How does one turn $5 billion into $8.75 billion over 5 years? With an investment that returns approximately 12% over that 15 year period, that's how. And that is roughly (and conservatively) what I expect Sirius XM (NASDAQ:SIRI) to achieve with its buyback plan. So how do I arrive at this number?

First, it's important to understand that when a company buys back its own shares, it is essentially investing in itself. By taking cash, or by taking on debt, and buying back its own shares it is saying that it expects the company to perform well enough that the return on capital will exceed the expense of that capital. In the case of cash, the return can simply be viewed as the company's appreciation moving forward. In the case of debt, the return is a bit more complex.

Because I expect Sirius XM to take on debt in order to fund at least part of its buyback, calculating the expected return was not as simple as plugging interest into a compound interest calculator and having a number handed to me. It's better to head back to that good old high school math that most people hate, and dust off the differential equations book to arrive at a formula.

The formula I came up with looks like this:

B(t)=((w/r)+(B(0)-w/r)*(exp(r*t))

B(0) = Expected buyback value

w = Withdrawal rate (expected interest on debt)

r = Expected rate of appreciation of Sirius XM

t = Number of years

You can graph the above equation yourself in Google (NASDAQ:GOOG) to give you an interactive graph of value in billions as y, in x years.

Or simply cut and paste the following :

graph y=((.1/.15)+(2-.1/.15)*(exp(.15*x))

In Google search.

Now, that's assuming $2 billion bought back. If you wish to go with the current buyback plan in place and not the $5 billion I am assuming will eventually be used, you should use the graph and equation above.

My assumptions for the above graph are underlying appreciation of Sirius XM of 15% per year, and an interest cost on debt of 5%. I used what I feel is a conservative estimate of growth for Sirius XM, and a reasonable expectation of borrowing costs at today's lower rates.

I am also assuming that the entire buyback is funded with debt. Again, this keeps my estimations on the conservative side of the fence as most expect the buyback to be funded in part with cash on hand and future cash flow. And a third assumption is that the debt is carried in full for five years, which is again conservative in that Sirius XM would likely pay down debt over time.

You will notice the withdrawal rate is derived from B(0),

w=0.05*B(0)

so if one wished to graph the value of what I expect will be a $5 billion buyback, your equation would be as follows:

graph y=((.25/.15)+(5-.25/.15)*(exp(.15*x))

Why is this important? Because if you assume that Sirius XM will be able to carry the cost of increased debt due to buybacks, and you are a longer term holder of Sirius XM, then you will want to know what the expected return might be on Sirius XM's investment in itself vs. simply piling up cash.

What does this all mean for the market cap at the end of 5 years following the completion of the buyback? If math holds true, and one expects Sirius XM to buy back roughly $5 billion in shares then a simple comparison can be made.

Since $5 billion in cash would simply be $5 billion in cash at the end of 5 years, Sirius XM might approach $40 billion in market cap at the end of that period given 15% company appreciation per year.

But, if Sirius XM borrows that future cash now to buy back shares, it receives the benefit of the underlying appreciation of the company on that $5 billion bought back. Using the equation above, $5 billion turns into nearly $8.75 billion in about 5 years. That $3.75 billion extra gets rolled into the market cap of the company, and investors should realize that amount of additional appreciation, which is nearly 19% increased appreciation from today's market cap. Instead of 100% return in 5 years, investors should see 119% return in 5 years. That's a nice little bonus.

Some may be quick to point out this is only an extra 3.8% per year when divided by 5 years, but it's significant enough to warrant attention. The benefit investors should realize from a buyback plan is long term, but becomes significant as time goes on. Longer term holders, or those with longer term objectives, should reap the benefits of Sirius XM's buyback today.

This is why it is important for investors to pay attention to buyback discussions or news within the Q1 conference call scheduled before market open April 30th. The lower the price that Sirius XM repurchases shares at, the greater the benefit down the road.

**Disclosure: **I am long SIRI. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure:** I am long SIRI January 2014 $2 and $2.50 calls.