During my teenage years, when I was contemplating my future academic path, my mother consistently encouraged me to pursue mathematics. She assured me that regardless of the industry I chose, I would always be highly employable because mathematics is universally disliked, and people would gladly pay someone else to handle it for them.

As a 16-year-old high school student, I found this notion difficult to comprehend, as my mathematical abilities did not exactly elevate my social standing. However, a mere decade later, I must concede that she was absolutely correct. Mathematics serves as a universal language, and even in fields like marketing, where creativity and language proficiency are highly valued, mathematical skills are becoming increasingly indispensable for data-driven marketers.

Finally, my moment of triumph has arrived! Armed with nothing more than a Master’s degree in Applied Mathematics and my trusty Ti-83 Plus calculator, I stride through the corridors of the nexus-security office, feeling like one of the cool, popular, and sought-after individuals that everyone desires to engage with. (Okay, I might be slightly exaggerating, but my colleagues appreciate my mathematical prowess, which is definitely a plus.)

It’s no secret that Erin Sagin and I are quite close. However, what most people are unaware of is that we had a major disagreement a couple of months ago. Erin had published a highly successful blog post entitled “Easy AdWords Bidding Strategies for Newbies & Math Haters.” Math haters? Seriously? I considered us friends! Needless to say, I couldn’t help but feel a sense of betrayal:

Rest assured, Erin and I quickly reconciled, and today (a mere 5 months later), I took the time to craft this blog post.

Math is undeniably awesome, folks! Not only can it boost your confidence when it comes to attracting potential romantic interests and impressing your relatives, but it can also significantly enhance your ability to consistently achieve your clients’ goals. The best part? You don’t need an advanced degree in mathematics or the expertise of a data scientist to harness its power. All you need are these handy equations and a basic understanding of algebra!

PPC Formula #1: Determining Your CPC Bid to Achieve Your Target CPA

Let’s start with the definition of CPA, which stands for cost per action (or cost per conversion or cost per lead):

Pretty straightforward, right? Now, let’s break it down further.

Your cost is calculated by multiplying the number of clicks your ads receive by your cost per click.

Conversions, on the other hand, are simply the number of clicks you get multiplied by your conversion rate.

Let’s plug these into our equation:

Interesting. Notice that the number of clicks appears in both the numerator and the denominator. Let’s simplify this equation:

Now, suppose we know our target CPA. What CPC would we need to set to ensure we hit that target CPA? Solving for CPC in this case gives us:

There you have it! As long as we know our conversion rate, we can guarantee that we achieve our CPA goal. For instance, if we know that our traffic converts at 2% and our target CPA is $25, we know we need to pay $25 x 0.02 = $0.50 CPC!

PPC Formula #2: Setting Your CPC Bid to Reach Your Target ROAS

Once again, let’s start with a simple definition of ROAS, or return on ad spend:

Let’s expand this a bit, just like we did with CPA.

Cost remains the same as before, calculated by multiplying the number of clicks your ads receive by your cost per click.

Revenue is slightly more complex. It’s the number of sales (or conversions) you generate multiplied by the average order value (AOV) of those sales.

Expanding this one step further, we previously established that conversions can also be expressed as the number of clicks you get multiplied by your conversion rate. Substituting this into our equation gives us:

Once again, we encounter clicks in both the numerator and the denominator. Let’s simplify:

Now, if we know our desired ROAS, we can solve for the cost we should be paying per click:

Voila! We now possess all the necessary information to set our bid and ensure we hit our ROAS goal. For example, if our target ROAS is 4.0, our average product sells for $150, and our traffic converts at 2%, then we know we need to pay 0.02 x $150 / 4.0 = $0.75 CPC.

PPC Formula #3: Setting Your CPC Bid to Maximize Profit

Maximizing profit - the holy grail of every CEO. This is where things get serious! Again, let’s start with a basic definition:

Now, let’s substitute our earlier expansions of Revenue and Spend:

Revenue is the product of your clicks, conversion rate, and average order value.

Spend is the product of clicks and CPC. This assumes your products are free to produce and ship; if you have additional production and shipping costs, be sure to factor them in here as well!

Since we’re dealing with maximizing a value, we’ll need to employ some calculus. Don’t worry, we’ll navigate this together! We’re simply going to take the derivative of profit with respect to the number of clicks our ads are receiving.

Now, let’s solve for the maximum profit. By setting this derivative to 0, we can determine the absolute maximum profit our campaign can potentially generate:

With a little algebra, we can easily find the optimal CPC bid that maximizes our profit:

Setting your CPC bids to maximize profit is truly that simple! It may have required a touch of calculus to get there, but all you ultimately need is the product of your conversion rate and your average order value. So, if your conversion rate is 2% and your average sale is $150, your campaigns will be most profitable at $3 CPC. Anything less than $3 CPC means more profit per sale but fewer sales overall. Anything more, and you’ll generate more sales but sacrifice profit on each additional one.

QED, PPC enthusiasts.

These formulas have been mathematically proven to deliver exceptional advertising results! Utilize them to establish and fine-tune your keyword-level bids and bid adjustments, and you’ll be well on your way to consistently hitting your goals and impressing your clients, bosses, and everyone else within your professional circle! Isn’t math truly remarkable?