The sum of perpetuities method (SPM) [1] is a way of valuing a business assuming that investors discount the future earnings of a firm regardless of whether earnings are paid as dividends or retained. SPM is an alternative to the Gordon growth model (GGM) [2] and can be applied to business or stock valuation if the business is assumed to have constant earnings and/or dividend growth. The variables are:
P
E
G
K
D
P=(
E*G | |
K2 |
)+(
D | |
K |
)
SPM is a generalized version of the Walter model.[3] The primary difference between SPM and the Walter model is the substitution of earnings and growth in the equation. Consequently, any variable which may influence a company's constant growth rate such as inflation, external financing, and changing industry dynamics can be considered using SPM in addition to growth caused by the reinvestment of retained earnings internally. Because
G
In a special case when a company's return on equity is equal to its risk adjusted discount rate, SPM is equivalent to the Gordon growth model (GGM). However, because GGM only considers the present value of dividend payments, GGM cannot be used to value a business which does not pay dividends. Also, when a firm's return on equity is not equal to the discount rate, GGM becomes highly sensitive to input value changes. Alternatively, SPM values dividends and retained earnings separately, taking into consideration the present value of the future income generated by retained earnings, and then summing this result with the present value of expected dividends held constant in perpetuity. Consequently, SPM can be used to value a growing company regardless of dividend policy. SPM is also much less sensitive to input value changes when a company's return on equity is different from the discount rate. An empirical test [1] shows that SPM is substantially more accurate in estimating observed stock market prices than the Gordon Growth Model.
The PEG ratio[4] is a special case in the SPM equation. If a company does not pay dividends, and its risk adjusted discount rate is equal to 10%, SPM reduces to the PEG ratio:
P=(
E*G | |
K2 |
)+(
D | |
K |
)
P=(
E*G | |
0.102 |
)+0
P=E*G*100
SPM is derived from the compound interest formula via the present value of a perpetuity equation. The derivation requires the additional variables
X
R
X
R
I:
II:
G=(
X | |
E |
)*R
Given by relationship II, a company with perpetual life which pays all of its earnings out as dividends has a growth rate of zero. It can therefore be valued using the present value of a perpetuity equation:
P=
E | |
K |
=
D | |
K |
P=
E | |
K |
=
(X+D) | |
K |
=
X | + | |
K |
D | |
K |
PVx
P=
PVx | |
K |
+
D | |
K |
PVx
X
X
PVx
R
X*R
X
PVx
PVx=
X*R | |
K |
X*R | |
K |
PVx
P=
| ||||
K |
+
D | |
K |
X*R
E*G
E*G
P=
E*G | |
K2 |
+
D | |
K |
The SPM equation requires that all variables be held constant over time which may be unreasonable in many cases. These include the assumption of constant earnings and/or dividend growth, an unchanging dividend policy, and a constant risk profile for the firm. Outside financing may not be considered unless the financing is perpetually recurring as capital structure must also be held constant.