# Draft article - Private annuity calculations

Suppose you want to sell a thing for a number of equal payments. How much should each payment be?

This is called a "sale for an annuity." This may sound jarring, if you associate the word "annuity" with "life expectancy." But in financial jargon, annuities can be for any finite term of years. For example, when you buy a house with an 80% mortage amortized over 30 years, you've given the bank a 30-year annuity, whose present value equals 80% of the house.

To calculate the amount of each annuity payment, you could use a mortgage calculator, or Excel, or tables published by the IRS. But you should also know how to do the math yourself. As one commentator has observed, "A private annuity has been described as the most talked about but least frequently used strategy in estate planning." Michael D. Mulligan, Sale to an IDIT for a Life Annuity: Confronting the Exhaustion Test and Other Challenges. Tax Management Memorandum, Vol. 57, No. 25, p. 519, 12/12/2016. Once I became comfortable with the math, I found I was much more comfortable selling the strategy.

Here's the equation: ____. You can undestand where this comes from by breakin ginto two steps.

- First, we modify the question: Instead of trying to solve for the payment in a finite annuity, let's assume the sale is for an infinite series of equal payments. In that case, each payment would be smaller. What is the size of each perpetual payment?
**Each perpetuity payment = the asset's FMV times the discount rate**. - Second, convert this perpetuity payment (i.e. for an infinite term) back into a annuity payment (i.e. for a finite term). The latter will be larger, because there are fewer payments. To get this larger number,
**multiply the perpetuity payment by a fraction: its numerator equals the FMV of the asset, and its denominator equals the sum of PVs of all payments through the term "n"]**. This latter term (i.e. the denominator) can also be expressed as**the perpetuity payment times [(1-1/(1+r))]/r.**

And we're done! If you're going to sell an asset for an "n" year annuity, the amount of each payment =

- amount of the perpetuity payment (i.e. asset's FMV times discount rate)
- times the ratio of
- FMV of an income interest in the asset for "n" years (i.e., perpetuity payment times [(1-1/(1+r))]/r)
- divided by asset's FMV.

This whole thing is pretty intuitive, once you grasp

- the linear relationship between the discount rate and the perpetuity;
- the cookie-cutter relationship between the perpetuity and the life estate/remainder, and
- the fancy relationship between the life estate and the annuity.

**Relationship of discount rate and perpetuity**. This is the** "perpetuity formula", **[perpetual payment = asset's present value * discount rate]. It's easy to remember, and even easier to derive. Ask yourself this: How much must I pay you, to make you indifferent about delaying receipt of $X from this year till next year? That's $X times the discount rate. (Where did I get that? It's the definition of the discount rate.) The same will be true next year, and the year after that, and the year after that ... all the way out to infinity. If at some point I confessed to you, "this whole thing has been a trick; you're never going to get $X," you shouldn't even mind. QED, I just sold you a thing for a perpetuity.

If you prefer, you could instead imagine this in terms of donkeys, sticks, and carrots on strings. Imagine that, with every footstep, the carrot grows at the discount rate, and the donkey gets to nibble off that little bit. If so, he should be willing to walk indefinitely--even if there is always a normal-sized carrot lying underfoot. In this analogy, the discount rate is the amount of carrot that a donkey must be allowed to nibble to justify delaying consumption of one normal-sized carrot by one footstep. This isn't something we derive; it is a function of the donkey.

**Relationship of perpetuity and life estate**. I call this a cookie-cutter relationship, because a life estate is the perpetuity payments prior to a stated term (the cookie cutout); the remainder is the perpetuity payments after that stated term (the leftover dough); and the life estate plus the remainder equals the perpetuity.

More generally, at step two of the equation above, notice the relationship between the numerator, [sum of PVs of all payments through the term "n"], and these three key actuarial concepts: term, remainder, and annuity.

- When we study a series of payments in isolation (i.e. to determine the present value of that series) we call it an annuity.
- When we study the exact same series of payments as the first of an infite series of similar payments (i.e. a perpetuity), we call it an income interest. It's specifically called a "term of years"/"term interest" where "n" is chosen arbitrarily, or a "life estate" where "n" is tied to a measuring life.
- The remainder interest is the cookie-cutout of these income and annuity interests: i.e. it represents the payments after the term "n," i.e. the parts of the perpuity which don't make it into the cookie. It's also called a "reversion" where it goes back to the seller.

**Relationship of life estate and annuity**. This last one, the coefficient [(1-1/(1+r))]/r, is a bit ugly. It comes from the **"annuity formula."** I won't derive this here; suffice it to say that it's a function of the discount rate, as it should be. This one you memorize.

So far, we've been discussing financial concepts. For tax law purposes, the question is: under what circumstances will the IRS accept this method of valuing an annuity? The answer is in the regulations under IRC 7520. The IRS has three main concerns.

**Choice of interest rate**. [in progress]**Unhealthy measuring lives**. [in progress]**Very healthy measuring lives**. [in progress]