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Adv Syst Sci Appl  2024; 02; 103-115 
Published online at https://ijassa.ipu.ru. 

Savage's Solution to the Problem of Three-Currency Deposit 
Diversification: Program Tools and Modeling Results 

Vitaly Molostvov* 

International Center of Decision Choice and Analysis of HSE University, Russia, Moscow 

Abstract: This paper presents the development of computing tools for finding the optimal structures of 
multi-currency deposits in terms of guaranteed risk under uncertain exchange rates. The approach 
utilizes Savage's minimax regret concept to calculate risk and guaranteed risk functions explicitly, 
assuming only the limits of possible changes in uncertain parameters are known. The Excel environment 
implements the algorithm for calculating the optimal solution that minimizes income loss due to 
incomplete information. Computational experiments analyzed the dependence of the optimal 
guaranteed risk on problem parameters, such as interest rates of currencies and boundaries of uncertain 
factors. The results can be used to analyze financial management problems in conditions of incomplete 
information. 

Keywords: optimization, incomplete information, minimax regret solution, deposit diversification, 
Excel  

1. INTRODUCTION 

The search of the optimal structure of a multi-currency deposit is complicated by the uncertainty 
of future values of economic parameters, in particular, the unknown values of the exchange rates 
of the currencies used. A typical situation is the so-called substantial uncertainty, in which there 
are not only no exact values of the uncertain parameters, but also no statistical characteristics. The 
Decision Maker (DM) knows only the limits of the possible values of these parameters. This leads 
to the emergence of risk, understood as a deviation of the result obtained from the desired or 
expected result. 

One approach to measuring and minimizing risk is Savage's concept of minimax regret. 
According to this , risk is interpreted as a loss from the ignorance of uncertain factors. The principle 
of the best-guaranteed outcome [2] is usually applied to the risk function to compute the smallest 
guaranteed risk. This is the approach used below in the problem of optimal deposit diversification 
across three currencies and  risk is understood according to Savage or regret risk [10] . The point 
of view used is that of the DM who allows risk and seeks to minimize it.  

The concept of Savage optimality [10] and other approaches to problems with uncertain 
parameters were considered in publications [5–7]. The paper [14] investigated the solution of 
Savage's minimax regret in the problem of deposit diversification for  two types of currencies. The 
article [8] is devoted to finding a solution with optimal guaranteed risk in a similar problem with 
three currencies. In it, an explicit form of the optimal solution is obtained and a constructive 
algorithm to compute it is proposed. The current paper continues this research.  

The proposed algorithm is implemented in Excel. An economic interpretation of the initial and 
derived parameters of the problem is given. Debugging and experimental calculations are carried 
out. The case of equal guaranteed yield of all three currencies (at minimum future rates) is 
investigated  analytically. Numerical modeling of the dependence of the minimum guaranteed risk 
and the structure of the optimal deposit on the parameters of the problem, such as the upper bounds 
of the future values of exchange rates and interest rates, is carried out.  

 
* Corresponding author: molostvov@list.ru 



V.S.MOLOSTVOV                        104 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

2. STATEMENT OF THE PROBLEM 

At the beginning of the deposit period, the DM  allocates one ruble among three types of deposits, 
buying dollars in the amount of 𝑥  rubles and euros in the amount of 𝑥  rubles at the initial rates 
𝐾  and  𝐾  respectively, leaving 1 − 𝑥 − 𝑥  rubles in a ruble deposit. At the end of the deposit 
period, the accrued interest (at the known interest rates 𝑑  and 𝑑 ) in  dollars and euros are 
converted into rubles at  exchange rates unknown at the beginning of the deposit period: 𝑦 ∈
[𝑎 , 𝑏 ],  𝑦 ∈ [𝑎 , 𝑏 ]. Here 𝑎  and  𝑏  (𝑎  and 𝑏 ) are the lower and upper limits of possible 
changes in the final dollar (euro) exchange rates. In accordance with these rules, the final income 
in ruble terms can be presented in the  form: 

𝑓(𝑥, 𝑦) = (1 + 𝑟)(1 − 𝑥 − 𝑥 ) + 𝑥
1 + 𝑑

𝐾
𝑦 + 𝑥

1 + 𝑑

𝐾
𝑦 .  (2.1) 

It is desirable to maximize this indicator by choosing a strategy 𝑥 = (𝑥 , 𝑥 ) ∈ 𝑋 =
{𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)} and the DM should take into account the possibility of realizing 
any uncertainty 𝑦 = (𝑦 , 𝑦 ) ∈ 𝑌 = [𝑎 , 𝑏 ] × [𝑎 , 𝑏 ]. 

Thus, the mathematical model of the diversification problem is represented by an ordered 
triplet 𝛤 =< 𝑋, 𝑌, 𝑓(𝑥, 𝑦) > where 𝑓(𝑥, 𝑦) is the income function of the depositor (DM), and the 
set X of DM's strategies x and the set Y of uncertainties y have the  form: 

𝑋 = {𝑥 = (𝑥 , 𝑥 )|𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)}, (2.2) 

𝑌 = {𝑦 = (𝑦 , 𝑦 )| 𝑦 ∈ [𝑎 , 𝑏 ] (𝑖 = 𝑑, 𝑒)}.  (2.3) 

The problem 𝛤 is a single-criteria decision-making problem under uncertainty: maximize a 
linear function of 𝑥 with uncertain coefficients on a polyhedron (triangle) 𝑋 given the uncertain 
factors. The presence of uncertainty leads to the concept of risk as a possibility of the deviation of 
some results from their desired or expected values. 

In the framework of Wald's maximin approach [13], this problem can be considered as an 
antagonistic game of DM against "Nature". Its solution is a "cautious" maximin strategy obtained 
according to the principle of the best-guaranteed result. Note that in the considered problem it has 
a rather trivial form: it is necessary to calculate the profitability of each type of currency at the 
most unfavorable possible combination of final exchange rates 𝑦 = (𝑎 , 𝑎 ) and invest all funds 
in the currency for which this indicator is maximized. If the maximum is reached on several 
currencies, the whole ruble amount is distributed among them in an arbitrary way. This result is 
true for any number of currencies [14]. 

As we know, the Wald criterion is very pessimistic and focuses on the worst case. It does not 
allow for risk and often gives extremely underestimated results. On the other hand, many DMs 
accept risk and need tools to minimize it. One of these is the principle of minimax regret (Savage's 
criterion). It also uses the notion of the best-guaranteed outcome, but applies not to the original 
quality indicator 𝑓(𝑥, 𝑦) but to the so-called risk (regret) function. 

3. SAVAGE'S PRINCIPLE OF MINIMAX REGRET 

Let 𝑓(𝑥, 𝑦) be  the income, 𝑋 be  the set of DM strategies, and 𝑌 be the set of uncertainties. Then 
𝑓 (𝑧, 𝑦) is the best possible income if there is an uncertainty 𝑦. However, the DM does not know 
in advance, what the value 𝑦 of the uncertain parameter will be. The difference 

Φ(𝑥, 𝑦) = max
∈

𝑓 (𝑧, 𝑦) − 𝑓(𝑥, 𝑦) (3.1) 

is called the Savage risk (regret) function. It represents the loss due to uncertainty which is  the 
difference between the best outcome that could have been obtained under known y and the actual 
outcome of some strategy 𝑥. This risk depends on both strategy x and uncertainty y. 

To minimize risk, the DM can use the concept of the best-guaranteed outcome (Wald's 
principle [13]). 

These considerations lead to the following definition (Savage's minimax regret principle [10]). 



105  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

Definition 3.1: 

Let us call the solution 𝑥  the  risk-guaranteed solution (RGS) of the problem 𝛤, if 

𝛷 = 𝑚𝑎𝑥
∈

𝛷(𝑥 , 𝑦) = 𝑚𝑖𝑛
∈

𝑚𝑎𝑥
∈

𝛷(𝑥, 𝑦),  (3.2) 

where the risk (loss) function 𝛷(𝑥, 𝑦) is defined in (3.1). 
For the sake of brevity, we also refer to the decision 𝑥  and risk  Φ  as the optimal decision 

and the optimal risk, respectively. 
According to  Definition 3.1, the construction of a RGS contains four steps.  
Step 1: Calculate 𝑓[𝑦] = max

∈
𝑓 (𝑧, 𝑦) for each uncertainty 𝑦 ∈ 𝑌. 

Step 2: Calculate the risk function 𝑓[𝑦] − 𝑓(𝑥, 𝑦).  
Step 3. Compute the internal maximum in (3.2) which is  the guaranteed risk  
max

∈
Φ(𝑥, 𝑦) = max

∈
(𝑓([𝑦]  − 𝑓(𝑥, 𝑦)) = Φ[𝑥] ≥ Φ(𝑥, 𝑦) for each strategy 𝑥 ∈ 𝑋. 

Step 4. Compute the external minimum in (3.2) which is  the best-guaranteed risk  Φ =
min

∈
Φ[𝑥] = Φ[𝑥 ].  

4. CONSTRUCTING A RISK-GUARANTEED SOLUTION 

The explicit form of the risk function for the diversification problem of a three-currency deposit 
was obtained in [8]: 

Φ(𝑥, 𝑦) = 𝑓[𝑦] − 𝑓(𝑥, 𝑦) = 

=

⎩
⎪⎪
⎨

⎪⎪
⎧

Φ (𝑥, 𝑦) = (1 + 𝑟) − 𝑓(𝑥, 𝑦) =

[(1 + 𝑟) − 𝜉 ]𝑥 + [(1 + 𝑟) − 𝜉 𝑦 ]𝑥 , 𝑦 ∈ 𝑌 ,

Φ (𝑥, 𝑦) = 𝜉 𝑦 − 𝑓(𝑥, 𝑦) =

(1 − 𝑥 )[𝜉 𝑦 − (1 + 𝑟)] + [1 + 𝑟 − 𝜉 𝑦 ]𝑥 , 𝑦 ∈ 𝑌 ,

Φ (𝑥, 𝑦) = 𝜉 𝑦 − 𝑓(𝑥, 𝑦) =

[1 + 𝑟 − 𝜉 𝑦 ]𝑥 + [𝜉 𝑦 − (1 + 𝑟)](1 − 𝑥 ), 𝑦 ∈ 𝑌  ,

 (4.1) 

where, for the sake of brevity, we used the notations 𝜉 =  , 𝜉 = . 

The sets 𝑌 , 𝑌 , 𝑌  are subsets of the set 𝑌 defined by additional linear inequations: 
𝑌 = {𝑦 ∈ 𝑌, 1 + 𝑟 ≥ 𝜉 𝑦 , 1 + 𝑟 ≥ 𝜉 𝑦 },  (4.2) 
𝑌 = {𝑦 ∈ 𝑌, 𝜉 𝑦 ≥ 1 + 𝑟, 𝜉 𝑦 ≥ 𝜉 𝑦 },  (4.3) 
𝑌 = {𝑦 ∈ 𝑌, 𝜉 𝑦 ≥ 1 + 𝑟, 𝜉 𝑦 ≥ 𝜉 𝑦 }.  (4.4) 

It is obvious that 𝑌 ∪ 𝑌 ∪ 𝑌 = 𝑌. Functions Φ (𝑥, 𝑦) (𝑖 = 1,2,3) are bilinear functions of 
the variables 𝑥 and  𝑦. Therefore, the risk function Φ(𝑥, 𝑦) for any fixed strategy 𝑥 is a piecewise 
linear function of uncertainty 𝑦. The sets 𝑌 , 𝑌 , 𝑌  are regions (polygons) of linearity on y of the 
risk function, i.e. the risk function Φ(𝑥, 𝑦) coincides with the linear (by x) function Φ (𝑥, 𝑦), 𝑦 ∈
𝑌 . In this case, it is said that the risk function Φ(𝑥, 𝑦) is defined at the point (𝑥, 𝑦) by the 
function Φ (𝑥, 𝑦). 

For each strategy 𝑥 ∈ 𝑋 the guaranteed risk Φ[𝑥] was calculated [8]: 
Φ[𝑥] = max{𝛼 𝑥 +𝛼 𝑥  , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 , 𝛽 (1 − 𝑥 ) + 𝛼 𝑥 }, 𝑥 ∈ 𝑋, (4.5) 
where the following notations are used:  

𝛼 = [(1 + 𝑟) − 𝜉 𝑎 ],  𝛼 = [(1 + 𝑟) − 𝜉 𝑎 ], (4.6) 
𝛽 = [𝜉 𝑏 − (1 + 𝑟)],   𝛽 =  [𝜉 𝑏 − (1 + 𝑟)]. (4.7) 

The secondary (derivative) coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽  depend on the above-mentioned initial 
parameters. The substantive interpretation for them is given below (see Section 6).  

The best (smallest) guaranteed risk Φ = min
∈

Φ[𝑥] = Φ[𝑥 ] is computed in [8] by considering 

the problem min
∈

Φ[𝑥] separately on the boundary and on the interior of the set of the depositor's 

strategies 𝑋 (triangle A(1;0)O(0;0)B(0;1)) followed by the selection of the best solution: 



V.S.MOLOSTVOV                        106 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

min
∈

Φ[𝑥] = min { min
∈[ , ]

Φ[𝑥], min
∈[ , ]

Φ[𝑥], min
∈[ , ]

Φ[𝑥], min
∈

Φ[𝑥]}.  #(4.8) 

The results  obtained can be restated in economic terms in the following form. 
Proposition 4.1: 
If the optimal guaranteed risk is achieved by a combination of dollar and ruble components, 

then the solution of the form 
𝑥 = (𝑥 , 0), 0 <  𝑥 < 1) 

wher 𝑥 =
, 𝑖𝑓     𝛽 ≥ 0,

, 𝑖𝑓    𝛽 ≤ 0,
 is optimal.  

The last two formulas can be combined into a single expression: 

𝑥 =
𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2

𝛼 + 𝛽
. 

Proposition 4.2: 

If the optimal guaranteed risk is achieved by a combination of euros and rubles, the solution 
of the form 

𝑥 = (0, 𝑥 ), 0 <  𝑥 < 1  

where  𝑥 =  
, 𝑖𝑓  𝛽 ≥ 0,

, 𝑖𝑓 𝛽 ≤ 0,
is optimal.  

Here also both formulas can be combined into  a single expression: 

 𝑥 =
𝛽 − 𝛽 (𝑠𝑖𝑔𝑛(𝛽 ) + 1)/2

𝛼 + 𝛽
. 

Proposition 4.3: 

If the optimal guaranteed risk is achieved by a combination of dollars and euros, then some 
solution from the set: 

{𝑥 = (𝑥 , 1 − 𝑥 ), 𝑥 = (𝑥 , 1 − 𝑥 ),  𝑥 = (𝑥 , 1 −  𝑥 )} ∩ 𝑋 

where 𝑥 = ,  𝑥 = ,  𝑥 =   is optimal. 

We speak about full diversification if the guaranteed risk function Φ[𝑥] reaches the minimum 
value Φ∗ on the set 𝑋 at some interior point 𝑋 and for all points of the boundary 𝑓𝑟𝑋 Φ[𝑥] > Φ∗. 

This condition means that the optimal diversification necessarily uses all three currencies. In 
the following, we restrict ourselves to this condition, since other possible situations are covered 
by Propositions 4.1–4.3. 

Proposition 4.4: 

If the optimal guaranteed risk is achieved only when all three currencies are used, then the 
optimal solution is the following: 

𝑥∗ = (𝑥∗ , 𝑥∗) = (
𝛽

𝛼 + 𝛽
,

𝛽

𝛼 + 𝛽
) 

provided that 0 < 𝑥 , 𝑥 < 1 and 𝑥 + 𝑥 < 1. 
Any point of a local minimum of a convex function on a convex set is simultaneously a point 

of global (on this set) minimum. Hence, if the point of internal minimum 𝑥∗ exists and 𝑥∗ =
(𝑥∗ , 𝑥∗) ∈ 𝑖𝑛𝑡𝑋, it will be the RGS. Whether it exists or not depends on the relationship between 
the parameters of the problem under consideration. Very often, the RGS is on the boundary of the 
set of admissible solutions. This means that the optimal diversification contains only two (or even 
one) of the three currencies. 

The list of all possible candidates for  the optimal solution specified in Propositions 4.1–4.4 
should be accomplished by three single-currency variants: (0;0) – only rubles, (1;0) – only dollars, 



107  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

(0;1) – only euros. As a result, we obtain an exhaustive set of nine solutions, at least one of which 
will be optimal. 

5. ALGORITHM FOR CONSTRUCTING A RISK-GUARANTEED SOLUTION 

Combining the results of Propositions 4.1–4.4 leads to the following scheme [8] for computing the 
risk-guaranteed diversification strategy and the smallest guaranteed risk. 

Step 1. Set numerical values of the interest rates 𝑟, 𝑑 , 𝑑  and the current exchange rates 𝐾 , 𝐾 ; 
set numerical values of the boundaries of possible changes in dollar and euro exchange rates 𝑎 , 𝑏  
and  𝑎 , 𝑏  , respectively. Any available information can be used, e.g. expert  estimates and 
forecasts, DM's own assumptions, etc. 

Step 2. Calculate the secondary parameters 𝛼 , 𝛼 , 𝛽 , 𝛽  from (4.6-4.7). 
Step 3. Compute the nine candidate points for the optimal solution. The corresponding 

formulas for these points are presented in Table 1 below. 
Step 4: Remove points that do not belong to the set 𝑋. 
Step 5: Calculate the values of the guaranteed risk for  the remaining points, select the solution 

with the best (minimum) guaranteed risk. 

Table 1. Candidates for solutions and verification of their admissibility conditions [8] 

N Candidates for solutions  Formulas 
Checking the 

conditions 
1 The point O=(0;0)  (rubles) (0;0) – 
2 The point A=(1;0) (dollars) (1;0) – 
3 The point   B=(0;1)   (euro) (0;1) – 

4 
The point (𝑥 , 0) 
(rubles and dollars) 

𝑥 =
𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2

𝛼 + 𝛽
 0 < 𝑥 < 1 

5 
The point (0, 𝑥 ) 
(rubles and euro) 

𝑥 =
𝛽 − 𝛽 (𝑠𝑖𝑔𝑛𝛽 + 1)/2

𝛼 + 𝛽
 0 < 𝑥 < 1 

6 
The point (1 − 𝑥 , 𝑥 ) 

(dollars and euro) 
𝑥 =

𝛼

𝛼 + 𝛽
 0 < 𝑥 < 1 

7 
The point (1 − 𝑥 , 𝑥 ) 

(dollars and euro) 
 𝑥 =

𝛽

𝛼 + 𝛽
 0 < 𝑥 < 1 

8 
The point (1 − 𝑥 , 𝑥 ) 

(dollars and euro) 
 𝑥 =

𝛼 + 𝛽

𝛼 + 𝛼 + 𝛽 + 𝛽
 0 < 𝑥 < 1 

9 
The internal point in X 

(rubles, dollars and euro) 

(𝑥∗ , 𝑥∗) = 

= (
𝛽

𝛼 + 𝛽
,

𝛽

𝛼 + 𝛽
) 

0 < 𝑥 , 𝑥 < 1 
𝑥 + 𝑥 < 1 

6. ECONOMIC AND GAME INTERPRETATION 

Let us explain the economic meaning of the derived parameters of the problem. Recall that 
𝑟, 𝑑 , 𝑑  are the interest rates on the corresponding deposits (ruble, dollar, euro); 𝐾  and  𝐾  are 
dollar and euro exchange rates against the ruble, respectively, at which the currency is purchased 
at the beginning of the year; 𝑎 , 𝑎  are the  lower and 𝑏 , 𝑏  are the  upper limits of the dollar and 
euro exchange rates, respectively, at the end of the deposit period. These parameters are assumed 
to be known to the DM at the beginning of the year. However, the values of uncertain parameters, 
i.e. the final values of the dollar and euro exchange rates are known only to the precision of the 
sets: 𝑦 ∈ [𝑎 , 𝑏 ],  𝑦 ∈ [𝑎 , 𝑏 ]. Income, measured in rubles, for a  one ruble allocation (1 −
𝑥 − 𝑥 , 𝑥 , 𝑥 ) is determined by the linear function (2.1) with uncertain coefficients.  



V.S.MOLOSTVOV                        108 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

The secondary coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽  depend on the above initial parameters and have the 

following meaningful interpretations. The expression 𝛼 = (1 + 𝑟) − 𝑎  defines the 

difference between the yield (1 + 𝑟) of one ruble in a ruble deposit and the yield when using it to  
buy  dollars at the beginning of the period at the rate of 𝐾  ruble/dollar and selling the total amount 

(1 + 𝑑 ) at the minimum rate 𝑎  at the end of the period. The parameter  𝛼 = (1 + 𝑟) − 𝑎   

defines the difference between the yields of the ruble and the euro if  the uncertain euro exchange 
rate takes the minimum possible value. 

Coefficient 𝛽  (𝛽 ) sets the difference between the yield of one ruble invested in a dollar (euro) 
deposit and the yield of a ruble in a ruble deposit at the highest final dollar (euro) exchange rate. 

As noted above, the problem of the best-guaranteed income can be regarded as an antagonistic 
game between DM and Nature. This game has, at the given uncertainty structure, a rather simple 
solution even in the case of any finite number of currencies [14]. Savage's minimax regret principle 
is related to the consideration of a more complex structure, the target risk function (3.1), to which 
Wald's minimax principle is then applied. Thus, the problem of computing optimal risk can also 
be interpreted as an antagonistic game between DM and Nature-uncertainty with the risk function 
as the payoff function. However, unlike the aforementioned DM-Nature game with income as a 
payoff function, this game does not typically have a saddle point [12] in pure strategies. Indeed, 
under the assumptions made, the maximin and minimax in this game exist but are not equal. The 
maximin is equal to zero, which  is the optimal value of the risk function under the uncertainty 
known to the DM in advance, i.e. actually without uncertainty. The minimax for a non-negative 
risk function is non-negative and equals zero only in degenerate cases, when there exists a strategy 
that is optimal under any uncertainty. Hence the absence of a saddle point [12]. 

In this paper, the DM's guaranteed strategy was found by direct computing ("by definition") 
the minimax of the risk function. 

In its original formulation, the problem under consideration is close to Markowitz's problem 
of optimal portfolio structure [3], first formulated  in 1952 and since  developed in numerous 
publications. Despite such a long history, it is still the case that "many investment efficiency 
indicators are based on Markowitz's portfolio theory" [1]. 

In Markowitz’s theory, the returns of individual portfolio assets are random variables with 
respect to which mathematical expectations and covariance matrices are assumed to be known [4]. 
The expected return of the whole portfolio and its variance are considered as two indicators to be 
optimized – return and risk. From a formal point of view, this is a two-criterion optimization 
problem [9] with linear (return) and quadratic (risk as variance of return) criteria and simple linear 
constraints. Various methods of finding Pareto-optimal solutions and ways  to  construct  efficient 
portfolios have been developed [11]. Most of them use techniques of  parameterization, for 
example, optimization by one criterion while restricting the values of another criterion, to construct 
the Pareto frontier.  

The subject of this paper is close to the topic of optimizing a portfolio with three assets, two of 
which are risky. However, there are two important differences – in the available information and 
in risk assessment. For a number of reasons, it is not always possible to obtain satisfactory 
estimates of the statistical characteristics of asset returns and one has to be satisfied with interval 
forecasts. The use of dispersion as a measure of risk is computationally convenient, but has  well-
known disadvantages. Among them is the need for information on the correlation of assets. The 
use of an alternative risk measure – Savage risk – develops additional directions of portfolio theory 
and practice. 

7. ANALYTICAL SOLUTION IN THE SPECIAL CASE 

Consider the special case when 𝛼 = 𝛼 = 0. This means that at minimum final values of dollar 
and euro exchange rates all three currencies give equal returns. Then, from the point of view of 
the Wald principle, any admissible diversification plan gives the same (and, therefore, optimal) 



109  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

result [14]. This is explained by the fact that this principle takes into account only the worst 
possible value 𝑦 = (𝑦 , 𝑦 ) ∈ 𝑌 for each strategy 𝑥 = (𝑥 , 𝑥 ) ∈ 𝑋. Thus, in our case, obviously, 
for all 𝑥 will be  𝑦 = (𝑎 , 𝑎 ). In contrast to Wald's principle, the Savage optimal solution is found 
taking into account both the lower and upper bounds of uncertainty 𝑏 , 𝑏  and the coefficients 
𝛽 , 𝛽  depending on them.  

The guaranteed risk function (4.5) takes the form 
Φ[𝑥] = max{0,  𝛽 (1 − 𝑥 ),  𝛽 (1 − 𝑥 )}, 𝑥 ∈ 𝑋  (7.1) 

From the condition 𝛼 = 𝛼 = 0 and the definition of the coefficients 𝛼 , 𝛼 , 𝛽 , 𝛽  it follows 
that 𝛽 > 0, 𝛽 > 0. In addition, (1 − 𝑥 ) ≥ 0, (1 − 𝑥 ) ≥ 0 for 𝑥 ∈ 𝑋. Therefore, 

Φ[𝑥] = max{ 𝛽 (1 − 𝑥 ), 𝛽 (1 − 𝑥 )}, 𝑥 ∈ 𝑋  (7.2) 
Recall that the set of admissible solutions has the form: 

𝑋 = {𝑥 = (𝑥 , 𝑥 )|𝑥 + 𝑥 ≤ 1, 𝑥 ≥ 0 (𝑖 = 𝑑, 𝑒)} (7.3) 
Obviously, a solution 𝑥 = (𝑥 , 𝑥 ) such that 𝑥 + 𝑥 < 1, cannot be the minimum point  of 

the function (7.2), since a small increase in both variables will result in a decrease in the function. 
Therefore, the point of minimum of the function Φ[𝑥] on the set 𝑋 lies on the segment [A,B] =
{𝑥 + 𝑥 = 1, 𝑥 ≥ 0, 𝑥 ≥ 0}, where 𝑥 = (1 − 𝑥 , 𝑥 ). In the particular case 𝛼 = 𝛼 = 0 taking 
into account the conditions 𝛽 > 0, 𝛽 > 0 we obtain from  4.3  as a corollary: 

Proposition 7.1: 

If 𝛼 = 𝛼 = 0 then the guaranteed risk function 𝛷[𝑥] reaches the minimum value on the set 
𝑋 at some point in the set {(1; 0), (0; 1), 𝑥 = (𝛽 /(𝛽 + 𝛽 ), 𝛽 /(𝛽 + 𝛽 )}. 

In the derivation of the corollary, it is taken into account that some points from proposition 4.3 
are duplicated here and that condition 𝑥 ∈ 𝑋 is fulfilled automatically in this case.  

For the final choice of the solution, we compute the value of the guaranteed risk function at the 
candidate points: Φ[(1,0)] = max{ 𝛽 (1 − 1), 𝛽 (1 − 0)} = 𝛽  , Φ[(0,1)] = 𝛽  , 
Φ[(𝑥 , 𝑥 )] = max{ 𝛽 𝛽 /(𝛽 + 𝛽 ), 𝛽 𝛽 /(𝛽 + 𝛽 ) } =  𝛽 𝛽 /(𝛽 + 𝛽 ). 

The last value is smaller than the values at the other two points. Hence, the RGS in  this  case  
is unique and has the form: 

𝑥 =
𝛽

𝛽 + 𝛽
,

𝛽

𝛽 + 𝛽
(7.4) 

It gives the minimum guaranteed risk Φ = Φ[𝑥 ] =  𝛽 𝛽 /(𝛽 + 𝛽 ). 
In the case of equal ruble profitability of  all three currencies (at minimum values of the final 

exchange rates), the optimal solution in terms of guaranteed risk is to distribute the ruble between 
the dollar and the euro in proportion to the upper possible values of their final exchange rates. 
If 𝛽 = 𝛽 = 𝛽, the optimal solution is 𝑥 = (0.5; 0.5).  

In the general case, the analytical search for the optimal solution of the diversification problem 
is difficult and requires the use of numerical methods and algorithms. 

8. COMPUTATIONAL TOOLS AND SOME RESULTS 

The suggested method is implemented in the Excel 2016 environment in two variants – user and 
research . The first  is designed for a user with minimal skills of working in a ready-made Excel 
spreadsheet. The work is reduced to the input of the above-mentioned initial data and the  analysis 
of the results. The corresponding spreadsheet is  consistent with Table 1 and contains the 
calculation formulas for the points, of which at least one is the desired solution.   

Table 2 reproduces part of the  spreadsheet to  input data.  

 

Table 2. Input of initial data. Input cells are highlighted in green 



V.S.MOLOSTVOV                        110 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

 
Table 3 shows a part of the Excel spreadsheet with calculations corresponding to the given 

initial parameters from Table 2. Column B contains the text of the formulas from  Table 1, columns 
C and D contain the corresponding calculations. The results are checked by logical operators for 
the fulfillment of the condition 𝑥 ∈ 𝑋 (descriptions of condition are in the column J). If the 
condition is not fulfilled, the text "NOT CANDIDATE" is written in the cell of column E, 
otherwise the result of the calculation of the guaranteed risk function at the corresponding point is 
given. Then the minimum of these results is calculated. The corresponding cell (or cells in case of 
non-singularity) is highlighted by means of conditional formatting, and the optimal solution is 
located to the left of it. According to the values of the target function at  the other acceptable 
candidate points, it is possible to estimate the increase in risk when reducing the number of 
currencies used.  

Table 3. Calculating the optimal solution 

 
The research variant contains a number of additional features. For example, instead of using 

the initial data, it is possible to directly specify the secondary   parameters 𝛼 , 𝛼 , 𝛽 , 𝛽  which also 
fully determine the solution. 

However, the direct assignment of the derived parameters cannot be arbitrary, since their values 
must be consistent with the original parameters (see formulas (4.6–4.7)). For example, as noted 
above, if 𝛼 = 𝛼 = 0, then the parameters 𝛽 , 𝛽  cannot be negative.  

Table 4 corresponds to the case considered above: 𝛼 = 𝛼 = 0 at 𝛽 = 0,2, 𝛽 = 0,3.  



111  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

Table 4. The case of equal minimal currency returns: 𝛼 = 𝛼 = 0; 𝛽 = 0,2, 𝛽 = 0,3 

 
The software implementation of the algorithm provides means for multivariate calculations, in 

which one or two parameters are varied. For example, by fixing the parameters 𝛼 , 𝛼 , it is possible 
to build a table showing the dependence of the optimal result on the varied values 𝛽 , 𝛽 . 
Alternatively, it is possible to study the  optimal risk dependence on the lower  bounds of rates 
𝛼 , 𝛼  when the values 𝛽 , 𝛽  are fixed . On the other hand, we can construct a table of the 
dependence of the optimal risk on the values of dollar and euro interest rates. There are also other  
experiments, for example, studying the dependence of the optimal deposit structure on the interest 
rate of one of the currencies.  

Table 5 shows the results of multivariate calculations of the optimal risk at all combinations  
of upper bounds of rates 𝑏   from 97 to 117 and  𝑏  from 108 to 128 respectively with step 1. Other 
parameters are fixed at the values from Table 2. 

Table 5. Dependence of optimal risk on upper limits of exchange rates 𝑏 , 𝑏 . 
In the upper line are dollar exchange rate values, in the left column – euro 

 
It is convenient to analyze the obtained data using Excel graphical tools. Thus, Fig. 1 shows 

graphs of the first 10 rows of Table 5. For the remaining eleven values of 𝑏  the graphs are similar. 
The graphs for the columns of the table also look similar. As one would expect, the optimal risk is 
an increasing nonlinear function for each parameter 𝑏 , 𝑏 . 

0,075541667 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
108 0,000 0,002 0,012 0,021 0,029 0,036 0,042 0,048 0,053 0,057 0,061 0,065 0,068 0,071 0,074 0,077 0,080 0,082 0,084 0,086 0,088

109 0,003 0,003 0,012 0,021 0,029 0,036 0,043 0,048 0,053 0,058 0,062 0,066 0,070 0,073 0,076 0,078 0,081 0,083 0,086 0,088 0,090

110 0,012 0,012 0,013 0,023 0,031 0,039 0,046 0,052 0,057 0,062 0,067 0,071 0,074 0,078 0,081 0,084 0,087 0,089 0,092 0,094 0,096

111 0,020 0,020 0,022 0,024 0,033 0,041 0,049 0,055 0,061 0,066 0,071 0,075 0,079 0,083 0,086 0,089 0,092 0,095 0,097 0,100 0,102

112 0,027 0,027 0,030 0,032 0,035 0,044 0,052 0,058 0,065 0,070 0,075 0,080 0,084 0,088 0,091 0,095 0,098 0,101 0,103 0,106 0,108

113 0,034 0,034 0,037 0,039 0,042 0,046 0,055 0,062 0,068 0,074 0,079 0,084 0,089 0,093 0,097 0,100 0,103 0,106 0,109 0,112 0,114

114 0,039 0,040 0,043 0,046 0,049 0,052 0,057 0,065 0,072 0,078 0,084 0,089 0,093 0,098 0,102 0,105 0,109 0,112 0,115 0,118 0,120

115 0,044 0,045 0,048 0,052 0,056 0,059 0,063 0,068 0,076 0,082 0,088 0,093 0,098 0,103 0,107 0,111 0,114 0,118 0,121 0,124 0,127

116 0,049 0,049 0,053 0,057 0,061 0,065 0,069 0,073 0,079 0,086 0,092 0,098 0,103 0,108 0,112 0,116 0,120 0,124 0,127 0,130 0,133

117 0,053 0,054 0,058 0,062 0,067 0,071 0,075 0,080 0,084 0,090 0,096 0,102 0,108 0,113 0,117 0,122 0,126 0,129 0,133 0,136 0,139

118 0,057 0,058 0,062 0,067 0,071 0,076 0,081 0,085 0,090 0,094 0,101 0,107 0,113 0,118 0,123 0,127 0,131 0,135 0,139 0,142 0,145

119 0,061 0,061 0,066 0,071 0,076 0,081 0,086 0,091 0,095 0,100 0,105 0,111 0,117 0,123 0,128 0,132 0,137 0,141 0,144 0,147 0,149

120 0,064 0,064 0,070 0,075 0,080 0,085 0,090 0,095 0,101 0,106 0,111 0,116 0,122 0,128 0,133 0,138 0,142 0,146 0,148 0,150 0,152

121 0,067 0,067 0,073 0,078 0,084 0,089 0,094 0,100 0,105 0,111 0,116 0,121 0,127 0,133 0,138 0,143 0,146 0,149 0,151 0,153 0,156

122 0,070 0,070 0,076 0,082 0,087 0,093 0,098 0,104 0,110 0,115 0,121 0,126 0,132 0,138 0,143 0,147 0,149 0,152 0,154 0,157 0,159

123 0,072 0,073 0,079 0,085 0,090 0,096 0,102 0,108 0,114 0,120 0,125 0,131 0,137 0,143 0,146 0,149 0,152 0,155 0,157 0,160 0,162

124 0,074 0,075 0,081 0,087 0,093 0,099 0,105 0,111 0,117 0,123 0,129 0,136 0,142 0,146 0,149 0,152 0,155 0,157 0,160 0,163 0,165

125 0,077 0,078 0,084 0,090 0,096 0,102 0,109 0,115 0,121 0,127 0,133 0,140 0,145 0,148 0,151 0,154 0,157 0,160 0,163 0,166 0,168

126 0,079 0,080 0,086 0,092 0,099 0,105 0,112 0,118 0,124 0,131 0,137 0,143 0,147 0,151 0,154 0,157 0,160 0,163 0,166 0,168 0,171

127 0,081 0,082 0,088 0,095 0,101 0,108 0,114 0,121 0,127 0,134 0,140 0,146 0,149 0,153 0,156 0,159 0,162 0,165 0,168 0,171 0,174

128 0,083 0,084 0,090 0,097 0,104 0,110 0,117 0,124 0,130 0,137 0,144 0,148 0,151 0,155 0,158 0,162 0,165 0,168 0,171 0,174 0,176



V.S.MOLOSTVOV                        112 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

 
Fig. 1. Graphs of the dependence of the optimal risk on the parameter 𝑏   

at different values of the parameter 𝑏 . Row 1: 𝑏 = 108, …, Row 10: 𝑏 = 117 

Table 6 shows the optimal deposit structures when the interest rate for the ruble changes from 
15% to 6% with  steps of -0.5%. Other parameters are fixed at the following values: dollar rate –
3%, euro rate – 3%, initial exchange rates – 90 rubles/dollar and 100 rubles/euro, dollar exchange 
rate limits – from 85 to 105, euro – from 95 to 115. The last line shows the corresponding values 
of the optimal guaranteed risk. As can be seen, it decreases monotonically with the growth of the 
ruble interest rate. Fig. 2 gives a visual representation of the dynamics. 

Table 6. Dependence of the deposit structure on the ruble interest rate 

 
 

 
Fig 2. Optimal deposit structure 

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

9 7 9 8 9 9 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 9 1 1 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7

M
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d 

ris
k

Upper value for the dollar exchange rate

Dependence of optimal risk min𝛷[𝑥] on bd

at different values of be

Row1

Row2

Row3

Row4

Row5

Row6

Row7

Row8

Row9

Row10

Ruble Rate 15,0% 14,5% 14,0% 13,5% 13,0% 12,5% 12,0% 11,5% 11,0% 10,5% 10,0% 9,5% 9,0% 8,5% 8,0% 7,5% 7,0% 6,5% 6,0%
Dollar 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,075 0,513 0,513 0,513 0,513 0,513
Euro 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,487 0,487 0,487 0,487 0,487
Ruble 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,925 0,000 0,000 0,000 0,000 0,000
Risk 0,048 0,052 0,057 0,062 0,066 0,071 0,076 0,080 0,085 0,089 0,094 0,099 0,103 0,108 0,109 0,109 0,109 0,109 0,109

 0.000

 0.200

 0.400

 0.600

 0.800

 1.000

 1.200
Deposit structure

Dollar Euro Ruble



113  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

Table 7 shows the results of calculations of the minimum guaranteed risk for dollar and euro 
interest rates in the range from 0,25% to 5% with steps of 0.25%. Other parameters are fixed at 
the following values: initial exchange rates – 90 rubles/dollar and 100 rubles/euro, dollar exchange 
rate limits – from 85 to 105, euro – from 95 to 115; ruble interest rate is assumed to be 12%. 

Table 7. Optimal guaranteed risk at different dollar (top row) and euro (left column) interest rates 

 
Fig. 3 shows graphs of the dependence of the optimal risk on the dollar interest rate 𝑑  at the 

first 10 fixed values of the interest rate 𝑑  for euro (the first 10 rows of Table 7). For the remaining 
11 values of 𝑑   the graphs are similar. The graphs for the columns of the table also look similar.  

 
Fig 3. Graphs of the dependence of the optimal risk on the interest rate 𝑑  for dollar at different values 

of the interest rate 𝑑  for the euro. Row 1: d = 0,25%, …, Row 10: d = 2,5% 

 

0,0755 0,25% 0,50% 0,75% 1,00% 1,25% 1,50% 1,75% 2,00% 2,25% 2,50% 2,75% 3,00% 3,25% 3,50% 3,75% 4,00% 4,25% 4,50% 4,75% 5,00%
0,25% 0,046 0,048 0,050 0,052 0,054 0,055 0,057 0,059 0,060 0,062 0,063 0,064 0,066 0,067 0,068 0,069 0,070 0,071 0,072 0,073

0,50% 0,047 0,049 0,051 0,052 0,054 0,056 0,058 0,059 0,061 0,062 0,064 0,065 0,067 0,068 0,069 0,070 0,071 0,072 0,073 0,074
0,75% 0,047 0,049 0,051 0,053 0,055 0,057 0,059 0,060 0,062 0,063 0,065 0,066 0,068 0,069 0,070 0,071 0,072 0,073 0,074 0,075
1,00% 0,048 0,050 0,052 0,054 0,056 0,058 0,059 0,061 0,063 0,064 0,066 0,067 0,069 0,070 0,071 0,072 0,074 0,075 0,076 0,076
1,25% 0,048 0,051 0,053 0,055 0,057 0,059 0,060 0,062 0,064 0,065 0,067 0,068 0,070 0,071 0,072 0,074 0,075 0,076 0,077 0,078
1,50% 0,049 0,051 0,053 0,055 0,057 0,059 0,061 0,063 0,065 0,066 0,068 0,069 0,071 0,072 0,073 0,075 0,076 0,077 0,078 0,079
1,75% 0,050 0,052 0,054 0,056 0,058 0,060 0,062 0,064 0,066 0,067 0,069 0,070 0,072 0,073 0,075 0,076 0,077 0,078 0,079 0,080
2,00% 0,052 0,053 0,055 0,057 0,059 0,061 0,063 0,065 0,067 0,068 0,070 0,071 0,073 0,074 0,076 0,077 0,078 0,079 0,081 0,082
2,25% 0,054 0,055 0,056 0,058 0,060 0,062 0,064 0,066 0,067 0,069 0,071 0,072 0,074 0,075 0,077 0,078 0,079 0,081 0,082 0,083
2,50% 0,056 0,057 0,058 0,059 0,061 0,063 0,065 0,067 0,068 0,070 0,072 0,073 0,075 0,077 0,078 0,079 0,081 0,082 0,083 0,084
2,75% 0,058 0,059 0,060 0,061 0,062 0,063 0,065 0,067 0,069 0,071 0,073 0,075 0,076 0,078 0,079 0,081 0,082 0,083 0,084 0,085
3,00% 0,060 0,061 0,062 0,063 0,063 0,064 0,066 0,068 0,070 0,072 0,074 0,076 0,077 0,079 0,080 0,082 0,083 0,084 0,086 0,087
3,25% 0,062 0,063 0,063 0,064 0,065 0,066 0,067 0,069 0,071 0,073 0,075 0,077 0,078 0,080 0,081 0,083 0,084 0,086 0,087 0,088
3,50% 0,063 0,064 0,065 0,066 0,067 0,068 0,069 0,070 0,072 0,074 0,076 0,078 0,079 0,081 0,083 0,084 0,085 0,087 0,088 0,089
3,75% 0,065 0,066 0,067 0,068 0,069 0,070 0,071 0,072 0,073 0,075 0,077 0,079 0,080 0,082 0,084 0,085 0,087 0,088 0,089 0,091
4,00% 0,066 0,067 0,068 0,070 0,071 0,072 0,073 0,074 0,075 0,076 0,078 0,080 0,081 0,083 0,085 0,086 0,088 0,089 0,091 0,092
4,25% 0,068 0,069 0,070 0,071 0,072 0,073 0,074 0,076 0,077 0,078 0,079 0,081 0,082 0,084 0,086 0,087 0,089 0,091 0,092 0,093
4,50% 0,069 0,070 0,071 0,073 0,074 0,075 0,076 0,077 0,078 0,079 0,081 0,082 0,084 0,085 0,087 0,089 0,090 0,092 0,093 0,095
4,75% 0,070 0,072 0,073 0,074 0,075 0,076 0,078 0,079 0,080 0,081 0,082 0,083 0,085 0,086 0,088 0,090 0,091 0,093 0,094 0,096
5,00% 0,072 0,073 0,074 0,075 0,077 0,078 0,079 0,080 0,081 0,083 0,084 0,085 0,086 0,087 0,089 0,091 0,093 0,094 0,096 0,097

  0.040

  0.045

  0.050

  0.055

  0.060

  0.065

  0.070

  0.075

  0.080

  0.085

  0.090

0 . 0 0 % 1 . 0 0 % 2 . 0 0 % 3 . 0 0 % 4 . 0 0 % 5 . 0 0 % 6 . 0 0 %

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Dollar interest rate

Dependence of the opt imal r isk on the dol lar interest rate
at d i f ferent values of the euro interest rate

Row1

Row2

Row3

Row4

Row5

Row6

Row7

row8

Row9

Row10



V.S.MOLOSTVOV                        114 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

The results give a partial idea of the capabilities of the proposed method and its software 
implementation, which gives an experienced Excel user other possibilities to analyze the problem 
under consideration by means of, for example, a "What-if" analysis. 

CONCLUSION 

The paper considers the problem of the optimal allocation of deposits to three of currencies (ruble, 
dollar, and euro) with uncertain future exchange rates of the dollar and euro. It is assumed that 
only the limits of possible changes in these uncertain parameters are known. The concept of 
Savage's minimax regret is used. According to this, the original problem is reduced to a minimax 
problem with the benefit lost due to uncertainty as the target function (the risk function). The risk 
function and the guaranteed risk function are found in explicit form. After that, the problem is 
reduced to finding the minimum point of a piecewise linear function under linear constraints.  

Explicit formulas are obtained for nine "representative" candidate points, at least one of which 
is the optimal solution. The final choice is made by direct comparison of the criterion values at 
these points.  

Computational tools have been developed in the Excel environment to find the structures of a 
three-currency deposit, optimal in terms of guaranteed risk, under conditions of unknown future 
exchange rates.  

The use of this computing environment has allowed the development of tools that combine 
ease of use for individuals with minimal spreadsheet skills and extensive data analysis tools for 
advanced users. 

The method is implemented in two variants – user and research. The first one is designed for 
those who have minimal skills of working in a ready-made Excel spreadsheet. This work is reduced 
to the input of the initial data and substantive analysis of the automatically obtained results. The 
second version contains a number of additional features: direct specification of the derived 
parameters of the profitability of individual currencies, multiple calculations varying one or two 
parameters of the problem, and a graphical visualization of the results.  

Debugging and experimental calculations were performed. In particular, the case of the equal 
guaranteed profitability of all three currencies (at minimum future exchange rates) was analytically 
investigated and used for debugging the program. Numerical modeling of the dependence of the 
minimum guaranteed risk on the maximum possible future values of currency rates was carried 
out. A monotonic increase in risk for each of these variables was revealed. 

The tables of dependence of the optimal deposit structure on the ruble interest rate (with other 
parameters fixed), as well as the tables of dependence of the optimal risk on the dollar and euro 
interest rates were constructed. 

The results can be used in analyzing problems of financial management under conditions of 
incomplete information. 

ACKNOWLEDGEMENTS 

This article is an output of a research project “Models of data analysis and decision making in the 
socio-economic sphere” implemented as part of the Basic Research Program at the National 
Research University Higher School of Economics (HSE University). 

The work was partially supported by the International Center of Decision Choice and Analysis 
of the National Research University Higher School of Economics. 

The author is grateful to the reviewers for their thorough and in-depth study of the paper, 
constructive comments, and valuable suggestions, which improved the paper significantly. 

 
 
 
 



115  SAVAGE'S SOLUTION TO THREE-CURRENCY DEPOSIT: PROGRAM TOOLS 

Copyright ©2024 ASSA  Adv. in Systems Science and Appl. (2024) 

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