What is an annuity? A fixed sum of money paid to someone each year.
Why is the present value of an annuity so important? You need to figure out how big this bag of money needs to be at t equals zero, today, in order to fund the equally sized recurring payments in future years.
There are multiple ways to calculate the present value of an annuity. There is the slow but easier to understand way, the fast but tougher to understand way, and the very fast and I really don’t care how it’s done way. In order to properly understand the present value of an annuity, start with the first method, and then progress to the faster methods.
⏱️TIMESTAMPS⏱️
00:00 What is an annuity
00:08 What is the present value of an annuity
00:23 Simple present value calculation
02:52 Present value of an annuity formula
04:42 Present value of an annuity in Excel
05:38 Present value of an annuity scenarios
You can calculate the present value of an #annuity the same way you calculate any other present value. Take those future amounts of money, one by one, and translate them back to today’s equivalent using the time value of money, the rate of return, or the interest rate, whichever applies. Now this calculation method of the present value of an annuity is doable for four payments, but it is a lot of work if you extend to 10, 20 or 30 years.
That’s why it’s good to know that there is a shortcut to the present value of an annuity calculation that can easily handle any number of periods for the recurring payments: the fast but tougher to understand way.
The present value of an annuity formula takes the calculations we used previously (where we translated each individual payment to today’s equivalent), and applies the mathematical logic of how to get to a sum of a geometric series. The resulting formula consists of PV for present value, P for the recurring payment of the same size, r for the interest rate per period (this can be per year, month, week or even day), and n for the number of periods (years, months, weeks, days). The present value of an annuity equals the value of each payment, times in the numerator one minus one plus r to the power minus n, divided by r in the denominator. That sounds complicated at first, but starts to make a lot of sense once you fill in the parameters. In our example, the value of each payment is $10,000, the interest rate 5% per year, and the number of periods 4 years. Plug those into the present value of an #annuityformula , and this is what that looks like. 1.05 to the power minus 4 equals 0.8227. Complete the next steps of the calculation, and you will get to the present value of the annuity of $35,460, which is the same amount we had before in the “step-by-step” method.
Some of you might complain that the textbook you studied or the online resource you consulted had a different formula. It might state in the numerator of the equation: one minus one divided by one plus r to the power n. Raising something to the power minus n is the same as dividing by that same something to the power n.
And now for the very fast and I really don’t care how it’s done way: calculating the present value of an annuity in Excel. Insert the PV function into your worksheet, and fill in the function arguments. Interest rate of 5% or in decimal notation 0.05, number of payment periods 4, and a payment of minus $10,000, lead to a present value of the annuity of $35,460, the same outcome as in the previous two methods. The Microsoft Excel help description states that PV, one of the financial functions, calculates the present value of a loan or an investment, based on a constant interest rate. So their thinking is primarily from the perspective of loans or mortgages, whereas mine is from the perspective of annuity payments as part of a pension plan.
The great thing about calculating the present value of an annuity in Excel is that you can quickly run some scenarios, to see the sensitivities of changing any of the input variables.
Philip de Vroe (The Finance Storyteller) aims to make accounting, finance and investing enjoyable and easier to understand. Learn the business and accounting vocabulary to join the conversation with your CEO at your company. Understand how financial statements work in order to make better investing decisions. Philip delivers #financetraining in various formats: KZfaq videos, livestreams, classroom sessions, and webinars. Connect with me through Linked In!
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