abstract valuation theory
Let M be a DCF (Discounted Cash Flow) valuation model. M could, for instance, be an Excel spreadsheet model constructed by an investment bank analyst for the purpose of determining the enterprise value of a company. More generally, M can be regarded as a function from S to R^∞ where S is the space of all possible configurations of assumptions and R^∞ is the infinite dimensional real vector space. Each point s∈S encodes all the assumptions needed for the valuation of the company, and each v∈R^∞ is an infinite vector of future cash flows where v_t is the cash flow at time t.
We will assume that S is large but nevertheless finite.
For an illustration of the new terminology, note the following two observations:
- The technique of scenario analysis can be broken down into two simple steps. First select a small subset S' of S, and then present the value of M at each s∈S'.
- The technique of sensitivity analysis consists of a depiction of what happens to M(s) as s∈S varies.
Analysts try hard to come up with the right discount rate for the company being valued. This calculation is important since the NPV (Net Present Value) of a company is determined uniquely by the discount rate d (a positive real number) and the projected stream of cash flows v.
Let R^e stand for the extended real number line and R^+ stand for the positive real numbers. Then NPV can be viewed as a function from (R^∞)x(R^+) to R^e. It maps a pair (v,d) to Σ(v_t/((1+d)^t)) where the infinite summation is taken from t=1 to ∞. (Note that this infinite series does not necessarily converge for a given (v,d). That is why we set the range of NPV as the extended real number line.)
For the determination of the appropriate discount rate, analysts either rely on garden-variety asset pricing models (such as CAPM) or create their own models. Each such model D can be viewed as a function whose domain is S and whose range is R^+. This makes sense since the calculation of the discount rate depends on the set of assumptions relevant to the valuation of the company.
Remark Both M and D are defined in such a generality that even intuition-based guesses qualify to be considered as "models".
Let P be a probability measure on S. Think of P as a function from S to the closed line segment [0,1], assigning probability P(s) to scenario s∈S. Moreover, P is chosen in such a way that the sum of all P(s) is 1. (i.e. It is believed that at least one of the scenarios in S will be realized.) Introduction of a probability measure into the picture allows us to utilize analyst's beliefs about the likelihood of each set of assumptions in S. Some scenarios, for instance, will be so nonsensical that they will be assigned zero probability.
Let's see what sorts of meaningful definitions we can make about the space of all valuation models M.
For the first five definitions, fix some arbitrary D.
Definition 1 We say that s∈S is M-equivalent to s'∈S if NPV(M(s),D(s)) = NPV(M(s'),D(s')). In other words M gives the same valuation under both scenarios.
Definition 2 We say that model M is more (s,s')-sensitive than model N if |NPV(M(s),D(s))-NPV(M(s'),D(s'))| is greater than |NPV(N(s),D(s))-NPV(N(s'),D(s'))|.
Definition 3 Expected-P-NPV of M, denoted by ∫(P,M), is Σ(NPV(M(s),D(s))*P(s)) where the finite sum ranges over s∈S. This quantity gives you the NPV expected by the analyst who assigns probability P(s) to scenario s∈S. Even if two analysts have wildly different probability measures P and P', they may end up agreeing on the expected NPV of M. In other words, ∫(P,M) may be equal to ∫(P',M) even if P≠P'. Although the sum involved in the definition is finite, ∫(P,M) may nevertheless be infinite because each NPV(M(s),D(s)) is an infinite sum. Of course, for reasonable M and P, either NPV(M(s),D(s)) will be finite or P(s) will be zero.
Definition 4 P-Covariance between M and N, denoted by Cov(P,M,N), is Σ((NPV(M(s),D(s))-∫(P,M))*(NPV(N(s),D(s))-∫(P,N)) where the finite sum ranges over s∈S. This definition corresponds exactly to what is called covariance in statistics.
Definition 5 P-Correlation between M and N, denoted by Corr(P,M,N), is Cov(P,M,N)/((√Cov(P,M,M))*(√Cov(P,N,N))). This quantity reflects the strength of a linear relationship between valuations given by M and N.
For the next three definitions, fix some arbitrary D and P.
Definition 6 We say that model M is equivalent to model N if NPV(M(s),D(s))=NPV(N(s),D(s)) for s∈S. If this is the case, we write M≅N. Since our goal is to value a company, it makes sense to not distinguish models that output the same NPV under every scenario.
Definition 7 We say that model M is probabilistically equivalent to model N if P({s|NPV(M(s),D(s))=NPV(N(s),D(s))})=1. If this is the case, we write M≈N. While deciding whether M and N are equivalent, we should not care about a valuation difference under an impossible scenario s where P(s)=0.
Definition 8 We say that model M is expectationally equivalent to model N, if ∫(P,M)=∫(P,N). If this is the case, we write M∼N. While comparing M and N, we should only look at the average NPV predicted by each model.
Proposition 1 M=N implies M≅N implies M≈N implies M∼N. The proof is obvious. Also, note that the reverse implications may not hold.
For the rest of the definitions, fix only an arbitrary P.
Definition 9 We say that model M is P-viable if, for all s∈S such that P(s)≠0, there exists a positive real number d such that NPV(M(s),d) is finite. This condition ensures that M(s)_t does not grow in a crazy fashion as t goes to infinity. Of course, there may still be an impossible scenario s such that there is no positive real number d rendering NPV(M(s),d) finite.
Definition 10 We say that model M is viable if, for all s∈S, there exists a positive real number d such that NPV(M(s),d) is finite. A viable M is necessarily P-viable.
Definition 11 We say that the pairs (M,D) and (N,D') are in P-agreement if NPV(M(s),D(s)) = NPV(N(s),D'(s)) for all for all s∈S such that P(s)≠0.
Proposition 2 Given a pair (M,D) and a P-viable N, there always exists a D' such that (M,D) and (N,D') are in P-agreement. For a proof, pick an arbitrary s∈S such that P(s)≠0. Then there exists a positive real number d such that NPV(N(s),d) is finite. This implies that the function f(x)=NPV(N(s),x) is surjective. In particular, there exists a positive real number d' such that f(d') = NPV(M(s),D(s)). Let D'(s) be this d'. The resulting function D' from S to R^+ may look nonsensical as a "discount rate model", but nevertheless it exists.
Definition 12 Generalized NPV function is the map from (R^∞)x((R^+)^∞) to R^e, which takes a pair (v,d) to Σ(v_t/((1+d_t)^t)) where the infinite summation is taken from t=1 to ∞. We denote this function NPV in bold. Here each d∈(R^+)^∞ is an infinite vector of discount rates where d_t is the discount rate that NPV applies to the cash flow coming at time t.
Proposition 3 Let M be viable model. Then, for each s∈S and d∈(R^+)^∞, there exists a positive real number d' such that NPV(M(s),d) = NPV(M(s),d'). In other words, introduction of generalized NPV function does not seem to yield any additional insight in the space of viable models. As in Proposition 2, the proof follows from the surjectivity of the function f(x)=NPV(M(s),x).
The above theoretical musings have no practical value. Nevertheless they do point out some of the intricacies involved in the art of discounted cash flow valuation.
We will assume that S is large but nevertheless finite.
For an illustration of the new terminology, note the following two observations:
- The technique of scenario analysis can be broken down into two simple steps. First select a small subset S' of S, and then present the value of M at each s∈S'.
- The technique of sensitivity analysis consists of a depiction of what happens to M(s) as s∈S varies.
Analysts try hard to come up with the right discount rate for the company being valued. This calculation is important since the NPV (Net Present Value) of a company is determined uniquely by the discount rate d (a positive real number) and the projected stream of cash flows v.
Let R^e stand for the extended real number line and R^+ stand for the positive real numbers. Then NPV can be viewed as a function from (R^∞)x(R^+) to R^e. It maps a pair (v,d) to Σ(v_t/((1+d)^t)) where the infinite summation is taken from t=1 to ∞. (Note that this infinite series does not necessarily converge for a given (v,d). That is why we set the range of NPV as the extended real number line.)
For the determination of the appropriate discount rate, analysts either rely on garden-variety asset pricing models (such as CAPM) or create their own models. Each such model D can be viewed as a function whose domain is S and whose range is R^+. This makes sense since the calculation of the discount rate depends on the set of assumptions relevant to the valuation of the company.
Remark Both M and D are defined in such a generality that even intuition-based guesses qualify to be considered as "models".
Let P be a probability measure on S. Think of P as a function from S to the closed line segment [0,1], assigning probability P(s) to scenario s∈S. Moreover, P is chosen in such a way that the sum of all P(s) is 1. (i.e. It is believed that at least one of the scenarios in S will be realized.) Introduction of a probability measure into the picture allows us to utilize analyst's beliefs about the likelihood of each set of assumptions in S. Some scenarios, for instance, will be so nonsensical that they will be assigned zero probability.
Let's see what sorts of meaningful definitions we can make about the space of all valuation models M.
For the first five definitions, fix some arbitrary D.
Definition 1 We say that s∈S is M-equivalent to s'∈S if NPV(M(s),D(s)) = NPV(M(s'),D(s')). In other words M gives the same valuation under both scenarios.
Definition 2 We say that model M is more (s,s')-sensitive than model N if |NPV(M(s),D(s))-NPV(M(s'),D(s'))| is greater than |NPV(N(s),D(s))-NPV(N(s'),D(s'))|.
Definition 3 Expected-P-NPV of M, denoted by ∫(P,M), is Σ(NPV(M(s),D(s))*P(s)) where the finite sum ranges over s∈S. This quantity gives you the NPV expected by the analyst who assigns probability P(s) to scenario s∈S. Even if two analysts have wildly different probability measures P and P', they may end up agreeing on the expected NPV of M. In other words, ∫(P,M) may be equal to ∫(P',M) even if P≠P'. Although the sum involved in the definition is finite, ∫(P,M) may nevertheless be infinite because each NPV(M(s),D(s)) is an infinite sum. Of course, for reasonable M and P, either NPV(M(s),D(s)) will be finite or P(s) will be zero.
Definition 4 P-Covariance between M and N, denoted by Cov(P,M,N), is Σ((NPV(M(s),D(s))-∫(P,M))*(NPV(N(s),D(s))-∫(P,N)) where the finite sum ranges over s∈S. This definition corresponds exactly to what is called covariance in statistics.
Definition 5 P-Correlation between M and N, denoted by Corr(P,M,N), is Cov(P,M,N)/((√Cov(P,M,M))*(√Cov(P,N,N))). This quantity reflects the strength of a linear relationship between valuations given by M and N.
For the next three definitions, fix some arbitrary D and P.
Definition 6 We say that model M is equivalent to model N if NPV(M(s),D(s))=NPV(N(s),D(s)) for s∈S. If this is the case, we write M≅N. Since our goal is to value a company, it makes sense to not distinguish models that output the same NPV under every scenario.
Definition 7 We say that model M is probabilistically equivalent to model N if P({s|NPV(M(s),D(s))=NPV(N(s),D(s))})=1. If this is the case, we write M≈N. While deciding whether M and N are equivalent, we should not care about a valuation difference under an impossible scenario s where P(s)=0.
Definition 8 We say that model M is expectationally equivalent to model N, if ∫(P,M)=∫(P,N). If this is the case, we write M∼N. While comparing M and N, we should only look at the average NPV predicted by each model.
Proposition 1 M=N implies M≅N implies M≈N implies M∼N. The proof is obvious. Also, note that the reverse implications may not hold.
For the rest of the definitions, fix only an arbitrary P.
Definition 9 We say that model M is P-viable if, for all s∈S such that P(s)≠0, there exists a positive real number d such that NPV(M(s),d) is finite. This condition ensures that M(s)_t does not grow in a crazy fashion as t goes to infinity. Of course, there may still be an impossible scenario s such that there is no positive real number d rendering NPV(M(s),d) finite.
Definition 10 We say that model M is viable if, for all s∈S, there exists a positive real number d such that NPV(M(s),d) is finite. A viable M is necessarily P-viable.
Definition 11 We say that the pairs (M,D) and (N,D') are in P-agreement if NPV(M(s),D(s)) = NPV(N(s),D'(s)) for all for all s∈S such that P(s)≠0.
Proposition 2 Given a pair (M,D) and a P-viable N, there always exists a D' such that (M,D) and (N,D') are in P-agreement. For a proof, pick an arbitrary s∈S such that P(s)≠0. Then there exists a positive real number d such that NPV(N(s),d) is finite. This implies that the function f(x)=NPV(N(s),x) is surjective. In particular, there exists a positive real number d' such that f(d') = NPV(M(s),D(s)). Let D'(s) be this d'. The resulting function D' from S to R^+ may look nonsensical as a "discount rate model", but nevertheless it exists.
Definition 12 Generalized NPV function is the map from (R^∞)x((R^+)^∞) to R^e, which takes a pair (v,d) to Σ(v_t/((1+d_t)^t)) where the infinite summation is taken from t=1 to ∞. We denote this function NPV in bold. Here each d∈(R^+)^∞ is an infinite vector of discount rates where d_t is the discount rate that NPV applies to the cash flow coming at time t.
Proposition 3 Let M be viable model. Then, for each s∈S and d∈(R^+)^∞, there exists a positive real number d' such that NPV(M(s),d) = NPV(M(s),d'). In other words, introduction of generalized NPV function does not seem to yield any additional insight in the space of viable models. As in Proposition 2, the proof follows from the surjectivity of the function f(x)=NPV(M(s),x).
The above theoretical musings have no practical value. Nevertheless they do point out some of the intricacies involved in the art of discounted cash flow valuation.
