How to discount cashflows

Clearly, some cash flows are riskier than others. Equally clearly, riskier cashflows are worth less than less risky cash flows. In order, therefore, to value risky cash flows, we need to know two things:

• How to measure risk, and
• how to discount risky cash flows.

Risky cash flows are, perforce, future cash flows. We know that we can value a future cash flow by discounting it at a given rate. Similarly, we can value risky future cash flows by discounting it at a higher rate than future riskless cash flows. The question, then, is: how much higher should a risky discount rate be than the corresponding riskless rate? Or, said differently, what is the risk premium per unit risk? But first, we need to explore the nature or risk, itself.

Risk, essentially, has to do with uncertainty. The riskiness of an asset has to do with the uncertainty regarding the quantity of future cash flows that it will generate. Although we will be interested in measuring the riskiness of individual assets, primarily we care about the riskiness of the entire portfolio of assets that we possess, i.e our entire wealth. We will see, in short order, that our focus on asset portfolios will affect our definition of the riskiness of individual assets.

Given our ultimate concern, we first define the riskiness of a portfolio. This is defined as the variance of returns on the portfolio. The variance of returns is the average squared deviation from the mean return. One justification for using this measure of risk is that historically, portfolios with higher return variances have yielded higher average returns. variance of returns have, in the past, yielded higher returns.

Portfolio Variance as a Measure of Portfolio Risk

The average return level is usually described by the expected value of the return and the risk is usually described by the variance of the return. The role of the expected return is clear: it gives us an idea of how high, the return is likely to be, on average. But how does variance measure risk?

Since the variance measures the spread of the distribution, we can learn something about the extent of uncertainty that exists regarding the return on the investment by looking at the variance. So variance measures risk in the sense of uncertainty.

However, some people intuitively associate risk with undesirable outcomes alone. If we think of the mean return as a neutral outcome, returns less than the mean would be bad outcomes and returns more than the mean would be good outcomes. If the return distribution is symmetric, this also means that good outcomes are as likely as bad outcomes. Since the variance measures the spread of the distribution around the mean, in this case, it would also give us a measure of how bad a bad outcome is likely to be, on average. However, if the distribution is not symmetric, then this is not true, and we need other measures such as skewness (the extent to which the distribution is skewed or tilted in one direction, instead of being symmetric) to give us this information. For symmetric distributions, like the normal distribution, the variance is a good measure of risk in this sense as well.

In what follows, we will interpret the concept of risk to mean uncertainty. However, the notion of variance as a measure of risk works only for portfolios containing the entire wealth of an individual —it does not work for the individual assets that make up the portfolio.

Risk Of Individual Assets Vs. The Risk Of Portfolios

The variance of returns can be used as a measure of risk if we are evaluating the entire portfolio of investments that an investor has. However, if the investment under consideration is meant to be an addition to an existing portfolio, then the variance of the asset return is not appropriate as a measure of risk of this investment. This is because we are interested primarily in the risk of our entire portfolio, and as far as this new asset is concerned, we would be interested in the incremental risk of the investment. The appropriate measure of the incremental risk of an investment differs from the variance, which is a measure of the total risk of the investment.

Consequently, we need to investigate how the variance of a portfolio is affected by the addition of a single asset. First, we need to define a portfolio in a more formal manner, and see how its variance is related to the variances of the assets that make up the portfolio.

Portfolios Defined

A set of assets is called a portfolio. If there are N assets in the economy, then we can define a portfolio by an N-vector containing the proportion of wealth invested in each of those N assets. Hence, if a particular portfolio consists of $500 invested in asset 4 and $500 invested in asset 2, the portfolio will be defined by the set of N numbers: (w₁. w₂. w₃. w₄. w_n ) = (0, 0.5, 0, 0.5, 0, 0. 0), i.e. w_i is the proportion of the investment in the entire portfolio that is put in the i th asset.

The return on a portfolio is related to the returns on the individual assets by the formula . and hence, we have , where r_i is the return on the i th asset, and m_i is the expected return on the i th asset. However, the expression for the variance, denoted by s 2. is more complicated, and is given by the following expression, where the subscript p refers to the portfolio, and s_ij refers to the covariance of returns between the i th and j th assets.

where the subscripts i. j in the second summation sign indicate that the sum is to be taken over all pairwise combinations of assets.

This expression can be visualized more easily in the following arrangement.