• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## 2.4.1 Valuing Probability Distributions

One possible method would be to estimate the relative probability of different future possibilities and then discount them to present value. For example, if we believed it was equally likely that an investment after one year would produce either \$5 or \$12 or \$20, and the current rate for a risk-free one-year investment were 6.3%, we could compute the investment's present value:

 Return Probability Expected return Present value \$5 33.33% \$1.67 \$1.67 / (1 + .063) = \$1.57 \$12 33.33% \$4.00 \$4.00 / (1 + .063) = \$3.76 \$20 33.33% \$6.67 \$1.67 / (1 + .063) = \$6.27 Total \$12.33 \$11.60

But this method depends on estimating the probabilities of many different results. If you assume a different probability distribution (let's say, 50-25-25) the expected returns change, and the present value is \$9.88 -- a significantly lower valuation.

In addition, and more problematic, this method does not take into account the variability of returns (the dispersion of the return distribution). If we assume an investment with returns more concentrated around the mean, but with the same overall expected return, we get the same present value.

 Return Probability Expected return Present value \$10 33.33% \$3.33 \$3.33 / (1 + .063) = \$3.14 \$12 33.33% \$4.00 \$4.00 / (1 + .063) = \$3.76 \$15 33.33% \$5.00 \$5.00 / (1 + .063) = \$4.70 Total \$12.33 \$11.60

But this cannot be right. We know that risk averse investors will value less variable returns more highly. There has to be some way to take into account risk volatility.

 2.4 Relationship of Risk and Return 2.4.2 Valuing Certainty Equivalents