# Price to Book: Rule of Thumb

In a post on MKL’s 2013 annual report, the Brooklyn Investor said something that caught my eye:

One thing people would do when looking at financials is to use a 10% hurdle; if a company grows book at 8%, it’s worth 0.8x book. If it grows at 12%/year, it’s worth 1.2x book. Never mind if that makes any sense or not, but that’s a quick rule of thumb thing that people sometimes look at.

One simple way to understand this rule is to assume that your target PE is 10 (target earnings yield = 10%). The earnings next year will be equal to the ROE times BVPS (say, 12% * BVPS). This is equal to the “E” in the target PE ratio of 10.  Thus, the target P/B = 12%/10% = 1.2, in this case, or 10*ROE in general.

I thought I’d explore this rule of thumb a little bit more. This is mostly in the context of compounding machines like MKL, where we expect to hold the security for a long time.

Suppose a firm has a book value per share of \$1 at time zero.

Let its return on equity (ROE) be r, and assume that it reinvests all its earnings. Suppose it is able to keep on reinvesting at this high ROE indefinitely.

Thus, the rate of increase in book value mirrors the ROE. After n years, the BVPS is:

$B_n = (1 + r)^{n}$

If the hurdle rate is d, then the net present value is the book value after n years discounted to the presented time:

$PV = \left(\dfrac{1+r}{1+d}\right)^n.$

A rational investor seeking a return equal to his or her hurdle rate would buy the security at a target P/B ratio of $PV/B_0 = PV$, since $B_0 = 1$ in this example.

Now let us suppose the hurdle is d = 10%, and n = 10 years.

I plotted the target P/B ratio as a function of the ROE, for r = 5% – 15%, using the equation above.

Next, I plotted the simple rule of thumb which suggests that the target P/B ratio may be approximated by 10r. As you see from the plot above, the rule of thumb holds up pretty well, especially for ROE near the discount rate of 10% (corresponding to the PE=10).

The “key” trick turns out to be using n = 10 years. If you use a smaller n (say n = 5) the rule of thumb makes you overpay for growth (r > d). On the other hand, for a larger (say n = 20 years), it makes your overpay for slow-growth companies.

## Math Aside

Mathematically, the rule of thumb can be understood/slightly generalized as follows: suppose $r = d + \Delta r$. Then the PV can be written as:

\begin{aligned} PV & = \left(\dfrac{1+r}{1+d}\right)^n\\ & = \left(\dfrac{1+d + \Delta r}{1+d}\right)^n\\ & \approx \left(\dfrac{1+d}{1+d}\right)^n \left(1 + \dfrac{n \Delta r}{1+d}\right)\\ & \approx 1 + n \Delta r \end{aligned}

Here, I performed a linearization in step 3, and assumed $1+d \approx 1$ in step 4. This simple math also provides a more general rule. If you want to hold something for n years, then target P/B ratio is

$P/B = 1 + n (r - d)$

For d = 10%, and n = 10 years, you see this generalized rule of thumb, the target $P/B = 1 + n (r - d) \approx 10 r$

As an example, if r = 12% and d = 10%, then $\Delta r = 0.02$. Hence, the target becomes,

$PV = 1 + 10 \times 0.02 = 1.2 = 10 \times r = 10 \times 0.12.$

### Target P/E ratio

If you have a calculator it is best to use the full formula,

$P/B = \left( \dfrac{1+r}{1+d} \right)^n$

to find out what multiple of book value to pay.

One can translate this into a PE ratio based on current earnings by simply multiplying the book value at t = -1, $B_{-1} = 1/(1+r)$ with the ROE, $E_0 = r/(1+r)$. Thus, the target P/E ratio based on DCF is,

$P/B = \left( \dfrac{1+r}{1+d} \right)^n\left(1 + \dfrac{1}{r}\right)$