260:
252:, and is also an efficient portfolio. With this portfolio, the investor will get highest satisfaction as well as best risk-return combination (a portfolio that provides the highest possible return for a given amount of risk). Any other portfolio, say X, isn't the optimal portfolio even though it lies on the same indifference curve as it is outside the feasible portfolio available in the market. Portfolio Y is also not optimal as it does not lie on the best feasible indifference curve, even though it is a feasible market portfolio. Another investor having other sets of indifference curves might have some different portfolio as their best/optimal portfolio.
494:
are extremely sensitive to small changes in the returns of the constituent assets and can therefore be extremely 'dangerous'. Positivity constraints are easy to enforce and fix this problem, but if the user wants to 'believe' in the robustness of the
Markowitz approach, it would be nice if better-behaved solutions (at the very least, positive weights) were obtained in an unconstrained manner when the set of investment assets is close to the available investment opportunities (the market portfolio) β but this is often not the case.
461:
498:
react to tiny return differences that are well within measurement error'. In the real world, this degree of instability will lead, to begin with, to large transaction costs, but it is also likely to shake the confidence of the portfolio manager in the model. Extrapolating this point further, among certain universes of assets, academics have found that the
Markowitz model has been susceptible to issues such as model instability where, for example, the reference assets have a high degree of correlation.
168:. All portfolios that lie below the Efficient Frontier are not good enough because the return would be lower for the given risk. Portfolios that lie to the right of the Efficient Frontier would not be good enough, as there is higher risk for a given rate of return. All portfolios lying on the boundary of PQVW are called Efficient Portfolios. The Efficient Frontier is the same for all investors, as all investors want maximum return with the lowest possible risk and they are risk averse.
145:
181:
237:
309:. The CML is an upward sloping line, which means that the investor will take higher risk if the return of the portfolio is also higher. The portfolio P is the most efficient portfolio, as it lies on both the CML and Efficient Frontier, and every investor would prefer to attain this portfolio, P. The P portfolio is known as the
497:
2. Practically more vexing, small changes in inputs can give rise to large changes in the portfolio. Mean-variance optimization suffers from 'error maximization': 'an algorithm that takes point estimates (of returns and covariances) as inputs and treats them as if they were known with certainty will
493:
1. Unless positivity constraints are assigned, the
Markowitz solution can easily find highly leveraged portfolios (large long positions in a subset of investable assets financed by large short positions in another subset of assets) , but given their leveraged nature the returns from such a portfolio
152:
As the investor is rational, they would like to have higher return. And as they are risk averse, they want to have lower risk. In Figure 1, the shaded area PVWP includes all the possible securities an investor can invest in. The efficient portfolios are the ones that lie on the boundary of PQVW. For
468:
Figure 5 shows that an investor will choose a portfolio on the efficient frontier, in the absence of risk-free investments. But when risk-free investments are introduced, the investor can choose the portfolio on the CML (which represents the combination of risky and risk-free investments). This can
176:
For selection of the optimal portfolio or the best portfolio, the risk-return preferences are analyzed. An investor who is highly risk averse will hold a portfolio on the lower left hand of the frontier, and an investor who isnβt too risk averse will choose a portfolio on the upper portion of the
505:
among assets in the market portfolio) needed to compute a mean-variance optimal portfolio is often intractable and certainly has no room for subjective measurements ('views' about the returns of portfolios of subsets of investable assets) . Furthermore, the information dependency and the need to
313:
and is generally the most diversified portfolio. It consists of essentially all shares and securities in the capital market (either long or short). The Market
Portfolio would not include a specific security if the correlation between the portfolio and the security is zero with negative return
255:
All portfolios so far have been evaluated in terms of risky securities only, and it is possible to include risk-free securities in a portfolio as well. A portfolio with risk-free securities will enable an investor to achieve a higher level of satisfaction. This has been explained in Figure 4.
35:
model; it assists in the selection of the most efficient portfolio by analyzing various possible portfolios of the given securities. Here, by choosing securities that do not 'move' exactly together, the HM model shows investors how to reduce their risk. The HM model is also called
287:
PX is more than the satisfaction obtained from the portfolio P. All portfolio combinations to the left of P show combinations of risky and risk-free assets, and all those to the right of P represent purchases of risky assets made with funds borrowed at the risk-free rate.
204:
are shown. Each of the different points on a particular indifference curve shows a different combination of risk and return, which provide the same satisfaction to the investors. Each curve to the left represents higher
317:
In the market for portfolios that consists of risky and risk-free securities, the CML represents the equilibrium condition. The
Capital Market Line says that the return from a portfolio is the risk-free rate
681:
646:
506:
calculate a covariance matrix introduces some, albeit manageable, computational complexity and constraint to model scalability for portfolios with sufficiently large asset universes.
248:. This point marks the highest level of satisfaction the investor can obtain. This is shown in Figure 3. R is the point where the efficient frontier is tangent to indifference curve C
114:
To choose the best portfolio from a number of possible portfolios, each with different return and risk, two separate decisions are to be made, detailed in the below sections:
209:
or satisfaction. The goal of the investor would be to maximize their satisfaction by moving to a curve that is higher. An investor might have satisfaction represented by C
715:
Benichou, R.; Lemperiere, Y.; Serie, E.; Kockelkoren, J.; Seager, P.; Bouchaud, J.-P.; Potters, M. (2017). "Agnostic Risk Parity: Taming Known and
Unknown-Unknowns".
457:
3. CML is always upward sloping as the price of risk has to be positive. A rational investor will not invest unless they know they will be compensated for that risk.
283:
PX shows a combination of different proportions of risk-free securities and efficient portfolios. The satisfaction an investor obtains from portfolios on the line R
509:
4. The expected returns are uncertain, and when we make this assumption, the optimization problem yields solutions different from those of the
Markowitz Model.
135:
A portfolio that gives maximum return for a given risk, or minimum risk for given return is an efficient portfolio. Thus, portfolios are selected as follows:
322:
risk premium. Risk premium is the product of the market price of risk and the quantity of risk, and the risk is the standard deviation of the portfolio.
291:
In the case that an investor has invested all their funds, additional funds can be borrowed at risk-free rate and a portfolio combination that lies on R
473:) and the purchase of efficient portfolio P. The portfolio an investor will choose depends on their preference of risk. The portion from I
259:
539:
161:, compared to T and U. All the portfolios that lie on the boundary of PQVW are efficient portfolios for a given risk level.
141:(b) From the portfolios that have the same risk level, an investor will prefer the portfolio with higher rate of return.
157:, there are three portfolios S, T, U. But portfolio S is called the efficient portfolio as it has the highest return, y
138:(a) From the portfolios that have the same return, the investor will prefer the portfolio with lower risk, and
502:
433:) is a measure of the risk premium, or the reward for holding risky portfolio instead of risk-free portfolio. Ο
454:
2. Only efficient portfolios that consist of risk free investments and the market portfolio P lie on the CML.
824:
834:
437:
is the risk of the market portfolio. Therefore, the slope measures the reward per unit of market risk.
481:. In this portion, the investor will lend a portion at risk-free rate. The portion beyond P is called
244:
The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the
829:
314:(gambling), or if the correlation is one (whichever has lower return would not warrant investment).
79:
49:
758:
32:
682:"Sherman ratio optimization: constructing alternative ultrashort sovereign bond portfolios"
647:"Sherman ratio optimization: constructing alternative ultrashort sovereign bond portfolios"
460:
8:
555:
301:
65:
775:
697:
662:
586:
448:
397:
275:
securities, as those securities are considered to have no risk for modeling purposes. R
245:
189:
45:
485:, where the investor borrows some funds at risk-free rate to buy more of portfolio P.
701:
666:
627:
590:
535:
263:
Figure 4: The
Combination of Risk-Free Securities with the Efficient Frontier and CML
144:
279:
PX is drawn so that it is tangent to the efficient frontier. Any point on the line R
803:
767:
689:
654:
617:
578:
86:
788:
217:
Thus, at any point of time, an investor will be indifferent between combinations S
180:
501:
3. The amount of information (the covariance matrix, specifically, or a complete
213:, but if their satisfaction/utility increases, the investor then moves to curve C
107:
89:
is concave and increasing, due to their risk aversion and consumption preference.
28:
622:
605:
306:
818:
631:
72:
582:
693:
658:
236:
44:
model due to the fact that it is based on expected returns (mean) and the
60:
Markowitz made the following assumptions while developing the HM model:
779:
272:
93:
469:
be done with borrowing or lending at the risk-free rate of interest (I
771:
807:
376:
41:
714:
569:
Scherer, B. (2002). "Portfolio resampling: Review and critique".
206:
20:
606:"Blockchain Token Economics: A Mean-Field-Type Game Perspective"
305:(CML). This line represents the risk-return trade off in the
48:(variance) of the various portfolios. It is foundational to
68:
is based on the variability of returns from said portfolio.
37:
734:
Valeyre, S. (2024). "Optimal trend-following portfolios".
447:, is the optimum combination of risky investments and the
99:
An investor either maximizes their portfolio return for a
121:
Selection of the best portfolio out of the efficient set.
103:
level of risk or minimizes their risk for a given return.
477:
to P, is investment in risk-free assets and is called
756:
Markowitz, H.M. (March 1952). "Portfolio
Selection".
603:
464:Figure 5: CML and Risk-Free Lending and Borrowing
816:
130:
786:
755:
118:Determination of a set of efficient portfolios.
171:
271:is the risk-free return, or the return from
148:Figure 1: Risk-return of possible portfolios
92:Analysis is based on single period model of
525:
523:
521:
488:
621:
184:Figure 2: Risk-return indifference curves
749:
604:Barreiro-Gomez, J.; Tembine, H. (2019).
534:. India: Taxmann Publications (P.) Ltd.
518:
459:
443:1. At the tangent point, i.e. Portfolio
440:The characteristic features of CML are:
258:
235:
192:for the investors. Indifference curves C
179:
143:
733:
708:
568:
529:
16:Portfolio optimization model in finance
817:
727:
679:
644:
13:
14:
846:
409:= standard deviation of portfolio
240:Figure 3: The Efficient Portfolio
796:The Journal of Political Economy
736:Journal of Investment Strategies
717:Journal of Investment Strategies
686:Journal of Investment Strategies
651:Journal of Investment Strategies
530:Rustagi, R.P. (September 2010).
387:= return on the market portfolio
164:The boundary PQVW is called the
78:An investor prefers to increase
188:Figure 2 shows the risk-return
787:Markowitz, H.M. (April 1952).
673:
638:
597:
562:
548:
503:joint probability distribution
366:= expected return of portfolio
125:
55:
1:
512:
131:Determining the efficient set
7:
623:10.1109/ACCESS.2019.2917517
325:The CML equation is :
172:Choosing the best portfolio
10:
851:
571:Financial Analysts Journal
489:Demerits of the HM model
153:example, at risk level x
789:"The Utility of Wealth"
583:10.2469/faj.v58.n6.2489
425:is the slope of CML. (R
400:of the market portfolio
50:Modern portfolio theory
759:The Journal of Finance
694:10.21314/JOIS.2023.001
680:Henide, Karim (2023).
659:10.21314/JOIS.2023.001
645:Henide, Karim (2023).
465:
264:
241:
185:
149:
33:portfolio optimization
750:Selected publications
463:
295:PX can be obtained. R
262:
239:
183:
147:
532:Financial Management
375:= risk-free rate of
825:Financial economics
483:Borrowing Portfolio
302:Capital Market Line
299:PX is known as the
835:Portfolio theories
466:
398:standard deviation
265:
246:indifference curve
242:
190:indifference curve
186:
166:Efficient Frontier
150:
108:rational in nature
46:standard deviation
541:978-81-7194-786-7
479:Lending Portfolio
27:β put forward by
842:
830:Financial models
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793:
783:
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635:
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566:
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559:
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311:Market Portfolio
87:utility function
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849:
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844:
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791:
772:10.2307/2975974
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747:
732:
728:
713:
709:
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639:
616:: 64603β64613.
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106:An investor is
85:The investor's
71:An investor is
58:
31:in 1952 β is a
29:Harry Markowitz
25:Markowitz model
17:
12:
11:
5:
848:
838:
837:
832:
827:
813:
812:
808:10.1086/257177
802:(2): 151β158.
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672:
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540:
516:
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343:
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335:
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307:capital market
296:
292:
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769:
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741:
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664:
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648:
641:
633:
629:
624:
619:
615:
611:
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600:
592:
588:
584:
580:
577:(6): 98β109.
576:
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526:
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522:
517:
510:
507:
504:
499:
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799:
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766:(1): 77β91.
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710:
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613:
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610:IEEE Access
556:"Markowitz"
451:portfolio.
126:Methodology
80:consumption
73:risk averse
56:Assumptions
819:Categories
513:References
273:government
177:frontier.
94:investment
64:Risk of a
702:259538567
667:259538567
632:2169-3536
591:154795184
66:portfolio
377:interest
42:variance
780:2975974
358:where,
207:utility
21:finance
778:
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589:
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449:market
225:, or S
23:, the
792:(PDF)
776:JSTOR
698:S2CID
663:S2CID
587:S2CID
229:and S
221:and S
200:and C
101:given
628:ISSN
536:ISBN
338:+ (R
320:plus
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804:doi
768:doi
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