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# Valuation of Intangibles Based on Lagged Investments The valuation of intangibles does not create any special problem as long as we keep track in a company’s general ledger the amounts of separate and distinct expenditures that for the purpose of valuation are treated as investments. Thus,

Capex → value of plant and equipment;

XRD → value of technology intangibles;

XAD → value of marketing intangibles; and

Computer software costs → value of copyrights.

Capex is capital expenditures; XRD is research & development expenses, and XAD is advertising expenses. One valuation method is based on investment costs — which we call the perpetual inventory method (PIM).

According to the perpetual inventory method, the value of any asset, including intangibles, depends on two factors: (i) the prior-period measure of the capital stock, and (ii) current investment (Capex, XRD, XAD, computer software expenses):

##### (1)     Kt = β Kt – 1 + It

where β = (1 − δ) is a capital accumulation coefficient, δ is a depreciation or amortization parameter, Kt – 1 is the prior-period measure of the capital stock, and and It is current investments.

We aim to eliminate the prior-period capital stock from equation (1) to facilitate computation of the current asset value. For this purpose, we lag (delay) the elements of equation (1) one period, and obtain:

(2)     Kt − 1 = β Kt – 2 + It – 1 et segue.

Next, we substitute equation (2) into (1), continue making further recursive substitutions, and obtain (after we stop four investment periods in the past):

##### (3)     Kt ≈ It + β It– 1 + β2 It– 2 + β3 It– 3 + β4 Kt – 4 + …

Since β < 1, the geometric series of capital accumulation coefficients 1, β, β2, β3, etc., in time become negligible as they decrease towards zero. As a result, among past investments, It – 1, It – 2, It – 3, etc., only the most recent investments have greater weights determining the current asset value, Kt.

We can continue to make recursive substitutions of the past capital stock into equation (1), and find that the current asset value depends on a historical series of lagged investments (dated quantities of investment):

##### (4)    Kt ≈ α0 It + α1 It – 1 + α2 It – 2 + α3 It – 3 + α4 It – 4 + …

where α0 = 1 and α1 = β, etc. We can stop this weighted sum in equation (4) after a few-periods (years) back-iteration because of the decay of the alpha reaction coefficients.

According to equation (4), which is another way of expressing equation (1), the value of any asset (including intangibles) is equal to a weighted sum of past investments, and the decreasing geometric weights are based on the depreciation rate. We notice that prior measures of the capital stock (Kt – h) disappear from the asset value equation (4). As a result, we are able to reduce the number of determinants of the value of any asset, again including intangibles, from two factors to investment as the only factor, measured by several years in the past. The number of lags depend on the depreciation or amortization rate.

By IFRS accounting convention and by legislative grace, current investments to create technology, marketing and software intangibles are expensed in the year they are incurred, and not capitalized on the company’s balance sheet. However, for the purpose of determining asset value based on PIM, we treat those expenses as investments. See a recent article on the WSJ — not questioning the improbable high value assigned to aggregate intangibles: http://www.wsj.com/articles/accountings-21st-century-challenge-how-to-value-intangible-assets-1458605126