Abstract
This paper studies the way to introduce risk into financial analysis. It suggests and analyzes metrics to improve the classic methods, like the Sharpe and Treynor ratios or the Net Present Value.
In particular, the paper suggests a linear penalization method developed by the authors, applying it to performance indexes and capital budgeting, and considering both total and systematic risk. The method shows obvious advantages in comparison with traditional methods.
JEL classification: G11; G31
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Notes
- 1.
Although some exceptions have also been pointed out.
- 2.
- 3.
- 4.
Although asset valuation models based on real options are also being increasingly applied for the last years, we are not entering here into their discussion.
- 5.
The NPV is considered, in a classic reference in finance, as is Brealey et al. (2008), as the first one of the seven most important ideas in finance. In order to introduce risk into the NPV, the most widely applied systems are the risk-adjusted rate of return valuation formula and the certainty equivalent valuation formula (Copeland et al., 2005, pp. 156–157). We will use an alternative system, derived from the NPV: the Penalized Present Value (PPV).
- 6.
Although both subjects have been and will be topic of discussion.
- 7.
Expected value.
- 8.
And we will explain later the meaning of the PIRR.
- 9.
It might be argued that somebody could, by borrowing money at the risk-free rate, start from “B” and reach an investment that outperformed “A”, and this is true. However, small investors cannot get themselves into debt at the risk-free rate. Besides, the performance of an investment fund is known ex-post, and by then, it is impossible for any investor to leverage his or her investment.
- 10.
As the market variance will be equal for all the portfolios.
- 11.
The classification obtained is the same as with M2 for beta (Modigliani, 1997).
- 12.
We will explain the PIRR for beta later.
- 13.
Dispensing here with the j subindex.
- 14.
For this reason we are employing the ≈ symbol.
- 15.
This would allow to measure risk with β and the use of Treynor.
- 16.
Out of the components of total risk, the diversifiable risk will disappear, and the market standard deviation is the same for every portfolio. Therefore, the only difference will be β.
- 17.
It could also be reasoned in this way: the market Sharpe, which is the slope of the CML, shows the "price of risk", which is the increase in the expected return that is demanded for a risk increase. Consequently, it is logical to give such a value to t in the formula (60.5).
- 18.
The use of indifference straight lines is a constraint of the system, which can be mitigated by the use of increasing t values as the risk grows. Sharpe also employs indifference straight lines, but instead of being parallels, they form a beam.
In Figure 60.1, following the Sharpe ratio, all the portfolios located on the same straight line starting from r0 would be indifferent. For example, if we focus on the straight line linking r0 and Ra, and according to the Sharpe ratio, all the portfolios on this straight line are indifferent to Ra, because their Sharpe value is the same: the tangent of αa. At the same time, these portfolios would be worse than those located on the straight line linking r0 and Rb, because the latter ones have a higher Sharpe. This is the consequence of Sharpe using a system of indifference straight lines forming a beam.
In the case of the PIRR system, indifference straight lines are drawn in parallel to the CML. The crossing point with the vertical axis is the PIRR, and its value is bigger the higher the straight line crosses with the vertical axis.
If we applied increasing t values, as it was mentioned before, we would get closer to obtain indifference curves and we could reach to a solution quite similar to the standard one.
- 19.
Discounted at the risk-free rate, as the risk penalization is applied afterwards.
- 20.
Use of indifference straight lines, statistical meaning of t… Regarding t value, it should grow with the investment period as well as the market Sharpe tends to do so.
- 21.
Here, as in some other occasions, we dispense with i subindex for NPV, not to make notation more complicated.
- 22.
This is the Jensen α.
- 23.
It can be adapted, for instance, from Copeland et al. (2005, p. 157).
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Gómez-Bezares, F., Gómez-Bezares, F.R. (2013). An Analysis of Risk Treatment in the Field of Finance. In: Lee, CF., Lee, A. (eds) Encyclopedia of Finance. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5360-4_60
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