Predicting Venture Cost of Capital with Failure Rate (or Visa Versa)

I just did a little more math and came up with an amazing (in my opinion) result.

Question: Can I use the failure rate to determine an appropriate cost of capital (or visa versa)?

Imagine our previous company, but let's simplify it:

Cash Flow: $10 (it doesn't matter - We'll see why shortly)
Regular discount rate: 8%
Venture discount rate: 20%
Failure Rate: ???

Let's look at a given cash flow in period (n) in isolation:

Well using Method 1 - Venture capital discounting @ 20%:
PV1 = $10 * (1 + g)^n / (1 + 20%)^n

Using Method 2 - Regular equity discounting @ 8%, but undetermined failure rate:
PV2 = $10 * (1 + g)^n * (1 - f)^n / (1 + 8%)^n

Note that PV1 = PV2 according to the law of one price.

Also note: for any given period, you can cancel the effect of:

• period because the failure and discount rate affect the PV in the same way,
• the value of the actual cash flow doesn't matter, and
• the growth factor

All of these factors cancel out algebraically.

So we now know that:
1 / (1.20) = (1 - f) / (1.08)

Solving for f = 10%

[Solution]

So we can see that there is a general relationship:

• ke, "normal" cost of equity
• kv, cost of venture capital
• f, failure rate

(1 + ke) / (1 + kv) = (1 - f)
f = 1 - [(1 + ke) / (1 + kv)]
f = (kv - ke) / (1 + kv)

Similarly:

kv = (ke + f) / (1 - f)