Backsolve Method

As discussed in the overview of the Option Pricing Model (OPM), one of the key inputs to the OPM is the Total Equity Value of the company.  If the valuator has the Total Equity Value of an enterprise, then he/she can input that (along with the other parameters) into the OPM to derive the fair market value of common stock; i.e., the strike price at which future options should be issued.

There are an assortment of methods used to determine the Total Equity Value of a firm, one of which is the backsolve method.


Backsolve Overview

In an OPM framework, the backsolve method is used to answer the singular question:

What would the total value of the enterprise need to be, in order for a third-party investor to invest at the given per-share price, accounting for all liquidation preferences and seniorities for all share classes in the enterprise?

In other words, given that an investment occurred, the backsolve method outputs the implied total value of the enterprise if the valuation accounts for all share class rights and preferences, as of the date of the latest financing.


Backsolve Requirements

According to the AICPA, the backsolve is the most reliable indicator of enterprise value for early-stage customers, provided the following:

  • transactions in the enterprise's shares have occurred at arm's length*

*Arm's length transaction: A transaction that was entered into by informed but unrelated market participants, simultaneously seeking the best terms possible.

Note: In many situations, the transactions are not done at arm's length.  It is still possible to perform the valuation in these cases, but additional considerations need to be made.


Backsolve Considerations

Similar to the OPM, the backsolve considers the various terms of an enterprise's stockholder agreements that would affect the distributions to each class of equity upon a liquidity event as of the future liquidation date, including:

  • the level of seniority among securities,
  • dividend policy,
  • conversion ratios,
  • and cash allocations.

Backsolve Inputs

Similar to the OPM, the backsolve relies on these inputs:

  • expected time to exit
  • the risk-free interest rate as of the valuation date
  • the volatility derived from similar publicly traded companies

However, while the total equity value is an input into the OPM, it is precisely the output of the backsolve.


Backsolve Example

eShares is engaged to value  ABC, Inc., a small early-stage company that expects to exit in 5 years (t=5).  Using a basket of similar public companies, eShares estimates the volatility to be 50%.  Additionally, the risk-free rate from the valuation date until the exit date is 1.08% as of the valuation date.  

ABC, Inc. Capitalization Table (as of valuation date):

  Series Seed Preferred Common Common Options
Number of shares: 1,000,000 500,000 125,000
Purchase (or exercise) Price: $1.00 - $0.20
Liquidation Multiplier 1.0x - -
Liquidation Preference $1,000,000 - -

First, eShares computes the  equity breakpoints to determine which share class(es) will have value at various liquidation amounts for the company:

From To Description
$0.00 $1,000,000.00 Series Seed Preferred only.
$1,000,000.00 $1,100,000.00 Common shares only.
$1,100,000.00 $1,600,000.00 Common options exercised
$1,600,000.00 Infinity Series Seed Preferred converts to common.

Next, compute the implied call-option value of each of these breakpoints, using the Black-Scholes-Merton model:

$??? - Total equity value

1.08% - Risk-free rate

50.00% - Volatility

5 years - Time to exit

Problem: 

What's the right total equity value?

Solution: 

By trial-and-error, 'guess' total equity values until the per-share price of the Series Seed Preferred equals the price paid in the most recent transaction ($1.00)


Iteration 1: Try a value of $1,000,000.

$1,000,000 - Total equity value

1.08% - Risk-free rate

50.00% - Volatility

5 years - Time to exit

From To Option Value Incremental Option Value
$0.00 $1,000,000.00 $439,426.46 $560,573.54
$1,000,000.00 $1,100,000.00 $411,966.06 $27,460.40
$1,100,000.00 $1,600,000.00 $307,497.12 $104,468.94
$1,600,000.00 Infinity - $307,497.12
- - Total: $1,000,000.00

The above implies a per-share value of:

From To Series Seed 
Preferred
Common
Shares
Common
Options
$0.00 $1,000,000.00 $560,573.54 $0.00 $0.00
$1,000,000.00 $1,100,000.00 $0.00 $27,460.40 $0.00
$1,100,000.00 $1,600,000.00 $0.00 $83,575.15 $20,893.79
$1,600,000.00 Infinity $189,229.00 $94,614.50 $23,653.62

Total: $749,802.54 $205,650.05 $44,547.41

Value per share: $0.75 $0.41 $0.36

Problem: 

The implied value per Series Seed Preferred share is too low (it should be $1.00 instead of $0.74).


Iteration 2: Try a value of $2,000,000.

$2,000,000 - Total equity value

1.08% - Risk-free rate

50.00% - Volatility

5 years - Time to exit

From To Option Value Incremental Option Value
$0.00 $1,000,000.00 $1,265,356.77
$734,643.23
$1,000,000.00 $1,100,000.00 $1,215,494.98 $49,861.79
$1,100,000.00 $1,600,000.00 $1,007,497.28 $207,997.70
$1,600,000.00 Infinity - $1,007,497.28
- - Total: $2,000,000.00

The above implies a per-share value of:

From To Series Seed 
Preferred
Common
Shares
Common
Options
$0.00 $1,000,000.00 $734,643.23 $0.00 $0.00
$1,000,000.00 $1,100,000.00 $0.00 $49,861.79 $0.00
$1,100,000.00 $1,600,000.00 $0.00 $166,398.16 $41,599.54
$1,600,000.00 Infinity $619,998.33 $309,999.16 $77,499.79

Total: $1,354,641.56 $526,259.11 $119,099.33

Value per share: $1.35 $1.05 $0.95

Problem: 

The implied value per Series Seed Preferred share is too high (it should be $1.00 instead of $1.35).


Iteration 3: Try a value of $1,407,116.15.

$1,407,116.15 - Total equity value

1.08% - Risk-free rate

50.00% - Volatility

5 years - Time to exit

From To Option Value Incremental Option Value
$0.00 $1,000,000.00 $756,202.27 
$650,913.88
$1,000,000.00 $1,100,000.00 $718,125.95 $38,076.31
$1,100,000.00 $1,600,000.00 $566,306.00 $151,819.95
$1,600,000.00 Infinity - $566,306.00
- - Total: $1,407,116.15

The above implies a per-share value of:

From To Series Seed 
Preferred
Common
Shares
Common
Options
$0.00 $1,000,000.00 $650,913.88 $0.00 $0.00
$1,000,000.00 $1,100,000.00 $0.00 $38,076.31 $0.00
$1,100,000.00 $1,600,000.00 $0.00 $121,455.96 $30,363.99
$1,600,000.00 Infinity $348,496.00 $174,248.00 $43,562.00

Total: $999,409.88 $333,780.28 $73,925.99

Value per share: $1.00 $0.67 $0.59

Success:

The implied value per Series Seed Preferred equals the price that was paid in the most recent investment ($1.00).


Thus, the Total Equity Value that would be used as input into the OPM would be $1,407,116.15.

For a refresher on how per-share values are calculated using the OPM, see the Option Pricing Model article.