A note on the valuation of asset management firms

Abstract

Market capitalization relative to assets under management is often used to value asset management firms. Huberman’s (2004) dividend discount model implies that cross-sectional variations in this metric are explained by cross-sectional differences in operating margins, and yet we find no evidence of this in our data set. We show that a superior model—inspired by the work of Berk and Green (2004)—includes also the level of fees as an explanatory variable. This approach dramatically increases the fit of our valuation model and casts doubt on the relevance of the so-called Huberman puzzle.

Appendix: Present-value model for asset management firms

We decompose the earnings of an asset management firm (for any given year) into the product of assets under management, percentage fees, and operating margins:

$$\begin{aligned} \hbox {Earnings}=\hbox {AuM}\times \frac{\hbox {Revenues}}{\hbox {AuM}}\times \frac{\hbox {Earnings}}{\hbox {Revenues}}=\hbox {AuM}\times f\times q. \end{aligned}$$

(9)

We now ask: What is the value of this earnings stream over n years? Let r denote the return on a given asset management mandate (e.g., a mutual fund) gross of fees but net of trading costs, and let R be the return on its “benchmark” (the comparable, or risk-adjusted, market return). First, we write the earnings for the (equity) owners of the asset management firm at the end of year 1—while assuming zero capital inflows and fees that are applied to the end-of-period net asset value—as follows:

$$\begin{aligned} \hbox {Earnings}_1 =\hbox {AuM}\left( {1+r}\right) fq. \end{aligned}$$

(10)

What is left to the investors in the asset management firm’s products then amounts to

$$\begin{aligned} \hbox {AuM}\left( {1+r}\right) \left( {1-f}\right) . \end{aligned}$$

(11)

A more general expression for earnings at the end of year i is given by

$$\begin{aligned} \hbox {Earnings}_i =\hbox {AuM}\left( {1+r}\right) ^{i}\left( {1-f}\right) ^{i-1}fq. \end{aligned}$$

(12)

The series of cash flows to discount can then be written as:

$$\begin{aligned} \hbox {Earnings}_1= & {} \hbox {AuM}\left( {1+r}\right) fq,\nonumber \\ \hbox {Earnings}_2= & {} \hbox {AuM}\left( {1+r}\right) ^{2}\left( {1-f}\right) fq,\nonumber \\ \hbox {Earnings}_3= & {} \hbox {AuM}\left( {1+r}\right) ^{3}\left( {1-f}\right) ^{2}fq,\nonumber \\&\vdots&\nonumber \\ \hbox {Earnings}_n= & {} \hbox {AuM}\left( {1+r}\right) ^{n}\left( {1-f}\right) ^{n-1}fq. \end{aligned}$$

(13)

At what rate should these cash flows be discounted? That is, what is the discount rate for cash flows containing the same form of systematic risks? A natural choice for such risk-adjusted returns are benchmark returns.⁷ These lead to the specification of the following discounted cash flow model:

$$\begin{aligned} P\left( n\right)= & {} \frac{\hbox {AuM}\left( {1+r} \right) fq}{1+R}+\frac{\hbox {AuM}\left( {1+r}\right) ^{2}\left( {1-f} \right) fq}{\left( {1+R}\right) ^{2}} \nonumber \\&+\cdots +\,\frac{\hbox {AuM}\left( {1+r}\right) ^{n}\left( {1-f}\right) ^{n-1}fq}{\left( {1+R} \right) ^{n}}. \end{aligned}$$

(14)

Further simplification results in

$$\begin{aligned} P\left( n\right)= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n \left( {\frac{\left( {1+r} \right) \left( {1-f}\right) }{1+R}}\right) ^{i}} \right] \nonumber \\= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\frac{\frac{\left( {1+r}\right) \left( {1-f}\right) }{1+R}-\left( {\frac{\left( {1+r}\right) \left( {1-f}\right) }{1+R}} \right) ^{n+1}}{1-\frac{\left( {1+r}\right) \left( {1-f} \right) }{1+R}}}\right] . \end{aligned}$$

(15)

To create more insight into Eq. (15), we assume $r=R$; that is, the performance gross of fees equals the benchmark performance (hence alpha equals zero). Then

$$\begin{aligned} P=P\left( n\right) =\hbox {AuM}\times q\left[ {1-\left( {1-f} \right) ^{n-1}}\right] . \end{aligned}$$

(16)

Assuming an infinite time horizon $\left( {n=\infty }\right)$, we arrive at Eq. (3) in the main text:

$$\begin{aligned} \frac{P}{\hbox {AuM}}=q. \end{aligned}$$

(17)

We can extend the model by adding alpha to portfolio returns; recall that $\left( {1+r}\right) =\left( {1+R}\right) \left( {1+a} \right)$. Thus,

$$\begin{aligned} P\left( n\right)= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n \left( {\frac{\left( {1+r}\right) \left( {1-f}\right) }{1+R}}\right) ^{i}}\right] \nonumber \\= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n \left( {\frac{\left( {1+R}\right) \left( {1-a} \right) \left( {1-f}\right) }{1+R}}\right) ^{i}}\right] \nonumber \\= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n \left( {\left( {1+a}\right) \left( {1-f}\right) } \right) ^{i}}\right] . \end{aligned}$$

(18)

We also assume that $\left( {1+a}\right) \left( {1-f}\right) =1$; that is, $f=a\left( {1-f}\right)$. Thus, fees are set so as to leave the client with no alpha. Adjusting Eq. (18) to reflect these equalities yields

$$\begin{aligned} P\left( n\right)= & {} \frac{\hbox {AuM}\times f\times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n 1^{i}}\right] =\frac{\hbox {AuM}\times a\times \left( {1-f}\right) \times q}{1-f}\left[ {\mathop {\sum }\limits _{i=1}^n 1^{i}}\right] \nonumber \\= & {} \hbox {AuM}\times a\times q\times n. \end{aligned}$$

(19)

Hence, it is clear that when alpha is not limited, the firm valuation goes to infinity.

Now suppose the asset management company receives a fixed level of earnings forever because its assets under management are fixed at full capacity, ${\mathbf{AuM}}^{*}$. In this case, the firm’s earnings are

$$\begin{aligned} \hbox {Earnings}=\hbox {AuM}^{*}\times f\times q. \end{aligned}$$

(20)

Discounting this cash flow to infinity creates a corresponding firm value, $p^{*}$, of

$$\begin{aligned} p^{*}= & {} \mathop {\lim }\limits _{n\rightarrow \infty } \left[ {\frac{\hbox {AuM}^{*}\times q\times f}{1+r}+\frac{\hbox {AuM}^{*}\times q\times f}{\left( {1+r} \right) ^{2}}+\cdots +\frac{\hbox {AuM}^{*}\times q\times f}{\left( {1+r}\right) ^{n}}}\right] \nonumber \\= & {} \hbox {AuM}^{*}\times q\times f\left[ {\frac{1}{1+r}+\frac{1}{\left( {1+r}\right) ^{2}}+\cdots +\frac{1}{\left( {1+r}\right) ^{n}}}\right] \nonumber \\= & {} \hbox {AuM}^{*}\times q\times f\left[ {\frac{\frac{1}{1+r}-\frac{1}{\left( {1+r} \right) ^{n+1}}}{1-\frac{1}{1+r}}}\right] \nonumber \\= & {} \hbox {AuM}^{*}\times q\times f\left[ {\frac{1}{r}}\right] . \end{aligned}$$

(21)

This expression is equal to Eq. (5) in the main text.