Value is an estimate: discounting, growth and the margin of safety

Every valuation is the same calculation: list the cash a thing will pay in the future, and pull each payment back to today. The whole art is in two guesses, and most of what follows is how much those guesses move the answer.

the estimated valuethe discount rate, the price of waitingthe cash received

A dollar later is worth less than a dollar now

Because a dollar now can earn interest. If money earns $r$ a year, then $100 today becomes $100(1 + r)$ in a year, so $100 due in a year is worth $100 / (1 + r)$ today. To bring a payment $n$ years away back to today, divide by $(1 + r)$ once per year:

A higher discount rate means a lower value today: the second row of Table 1 is the first row with the rate two points higher, and it shaved 3.5%. The last row, and the first panel of Figure 1, is why distant cash barely matters: fifty years out at 8%, a dollar is two cents.

The discount rate is what the money could earn elsewhere at similar risk: a broad index fund for a stock, a government bond for something safe. Pick it too low and everything looks cheap.

A stream of payments

A stock is a stream of dividends. Add up the present value of each (Table 2):

The forever case has a one-line answer, $D / r$, and it is the limit of the ten-year sum as the years pile up: the payments beyond year ten are worth $5.79 in total, most of it from the first few of them. If the dividend also grows at a steady rate $g$, the Gordon growth formula gives:

Price equals next year’s dividend divided by the gap between the discount rate and the growth rate. The gap in the denominator is the whole story (Table 3):

As $g$ creeps towards $r$ the denominator shrinks towards zero and the price explodes; at $g \ge r$ the formula gives nonsense, because a dividend growing as fast as the discount rate is worth an infinite amount. The second panel of Figure 1 draws the curve. No company grows faster than the economy forever, so the long-run $g$ is always a few percent, and a model that needs 8% to justify a price is a model that has already failed.

A discounted cash flow, and where the answer lives

A DCF forecasts a few years of cash flows explicitly, then handles everything after with a Gordon-style terminal value. A small one: $100 of cash this year, growing 8% a year for five years, then 3% forever, discounted at 10% (Table 4).

Three quarters of the value is the terminal value, the part that rests on one long-run growth guess and one discount rate. Change either and the answer moves more than the whole explicit forecast is worth (Table 5):

The honest range is $1,144 to $4,000 for the same business, depending on two numbers nobody can verify. That is why a practitioner works with a range and a margin of safety rather than a point: estimate the value, then only buy well below it.

The last row of Table 6 is the expected value of a valuation under several futures, and it is still one number: one of the three scenarios will happen, not the average.

Book value is accounting, not value

A company’s book value is its assets minus its liabilities on the balance sheet, which carries assets at cost less depreciation. Assets $500M, liabilities $320M, 30 million shares: book value $180M, $6 a share. At a price of $9 the price-to-book ratio is 1.5. A ratio above 1 is normal, because a working business is worth more than its parts sold off: the brand, the customers and the growth are not on the balance sheet. A ratio below 1 can mean a bargain or a business in trouble; the number alone does not say which. In a forced sale, book value can overstate what the assets fetch.

Being right about value is not a good trade

The gap between price and value pays only if it closes, and only if you are still there when it does. A stock can stay mispriced for years; a short position (borrowed shares sold, to be bought back later) loses without limit while the price keeps rising; a margin call, the broker’s demand for more cash when a position bought with borrowed money falls, can force a sale at the worst moment. Value tells you what to own. It does not tell you when the market will agree, and the market is under no obligation to.

Traps

Table 7 is the list of mistakes, with the fix beside each.

The code

Every number above, in plain Python with no libraries, so it runs as pasted. Each function is one formula from above; the print lines reproduce the tables.

def present_value(cash, rate, years):
    return cash / (1 + rate) ** years                     # divide by (1 + rate) once per year

def gordon(next_dividend, rate, growth):
    assert growth < rate, "the formula only works when growth is below the discount rate"
    return next_dividend / (rate - growth)

def dcf(cash0=100, growth=0.08, rate=0.10, g_term=0.03, years=5):
    flows = [cash0 * (1 + growth) ** k for k in range(1, years + 1)]           # years 1 to 5
    explicit = sum(cf / (1 + rate) ** k for k, cf in enumerate(flows, 1))      # their value today
    terminal = flows[-1] * (1 + g_term) / (rate - g_term) / (1 + rate) ** years  # years 6 onward
    return explicit, terminal, explicit + terminal

margin = lambda value, price: (value - price) / value     # divide by the value, not the price

print(round(present_value(121, 0.10, 2), 2), round(present_value(121, 0.12, 2), 2))   # 100.0  96.46
print(round(sum(present_value(1, 0.08, k) for k in range(1, 11)), 2), 1 / 0.08)   # 6.71  12.5
print(gordon(2, 0.10, 0.05))                              # 40.0
print([round(x) for x in dcf()])                          # [473, 1342, 1816]
for r in (0.08, 0.10, 0.12):                              # the sensitivity table
    print(r, [round(dcf(rate=r, g_term=g)[2]) for g in (0.0, 0.02, 0.03, 0.05)])
print(margin(62.50, 50))                                  # 0.2

It prints 100.0 and 96.46, then 6.71 and 12.5, then 40.0, then the DCF’s three parts [473, 1342, 1816], then the sensitivity table row by row, then 0.2. round(x) drops the decimals, and enumerate(flows, 1) numbers the cash flows from year 1. dcf(rate=r, g_term=g) changes only those two inputs and keeps the others at their defaults.

Summary

Table 8 collects the formulas.

References

• Myron J. Gordon and Eli Shapiro, “Capital Equipment Analysis: The Required Rate of Profit”, Management Science 3(1), 102–110, 1956, the growth formula
• Philip J. Romero and Tucker Balch, What Hedge Funds Really Do: An Introduction to Portfolio Management (Business Expert Press, 2014), chapters 5 and 6: present value, intrinsic value as the sum of discounted dividends, book value against market value, and Graham’s margin of safety (chapter 5)
• Benjamin Graham, The Intelligent Investor (Harper, revised edition 2006), chapter 20: the margin of safety