1、 投資的收益和風(fēng)險(xiǎn) 摘要 投資的收益與風(fēng)險(xiǎn)作為高科技產(chǎn)業(yè)化的催化劑和孵化劑，日益引起了任夢(mèng)的廣泛關(guān)注和重視。在我國(guó)，風(fēng)險(xiǎn)投資剛剛起步，但對(duì)國(guó)民經(jīng)濟(jì)發(fā)展和社會(huì)進(jìn)步意義十分重大，因而越來(lái)越引起人們的重視。然而，總體收益盡可能大和總體風(fēng)險(xiǎn)盡可能小兩個(gè)目標(biāo)在一定意義上市對(duì)立。即建立收益與風(fēng)險(xiǎn)的雙目標(biāo)優(yōu)化模型，尋求最佳投資組合。模型一：固定風(fēng)險(xiǎn)水平，優(yōu)化收益；模型二：固定盈利水平，極小化風(fēng)險(xiǎn)；模型三：將前量目標(biāo)加權(quán)，引入風(fēng)險(xiǎn)偏好系數(shù)。然后使用matlab分別求解三個(gè)模型。關(guān)鍵詞：投資組合 雙目標(biāo)優(yōu)化模型 風(fēng)險(xiǎn)偏好 風(fēng)險(xiǎn) 收益一、 問(wèn)題重述 隨著市場(chǎng)經(jīng)濟(jì)的快速發(fā)展，投資各個(gè)項(xiàng)目進(jìn)行贏利，已是許多公司取得利

2、潤(rùn)的的主要途徑。但是這樣的投資又存在著一定的風(fēng)險(xiǎn)。不是每一次投資都能百分之百獲得利潤(rùn)，所以怎樣緩解與解決贏利與風(fēng)險(xiǎn)之間的矛盾，是每一個(gè)投資商及待解決的問(wèn)題。本題就是要通過(guò)一個(gè)實(shí)例，建立數(shù)學(xué)模型，用數(shù)學(xué)的眼光來(lái)看待及解決這個(gè)問(wèn)題。 假定市場(chǎng)上有n種資產(chǎn)（i=1,2，n）可以選擇，現(xiàn)用數(shù)額為M的相當(dāng)大的資金作一個(gè)時(shí)期的投資.這n種資產(chǎn)在這一時(shí)期內(nèi)購(gòu)買(mǎi)的平均收益率為，風(fēng)險(xiǎn)損失率為，投資越分散，總的風(fēng)險(xiǎn)越小，總體風(fēng)險(xiǎn)可用投資的中最大的一個(gè)風(fēng)險(xiǎn)來(lái)度量. 購(gòu)買(mǎi)時(shí)要付交易費(fèi)，(費(fèi)率)，當(dāng)購(gòu)買(mǎi)額不超過(guò)給定值時(shí)，交易費(fèi)按購(gòu)買(mǎi)計(jì)算.另外，假定同期銀行存款利率是，既無(wú)交易費(fèi)又無(wú)風(fēng)險(xiǎn).（=5%）二、 已知n=4時(shí)相關(guān)

3、數(shù)據(jù)如下：（%）（%）（%）（元）S1282.51103S2211.52198S3235.54.552S4252.66.540 試給該公司設(shè)計(jì)一種投資組合方案，即用給定的資金M，有選擇地購(gòu)買(mǎi)若干種資產(chǎn)或存銀行生息，使凈收益盡可能大，使總體風(fēng)險(xiǎn)盡可能小.三、 問(wèn)題分析 從題中得知，本道題的目標(biāo)就是如何投資，使得凈收益盡可能大，使總體風(fēng)險(xiǎn)盡可能小。如何使用這N種投資是關(guān)鍵，并且在計(jì)算收益的同時(shí)還要兼顧投資的風(fēng)險(xiǎn)度，只有在收益盡可能大，風(fēng)險(xiǎn)盡可能小的條件下，才符合本題的主旨。并且在這個(gè)過(guò)程中收益率與損失率不能受外界條件的干擾，這樣才能夠穩(wěn)定的制定出最佳投資方案。當(dāng)然還從題中還得出了，本題的三大關(guān)鍵的

4、比較，那就是交易費(fèi)的多少，這也會(huì)影響到盡收益的多少。就要看購(gòu)買(mǎi)額的多少，還要計(jì)算如果不搞投資，全都存銀行是收益。最后再同等的比較這三鐘可能的收益和之前的綜合風(fēng)險(xiǎn)相比較，最后得出最佳答案。四、 模型假設(shè)1、 投資數(shù)額相當(dāng)大，為了便于計(jì)算，假設(shè)=1；2、 投資越分散，總的風(fēng)險(xiǎn)越小；3、 總體風(fēng)險(xiǎn)投資項(xiàng)目中最大的一個(gè)風(fēng)險(xiǎn)來(lái)度量；4、 中資產(chǎn)之間是相互獨(dú)立的；5、 在投資的這一時(shí)期內(nèi)，為定值，不受意外因素影響；6、 凈收益和總體風(fēng)險(xiǎn)只受，影響，不受其它因素干擾。五、 符號(hào)說(shuō)明第中投資的項(xiàng)目，如股票、債券第中投資的平均收益第種投資的交易費(fèi)率第種投資的風(fēng)險(xiǎn)損失率同期銀行利率第種投資的交易定額投資第種投資項(xiàng)

5、目的資金投資風(fēng)險(xiǎn)度總體收益總體收益的增量 六、模型的建立與分析1、 總體風(fēng)險(xiǎn)用所投資的中最大的一個(gè)風(fēng)險(xiǎn)來(lái)衡量，即max/=1,2,.n2、 購(gòu)買(mǎi)所付交易費(fèi)是一個(gè)分段函數(shù)，即交易費(fèi)= 而題目中所給定的定值（單位：元）相對(duì)總投資M很小，更小，可以忽略不計(jì)，這樣購(gòu)買(mǎi)的凈收益為3、 要使凈收益盡可能大，總體風(fēng)險(xiǎn)盡可能小，這是一個(gè)多目標(biāo)規(guī)劃模型： 目標(biāo)函數(shù) 約束條件4、 模型簡(jiǎn)化：a.在實(shí)際投資中，投資者承受風(fēng)險(xiǎn)的程度不一樣，若給定風(fēng)險(xiǎn)一個(gè)界限，使最大的一個(gè)風(fēng)險(xiǎn)，可找到相應(yīng)的投資方案。這樣把多目標(biāo)規(guī)劃變成一個(gè)目標(biāo)的線性規(guī)劃。 模型1 固定風(fēng)險(xiǎn)水平，優(yōu)化收益 目標(biāo)函數(shù)： 約束條件：b.若投資者希望總盈利至

6、少達(dá)到水平以上，在風(fēng)險(xiǎn)最小的情況下尋找相應(yīng)的投資組合。模型2 固定盈利水平，極小化風(fēng)險(xiǎn) 目標(biāo)函數(shù)： 約束條件：c.投資者在權(quán)衡資產(chǎn)風(fēng)險(xiǎn)和預(yù)期收益兩方面時(shí)，希望選擇一個(gè)令自己滿意的投資組合，因此對(duì)風(fēng)險(xiǎn)、收益權(quán)衡賦予權(quán)重,成為投資偏好系數(shù)。模型3 目標(biāo)函數(shù)： 約束條件 模型1的求解模型1為： 由于是任意給定的風(fēng)險(xiǎn)度到底怎樣給定沒(méi)有一個(gè)準(zhǔn)則，不同的投資者有不同的風(fēng)險(xiǎn)度，我們從開(kāi)始，以步長(zhǎng)進(jìn)行循環(huán)搜索，編制程序如下：a=0;while(1.1-a)1 c=-0.05 -0.27 -0.19 -0.185 -0.185; Aeq=1 1.01 1.02 1.045 1.065; beq=1; A=0 0

7、.025 0 0 0;0 0 0.015 0 0;0 0 0 0.055 0;0 0 0 0 0.026; b=a;a;a;a; vlb=0,0,0,0,0; vub=; a x=x Q=-val plot(a,Q,.),axis(0 0.1 0 0.5), hold on a=a+0.001;endxlabel(a),ylabel(Q)計(jì)算結(jié)果a = 0x = 1.0000 0.0000 0.0000 0.0000 0.0000Q = 0.0500Optimization terminated.a = 1.0000e-03x = 0.8316 0.0400 0.0667 0.0182 0.0

8、385Q = 0.0755Optimization terminated.a = 0.0020x = 0.6633 0.0800 0.1333 0.0364 0.0769Q = 0.1011Optimization terminated.a = 0.0030x = 0.4949 0.1200 0.2000 0.0545 0.1154Q = 0.1266Optimization terminated.a = 0.0040x = 0.3266 0.1600 0.2667 0.0727 0.1538Q = 0.1521Optimization terminated.a = 0.0050x = 0.1

9、582 0.2000 0.3333 0.0909 0.1923Q = 0.1776Optimization terminated.a = 0.0060x = 0.0000 0.2400 0.4000 0.1091 0.2212Q = 0.2019Optimization terminated.a = 0.0070x = 0.0000 0.2800 0.4667 0.1273 0.1016Q = 0.2066Optimization terminated.a = 0.0080x = 0.0000 0.3200 0.5333 0.1271 0.0000Q = 0.2112Optimization

10、terminated.a = 0.0090x = 0.0000 0.3600 0.6000 0.0233 0.0000Q = 0.2155Optimization terminated.a = 0.0100x = 0.0000 0.4000 0.5843 0.0000 0.0000Q = 0.2190Optimization terminated.a = 0.0110x = 0.0000 0.4400 0.5447 0.0000 0.0000Q = 0.2223Optimization terminated.a = 0.0120x = 0.0000 0.4800 0.5051 0.0000 0

11、.0000Q = 0.2256Optimization terminated.a = 0.0130x = 0.0000 0.5200 0.4655 0.0000 0.0000Q = 0.2288Optimization terminated.a = 0.0140x = 0.0000 0.5600 0.4259 0.0000 0.0000Q = 0.2321Optimization terminated.a = 0.0150x = 0.0000 0.6000 0.3863 0.0000 0.0000Q = 0.2354Optimization terminated.a = 0.0160x = 0

12、.0000 0.6400 0.3467 0.0000 0.0000Q = 0.2387Optimization terminated.a = 0.0170x = 0.0000 0.6800 0.3071 0.0000 0.0000Q = 0.2419Optimization terminated.a = 0.0180x = 0.0000 0.7200 0.2675 0.0000 0.0000Q = 0.2452Optimization terminated.a = 0.0190x = 0.0000 0.7600 0.2278 0.0000 0.0000Q = 0.2485Optimizatio

13、n terminated.a = 0.0200x = 0.0000 0.8000 0.1882 0.0000 0.0000Q = 0.2518Optimization terminated.a = 0.0210x = 0.0000 0.8400 0.1486 0.0000 0.0000Q = 0.2550Optimization terminated.a = 0.0220x = 0.0000 0.8800 0.1090 0.0000 0.0000Q = 0.2583Optimization terminated.a = 0.0230x = 0.0000 0.9200 0.0694 0.0000

14、 0.0000Q = 0.2616Optimization terminated.a = 0.0240x = 0.0000 0.9600 0.0298 0.0000 0.0000Q = 0.2649Optimization terminated.a = 0.0250x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0260x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0270x =

15、 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0280x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0290x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0300x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimizat

16、ion terminated.a = 0.0310x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0320x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0330x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0340x = 0.0000 0.9901 0.0000 0.00

17、00 0.0000Q = 0.2673Optimization terminated.a = 0.0350x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0360x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0370x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0380x

18、 = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0390x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0400x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0410x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimiz

19、ation terminated.a = 0.0420x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0430x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0440x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0450x = 0.0000 0.9901 0.0000 0.

20、0000 0.0000Q = 0.2673Optimization terminated.a = 0.0460x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0470x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0480x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.049

21、0x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0500x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0510x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0520x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optim

22、ization terminated.a = 0.0530x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0540x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0550x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0560x = 0.0000 0.9901 0.0000

23、0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0570x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0580x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0590x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0

24、600x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0610x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0620x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0630x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Opt

25、imization terminated.a = 0.0640x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0650x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0660x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0670x = 0.0000 0.9901 0.000

26、0 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0680x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0690x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0700x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0

27、.0710x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0720x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0730x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0740x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673O

28、ptimization terminated.a = 0.0750x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0760x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0770x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0780x = 0.0000 0.9901 0.0

29、000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0790x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0800x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0810x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a =

30、 0.0820x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0830x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0840x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0850x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.267

31、3Optimization terminated.a = 0.0860x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0870x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0880x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0890x = 0.0000 0.9901 0.0000 0.0000 0.0000Q = 0.2673Optimization terminated.a = 0.0900x = 0.0000 0.9901 0.0000 0.0000 0.