V. Reznik and U. Spreitzer Dr. Dr. Heissmann GmbH, Abraham-Lincoln-Str. 22, Wiesbaden.

Transcription

1 n investigation of a portfolio-loss uner the CPM V. eznik an U. Spreitzer Dr. Dr. Heissmann GmbH, braham-incoln-str., 6589 Wiesbaen. bstract: We consier a portfolio built accoring to the Capital Market ine of the Capital-sset-Pricing Moel. The universe of asset classes inclue marketable shares an bons only. We investigate losses that emerge when the rate of return of the portfolio is lower than that require to fulfil a efine obligation. We will classify these losses an calculate upper limits for them. ntrouction n Germany, pension liabilities that are finance internally by so calle book reserves, support fun or pension funs are almost entirely unerwritten by the Pensionssicherungsverein auf Gegenseitigkeit PSV ag. This institution levies an insurance premium proportional to the value measure by the accrue actuarial liability using the Entry ge Normal Valuation Metho with fixe assumptions of the legally veste benefit obligation. The funing metho use by the PSV ag can be escribe as a Pay-s-You-Go metho until retirement commencement. For retirees, however, the full present value on very conservative assumptions is provie to a consortium of insurers, that account for erience gains an losses in an own profit an loss statement towars the PSV ag Terminal Funing. The central question to ask therefore is: what percentage of the obligation is use as a premium over the last 3 years, the percentage has been fairly volatile, an is this premium a goo measure of the risk not to fulfil the obligation? Several reports still suggest a premium which is proportional to the value of the legally veste obligation but that this premium shoul be moifie in accorance with the ifferent pension vehicle pension fun, support fun, book reserve, etc. in operation. n this publication we give a efinition of this risk or loss, base on the so calle lower partial moments as publishe by Fishburn. We consier a portfolio consisting of bons an shares. Portfolios containing erivatives an other instruments such as futures, options, etc. are beyon the scope of this paper. We escribe the shares as assets with a variable rate of return. We assume that the capital market can be escribe with the Capital sset Pricing Moel CPM 3, an that the investment of the portfolio correspons to the Capital Market ine of the CPM. Our moel is a one perioic moel. We give a measure of this loss as a function of ecte rate of return, variance, etc. of the assets. CP_oss_f p...4

2 Selection of losses We have selecte three ifferent losses: the average loss, the maximum loss M an the valueat-risk-loss E. We assume that the rate of return an the probability ensity p of the portfolio are ranom variables with real values. From this we conclue that the following integrals which efine the losses are well efine.. The average loss We efine p is the rate of return, which is essary to fulfil obligations. is the amount of the investment. The probability to obtain a loss larger than is: p with 3 p. We use instea of: 4 p because the yiel above,is not of immeiate interest. ather, we focus on the value of the loss.. The maximum loss M We efine M min 5 M p. We call min the minimum rate of return. This means that for a rate of return above min we assume that the loss is not relevant..3 The value-at-risk loss V V is the loss which is cause by the very low rates of return from the portfolio. We efine it similar to the Value-at-isk 4. Therefore we consier the -γ-quantil, which is efine as follows: 6 P Va γ. Va is the rate of return, for which only γ of all are below. CP_oss_f p...4

3 V is therefore Va 7 V p Concerning value-at-risk there is also the following efinition 4 : 8 Va z γ with z γ is the γ -Quantil of stanar normal istribution. 3 Estimation of loss of the portfolio We consier a portfolio whose investment is a combination of the bons an the so calle tangency portfolio which consists of assets only. The rate of return of this pension fun portfolio is: 9 the ecte rate of return is E or an the stanar eviation with is the rate of return, the ecte rate of return, the stanar eviation of the Tangency portfolio an the rate of return of the bons. lso we assume. This investment on this tangent is also calle an investment on the capital market line 3. Using we obtain for the loss : We efine 3 p p. p. n the following we will use only p an we call this p. From this we conclue: 4 p. M an V we can calculate in a analogue manner. 4 oss of the portfolio, when the rate of return of the portfolio is normal istribute We assume, that is normally istribute: CP_oss_f p.3..4

4 CP_oss_f p p. Calculations with other ensities of probability may be one in other publications. From this we conclue: 6 using 7 x x we obtain 8 an 9. Using

5 CP_oss_f p.5..4 a a x x x a a we conclue Using 9 an we can give an upper limit for : This can be written as: 3 f f with 4 f or as 5 We efine the function g, : 6 g. an therefore follows: 7 g. M an V we can measure similarly.

6 We want to know how the investment on the Capital Market ine changes from bons to assets an vice versa when the investor emans that the losses shoul not excee a value which he efine a priori. We efine, which is an upper limit for, M or V, as proportional to the investment, with a rate λ: 8 λ with 9 λ g. We have limite as follows 3. When i.e. that agent invests only in bons we obtain: 3. When, i.e. investor invests only in the tangency portfolio an we obtain: 3 λ g. Using or 7 we can ecie whether a loss which is given a priori can be reache with the characteristics of the portfolio as, an. ssuming, that the tangency portfolio also an is fixe e.g. DX 3 we can ecie whether changing can reuce the loss. 5 Conclusion Using a one-perio moel we have efine the loss which an investor can suffer when he wants to fulfil obligations by using the returns on his portfolio. We measure these losses as a function of the composition of the portfolio, of the ecte rate of return of the assets, of the rate of return of the bons an of the variance of the asset. The moel prepare in this paper has limitations. n aning this moel the following points shoul be taken into consieration: More than one perioic moel. ifferent composition of the portfolio e.g. incluing futures, options, etc. Different istributions of probabilities in rates of return. Gerke W., Heubeck K., Gutachten zur künftigen Funktionsfähigkeit er nsolvenzsicherung urch en Pensions- Sicherungs-Verein VVaG, BetrV 5, CP_oss_f p.6..4

7 Fishburn, P.C., Mean risk analysis with risk associate with below-target returns, merican Economic eview 67, Sharpe, W.F., Capital sset Prices: Theory of Market Equilibrium uner conitions of risk, The Journal of Finance 9, Klüppelberg, C., Korn., Optimale Portfolios mit beschränktem Value-at-isk, Solutions 3, CP_oss_f p.7..4