Excerpt

## Table of Contents

Illustration Overview

Table Overview

Equation Overview

Abbreviation Overview

1. Introduction

2. Theoretical Foundation of Beta

3. Concept of Duration

4. Portfolio Management with Beta and Duration

4.1. Beta

4.2. Duration

5. Limitations of these models

5.1. Beta

5.2. Duration

6. Conclusion

References

## Illustration Overview

Figure 1: Diversification of unsystematic risk

Figure 2: Linear regression Daimler DAX

Figure 3: Effect of Duration

## Table Overview

Table 1: Duration example

Equation Overview

Equation 1: Systemic Risk Measured by Beta

Equation 2: Duration

Equation 3: Modified Duration

## Abbreviation Overview

illustration not visible in this excerpt

## 1. Introduction

Since the financial crisis of 2007/2008 risk management become a boost in financial institutions. The crisis has shown that the risk management of most institutions are inefficient, their models inadequate and that regulation failed their aim to avoid such a major crisis (Bessis, 2010).

To identify, measure, control and price risk and to estimate the effect on a portfolio is a hard task because it is a look towards the future. But it is essential because it has an impact on the profitability, the solvency and so on to the future survival (Sironi and Resti, 2007, p. xxii).

This paper describes two models of measuring risk, the theoretical foundation of Beta and the concept of Duration. Furthermore a quantified demonstration of these models is provided to show the practical implementation. However, every model has limitations which are critical shown in the last chapter and in the last chapter a general conclusion is stated.

## 2. Theoretical Foundation of Beta

Beta is a measure of the volatility of a single asset or a portfolio compared to the whole market. Hereby Beta describes the systematic risk, i.e. market risk, which cannot be diversified. Markowitz (1952, pp. 77 ff) found that it is possible to diversify a part of the risk of a portfolio which is the risk of a particular share or rather the unsystematic risk. Depending on the correlation coefficient be- tween the assets in the portfolio the total risk decreases with the number of in- vestments as shown in Figure 1.

Figure 1: Diversification of unsystematic risk Source: (Watson and Head, 2007, p. 214)

illustration not visible in this excerpt

The market portfolio has a Beta of exactly one. A Beta lower than 1 means that the portfolio is less volatile than the market and a Beta above 1 means that the asset is more volatile than the market (Scheld, 2013). At a Beta of 1 the asset is as volatile as the market. Also negative Betas imaginable in theory what means that the asset moves in the opposite direction as the market. Conceivable would be gold but in empirical studies the proof failed. Beta is calculated by the co- variance between the portfolio and the market and the variance of the market.

illustration not visible in this excerpt

Equation 1: Systemic Risk Measured by Beta

Source: Author’s Illustration based on (Steiner et al., 2012, p. 25)

Beta is also used in the Capital Asset Pricing Model (CAPM) founded by Sharpe (1964), Treynor (1961) and Lintner (1965) to calculate the required return. Hereby an investor gets only compensated for the systematic risk he takes because the unsystematic risk can be diversified.

Beta is calculated through Equation 1 or rather through a linear regression as shown in Figure 2. Both are coming to the same numbers. To do these calcula- tions the data basis is crucial. Comparable are performance market indices and performance stock data or price indices with stock prices. For the example the DAX as a performance index and adjusted data for Daimler to take dividends into account are used from the 28.01.2000 until the 10.11.2014 on a weekly basis. Therefore the covariance is 0.0013085 and the variance is 0.00111587. The covariance divided by the variance produce a Beta of 1.17. The same number is provided by a chart and a trend line through the data points. The slope (1.17) of the function is Beta. Beta can be interpreted that Daimler is more volatile than the market. If the market moves 1 % Daimler moves on average 1.17 %. Furthermore with the standard error a confidence interval can be built in which Beta lies with a probability of 95 %. Therefore the standard error (0.034481) is subtracted and added twice. So the statement can be formed, that with a probability of 95 % Beta is between 1.11 and 1.24.

Figure 2: Linear regression Daimler DAX

illustration not visible in this excerpt

Source: Author’s Illustration based on data from (Yahoo Finance)

## 3. Concept of Duration

The concept of Duration is to calculate a point in time where the effect of changing interest rates is immunized (Bierwag, G. O., 1977) (Reilly and Sidhu, 1980). At this point in time the effect of the changed value and the changed interest rate for reinvested interest payments are in balance. Furthermore with the modified Duration it is possible to measure how sensitive an asset is if the market yield rate changes (Macaulay, 1938, pp. 45 ff).

The Duration of financial instruments is calculated with the weighted average of the times when cash flows are received (Brealey and Myers, 2003, p. 239).

illustration not visible in this excerpt

Equation 2: Duration

Source: Author’s illustration based on (Brealey et al., 2011, p. 79)

In Table 1 the Duration is calculated for a bond with a maturity of five years, a face value of 1000 Euro, a coupon of 8 % and a discount rate of 10 %. First the cash flows are calculated and then discounted. In the third row all present val- ues (PV) are weighted from the total PV. Next the weighted PVs are multiplied by the years and then summarized. So the outcome for the Duration is 4.28 years.

illustration not visible in this excerpt

Table 1: Duration example Source: Author’s illustration

In Figure 3 the Duration is visualized and both mentioned effects as well. So in the example in 4.28 years (4 years and 102 days) both effects are in balance (immunization function). The graph is for an increasing yield. For sinking inter- est rates it is vice versa. The present value increase and through lower interest rates on the reinvestments of the cash flow the end value decreases.

illustration not visible in this excerpt

Figure 3: Effect of Duration

Source: Author’s illustration based on (Callsen-Bracker, 2013)

To calculate the modified Duration as a measurement for the sensitivity to interest rate changes the following equation is used.

illustration not visible in this excerpt

Equation 3: Modified Duration

Source: Author’s illustration based on (Brealey et al., 2011, p. 80)

Ergo the statement can be made that if the market interest rate changes 1 % the bond change 3.89 % in the opposite direction. Generally can be said that the shorter the maturity and/or the higher the coupon that leads to a smaller Duration and modified Duration. For zero bonds the maturity is equal to the du- ration because there is only one cash flow at the end (Sironi and Resti, 2007, pp. 39 ff).

**[...]**

- Quote paper
- Christoph Schubert (Author), 2014, Beta and Duration as Measurements of Future Risk and Returns, Munich, GRIN Verlag, https://www.grin.com/document/299992

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