Jonathan Landy – LTV and the time value of money

jslandy's notes on math, data, etc.

Here, we review the time value of money concept and its application to<br>optimizing marketing (or other asset investment) spend.

Time value of money

A popular lifetime value (LTV) variant is built on top of the "time value of<br>money" concept. Essentially this says that we'd rather have a dollar now than<br>in the future. To formalize this, we imagine we have a source of capital that<br>comes with an associated interest rate. This capital source could be a line of<br>credit, or it might be our own capital that would otherwise be allocated to<br>some other "baseline" investment. As we explain below, the capital source's<br>interest rate sets the time value of money.

A common application of LTV is to set marketing budgets: We can think of LTV as<br>the effective value of a revenue stream, and we will want to acquire the stream<br>provided its cost is less than this effective value. When this holds:

If the capital source is a loan, we should borrow the money to purchase the<br>asset. Its resulting revenue will then be enough to both pay back the loan<br>and generate excess profit on top of that.

If the capital would instead come from reallocating funds from a baseline<br>investment, the asset here offers a higher rate of return. We should<br>therefore shift capital toward this asset if possible, continuing until<br>diminishing returns cause the marginal returns of the two investments to<br>become equal.

Evaluating LTV as a discounted sum of profits

We'll demonstrate how to evaluate LTVs by example.

Example 1

Suppose we can borrow capital at a \(20\%\) interest rate, say, and are<br>considering purchasing an asset for $\(C\) that will deliver $ \(p_T\) in<br>\(T\) years. Is this a profitable thing to do?

To check, suppose we do borrow the $\(C\) now to buy the asset. Then in \(T\)<br>years when the profit \(p_T\) comes in, we check if we have enough to pay back<br>the loan. The amount we owe after \(T\) years is

\begin{equation}<br>\text{compounded debt owed} = \$C \cdot 1.2^T \tag{1}\label{1}<br>\end{equation}

Therefore, if

\begin{eqnarray}<br>\tag{2}\label{2}<br>p_T = \$C \cdot 1.2^T,<br>\end{eqnarray}

the asset will return just enough money to pay back the loan. With this, we<br>then define

\begin{eqnarray} \tag{3} \label{3}<br>\text{LTV} \equiv \frac{ p_T}{ 1.2^T}<br>\end{eqnarray}

and note that if the asset's cost \(C , the asset will generate enough<br>revenue to (i) pay back the compounded debt (\ref{1}), and then (ii) have some<br>excess that we can pocket as profit. On the other hand, if \(C > LTV\), we won't<br>be able to pay back the loan from the asset's revenue. We see then that<br>(\ref{3}) is precisely the amount we should be willing to pay for the asset:<br>below this the asset will make a profit, above it will not (or in the case of a<br>baseline investment, this determines whether or not we'd have been better off<br>sticking with that baseline).

The factor of \(1.2^{-T}\) in (\ref{3}) is called the discount factor. This<br>effectively reduces the amount of value we place on revenue that comes in at<br>time \(T\). Higher interest rates force us to more strongly discount future<br>earnings, and the larger \(T\) is, the less we value that future revenue. This<br>is the time value of money.

Example 2

Suppose now we pay $\(C\) and get back a profit stream with \(p_t\) coming in<br>at year \(t\), for \(t = 0, 1, 2, \ldots\). Is this profitable?

In this case, we can write the LTV of the full stream as a sum of the LTVs of<br>the individual payments. These can be read out using (\ref{2}), so the full LTV<br>is

\begin{equation}<br>\text{LTV} = \sum_t \text{LTV}_t = \sum_{t=0}^{\infty} \frac{p_t}{1.2^t} \tag{4}\label{4}<br>\end{equation}

This is the full "discounted sum of profits" formula for LTV. If this is larger<br>than $C, we'll again have enough to gradually pay back the loan and then<br>pocket the excess as profit (or obtain a better rate of return than a baseline<br>investment).

Example 3

Suppose a bond pays out $1 every year off to infinity. How much is it<br>worth to us if our discount rate is again 20%? From (\ref{4}) this is

\begin{equation}<br>\text{LTV} = \sum_{t=0}^{\infty} \frac{1}{1.2^t} = \frac{1}{1 - 1/1.2} = 6 \tag{5}\label{5}<br>\end{equation}

Although the bond will eventually pay out an infinite amount of money, we have<br>to pay up front to get it. It's a profitable thing for us to do, provided the<br>cost of the bond is less than $6.

Practical application challenges

In practice one is often concerned with marginal returns: As you ramp up spend<br>on a given acquisition channel you may see diminishing returns. In this case,<br>early spend might allow you to acquire revenue streams whose LTV is larger than<br>their costs, but as you spend more, the LTV of the new acquisitions go down.

In a situation like this, you need to calculate the return on each individual<br>dollar invested, setting spend to the value where the marginal LTV captured is<br>exactly equal to one dollar. That is, at this point, if you spent one more<br>dollar...