An investment strategy with optimal sharpe ratio

Transcription

1 The 22 n Annual Meeting in Mathematics (AMM 2017) Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai, Thailan An investment strategy with optimal sharpe ratio S. Jansai a, an D. Thongtha a,b, a Department of Mathematics, Faculty of Science King Mongkut s University of Technology Thonburi, Bangkok 10140, Thailan b Theoretical an Computational Science Center (TaCS), Science Laboratory Builing, Faculty of Science King Mongkut s University of Technology Thonburi, Bangkok 10140, Thailan Abstract New investment moel, involving buying an selling points, is propose an Sharpe ratio is now consiere as a performance measure of the investment. With this moel, the strategy which makes a maximal Sharpe ratio is investigate. Furthermore, numerical results are also provie in this work. Keywors: Investment Moel, Sharpe Ratio, Performance Measure MSC: 49K99, 60A99. 1 Introuction Nowaays, a number of people who spen their free time on an investment increases. More people pay their attention to stuy on financial plan in orer to get stability in their life an have financial freeom after retirement. To fulfill those reams, making more income is an important choice. Some investors try to fin an investment strategy that provies high return. However, risk level is a main factor having a negative correlation to the return. High return with high risk level may not help people to reach their goals. On the contrary, low return with low risk level may provie a better alternative. This leas to a question about the tools that help the investors to get the best choice. Therefore, an appropriate way to invest an a performance measure have been iscusse in various sector of financial market. In orer to get higher return, investment moel plays an important role. Therefore, more investors pay their attention to stuy on investment strategies an then fin their own ways to invest in risky assets. Moreover, this topic also attract many researchers such as Piotrowski an Slakowski [1], Scafetta et al. [4], Zhang an Li [5] an Kotter an Bauerle [6]. Some researchers attempt to investigate the strategies that can be use to make more profit (equip with a better performance). For example, in 2003, Piotrowski an Slakowski [1] introuce an investment moel which is calle the Merchanising Mathematician moel (MM moel). Their moel is focuse on two transaction activities which are buying an asset (exchanging money for the asset) an selling an asset (exchanging asset for the money). They set conition on exchanging base on the logarithmic quotation. In their work, the profit intensity is use as an This research was financially supporte by Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi Corresponing author. Speaker. aress: supharut.gago8@mail.kmutt.ac.th (S. Jansai), awu.tho@kmutt.ac.th (D.Thongtha). MIS-02-1

2 investment performance measure. By this measure, the performance is evaluate by comparing profit per an investment uration. The main result of their research is fining out a conition on buying activity that make maximal profit intensity uner the MM moel. For an investment performance measure, there are various type of well-known measures in an investment sector such as Sharpe ratio, Treynor ratio an Jansen s Alpha. Each measure is create to evaluate return per unit of risk in ifferent views. The Sharpe ratio, which is efine simply in term of the average of excess return an the stanar eviation of return, are wiely use as criteria for ranking mutual fun. The main benefit of this measure, when it is compare with other measures, is that Sharpe ratio can be applie to ifferent types of asset. Therefore, there are many research in financial market that Sharpe ratio is use as a tool. To illustrate, in 2012, Bansal et al. [2] use Sharpe ratio to stuy the performance of twelve selecte mutual funs in Inian mutual fun inustry from May 2005 to April In 2014, Jani an Jain [3] provie insights of risk an return in three mutual funs. They applie the Sharpe ratio to etermine the performance of such mutual funs with ata of aily return. By comparing with market return as benchmark, their result inicates that which funs outperforme the market. In this work, we introuce new investment moel consisting of two transaction activities. Sharpe ratio is consiere as an investment performance measure. We also investigate the conition on transaction activity that provies a maximal Sharpe ratio uner the propose moel. Furthermore, an application of our results, equip with statistical an mathematical tools, to real ata is also illustrate. The content in this paper is organize into 5 sections. In section 2, The Sharpe ratio is escribe. New investment moel, which is our main result, is propose in section 3. Numerical application an conclusion are provie in section 4 an section 5, respectively. 2 Sharpe Ratio Sharpe ratio, which was evelope by William F. Sharpe in 1994, is consiere as a well-known an highly useful measure of investment performance. It is efine by the proportion of expecte of return an stanar eviation of the return, which is a risk on an investment. Therefore, the high stanar eviation becomes a rawback because an uncertainty of outcome increases. Let R be a return rate of an investment an r f a risk-free rate. The Sharpe ratio is efine as Sharpe ratio = E(R) r f σ R. By the efinition, the Sharpe ratio inicates investment s excess return per unit of risk. This ratio is wiely use to evaluate the performance of an investment in both iniviual portfolio an fun. The avantage of this measure is that it can use to compare the performance of funs emphasizing on investment in various sectors of the market (see [2], [3]). For example, If fun A, investing on bon, provies yiel 5% while fun B, investing on gol, provies 7%. If the risk free rate is 2% an stanar eviation of return on fun A an B are 10% an 25%, respectively. Thus, the Sharpe ratio of fun A an B are 0.3 an 0.2, respectively. This can be interprete that fun A performs better than fun B even they are invest in ifferent assets. 3 Investment Moels In this section, we will introuce an investment moel an investigate a conition on the moel for which it provies the best performance. In 2003 [1], Piotrowski an Slakowski propose the merchanising mathematician moel (MM moel) as an investment moel. This moel consists of two transactional activities which are rational buying an ranom selling. The first activity involves rational buying of the asset Θ. This step is exchanging the money $ for the asset Θ. The secon one focuses on ranom selling such an asset Θ which is exchange for money $. Let V Θ an V $ be some given amounts MIS-02-2

3 of the asset Θ an the money $, respectively. At time t, efine the logarithmic quotation for the asset Θ by p t ln(v $ ) ln(v Θ ) = ln( V $ V Θ ). (3.1) If the first transaction happens at time t 1 with the logarithmic quotation p t1 an the secon activity occurs at time t 2 with the logarithmic quotation p t2. Therefore, the profit, which now is the logarithmic rate of return, is equal to r t1,t 2 = p t2 p t1. (3.2) In MM moel, Piotrowski an Slakowski set conitions on purchasing an selling the asset by using the logarithmic quotation. The rational buying is a purchase boune by a fixe logarithmic quotation. For the ranom selling, the logarithmic quotation is set to be. In this work, a new investment moel is introuce by moifying an iea in [1]. In this moel, the transaction activities, which is buying an selling, are assume to happen at logarithmic quotations a an b, respectively. Furthermore, after buying the asset, the investors are able to sell their asset in various urations with ifferent percentages of total asset. Suppose that the investors buy an asset when the logarithmic quotation p = a an the selling interval is [a 0, b], for some a 0 R, being partitione into k subintervals. Let l 0 = 0 an l 1, l 2,..., l k be an increasing sequence in [0, 1] with l k = 1, θ i = a 0 +l i (b a 0 ) for i = 0, 1, 2,..., k an I 1 = [a 0, θ 1 ] an I i = (θ i 1, θ i ] for i = 2,..., k. For i = 1, 2,..., k, let w i [0, 1] with w i = 1. If the investors sell their asset uniformly by weight w i when p I i, the logarithmic quotation for the selling activities can be written as p(b) = w i pi (p I i ) (3.3) where I is an inicator function. Therefore, with this moel, their profit is E(p(b)) a an the Sharpe ratio of the portfolio is ρ(b) = E(p(b)) a r f = V ar(p(b) a) E(p(b)) a r f E (p(b) 2 ) (E (p(b))) 2 (3.4) where r f is the risk free rate. In orer to investigate the value of b which makes a maximal Sharpe ratio, we nee some mathematical properties of ρ which is escribe in the following proposition. Proposition 3.1. Let p be a continuous ranom variable with probability ensity function of f an ρ the Sharpe ratio efine by equation (3.4). Then b E (p(b)) = b E ( p(b) 2) = w i [l i θ i f (θ i ) l i 1 θ i 1 f (θ i 1 )], w 2 i [ li θ 2 i f (θ i ) l i 1 θ 2 i 1f (θ i 1 ) ]. Proof. By the efinition of expectation, we have E (p(b)) = θi θ i 1 w i pf (p) p. MIS-02-3

4 By applying the funamental theorem of calculus, we get that Note that p(b) 2 = b E (p(b)) = b = [ θi w i 0 θi 1 ] pf (p) p pf (p) p 0 w i [l i θ i f (θ i ) l i 1 θ i 1 f (θ i 1 )]. wi 2 p 2 I (p I i ). Therefore, the secon result can be prove irectly by using the same argument. Our main purpose is to fin the selling interval [a 0, b] in our propose moel that makes the best performance. The next theorem provies a conition that can be use to solve for b. The result from this conition can be consiere as an caniate of least upper boun on the selling interval. However, after getting values of b from the theorem, we have to etermine whether such values of b provie a maximum value of ρ. This can be checke by using several techniques such as sketching its graph or using the secon erivative test. Theorem 3.2. Let p is a continuous ranom variable with probability ensity function of f an ρ the Sharpe ratio efine by equation (3.4). Assume that V ar(p(b)) > 0. Then, b 0 is a critical point of ρ if b 0 satisfies the conition: (E (p(b)) (a + r f )) b E ( p(b) 2) = 2 ( E ( p(b) 2) (a + r f )E (p(b)) ) E (p(b)). (3.5) b Proof. The theorem can be prove by setting ρ (b) = 0. Then, we get an equation 0 = 2V ar(ρ(b)) b E(p(b)) (E(p(b)) (a + r f )) b V ar(p(b)). Using the relation that V ar(ρ(b)) = E ( p(b) 2) (E (p(b))) 2, the above equation becomes (E(p(b)) (a + r f )) b E(p(b)2 ) = 2 ( E(p(b) 2 ) (a + r f )E(p(b)) ) b E(p(b)). Hence, the proof is complete. 4 Numerical Application In section3, our propose moel is esigne to purchase an asset with two transaction activities. These two activities have conitions base on the logarithmic quotation. We also give a conition which make a maximum Sharpe ratio in theorem 3.2. In this section, we will illustrate an application of the theorem to real ata in the Stock Exchange of Thailan (SET). The ata of stock TOP, Thai Oil Public Company Limite, between 2013 an 2015 is consiere as observation ata. The logarithmic quotation for the asset the asset Θ is efine by p ln( V $ V Θ ) where V $ an V Θ are the amounts of the money $ an the asset Θ, respectively. Therefore, in case of stock, the amounts V $ an V Θ are the prices (P ) an the book values (BV ) of the stock. Thus, the The logarithmic quotation of the stock becomes p = ln( P BV ) To fin the optimal selling point (b 0 ) in our moel by using theorem 3.2, there are several steps of computing which are : 1. Collecting aily ata P an BV an computing p. MIS-02-4

5 2. Testing for istribution of p. 3. Evaluating the moments of p an their erivatives. 4. Solving for b 0 by using Theorem Determining an optimality of b 0. In the first step, we collect the aily ata P an BV of stock TOP. There are 733 ata of P an BV, between 2013 an 2015, which can be ownloae from After that, we calculate the logarithmic quotation p = ln ( ) P BV of aily ata. The graph of p is shown in the following figure: Figure 1. Data of ln( P BV ) of TOP between 2013 an Accoring to theorem 3.2, we have to fin the istribution of the logarithmic quotation p. In this step, the istribution of p is investigate by using Kolmogorov-Smirnov test (KS - test) in ARENA. The result reveal that the istribution of logarithmic quotation p of stock TOP is a normal with parameter mean (µ) = an stanar eviation (σ) = with significant level α = That is, p N(0.238, ). Therefore, the probability ensity function of p is f(p) = (p 0.238) 2 2π e 2(0.123) 2, < p <. Figure 2. Graph of the istribution of ln( P BV ). Following our investment moel, suppose that an investor buys the stock TOP when p = 0. This means that the investor buy the asset when stock price is equal to its book value. If we partition the selling interval into k = 3 subintervals with a 0 = 0.01, l 1 = 0.5, l 2 = 0.8 an l 3 = 1. Then, θ 0 = 0.01, θ 1 = (b 0.01), θ 2 = (b 0.01), θ 3 = b an the 3 subintervals are I 1 = [0.01, 0.5b ], I 2 = (0.5b , 0.8b ] an I 3 = (0.8b , b]. If we set the weights w 1 = 0.4, w 2 = 0.3 an w 3 = 0.3, p(b) in equation (3.3) for stock TOP becomes p(b) = 0.4pI (p I 1 ) + 0.3pI (p I 2 ) + 0.3pI (p I 3 ). MIS-02-5

6 Therefore, E(p(b)) = E(p(b) 2 ) = b 0.01 b b b 0.01 b b pe (p 0.238) p xe (p 0.238) p an p 2 e (p 0.238) p p 2 e (p 0.238) p b b b b pe (p 0.238) p By Proposition 3.1, we get the erivatives of E(p(b)) an E(p(b) 2 ) as follows: p 2 e (p 0.238) p (0.5b 0.233) 2 E (p(b)) = (0.08b )be be (b 0.238) an (4.1) b b E ( p(b) 2) = ( b b )e (0.5b 0.233) b 2 e (b 0.238) (4.2) Next, by setting r f = 0.02, we will fin the en point of the last subinterval, say b 0, that make maximal Sharpe ratio of our investment. The point b 0 is first consiere as a critical point of ρ which can be calculate by solving an equation (3.5) in theorem 3.2 together with equations (4.1)-(4.2). For such a computation, a bisection metho on the boune interval [0.01, 10] with limite error is use to solve for b 0. The limite value of b is set to 10 because, in stock market, it is sufficiently large to consier the bounary of as e10. After computing, we get b 0 = 0.6. Finally, the b 0 is etermine whether it provies a maximum of ρ on [0.01, 10]. We check this property by sketching its graph which is shown in the following figure: P BV Figure 3. Graph of Sharpe ratio. The figure shows that b 0 = 0.6 make a maximum Sharpe ratio on [0.01, 10]. Therefore, we get the strategy for investment accoring to the propose moel as follows: Suppose an investor buys the stock TOP when its price is equal to its book value an plans to sell the asset into 3 intervals. Then, he shoul sell 40% of his asset when b 0 [0.01, 0.305].The other intervals are (0.305, 0.482] an (0.482, 0.6] for which he sells 30% of their portfolio on each interval. With this strategy, the maximal Sharpe ratio of this investment is Conclusion In this work, new investment moel, base on buying an selling activities, is propose. The conition on these two transactions is efine in term of the logarithmic quotation. The Sharpe MIS-02-6

7 ratio is consiere as a measure of investment performance. Moreover, with the new moel, we investigate a conition on the selling point that possibly makes a maximal Sharpe ratio. An application of our results to real ata is also provie in the last section. Acknowlegment. The authors woul like to thank all anonymous referees for their careful reaing of the manuscript an many helpful comments. The first authors woul like to thank the Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi for financial support. References [1] E.W. Piotrowski, J. Slakowski, The merchanising mathematician moel: profit intensities, Physica A 318 (3)(2003) [2] S.Bansal, S. Kumar, S.K. Gupta, Test of Sharpe ratio on selecte mutual fun schemes. International Journal of Marketing, Financial Services an Management Research (1)(2012) [3] D.J. Jani, R. Jain, Measuring Risk Ajuste Return (Sharpe Ratio) of the Selecte Mutual Funs A case of Daily Returns, Journal of Business Management & Social Sciences Research (JBM&SSR) (3)(2014) [4] N. Scafetta, S. Picozzi, B.J. West, A trae-investment moel for istribution of wealth, Physica D, (193)(2004) [5] T. Zhang,H. Li, Buying on margin, selling short in an agent-base market moel, Physica A, vol. 392, pp , [6] M. Kotter,N. Bauerle, The perioic risk moel with investment, Insurance: Mathematics an Economics 42, vol.42, pp , MIS-02-7