MBA 8415
Derivations for Levered and Unlevered Equity Betas (β)
Dr. D.A. Stangeland
Converting between unlevered and levered equity betas is a straightforward process, but
attention must be paid to assumptions regarding taxes paid by the firm and its investors. Three
cases are presented below.
Definitions:
Let
VU = the total value of an unlevered firm (no debt in the capital structure)
VL = the total value of an levered firm
Define VL = D + E
(Equation 1)
Where D = Debt value
E = Levered equity value
Note: VU = Unlevered equity value
Also, define the risk of the equity as follows:
βU = βUnlevered equity
βL = βLevered equity
Case 1: No Tax Case
From M&M we know that VL = VU which results because the total cash flows to debt and equity
for the levered firm are the same as the total cashflows to equity in the unlevered firm. This also
implies that the overall risk of the levered firm must equal the overall risk of the unlevered firm.
Therefore, βU, the risk of unlevered firm, must equal the risk of the levered firm. The risk of the
levered firm is the weighted average of the risk of the debt and the levered equity.
D
E
 D 
 L
Thus, βU must equal
VL
VL
Rearranging terms we get L 
 V
D
VL 
D
 u 
D   L u  D
E
E
E 
VL 
Substitute in Equation 1 for VL and we get  L 
DE
D
u  D
E
E
D
D
 D
Rearranging terms we get L  1  u  D   u   u   D 
E
E
E

D
 u   D 
E
No tax case
L  u 
Copyright © 2002 David A. Stangeland
MBA 8415
Notes and Derivations for Determining Levered Equity Betas (β)
Dr. D.A. Stangeland
Page 2 of 4
Case 2: Corporate Taxes
From M&M with corporate tax we have VL = VU + TC ∙ D.
(Equation 2)
Since VL = D + E is also true, the following equality holds:
D + E = VU + TC ∙ D
(Equation 3)
The weighted average risks of the components on both sides of Equation 3 must be equal and
these represent the overall risk of the levered firm. Thus we get
D
E
V
T D
D 
L  u u  c
D
VL
VL
VL
VL
Multiplying both sides by VL, we get
D  D  E  L  Vu  u  Tc  D  D
Rearranging terms we get L 
1
Vu  u  Tc  D  D  D  D 
E
Rearrange Equation 2 and substitute in VL  Tc  D for Vu and we get
L 
1
VL  U  Tc  D  U  Tc  D  D  D  D 
E
Now multiply through by
L 
1
and substitute in Equation 1 to get
E
D
DE
D
D
D
u  TC  u  D  Tc  D  u  1  Tc u  D 
E
E
E
E
E
L  u 
D
1  Tc u  D 
E
Corporate Tax Case
Copyright © 2002 David A. Stangeland
MBA 8415
Notes and Derivations for Determining Levered Equity Betas (β)
Dr. D.A. Stangeland
Page 3 of 4
Case 3: Corporate and Personal Taxes
From Miller’s Debt and Taxes article we know that
 1  Tc 1  TPE 
VL  Vu  D  1 
1  TPD  

(Equation 4)
Where TPE is the personal tax rate on equity income for the marginal investor and TPD is the
personal tax rate on debt income for the marginal investor.
Substitute in Equation 1 for VL and we get D  E  Vu  D  

(Equation 5)
 1  Tc 1  TPE 
Note: to shorten the text of the derivation, let 
  1 
1  TPD  

(Equation 6)
The weighted average risks of the components on both sides of Equation 5 must be equal and
these represent the overall risk of the levered firm. Thus we get
D
E
V
D  

D 
L  u u 
D
VL
VL
VL
VL
Multiplying both sides by VL, we get D  D  E  L  Vu  u  D  
  D
Rearranging terms we get L 
1
Vu  u  D    D  D  D 
E
Rearrange Equation 4 and substitute in VL  D  
 for VU and we get
1
VLu  D u  D  D  D  D  and now multiply through by 1 and substitute in
E
E
D
DE
D
D
D
u  
  u  
  D  D   u  1  
  u   D 
Equation 1 to get L 
E
E
E
E
E
L 
Finally, insert Equation 6 to get L  u 
L  u 
D   1  Tc 1  TPE   
   
1 1
1  TPD    u D
E  
D  1  Tc 1  TPE 
u  D 
E  1  TPD  
Corporate and Personal Tax Case
Copyright © 2002 David A. Stangeland
MBA 8415
Notes and Derivations for Determining Levered Equity Betas (β)
Dr. D.A. Stangeland
Page 4 of 4
Summary
To lever or unlever equity betas, use the appropriate formulas below depending upon the
assumptions regarding taxes.
L  u 
D
 u   D 
E
No Tax Case
L  u 
D
1  Tc u  D 
E
Corporate Tax Case
L  u 
D  1  Tc 1  TPE 

 u   D 
E  1  TPD  
Corporate and Personal Tax Case
Copyright © 2002 David A. Stangeland