p = Sl.6303m
20.2/(1 + 0.16000)‘” = $4.579Om
Time f = ONPV of Table 1 cash flows = NPV = I$5.2320m 1 Case c
t=o $1.oOOO/$1.00 = 1.0000 1 $1.1115/S1.00= 1.1115 2 Sl.2355/$1.00 = 1.2355 3 %1.3733/S1.00 = 1.3733
n: $1.5264/$1.00 = 1.5264
$6 $1.6967/$1.00 = 1.6967
6;
$1.8859/$1.00 = 1.8859 $1.0962/$1.00 = 1.0962 $1.3300/S1.00 = 1.3300
9 $1.5899/$1.00 = 1.5899 10 $1.8787/$1.00 = 1.8787
O.OiW% 12.491% 12.491% 12.491% 12.491% 12.491% 12.491% 12.491% 12.491% 12.491% 12.491%
+ 12.000% I2.o00% 12.OOQ% 12.ooo% I2.c00% 12.000% 12.@xl% 12.000% 12.OcKwa 12.000% I2.o00%
= 12.000% 24.491% 24.491% 24.491% 24.491% 24.491% 24.491% 24.491% 24.491% 24.491% 24.491%
- S6.0m - E0.7m - S0.8m -$l.Om
S2.2m $2.7m $3.3m S4.lm 05.lm 66.2111
%20.2m
-6.0/(1 + 0.12000)” = -S6.OOOm ~0.7/(1 + 0.24491)’ = -$0.5623m -O.S/(l + 0.24491)* = -S0.5162m -l.O/(l + 0.24491)’ = -$0.5183m
2.2/(1 + 0.24491)’ = $0.9159m 2.7/(1 + 0.24491)s = $09030m 3.3/(1 + 0.24491)6 = $0.8865111 4.1/(1 + 0.24491)’ = $0.8847m 5.1/(1 + 0.24491)’ = 30.844Om 6.2/(1 + 0.24491)9 = %0.8633m
20.2/(1 + 0.24401)“’ = 52.2593111
Time t = ONPV of Table I cash flows = NPV = 1 $O.C@OO 1 Case D
‘rjt(f) = [l + b(f) + pjt(f)]‘[l + ir(t)I
Risk analysis for multinational corporations 215
of capital rate. For example, in Year 10, the X,(10) = $20.2m Mexican dividend has a risk-adjusted value of
rl2(lw2uo) = cw21uow2(1o) = (1/2.8787)($20.2m)
= $7.0171m.
Its NPV is as follows:
$7.0171m $20.2m 20.2m
(1 + 0.12)‘O = (1 + 0.12 + 0.12491)” = 2.X787(1.12) 10
,:,,, = &, = &,
Setting pjk(t) = 0.12491 [such that ik(t) + pjk(t) = 0.24491 is the IRR value] may be useful if analysts also derive the corresponding rjt(t) risk differential equivalencies shown in Case D of Table 4. These rjk(t) values may then be utilized by MNC managers to judge what dollar risk differentials are implicitly assumed ly project IRR values are to be compared for various foreign investment proposals.
8. MULTIPLE-COUNTRY PAIRED COMPARISON RESPONSE INCONSISTENCIES
The major purpose of this paper is to propose a means whereby pjJt) relative risk discount premiums may be related to expert judgment regarding rj~(t) perceived multiple-country foreign investment risk differentials. When these rjt(t) responses are elicited pairwise for n countries, then n(n - 1)/2 judgments must be elicited. These, in turn, comprise the reciprocal response matrix [R(t)] mentioned earlier in this paper. When making paired country comparisons, however, respondents may be inconsistent in the sense discussed by Saaty [9,pp. 17-251. Perfect response consistency implies that throughout the response matrix, r+(t) = r,,(t)rjk(t). Hundreds of applications of paired comparison judgmental responses [e.g. 4,9-561 indicate that respondents are rarely perfectly consistent unless their judgment freedom is greatly constrained to force them to be consistent. Saaty [9,p. 213 derives a consistency ratio (CR.) based upon the maximum eigenvalue of the response matrix and average randomized responses and applies a rule of thumb that whenever C.R. < 0.10 response inconsistency can be adjusted ex post rather than forced by greatly reducing response freedoms. Saaty and others in numerous references [e.g. 9,22-61 advocate consistency adjustments based upon replacing the rjk(t) original responses with uj/ul, ratios of components of the normalized principal right (PR) eigenvector corresponding to the maximum eigenvalue (2) of [R(t)]. In more recent papers [17,18] it is argued that even better consistency adjustments can be attained using wf(t)/w:(t) replacements for rjk(t) responses, where wf(t) values are the least-squares adjustments optimizing the model:
minimize TC ('jr - wj/w32
subject to TwjE1
wj > 0, j = l,...,n.
This forces consistency adjustments to be as close as possible to original response judgments in a
276 R. E. JENSEN
least-squares sense. Additional justifications for such adjustments relative to PR-eigenvector adjustments are given in Refs [17, IS].
By way of illustration, suppose that for Year t = 10, management of an MNC provides the following risk differential responses:
country to country rjr( 10) elicited responses
Mexico to U.S.A. r,,(lO) = 4.00
Korea to U.S.A. r3r(10) = 3.00
Canada to U.S.A. r4r(10) = 4.00
Mexico to Korea r,,(lO) = 2.00
Mexico to Canada r,,(lO) = 3.00
Korea to Canada r&10) = 2.00
This gives rise to the response multiple-country matrix:
l/l l/4 l/3 l/4
original [R(lO)] = 4/l l/l 2/l 3/l I I 3/l l/2 l/l 2/l
4/l l/3 l/2 l/l
maximum eigenvalue 2 = 4.154
C.R. = 0.057.
Since C.R. > 0, some response inconsistency is present. For example, if r,,(lO) = 4.00 and r,,(lO) = 3.00 then r,,(lO) should have been 4/3 = 1.33 instead of 2.00. Similarly, if r,,(lO) = 2 and r,,(lO) = 4 then rz4(10) should have been 2/4 = 0.50 instead of 3.00 for perfect consistency.
Applying least-squares consistency adjustments, the optimal least-squares weights of
wT(10) = 0.0087, wf(l0) = 0.408, w$( 10) = 0.282 and wX( 10) = 0.232 derived by procedures described and illustrated in Refs [ 17,181 yield the following consistency adjusted response matrix:
consistency adjusted [ W*( lo)] =
r 0.087/0.087 0.087/0.408 0.087jO.272 0.087fO.232
0.408/0.087 0.408jO.408 0.408jO.272 0.408jO.232
0.272/0.087 0.272/0.408 0.27210.272 0.27210.232
L 0.23210.087 0.23210.408 0.23210.272 0.23210.232
maximum eigenvalue 1 = 4.000
C.R. = 0.000.
These consistency adjusted risk differential responses and corresponding relative risk premiums are shown in Table 5. The calculations in Table 5 proceed sequentially as follows:
Elicit rj,(lO) risk differentials for Year 10
Step 1 Derive wT(lO) least-squares adjusted weights ;
Set i,(lO) = 0.1200 for Year 10 I
Step 2 Calculate ~~~(10) value from w~(lO)/w~(lO) and i,(lO)
Derive i,(lO) = i,(lO) + ~~~(10) I. 9
Risk analysis for multinational corporations 211
Step 3 Calculate ~~~(10) value from w:(lO)/w:(lO) and i,(lO)
Derive i,(lO) = i,(lO) + ~~~(10) 1:
Step 4 Calculate ~~~(10) value from w$(lO)/w:(lO) and i,(lO)
Derive i,(lO) = i,(lO) + ~~~(10) I; Step 5 [Derive all other pjk( 10) values from wT( lO)/w:(lO)’ ratios and &(lO)].
Table 5 is a multiple-country extension of the two country (U.S.A. vs Mexico) cases illustrated previously. In Table 5, it is assumed that MNC managers supplied only the [R( lo)] response matrix shown above and an iI = 12.00% discount rate for U.S. investments in Year 10. The least- squares risk differentials, relative risk premiums and discount rates for Mexico [i,(lO) = 30.72%], Korea [i3( 10) = 25.52%] and Canada [i4( 10) = 23.54%] may be computed sequentially from those initial inputs.
For example, suppose an X,( 10) = $20.2m dividend is anticipated from Mexico in Year 10. This has a $1.39m NPV computed from risk differentials and discount rates from other countries as follows (using Table 5 outcomes):
(1) Mexico to U.S. present value of $20,2m in Year 10
$20.2m $20.2m $4.3073m
(1 + 0.3072)” = 4.68966(1 + 0.12)” = (1 + 0.12)” =
(2) Mexico to Korea present value of $20.2m in Year 10
$20.2m $20.2m $13.4667
(1 + 0.3072)” = 1.5(1 + 0.2552)” = (1 + 0.2552)” =
(3) Mexico to Canada present value of $20.2m in Year 10
S20.2m S20.2m $11.4863m
(1.3072)” = 1.75862(1 + 0.2354)” = (1 + 0.2354)” =
j $1.39m 1
$1.39m .
In other words, $20.2m from Mexico in Year 10 has a risk-equivalent value of $4.3073m from the U.S.A., $13.4667m from Korea and S11.4863m from Canada. All of these have an NPV of S1.39m using relative risk premium discount rates of ~~~(10) = 0.1872, pz3( 10) = 0.0520 or ~~~(10) = 0.0718 to be added to i,(lO) = 0.1200, i,(lO) = 0.2552 or i,(lO) = 0.2354, respectively. These would not, however, have yielded the same $1.39m present value if the initial rjk(t) risk differential responses were not adjusted for response inconsistency.
9. ELICITATION OF RISK DIFFERENTIAL JUDGMENTS AS DYNAMIC TIME FUNCTIONS
When dealing with multiple countries and multiple time periods, the elicitation of all rjr(t) risk differentials may become impractical due to a large number of input parameters. One means of simplification is to set rjt(t) at some average value rjk(o over all time periods. For example, a risk
differential between the Mexican vs U.S. investments might be deemed to be $3.00 to S1.00 over all 10 years of the Mexican investment. When dealing with the future, however, it is usually important to allow for variability in risk over time, e.g. increasing risk over time. Saaty [9, pp. 95- 1051 introduces a “dynamic judgments” approach that will now be adapted to the derivation of relative risk premiums to be added to discount rates. The approach not only allows for functional
Tabl
e 5.
Der
ivat
ion of
rel
ativ
e ris
k pr
amur
ns
and
coun
try
disc
ount
rat
es for
Yea
r IO
usi
ng lea
st-s
quar
es cons
iste
ncy ad
lust
ed res
pons
es: mu
ltip
le-c
ount
ry ana
lysi
s
Year
IO
Le
ast-
squa
res co
nsis
tenc
y Ye
ar IO
Ye
ar IO
elic
ited
risk
ad
just
ed ris
k di
ffer
enti
als
Year
10
rel
atwe
ris
k pr
emun
s Co
untr
y k di
scou
nt
Coun
tryj
di
scou
nt
dii%
rent
ials
fo
r Ye
ar lO
cash
Ro
ws"
-1 -
i,(l
O) + ex
p[(l
ogr,
,(lO
)+
lOlo
g[l
+ i,
(lO)
]]/l
O~
rate
ra
te
r,,(
lO)=
4/l
w:
/w:
= 0.
408/
0.08
7 =
4.68
966
Mexi
co to
U.S
.A. =
p2,(
10)=
0.
1872
i,
(lO)
= 0.
1200
0.
1872
+ 0.
1200
= 0.
3072
= i,
(lO)
Mexi
co
r,,(
lO)=
3/
l w:
/w:
= 0.
272/
0.0X
7 =
3.12
644
Kore
a to
U.S
.A. =
pz,(
lO) =
0.13
52
i,(l
O) = 0.
1200
0.
1352
+ 0.
1200
= 0.
2552
= i,
(lO)
Kore
a
r,,(
lO)=
4/l
W.
/W:
= 0.
232/
0.0X
7 =
2.66
667
Cana
da
to U.S
.A. =
p,,(
lO)=
0.
1154
i,(
iO)
= 0.
1200
0.
1154
+ 0.
1200
= 0.
2354
=
i,(lO
) Ca
nada
r23(
10) =
211
w:
/'w:
= 0.
408/
0.27
2 =
1.5C
000
Mexi
co to
Kor
ea =
&lo)
= 0.
0520
r,
(lO)
= 0.
2552
0.
0520
+ 0.
2552
= 0.
3072
= i,
(lO)
Mext
co
rlJI
O) = 3/
l w:
/w:
= 0.
408/
0.23
2 =
1.75
862
Mex~
coto
Ca
nada
=
pJlO
)=
0.07
18
I‘JI
O) = 0.
2354
0.
071X
+ 0.
2354
= 0.
3072
= IJ
IO) Me
xtco
54(1
0) =
2/l
w:/w
: =
0.27
2/0.
232 =
1.17
241
Kore
a to
Can
ada
= I,
,,
= 0.
0198
i,
(lO)
= 0.
2354
00
198
+ 0.
2354
= 0.
2552
= i,
(lO)
Kore
a
r,,(
lO) = 1,
'4
w:iw
: =
0.08
7/0.
408 =
0.21
324
U.S.
A. to Me
xico
=
~~~(
10) =
PO.1
872
i,(l
O) = 0.
3072
-0
.187
2 +
0.30
72 = 0.
1200
= i,
(lO)
U.S.
A
r,,(
lO) = 11
3 w:
/w:
= 0.
087/
0.27
2 =
0.31
985
U S.
A. to Ko
rea
= pl
,(lO
)= PO
.135
2 r,
(lO)
= 0.
2552
PO
.135
2 +
0.25
52 = 0.
1200
= i,
(lO)
U.S.
A
rlJI
O)=
l/4
w:,w
: =
0.08
7/0.
232 =
0.37
500
U.S
.A.
to Ca
nada
=
p,Jl
O)=
-011
54
IJIO
) =
0.23
54
PO.1
154
+ 0.
2354
= 0.
1200
= r,
(lO)
U.S.
A
r,,(
iO) = l/
2 w:
;w:
= 0.
272/
0.40
8 =
0.66
667
Kore
a to
Mex
tco
= p,
,(lO
) =
PO.0
520
i,(l
O) = 0.
3072
-0
.052
0 +
0.30
72 = 0.
2552
= i,
(lO)
Kore
a
r,,(
lO)=
Ii
3 w:
/w:
= 0.
232/
0.40
8 =
0.56
863
Cana
da
to Mex
ico
= p4
J10)
=
-007
18
iI
= 0.
3072
PO
.071
8 +
0.30
72 = 0.
2354
= ~~
(lO)
Cana
da
r,,(
lO) = I/
2 w:
iw:
= 0.
232/
0.27
2 =
0.85
294
Cana
da
to Kor
ea =
/)Jl
O)=
-0.0
198
i,(l
O)- 02
552
-0.0
198
+ 0.
2552
= 0.
2354
= 1~
(10)
Cana
da
*The
se ar
e de
rive
d by
an
algo
rith
m de
scri
bed m
Refs
117
, IS
].
_.
Risk analysis for multinational corporations 279
changes in judgment, but it greatly reduces the number of elicited judgments necessary to calculate relative risk premiums over multiple time periods.
For example, if r,,(t) = $0.3t + 1 is derived from a single time functional judgment of an MNC manager in the Mexican project for the IMC case in Table 1, then all r,,(t) values shown in Case A in Table 3 can be derived from that single input rather than 10 input judgments for individual r,,(t) values. At the same time, r,,(t) = 0.3t + 1 allows management to specify a functional increase of risk at a 0.3t rate over time. Saaty presents illustrative dynamic judgment time formulas, reproduced in Table 6. These are quite general in nature and were first applied in a stochastic
capital budgeting model in Ref. [4]. Suppose that for four countries, the following dynamic time functionals are elicited from MNC
managers:
Mexico to U.S.A.
Korea to U.S.A.
Canada to U.S.A.
Assuming response consistency, the following
Mexico to Korea
Mexico to Canada
Korea to Canada
rzl(t) = 0.30t + 1
r,,(t) = 0.25t + 1
r,,(t) = 0.20t + 1.
time functionals can be derived:
r23@) = r21W31(t)
r24@) = r2 rWdl(t)
r&J = r31@Yr41(t).
The other rkj (t) = l/rp(t) values are reciprocals of the above functions. Recall that the relative risk premium formula is
PjlJt) = - 1 - it(t) + exp [(logrjdt) + t lOg[l + i&t)]j/tl.
Using rjk(t) time functionals shown above and iI = 0.12, the pjk(t) relative risk premiums and i,.(t) discount rates are as shown in Table 7. The derivation proceeds sequentially in Table 7 as follows:
Step 1 Elicit rjr(t) risk differential judgments
Elicit iI = 0.1200 judgment I;
Step 2 Calculate p21(t) values from r21 and iI
Calculate i2(t) = iI + pzl(t) I. 9
Time-dependent importance intensity
Table 6. Some alternative dynamic judgment response formulas (source: Ref. [9, p. 961)
DescriDtion ExDhnatiOn
a
OL,t + clI
b, log(t + 1) + b,
Constant for all f, 1 C x < 9 an integer
Linear relation in f, increasing or decreasing to a point and then a constant value thereafter. Note that reciprocal is a hyperbola
Logarithmic growth up to a certain point and constant thereafter
c,exp(c,t) + c3
d,t’ + d,t + d,
e,t”sin(t + e2) + e3
Exponential growth (or decay if c1 is negative) to a certain point and umstant thereafter (note reciprocal of case c1 negative is the logistic S-curve)
A parabola giving a maximum or minimum (depending on d, being negative or positive) with a constant value thereafter. (May be modified for skewness to the right or left.)
Oscillatory
No change in relative standing
Steady increase in value of one activity over another
Rapid increase (decrease) followed by slow increase (decrease)
Slow increase (decrease) followed by rapid increase (decrease)
Increase (decrease) to maximum (minimum) and then decrease (increase)
Oscillates depending on n > 0 (n < 0) with decreasing (increasing) amplitude
Catastrophies Discontinuities indicated Violent change in intensity
280 R. E. JENSEN
Step 3 Calculate pX1(t) values from rjl(t) and ii(t)
Calculate i3(t) = ii(t) + psi(t) I; Calculate
Step 4 p4i(f) values from rdl(t) and ii(t)
Calculate i4(t) = ir(t) + P4i(r) I. The resulting relative risk premiums and foreign discount rates that are consistent with the elicited risk differential functions are shown in Table 7. In Step 1, i,(t) = 0.1200 is specified for the U.S.A. and in succeeding steps pjl(t) values can be derived for allj countries and t time periods from their corresponding rjt(t) ratios of time functions. For example, in Year 2 the relative risk premium for Korea to the U.S.A. is p3i(2) = 0.28000 (28.0%) such that the Korean discount rate is
id4 = G(2) + ~~~(2)
= 0.12000 + 0.28000
= 0.40000 (40.0%).
Similarly, the relative risk premium for Korea to Canada is ~~~(2) = 0.05604 such that the Korean discount rate is also
W = LO) + ~~~(2)
= 0.34400 + 0.05600
= 0.40000 (40.0%).
10. SUMMARY AND CONCLUSIONS
Surveys of MNC managers reveal that adding risk premium to discount (target, hurdle, return) rates in discounted cash-flow NPV calculations is a very popular means of allowing for foreign investment risk. However, the “premium or increase in the discount rate is usually arrived at subjectively and is rarely justified” [l, p. 1331. Also, the implicit assumption that dividends from different countries sent to MNC headquarters are reinvested at different rates (i.e. the hurdle rates of the countries from whence they came) is highly unrealistic and gives rise to inevitable differences between planned and realized investment returns. This paper takes the position that the same reinvestment rate for all dividends received is more appropriate in the spirit that this rate discounts at estimated average cost of funds when ultimate reinvestment rates are impossible or impractical to predict with sufficient accuracy for utilizing dynamic reinvestment rate models.
In spite of using a single reinvestment rate for all foreign dividends sent to MNC headquarters, this paper derives relative risk premiums that are implicit in risk adjustments of cash flows from foreign investment cash-flow projections. The approach can be applied to two or more countries simultaneously and is based upon subjective expert judgment as to multivariate political risks of foreign investments. The proposed analysis procedure is provided below:
(1) The first step is to project the dividend cash-flow stream that adjusts for inflation differences in the manner illustrated in the Mexican IMC case summarized in Table 1.
(2) Respondents (e.g. MNC managers) are then asked to evaluate multivariate political and other risks of each country having a proposed investment project. For every future Year t of a project, each respondent supplies an indifference point at which an amount returned from Country j is risk equivalent per dollar returned from Country k. For example, in Year 10 the respondent might conclude that it would take at least $4.00 from Mexico to be risk indifferent per $1.00 returned from U.S. investments. Then the expert’s rjk(t) risk differential equivalency ratio in this instance becomes $4.00/$1.00 = 4.00. These ratios comprise each [R(t)] risk differential response matrix. There is one matrix for each Year t and each respondent. Use of multiple-country risk differentials is illustrated in Table 5.
Tab
le
7. D
eriv
atio
n of
rel
ativ
e ri
sk
prem
ium
s an
d fo
reig
n di
scou
nt
rate
s fr
om
dyna
mrc
tim
e fu
nctio
nal
resp
onse
s of
ris
k di
t%re
ntla
l eq
uiva
lenc
ies
Yea
r
Ris
k di
ffer
entia
l eq
uiva
lenc
ies
Rel
atw
e ri
sk
prem
lum
s”
Dis
coun
t (t
arge
t. hu
rdle
) ra
te?
Mex
ico
to
Kor
ea
to
Can
ada
to
Mex
ico
to
Kor
ea
to
Can
ada
to
U.S
.A.
Mex
ico
Kor
ea
Can
ada
U.S
.A.
U.S
.A.
U.S
.A.
U.S
.A.
U.S
.A.
U.S
.A.
i,(t)
i,(
t)
i,(t)
i,(
t)
(0.3
0t +
I)
/(ot
+ I)
(0
.25t
+
l)/(
ot
+
I)
(0.2
01 +
I)/
(01
+ 1)
O
.ooo
oO
O.O
OO
OO
O
.OO
KKl
0.12
ooo
0.12
000
0. I
2OO
a O
.l2O
@l
t=O
1.
00
I.00
1.
00
0.33
600
0.28
GQ
o 0.
2240
0 0.
1200
0 0.
4560
0 0.
4oO
oo
0.34
400
1 1.
30
1.25
1.
20
0.29
670
0.25
171
0.20
520
0.12
Otx
l 0.
4167
0 0.
3717
1 0.
3252
0 2
1.60
1.
50
I .40
0.
2671
9 0.
2296
8 0.
1899
6 0.
12O
clI
0.38
719
0.34
968
0.30
996
3 I .
90
1.75
1.
60
0.24
403
0.21
191
0. I7
729
0.12
ooo
0.36
403
0.33
191
0.29
729
4 2.
20
2.00
1.
80
0.22
526
0.19
721
0.16
654
0.12
Goo
0.
3452
6 0.
3172
1 0.
2865
4 5
2.50
2.
25
2.00
0.
2096
7 0.
I847
9 0.
1572
9 0.
12oo
o 0.
3296
7 0.
3047
9 0.
2772
9 6
2.80
2.
50
2.20
0.
1964
7 0.
1741
4 0.
1492
1 0.
1200
0 0.
3164
7 0.
2941
4 0.
2692
1 7
3.10
2.
75
2.40
0.
1851
3 0.
1648
7 0
1420
9 0.
12co
o 0.
3051
3 0.
2848
7 0.
2620
9 8
3.40
3.
Gil
2.60
0.
1752
4 0.
1567
1 0.
1357
5 0.
12O
cnl
0.29
524
0.27
671
0.25
575
9 3.
70
3.25
2.
80
0.16
654
0. I4
948
0.13
006
0.12
Ocn
3 0.
2865
4 0.
2694
8 0.
2500
6 10
4.
00
3.50
3.
00
Mex
ico
to
Mex
ico
to
Kor
ea
to
Mex
ico
to
Mex
ico
to
Kor
ea
to
U.S
.A.
Mex
ico
Kor
ea
Can
ada
Yea
r U
.S.A
. U
.S.A
. U
.S.A
. K
orea
C
anad
a C
anad
a i,(
t)
l*(f
) i,(
t)
i4w
(0.3
0t
+ 1)
/025
t +
1)
c0.3
0t
+ 1)
/(0.
201
+
I)
(0.2
5t
+ I)/
(O.Z
Ot
+ I)
O.o
oooO
O
SIO
OO
O
O.o
oooO
0.
1200
0 0.
1200
0 0.
1200
0 0.
1200
0 r=
O
l.OlH
o l.w
OO
1.
OoO
O
0.05
600
0.11
200
0.05
600
0.12
000
0.45
600
0.4G
Qoa
0.
3440
0 1
I .040
0 1.
0833
1.
0417
0.
0450
1 0.
0915
2 0.
0464
9 0.
1200
0 0.
4167
2 0.
3716
9 0.
3252
0 2
I .066
7 1.
1429
1.
0714
0.
0375
0 0.
0772
2 0.
0397
4 0.
1200
0 0.
3871
8 0.
3497
0 0.
3099
6 3
1.08
57
I.18
75
1.09
38
0.03
2 12
0.
0667
4 0.
0346
2 0.
1200
0 0.
3640
3 0.
3319
1 0.
2972
9 4
1.10
00
1.22
22
1.11
11
0.02
805
0.05
872
0.03
067
0.12
OiI
O
0.34
526
0.31
721
0.28
654
5 1.
1111
1.
2500
1.
1250
0.
0248
8 0.
0523
8 0.
0275
1 0.
1200
0 0.
3296
7 0.
3048
0 0.
2772
9 6
1.12
00
1.27
27
1. I3
64
0.02
234
0.04
727
0.02
492
0.12
OO
Q
0.31
648
0.29
413
0.26
921
7 1.
1273
I.
2917
I.
1458
0.
0202
6 0.
0430
4 0.
0227
7 0.
12O
Gil
0.30
513
0.28
486
0.26
209
8 1.
3333
1.
3077
I.
1538
0.
0185
3 0.
0394
9 0.
0209
7 0.
12oo
o 0.
2952
4 0.
2767
2 0.
2557
5 9
I.13
85
1.32
14
I.16
07
0.01
707
0.03
649
0.01
942
0.12
000
0.28
655
0.26
948
0.25
006
IO
1.14
29
1.33
33
1.16
67
“p,k
(t) = -
1 ~
i,(t)
+
exp[
{logr
,,(t)
+
tlog[
l +
iJt)
]}/tJ
=
([(r
,,(t)
]‘”
- l]
[l
+ i
x(t)
].
b JS
tep
t,(r)
= 0
.120
0; lS
tep
i,(t)
= i
ll0
+ p
i,(t)
; [S
tep
i,(t)
=
i,(t)
+
p,,(
t);
[ ia
=
l,(t)
+
ljdl(t
)
---I
--
.,-
, _
-
._
^.
-.
I -
-.
_
-.
._
,,
. -
. .
.-
-.-
-
282 R. E. JENSEN
(3) The tedium of having separate responses for each Year t can be reduced by utilizing dynamic time functional responses proposed by Saaty [9, pp. 95-1051 and first applied to capital budgeting problems by Jensen [4]. The procedure is illustrated in Table 7.
(4) Responses from multiple experts can be combined in a number of ways. One approach is the use of geometric means of risk differential responses. Such an approach to multiple-respondent aggregation was first proposed by Saaty
C9, P. 631. (5) MNC management must specify the i,(t) discount rate for its home base, e.g. the
average opportunity reinvestment rate in its home country. (6) The rjk(t) risk equivalency judgments might have internal response inconsistencies.
These can be adjusted for consistency by either eigenvector component ratios proposed by Saaty [e.g. 9,22-511 or least-squares approaches proposed by Jensen [17, IS]. In this paper least-squares adjustments are illustrated in Table 5. In
other words, rj~(t) response ratios are replaced by wf(t)/wz(t) ratios having internal response consistency as close as possible to rjk(t) responses under a least- squares criterion.
(7) Since managers in many MNC firms prefer to conceptualize foreign risk in terms of discount rate premiums, this paper illustrates how pj,,(t) relative rate premiums can be derived from their rjk(t) risk differential equivalencies and iI input values for the home country. The discount rate of any Country j may be derived from a Country k discount rate as &(t) + pjk(t). This is illustrated in Tables 5 and 7. It is also illustrated in Cases A and B of Table 3.
(8) In the case where MNC managers prefer to specify pjr(t) risk premiums rather than rjdt) risk differential ratios, the risk differential equivalencies may be derived as illustrated in Cases C and D in Table 4. This may help to show MNC managers what levels of risk indifference are implicit in their chosen risk premium discount rates. The major purpose of this paper, however, was to propose that relative risk premium discount rates instead be derived from elicited risk d@erential responses that are based upon perceived points of indzrerence between foreign
investments having direrent types and levels of risk. (9) The analysis then may or may not proceed formally into weighing risk-adjusted
NPV outcomes against other financial and nonfinancial investment criteria. One means of doing so using AHP analysis has been illustrated in this paper.
In conclusion it should be pointed out that the analysis proposed in this paper assumes risk independence between investment projects in different countries. For example, the risk differential equivalency r,,(t) values between Mexico and the U.S.A. were assumed to be independent of whether or not projects in Korea and Canada were undertaken or not undertaken. In a more complicated MNC world, this type of analysis would require MNC managers to “bundle up” interdependent (interactive) investment proposals in multiple countries into more complex “independent” proposals.
Alternatively, the extended analysis procedures termed combinatorial (interdependent, interactive) elicitation approaches in Refs [4, 1 l-1 31 or the interactive feedback systems in [9, pp.20442101 may be formulated, but they are not yet practical for complex real-world problems because of the tremendous amount of judgmental input data required. In general, however, it is usually unwise to undertake separable NPV analyses of interdependent projects. The approach of “bundling up” these components into a composite proposal is generally required for any type of NPV analysis, including procedures proposed in this paper.
Risk analysis for multinational corporations 283 *, .
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