Excerpt

## Content

1.) Introduction

2.) Risk of an Investment

2.1.) Definitions of Risk

2.2.) Methods for the Determination of the Shortfall Risk

3.) Results of the Simulation

3.1.) The Investigation Field

3.2.) Benchmark: nominal Capital Maintenance

3.2.1.) Group A

3.2.2.) Group B

3.2.3.) Group C

3.2.4.) Group D

3.3.) Benchmark: real Capital Maintenance

3.3.1.) Group A

3.3.2.) Group B

3.3.3.) Group C

3.3.4.) Group D

4.) Conclusions

## 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.

## 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.

### 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 rn and the target yield is specified by rt.

illustration not visible in this excerpt^{7}

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.

illustration not visible in this excerpt

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.

## 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

7

- Fund C2

1 - 5 years 100% stock fund

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

- Fund C3

1 - 5 years 100% stock fund

6 - 10 years 90% stock fund, 10 % bond fund

11 - 15 years 80% stock fund, 10% bond fund, 10% real estate fund

16 - 20 years 70% stock fund, 20% bond fund, 10% real estate fund

21 - 25 years 60% stock fund, 20% bond fund, 20% real estate fund

26 - 30 years 50% stock fund, 30% bond fund, 20% real estate fund

- Fund C4

1 - 5 years 100% stock fund

6 - 10 years 90% stock fund, 10 % real estate fund

11 - 15 years 80% stock fund, 10% bond fund, 10% real estate fund

16 - 20 years 70% stock fund, 10% bond fund, 20% real estate fund

21 - 25 years 60% stock fund, 20% bond fund, 20% real estate fund

26 - 30 years 50% stock fund, 20% bond fund, 30% real estate fund

D) In group D, there is also a shift from the stock fund proportion into the bond and the real estate fund. But in contrast to group C, the part of the stock fund shrinks to 0% of the invested capital at the end.

- Fund D1

1 - 5 years 100% stock fund

6 - 30 years all 5 years 20% less stock fund, 10 % more bond fund and 10% more real estate fund

- Fund D2

1 - 10 years 100% stock fund

11 - 30 years annually 5% less stock fund and 2,5% more bond and real estate fund each.

- Fund D3

1 - 20 years 100% Stock fund

21 - 30 years annually 10% less stock fund and 5% more bond and real estate fund each.

- Fund D4

1 - 25 years 100% stock fund

26 - 30 years annually 20% less stock fund and 10% more bond and real estate fund each.

### 3.2.) Benchmark: nominal Capital Maintenance

#### 3.2.1.) Group A

As described the funds in group A are representative for an investment in either 100% of a stock fund (Fund A1), a bond fund (Fund A2) or a real estate fund (Fund A3). The results are represented in the following table 1.

illustration not visible in this excerpt

Table 1: nominal capital maintenance, group A, all values in % to the benchmark, n.a. = not available, because of no shortfall

The table above shows clearly the profitability of the stock markets. After 30 years the expected value (EV) of Fund A1 reaches 926% of the invested amount. In comparison Fund A2 increases to 352% and the investment in Fund A3 increases to 298%. The shortfall risk measurements of Fund A1 (the stock fund) have a higher level than the one’s of Fund A2 (the bond fund) or Fund A3 (the real estate fund). In addition the shortfall measurements for Fund A2 and Fund A3 reduce to 0% after 10 years. Both the shortfall probability (SP) and the shortfall expected value (SEV) show a decreasing trend for all three funds. The reason for this, is the timely risk diversification effect, which grows over the entire term of the savings plan. It is noticeable that the shortfall probabilities of Fund A2 and Fund A3 are relatively high in the first year and sharply decrease in the next years.^{10} This is an effect of the relatively high loading charge, which affects the performance of a short-term investment in both funds.

#### 3.2.3.) Group C

The funds of group B have shown that a mixture of stock, bond and real estate funds reduce the risk without a relinquishment of the strength in earnings of the stock markets. The funds of group C are following this approach. Their ratio of the stock fund is decreasing gradually until 50% at the end of the savings plan. The next table 3 presents the results of the simulation for group C.

illustration not visible in this excerpt

Table 3: nominal capital maintenance, group C, all values in % of the benchmark, n.a. = not available, because of no shortfall

The table shows that the expected final value after 30 years varies from 659% (C2) to 692% (C1). These results have a higher level than the Funds B1 and B2. It is remarkable that the risk measurements of all funds in group C shrink to 0% during the investment period. The shortfall probability and the shortfall expected value are decreasing continually over the savings plan. But in contrast to the funds of group B, they need almost 30 years to reach a value of 0% (B1: 10 years, B2 20 years). For the years 1 till 15, the measurement ALS has an increasing trend, which is typical for the stock fund. In the years 15 till 20, the reduction of ALS by the admixture of the bond and the real estate fund starts. In these years the ratio of the stock fund is still 70%.

The following illustrations shows the process of the measurement ALS for all funds of group C:

illustration not visible in this excerpt

3.2.4.) Group D

In group D there is like in group C a shift from the stock fund proportion into the bond and the real estate fund. The part of the stock fund shrinks to 0% of the invested capital at the end. The shift starts in the Funds D1 and D2 after 5 and 10 years respectively and in the Funds D3 and D4 after 20 and 25 years respectively. The following table 4 shows the result of the simulation.

illustration not visible in this excerpt

Table 4: naminal capital maintenance, group D, all values in % of the benchmark, n.a. = not available, because of no shortfall

The risk measurements of Fund D1 and D2 follow also a decreasing process and reach at the latest after 25 years a value of 0%. By comparison with Fund C1 (692%) the expected final value of the Funds D1 (486%) and D2 (582%) are on a lower level. The Fund D3 reaches after 30 years an expected final value of 724% and a value of 0% for all risk measurements.

At the end of the savings plan the shortfall probability and the shortfall expected value of Fund D4 are still positive on a low level. In contrast to these measures, the average loss in case of a shortfall (ALS) reach a high level (10%) after 30 years. The effect of a risk reduction, by admixing the bond and the real estate fund to the stock fund is not valid anymore, because the switch from the stock fund to the bond and the real estate fund starts too late. The following illustration shows the process of the measure ALS for all funds of group D. It is recognizable that this measure decreases for the funds D1, D2 and D3 at the end of the savings plan.

illustration not visible in this excerpt

### 3.3.) Benchmark: real Capital Maintenance

#### 3.3.1) Group A

The benchmark real capital maintenance affects all results of the simulation in a bad way. This is also visible in the following table 5 which contains the results of the simulation for group A.

illustration not visible in this excerpt

Table 5: real capital maintenance, group A, all values in % of the benchmark, n.a. = not available, because of no shortfall

The expected final value after 30 years shrinks to 556% (A1), 220% (A2) respectively 298% (A3). All risk measurements of Fund A2 and A3 reach only after 20 years (instead of 10 years) respectively after 10 years (instead of 5 years) a value of 0%. Again it is noticeable that the shortfall probabilities of Fund A2 (38,4%) and Fund A3 (64,9%) are relatively high in the first year and sharply decrease in the next years. In addition the expected final value after the first year decreases to 99,06% for Fund A3. This is an effect of the relatively high loading charge, which affects the performance of a short-term investment in both funds. In the following illustration, it is visible that the measurement ALS for Fund A1 shows an increasing trend which is typical for stock funds.

illustration not visible in this excerpt

#### 3.3.2.) Group B

The following table 6 shows the results for group B. It is noticeable that only the risk

measurements for Fund B1 decrease to a value of 0% at the end of the savings plan. But this takes place after 20 years (instead of 10 years).

illustration not visible in this excerpt

Table 6: real capital maintenance, group B, all values in % of the benchmark, n.a. = not available, because of no shortfall

illustration not visible in this excerpt

For the Funds B2 and B3 the average loss in case of a short (ALS) stays on a high level for the whole investment period. After 30 years, this risk measure still has a value of 10% (B2) respectively 15% (B3). These are bad results for products which are used for the private retirement. The following illustration shows that the ALS of Fund B2 is relatively constant for the years 1 to 15 and increases with a large amplitude at the end of the savings plan.

#### 3.3.3.) Group C

The following table 7 presents the results for group C. In contast to the benchmark nominal capital maintenance all risk measurements stay on a positive level. After 30 years the shortfall probability varies only between 0,3% and 0,4% and the shortfall expected value reaches a low level of 0,4%.

illustration not visible in this excerpt

Table 7: real capital maintenance, group C, all values in % of the benchmark, n.a. = not available, because of no shortfall

illustration not visible in this excerpt

For all funds the average loss in case of a shortfall (ALS) oscillates around the value of 15%. At the end of the savings plan, the ALS has a large amplitude. The following illustration shows that there is no effect of a risk reduction, by admixing the bond and the real estate fund to the stock fund.

3.3.4.) Group D

illustration not visible in this excerpt

Table 8: real capital maintenance, group D, all values in % of the benchmark, n.a. = not available, because of no shortfall

The risk measurements of the Funds D1 and D2 (early switch into the bond and the real estate fund) reach only at the end of the savings plan a value of 0%. The shortfall probabilities of the Funds D3 and D4 still have after 30 years a value of 0,8% respectively of 2,1%. The following illustration shows the process of the average loss in case of a shortfall. It is visible that this measure has a decreasing process after the shift into the bond and the real estate fund is started. The admixing of the bond and the real estate fund to the stock fund achieves this risk reduction effect. The extent of this effect depends on the period (beginning, middle or end) of the savings plan, in which the shift is started. For the Funds D3 and D4, this effect can not fully compensate the increasing process for the ALS of the stock fund.

illustration not visible in this excerpt

## 4.) Conclusions

In the simulation it is visible that the results of a continuous investment in stock, bond and real estate funds depend on the chosen benchmark. The German legislation requires the nominal capital maintenance and most of the mentioned investment funds reach this benchmark. By considering this benchmark the Fund D3 is the best one. After 30 years, the expected value of this funds increases to 724,96% and all risk measurements decrease to 0%. In the first 20 years, this fund is totally invested in the stock fund. After the 20th year the part of the stock fund shrinks gradually to 0% of the invested capital at the end of the savings plan.

Only five of fourteen funds reach the real capital maintenance as benchmark. Fund D2 is the best fund. After 30 years the expected value increases to 349,60% of the benchmark and all risk measurements decrease to 0%. For the first 10 years this fund is totally invested in the stock fund and then the shift into the bond and the real estate fund starts. Altogether this simulation shows that in the beginning of the savings period the investor should be invested for 100% in the stock fund. In the middle of the investment period the investor should switch his capital into the bond and the real estate fund. The level of the loading charge is important for the results of the investment in funds. This simulation considers the usual loading charges of the German mutual fund market for each savings ratio. But this simulation doesn’t consider the loading charges which have to be paid for a shift into another fund. Besides, it is possible to invest in funds at an internet bank (direct bank), which has a reduced loading charge (mostly 50%). It is noticeable that the risk measurement ALS of the stock fund has a high amplitude at the end of the investment period. This high amplitude arises because of the relative low numbers of simulation operations. By raising this number of simulation operations, the high amplitude would decrease and the increasing trend of the measure ALS would be clearly visible.

Altogether the investment in stock funds has the highest prospects but it follows the “law of efficient markets”^{11} which means that higher prospects always belong to higher risk.

## References

1.) Albrecht, Peter; Maurer, Raimond; Schradin, Heinrich R. (Albrecht 1999): Die Kapitalanlageperformance der Lebensversicherer im Vergleich zur Fondsanlage unter Rendite- und Risikoaspekten, Karlsruhe, Verlag Versicherungswirtschaft GmbH, 1999

2.) Gadient, Jürg (Gadient 2001): Life Cycle Asset Allocation, Bern, Stuttgart, Wien, Publikation der Swiss Banking School, 2001

3.) Fama, Eugene F.; Schwert, G. William (Fama 1977): Asset Returns and Inflation, Journal of Financial Economics 5, 1977

4.) Hollidt, Stefan (Hollidt 1999): Der Einsatz von Shortfall-Maßen im

Portfoliomanagement, Frankfurt am Main, Bankakademie-Verlag, 1999

5.) Kaduff, Jochen V. (Kaduff 1968): Shortfall risk basierte Portfolio-Strategien, Bern, Stuttgart, Wien, Bank- und finanzwirtschaftliche Forschungen Bd. 239, 1996

6.) Laux, Helmut (Laux 1995): Entscheidungstheorie, Berlin, Heidleberg, New York, Springer-Verlag, 1995

7.) Maurer, Raimond; Schlag, Christian (Maurer 2001): Investmentfonds-Ansparpläne: erwartetes Versorgungsniveau und Shortfall-Risiken, aus Sonderdruck: Der Langfristige Kredit 12-2001, S. 440-445, 2001

**[...]**

^{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

^{7} See Maurer (2001), page 441

^{8} See Maurer (2001), page 441

^{9} See Maurer (2001), page 444

^{10} The shortfall probability of the real estate fund is already after the second year low.

^{11} See Maurer (2001), page 440

- Quote paper
- Benjamin Wolf (Author), 2002, Life-Cycle Investing, Munich, GRIN Verlag, https://www.grin.com/document/25232

Comments