THEORETICAL ASPECTS OF THREE-ASSET PORTFOLIO MANAGEMENT

Transcription

1 THEORETICAL ASPECTS OF THREE-ASSET PORTFOLIO MANAGEMENT Michal ŠOLTÉS ABSTRACT: The aer deals with three-asset ortfolio It focuses on ordinary investor for whom the Marowitz s theory of selection of otimal ortfolio is often too difficult to use in ractise In the aer new formulas for calculation of the weights of the assets in three-asset ortfolio otimised according to the ris measured by standard deviation are being derived The aer also deals with comarison of otimisation of two- and three-asset ortfolio Also the formulas for calculation of weights of assets in three-asset ortfolio which is otimised under the re-defined rate of return are derived in the aer KEY WORDS: ortfolio ortfolio management otimisation ortfolio s analysis standard deviation JEL CODE: G INTRODUCTION Basic aim of every investor is rofit However rofit maximization is not so easy aim to achieve as every investor has certain level of ris aversion at the same time Of course the level of aversion is different for each investor Profit and ris are two asects that cannot be searated and between which indirect roortion exists The higher yield investor wants to gain the higher ris has to be undertaen Regular investor refers the highest yield and the lowest ris at the same time Theory of ortfolio management deals with these questions Ris of ortfolio can be lowered through diversification e i slitting the investment to several assets Not all assets can be however used for diversification It is interesting to observe which assets are suitable for effective diversification i e diversification which leads to lowering the whole ris of ortfolio It is also interesting to now how to create a ortfolio with the minimum ris level and with the minimum ris level under the condition of required rate of return The aim of this aer is to give answers to the questions stated above in case of three assets Assoc rof Michal Šoltés Technical university of Košice Faculty of Economics Nemcovej Košice Slovaia MichalSoltes@tues

2 CURENTUL JURIDIC PORTFOLIO WITH THE MINIMUM RISK LEVEL Let us suose that there is ossibility to invest our financial caital to three different assets X X X with exected rates of return r r r standard deviations and their covariance s If weights of these assets in the ortfolio created will be then variance of such created ortfolio will be calculated from the formula () As then () Theoretically it is ossible to create an infinite number of ortfolios from three different assets each with different weights of assets Our tas is to find the one with the minimum ris level so weights of assets must meet these equations 0 0 from which we get ) As ortfolio with weights which meet the system of equations () will have the minimum ris level if

3 Michal ŠOLTÉS D 0 Moreover > 0 and > 0 must be valid It can be roved that if min ij i (4) j and if we order the assets so that (5) then all conditions stated above are met If system () is solved using Cramer s rule then after modification we get the formulas for weights of assets in ortfolio with the minimum ris level as these (6) We can formulate these statements: If conditions (4) are met i e min ij i j then assets are suitable for diversification In such case we order the assets so that i e so that relation (5) was valid The weights of assets in the ortfolio will be calculated using formulas (6) 4 Exected rate of return of such created ortfolio can be calculated from the relation r r r r and its standard deviation from the formula () 5 Such created ortfolio has the minimum standard deviation from all the ossible ortfolios that can be created using assets X X X

4 CURENTUL JURIDIC Examle Let us suose that there is a ossibility to invest money to three different assets with the exected rates of return r 5% r 9% r 6% standard deviations % 44% 6% and covariances = 75 = 0 = 90 Find the ortfolio with the minimum ris level Solution As conditions (4) are met assets are suitable for diversification The relationshi (5) is also valid so the weights can be calculated using formulas (6) as follows The standard deviation of this ortfolio is and exected rate of return is r COMPARISON OF INVESTMENT TO THREE-ASSET PORTFOLIO WITH TWO-ASSET PORTFOLIO If assets are suitable for diversification then three-asset ortfolio with minimum rate of ris is more effective than two-asset ortfolio created from any two chosen assets Šoltés M and Šoltés V (00) roved that the weights of two-asset ortfolio with minimum ris level can be calculated using formulas standard deviation of this ortfolio from the formula and Using assets X and X from examle we get ortfolio with weights and standard deviation a r Using assets X and X we get ortfolio with weights and standard deviation a r 59 0 Using assets X and X we get ortfolio with weights and standard deviation a r 666 0

5 4 Michal ŠOLTÉS As we can see diversification to two-asset ortfolio is not as effective as to threeasset ortfolio PORTFOLIO WITH THE MINIMUM RISK LEVEL AT THE REQUIRED RATE OF RETURN Let us consider three assets X X X with exected rates of return r r r standard deviations and covariance while min ij i j We have roved that these assets are suitable for diversification Let us find the ortfolio which has the created using the assets X X X with required rate of return r r r minimum standard deviation We have r r r r a Solving this system of two equations with three unnown quantities we have r r r r r r (7) r r r r r r As >0 >0 >0 if will be stated arbitrary under condition of r r r r max 0 then such ortfolios will have required rate of return r r r r r but different standard deviations Let us mar r r a r r (8) r r b r r r r c r r r r d r r Then relation (7) can be modified as follows a b c d (9) Let us find the ortfolio with required rate of return r and with minimum ris level at the same time Using formula () for variance of three-asset ortfolio and formulas for and (9) we have a b c d a b c d a b c d

6 CURENTUL JURIDIC 5 from which a b b c d d a b b c d d bc d a b d b d b d db ab cd a c bc ad b d b d db b d b d db b d 0 as correlation coefficients have to always be within interval above it is obvious that ortfolio with weights From stated ab cd b a d c bc ad b d db a b c d (0) time will be the one with required rate of return and the minimum ris level at the same Examle Let us suose that we have three assets with exected rates of return r 0% r 4% r 8% standard deviations σ % σ 0% σ 8% and covariances = 70 = 80 = 0 Let us find the ortfolio created from these assets which has rate of return 6% Solution From formula (8) we have a = 05 b = c = 05 d = Weights of our ortfolio with required rate of return 6% can be calculated using formulas (0) Portfolio with weights of = 00 = 0456 a = 05 has exected rate of return r % Variance of ortfolio calculated from

7 6 Michal ŠOLTÉS formula () is and & standard deviation is 0 CONCLUSION From stated above we can obtain the algorithm of diversification to three assets consists as follows: REFERENCES For diversification we use only assets that meet the condition ij min i j is met If so we order assets so that Find the ortfolio with the lowest standard deviation i e ris level while weights of assets in the ortfolio can be calculated from formulas (6) or solving the system of equations () Then calculate exected rate of return and standard deviation of this ortfolio Portfolio with lower exected rate of return does not have any ractical sense 4 If we require higher rate of return then we create the ortfolio with weights calculated using formulas (0) The level of ris will be higher as well Jindřichovsá I Blaha Z: Podniové finance Management Press Praha 00 ISBN Kelbel M Nováče E: Analýza ortfólia Elfa sro Košice 998 Šoltés V Penja V: Finančná matematia Košicá tlačiarensá as Košice 00 ISBN X Šoltés V - Šoltés M: Analysis of two-asset ortfolio In: Eonomie a management roč 6 secial issue (00) s 6-65 ISSN -609