# Investing in stagnant firms

This article has two purposes:

• to calculate the value of stagnant firms,
• to formulate a strategy for investing in such firms.

## Valuation of stagnant firms

The liquid value (LV) of a firm is the net amount of cash left by liquidating the firm. This is the amount of cash that remains after paying all liquidation costs, selling all assets, and paying all debts. This is the amount distributed to shareholders if the firm is liquidated. This is a lower bound of the value of that firm.

The market capitalization (MC) of a firm is its number of shares times its share price. It is the market’s opinion about the firm’s value.

A share is fractional ownership. If a firm has $$n$$ outstanding shares, then the its share price is $$MC/n$$.

Defining stagnant: Let there be a firm; its LV is constant; it also pays out dividends with constant amount and period. We call such firm stagnant.

The no-arbitrage principle (NAP) suggests that if the MC of a firm is below its LV, then one can profit by buying a majority of shares and liquidating the firm. If MC exceeds LV, then one can profit by borrowing a majority of shares and liquidating the firm. However, in reality, is prohibitively expensive to acquire a firm and liquidate it.

Stocks tend to be overvalued because buying shares is easier than short-selling them.

Owning a stagnant firm is equivalent to owning a constant amount of cash plus a perpetuity: $F~t = LV + delay~t~D$ where $$LV$$ is the constant liquid value of the firm, $$t$$ is the time to nearest future dividend payment, and $$D$$ is the perpetuity equivalent to the constant dividends: $D = \sum_{k=0}^{\infty} delay~(k \cdot T)~d$ where $$d$$ is the individual dividend amount and $$T$$ is the dividend period. The plot of $$F~t$$ against $$t$$ resembles sawtooth with period $$T$$.

Reminder: the present value of a constant perpetuity is: $\sum_{k=0}^{\infty} \frac{x}{(1+r)^{k \cdot T}} = \frac{x}{1-\frac{1}{(1+r)^T}}.$

The NAP suggests that two entities having the same cash flow should have the same value: \begin{align*} value~(F~t) &= value~(LV+delay~t~D) \\ MC~t &= LV + \frac{d}{(1+r)^t \cdot \left(1-\frac{1}{(1+r)^T}\right)} \end{align*} where $$r$$ is the risk-free rate. You must use the same unit for $$t$$, $$T$$, and the time unit of $$r$$.

For an average person, $$r$$ is the maximum interest rate that person can get from government-insured time deposits at banks or from government bonds. In Indonesia, in May 2015, the highest interest rate that the Indonesia Deposit Insurance Corporation will guarantee was 7.75%/year.

Risk-free rate greatly affects valuation. Suppose that there is a firm; it has 1,000,000 outstanding shares; its LV is $1,000,000; it gives$50,000 dividend every quarter (5% dividend yield). The following table illustrates the effect of risk-free rate on the valuation of that firm:

$$r$$ (%/year) MC ($) 9 3,343,876 6 4,457,423 1 21,124,845 0.25 81,124,961 Thus if the Fed cut the federal funds rate from 6%/year to 0.25%/year (which indeed began in 2008), then a bank’s valuation for that share would jump from$4.46 to \$81.12. If the price doesn’t change, then the firm’s valuation has shrink to an eighteenth of its original. As the risk-free rate approaches zero, the firm valuation grows without bound.

## Strategy for investing in stagnant firms

The strategy is a combination of periodically doing these:

1. start searching from the firm with the highest dividend yield;
2. keep transaction costs low; pick the cheapest discount broker;
3. calculate LV and see price history to ensure that the share price is not exuberantly overpriced, and to ensure that the firm, profit, and dividend are sustainable; avoid catching falling daggers;
4. build position gradually by dollar-cost averaging (DCA).

In such investment, the only exit is when the company is no longer sustainable, but you should have done your fundamental analysis to avoid having to exit.