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Content

1.)  Introduction  03

2.)  Risk of an Investment  04

2.1.)  Definitions of Risk  04

2.2.)  Methods for the Determination of the Shortfall Risk  05

3.)  Results of the Simulation  06

3.1.)  The Investigation Field  06

3.2.)  Benchmark: nominal Capital Maintenance  08

3.2.1.)  Group A  08

3.2.2.)  Group B  09

3.2.3.)  Group C  11

3.2.4.)  Group D  12

3.3.)  Benchmark: real Capital Maintenance  14

3.3.1.)  Group A  14

3.3.2.)  Group B  15

3.3.3.)  Group C  16

3.3.4.)  Group D  17

4.)  Conclusions  18

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1.) Introduction

In May 2001, the “Gesetz zur Reform der gesetzlichen Rentenversicherung und zur Förderung eines kapitalgedeckten Altersvorsorge Vermögens (AvmG)” was passed by the German legislation. The target of this law is to encourage the private retirement provision with additional governmental extra pay or with tax deductions in terms of special expenses.

The purpose of this paper is to give an overview of some possible strategies for the capital spending in investment funds. These strategies are both partly static and dynamic.

A basic method to measure the risk of such investment strategies is the volatility. Another approach of measuring risk is explained in chapter 2 and then used in chapter 3 for a simulation. This simulation considers the nominal capital maintenance which is required by the German legislation to receive the governmental relief (encouragement). Furthermore in this chapter an analysis with the objective of the real capital maintenance is held.

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2.) Risk of an Investment

2.1.) Definitions of Risk

Often decisions can only be made under uncertainty. Uncertainty means, that a decisionmaker supposes at least two possible results for a future event whereupon one will certainly happen. In the literature, two cases of uncertainty are distinguished. ¹ On the one hand, if the decision-maker has no probabilities for the future results then this is called uncertainty in a narrower sense. On the other hand, if the decision-maker has a probability for all possible future results this is designated risk. Now the risk of an investment decision is to miss a desired financial target in the future. The volatility of an asset is a possible risk measurement which specifies the fluctuation margin over the mean value of this asset. This deviation can be over or underneath the mean value. Many investors only assess a deviation underneath the mean value as a risk, because only in this situation they miss their financial target. Roy (1952) ² has described for the first time this idea of risk and designated it as “Probability of Disaster” or “Chance of Disaster”. He quantified this with the probability to undercut the defined minimum yield. Later on this risk measurement was characterized as “Shortfall Probability” by Leibowitz and Henriksson (1989) ³ . According to Roy investors are searching for the portfolio which minimizes the shortfall probability and generates their defined minimum yield. He called this principle as “Safety First”. ⁴ Kataoka (1963) ⁵ varied this principle by defining the shortfall probability and then searching for the portfolio with the maximum yield which will fall short of this probability. The approach of Telser (1955) ⁶ was chronologically prior to Kataoka (1963), but is a combination of the procedures of Roy and Kataoka. He fixed the minimum yield and the shortfall probability and searched the portfolio with the maximum yield which fulfills both preconditions.

¹ See Laux (1995) page 25

² See Kaduff (1996) page 85

³ See Kaduff (1996) page 85

⁴ See Kaduff (1996) page 85

⁵ See Kaduff (1996) page 86

⁶ See Kaduff (1996) page 86

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2.2.) Methods for the Determination of the Shortfall Risk

Depending on his individual risk attitude, an investor defines a target yield as a benchmark which is the starting point for the following calculations. Each undercut of the fixed target yield is an undesirable result of the investment and therefore a shortfall. First of all, there has to be a definition of an indicator function in order to estimate the theoretical shortfall measurements. In this function the achieved yield in year n is specified by r n and the target yield is specified by r t .

(1)

The measurement to quantify the shortfall risk is the shortfall probability (SP). ⁷ It is calculated by the shortfall frequency.

1 N

(2)

This measurement declares the probability to miss the defined benchmark. The shortfall probability gives no information about the level of the loss. This is determined by the average loss in case of a shortfall (ALS). ⁸ This index gives the average level of the loss, if there is a shortfall.

(3)

The last measurement is the shortfall expected value (SEV). It is calculated by the multiplication of ALS and SP.

(4) SEV := SP * ALS

The shortfall expected value gives the average level of the loss.

⁷ See Maurer (2001), page 441

⁸ See Maurer (2001), page 441

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3.) Results of the Simulation

3.1.) The Investigation Field

Starting point of the investigation is an investor who spends a certain amount each year in advance in a stock fund, in a bond fund, in a real estate fund or in several mixtures of these three funds. 1000 alternative future performances, for each of these three funds, were created with the help of a simulation and the relevant risk measurements were then calculated using these performances.

In line with the German mutual fund market the investor also has to pay a loading charge of 5% for the stock and the real estate fund and 3% for the bond fund for each savings installment. In addition all funds are reinvesting, that means that return of the funds is immediately reinvested into the fund, without consideration of any loading charge. Furthermore, no loading charge has to be paid when restructuring the fund’s investment. The length of the accumulation plan is 1 to 30 years. The expected final value (EV) and the risk measurements presented in chapter 2.2 are used to compare the investment strategies.

The examination includes four different groups, whereby each contains three or four different strategies of investing in funds. The benchmark for all strategies is the nominal or the real capital maintenance. The examinations refers to the following groups:

A) The first group consists of a 100% investment in each a stock fund (Fund A1), in a bond fund (Fund A2) or in a real estate fund (Fund A3). B) The funds of group B consider the investment boundaries for Altersvorsorge-Sondervermögen as of § 37i KAGG. 9

• Fund B1 (conservative): 21% stock fund, 49% bond fund, 30% real estate fund

• Fund B2 (balanced): 50% stock fund, 35% bond fund, 15% real estate fund

• Fund B3 (yield-oriented): 75% stock, 20% bond and 5% real estate fund C) In each fund of group C, a shift takes place so that the part of the stock fund is 50% of the invested capital at the end of the savings plan.

• Fund C1 1 - 5 years 100% stock fund

6 - 30 years all 5 years 10% less stock fund and 10 % more bond fund

⁹ See Maurer (2001), page 444