Investment Return Requirement

Joachim Kuczynski, 02 July 2022

In this post I want to give a derivation of the return requirement of an additional investment opportunity for an investor having an existing investment / security portfolio. In my point of view this is the key point of portfolio theory to understand the discounting of cash flows in a DCF analysis.

Let us assume an investor which owns a portfolio of investments or securities with relative shares x_i having annual return rates R_i and standard deviations of the annual return rates \sigma\left( R_i \right). The variance of the portfolio return rate is given by:

    \[var\left( R_P \right)=\sum_{i}^{}x_i\text{cov}\left( R_i, R_P \right) \text{, or}\]

    \[ var\left( R_P \right) =\sum_{i}^{}x_i\sigma\left( R_i \right)\sigma\left( R_P \right)\text{corr}\left( R_i, R_P \right)\]

Dividing both sides by standard deviation \sigma\left( R_P \right) gives the standard deviation \sigma\left( R_P \right):

    \[\sigma\left( R_P \right)=\sum_{i}^{}x_i\sigma\left( R_i \right)\text{corr}\left( R_i, R_P \right)\]

That means that the incremental risk contribution of each investment to the risk of the portfolio ist \sigma\left( R_i \right)\text{corr}\left( R_i, R_P \right).

Instead of including a new investment into the portfolio the investor can also increase the return of the protfolio by increasing the risk of the portfolio. This reward-to-volatility ratio of the tangential portfolio is given by the Sharpe Ratio:

    \[\frac{E\left( R_P\right)-r_f}{\sigma \left( R_P \right)}\]

E\left( R_P\right) is the expected value of R_P and r_f is the risk-free or default-free rate. The investor wants to invest in the new opportunity, if the additional return rate of this investment is higher than an investment in the existing portfolio with the same risk changes. Hence we obtain the requirement to invest in the new investment opportunity:

    \[\text{E}\left( R_i \right)-r_f > \sigma\left( R_i \right)\text{corr}\left( R_i,R_P \right)\frac{E\left( R_P\right)-r_f}{\sigma\left( R_P \right)}\]

With that we can define the sensitivity \beta_i^P of the new investment to the existing portfolio:

    \[\beta_i^P=\frac{\sigma\left( R_i \right)\text{corr}\left( R_i,R_P \right)}{\sigma\left( R_P \right)}\]

Substituting with \beta_i^P the requirement for the new investment becomes the well-known equation:

    \[\text{E}\left( R_i \right) > r_f+\beta_i^P\left( E\left( R_P\right)-r_f \right)\]

With that we can define a minimal annual return rate of the investment r_i:

    \[ r_i = r_f+\beta_i^P\left( E\left( R_P\right)-r_f \right) \]

This is the right (leveraged) discount rate for cash flows financed by equity. It is the minimum rate at which an investor would decide to allocate the new investment opportunity in his portfolio, because the expected risk-adjusted return rate is higher that the risk-adjusted rate of the existing portfolio. It is easy to see that each cash flow has to be discounted with its specific risk-adjusted rate, when they have different risks. Because of the additivity of net present values the investor can discount each cash flow seperately and sum up the NPV of all cash flows. This is called component cash flow procedure, see this post.

If cash flows are financed by debt and equity, the discount rate is the weighted average of debt and equity return rate requirements (WACC).

Quite often the portfolio of a so called marginal investor is not known in detail. One possibility is to assume that his portfolio consists of all available securities in the market with its specific weighted shares. With the assumptions of the Capital Asset Pricing Model (CAPM) the efficient tangential portfolio is the market protfolio and the expected portfolio return rate is the expected return rate of the market. In most cases the S&P500 is taken as reference portfolio.

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