Pay It Again, Sam

In my post on present value, I promised to explain how to turn a series of payments into a present value. This is the promised follow-up post.

Nearer and Farther Futures

I'm going to take the existence of a "discount rate" as a given, though I'll discuss it in a later post. For now, it's just the conversion factor you use between the value of something now and the value of something in a year.

I'm going to deal with it in decimal form, so for example if your annual discount rate is 5%, that's the same as 0.05

Chris has a bond that could pay them $1,000 today, or $1,100 in two years. Their discount rate is 4%. How do they use present value to compare these two amounts?

We already know how to compare the value in a year with the value in two years. Just imagine Chris a year in the future. Then the $1,100 payment is only a year away, and dividing by the discount rate,  the present value as of a year in the future is $1,100/1.04

Now we have the value as of a year from today, and we can apply the same conversion; the present value today is the value in a year, divided by the discount factor, or ($1,100/1.04)/1.04=$1,100/(1.04^2)

We can generalize this as

present_value = future_value / ( (1 + discount_rate) ^ time_difference )

A Note on Units

Discount rates are usually given in annual terms. If you are trying to discount over days or months instead of years, just use the appropriate fraction of the year as the time_difference. For example, a payment in one month has a time_difference of 1/12. Many common calculating tools, such as Excel, WolframAlpha, and Google, can handle fractional exponents just fine.

I'm not going to explain what they mean exactly, but basically if you multiply 1.04^(1/12) by itself 12 times, you'll get 1.04. So that's the rate you'd have to compound interest at monthly, to get an annual rate of 4%.


Discounting Multiple Payments

OK, so what if there's a stream of more than one payment? Well, you just calculate the present value of each, and add them up. So if you're trying to figure out how much an annual income of $50,000 for 30 years is worth to you today, you can just calculate the present value of each year's income, and add them all up.

This sounds like a lot of work, but there are calculating tools that can help you. Here is the annual calculation in Wolfram Alpha, using mathematical summation notation. I assumed that you only get one paycheck per year, at the end of the year. If you want to assume you get paid at the beginning of the year, just sum from 0 to 29 instead of 1 to 30.

If (more realistically) you get payments twice a month, just proceed by 24ths.

Excel also has functions you can use, PV() and NPV(), that ask for your discount rate, the payment amount, and the number of payment periods. and return a present value.


Perpetuities: A Neat Shortcut

What happens if you want to value a stream of payments continuing forever? You might think that this would eventually add up to infinite present value even if your discount rate is high. But if you're using a fixed discount rate, that's not actually true.

It turns out that there's a simple formula for the value of a "perpetuity," a set of periodic payments continuing forever, starting one period in the future. It's the payment amount, divided by the discount rate. If you like math puzzles, you can work out for yourself why this is the case - or look up the explanation - or if that's not clear to you, let me know and I'll write something up.


present_value_of_perpetuity = payment_amount / discount_rate

For example, let's say Dana owns preferred stock that pays  $1,000 in a year, and every year thereafter, forever.  Dana's discount rate is 5%. The present value of this is:

$1,000 / 0.05 = $20,000.

The first important thing about this is that since nothing takes longer than forever, any finite series of constant payments will never be worth as much as the perpetuity with the same payment size. So the value of a perpetuity is an upper bound, and one you can calculate very fast.

Suppose Dana's payments only lasted for 50 years. She did a PV calculation and got a value of $30,000, but the PV calculation is tricky and she wants to know if this is right. She can do the perpetuity calculation in a few seconds and see that her number is impossibly high. She rechecks her numbers and finds that the present value is $18,255.93

The second way this is useful is that many long series of payments are pretty close to the value of a perpetuity. You may have noticed that the value of the 50-year series of payments was close to the perpetuity value Dana first calculated. The reason for this is the same reason a perpetuity has a finite value in the first place: because discounting compounds, most of the value of a perpetuity is in the first few years.


In sum:

  • present_value = future_value / ( (1 + discount_rate) ^ time_difference )
  • The present value of a series of payments is the sum of the present values of each payment
  • There are lots of calculating tools that will take care of tedious calculations like this for you
  • present_value_of_perpetuity = payment_amount / discount_rate
  • The present value of a long series of constant payments is close to the present value of the perpetuity with the same payment.

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