The time-value of money is the principle that money today has a different value than money tomorrow. At the centre of this concept is the annuity and its formula, which I will not derive or prove here because I’m the only one who will find it interesting. An annuity is a set of fixed payments that are made over a specified period of time. I’m going to focus on a specific type of annuity here: a *life annuity*. A life annuity is an insurance policy sold by an insurance company that pays a (usually monthly) payment to the buyer until the buyer (or the buyer’s surviving partner in some cases) dies. Sometimes you just want to be able to know how much money you’re going to have in the future and the life annuity is one way to do that. Life annuities can seem expensive if you’re not used to seeing the numbers. The high price is the price of certainty. I’m going to talk about annuities in the context of retirement, but the math involved translates into pretty much every area of finance so pay attention. The words annuity and life annuity will be used interchangeably depending on context. Get ready to bite off more than most of us can chew, myself included.

### An annuity example

Dave is 60 years old, unmarried, about to retire and has $250,000 in unregistered savings. He wants to have a steady stream of income for the rest of his life. He decides to buy a life annuity. The insurance company offers him $1232 per month for the rest of his life in exchange for the $250,000 payment. When Dave dies, the insurance company is off the hook. If Dave lives until 100 he’s going to receive $591,360 in total over that time (that’s 480 months). If Dave dies at 66 he will have only received $88704, less than he put in. Is the annuity a good deal?

### What’s an annuity really worth?

Good question. It’s probably becoming obvious to you that this could go either way. If you die young and you just bought an annuity, you could get hosed out of all that money in two ways. First, your estate will have nothing to give to possible heirs. Second, you won’t have had access to that pile of money you used to buy the annuity you didn’t use. But what if you live a long time? This is why insurance companies are the ones selling annuities. Actuaries (people who take piles of statistics and crunch the numbers to calculate insurance premiums) have figured out the price to sell you an annuity and still make money. They are betting that *you will die early enough that they’ll make a profit*. Will you? Not sure. But the stats say the insurance company will make money. In addition, the insurance company is likely making money on *spread*. That is they have calculated a rate that they will pay you that is *lower* than the rate they think they can achieve through investing your money. To calculate the present value (PV, the value *today*, when you’re making the decision) of an annuity you need to know how long you’re going to live. Once you’ve got that amount, the math is fairly straightforward:

Where *PMT* is the periodic payment, *n* is the number of payments (generally months you think you’ll live) and *i* is the annual interest rate.

Wait, what annual interest rate?? Ahhh. That’s the thing. There is a rate that the insurance company selling the annuity assumes they can make money on. That is, they can earn *more* money investing your money than what they pay out. So really, you should figure out whether you could *reasonably replicate the performance* of the annuity you’re looking to buy to find out if it’s worth it. Perhaps then the best way to find out if an annuity is worth it is to find out what that rate is! Great. Remember algebra? In the above formula, we were calculating the present value. But if we’re buying an annuity we already know the present value. That’s the price the insurance company is selling the annuity for. What we really want to know is *i*, the rate the insurance company needs to receive investing your money to break even. Let’s re-arrange the equation to solve for *i*! Not happening. This is pretty much impossible because of the derivation of the original equation. I won’t explain the math past that but the general approach to solving for the interest rate of an annuity is to guess and check, usually with a computer. Microsoft Excel will do this for you with the RATE function, for example. Here’s a table using the above example for Dave:

Years | Break-even rate (monthly compounding) |
---|---|

17 | 0.06% |

20 | 1.72% |

25 | 3.35% |

30 | 4.26% |

35 | 4.81% |

40 | 5.16% |

45 | 5.39% |

What does this table mean? Simply put, as long as the insurance company can invest at an annual interest rate greater than the rate in the right hand column, they will make money. So if 60-year old Dave lives to be 85 (25 year annuity), The insurance company has to make at least 3.35% annualized on Dave’s $250,000 premium to break even. Conversely if Dave dies at 77, the insurance company can earn only 0.06% annually on Dave’s premium and still break even. If you’re paying attention, you may have noticed that if Dave dies before 77, the insurance company makes money at rates less than 0%. That’s quite the hedge. To add to the confusion, there are several different types of annuities. I’ll break them down very concisely here.

Type | Description |
---|---|

Life | income for life |

Life plus guarantee of X years | income for life with at least X years of income to your estate if you die before X years |

Life plus joint-and-last-survivor | income for life for you and your spouse, payments end when both of you have died |

Indexed life | income for life and payments increase with inflation |

### The perpetuity: a special case

Imagine if the money could last *forever*! Such a tool exists, my friend. It’s called a perpetuity. In a perpetuity, you pay once and get money forever. As you can imagine, they’re not very popular since people don’t live forever so they’re a huge waste of money. Generally they’re used for things like college and university scholarship endowment funds— a situation where the money can’t run out. The math of a perpetuity is simple: the payment is equal to the premium times the interest rate. Many people who can do math suggest that the best inflation-indexed interest rate is about 4% and thus you can effectively replicate a perpetuity by taking out 4% of your money every year provided your investments roughly track a global index. That means that for every $100,000 you have invested you can take out $4000 per year ($333/month) and *never run out of money *because it will all be there when you’re dead. This is good math for the very-early-retirement crowd. Of course if you take out more than that, the money won’t last forever in this model.

### Risk

The major risk of buying an annuity is that the insurance company will default or go bankrupt. There’s no rate risk, which is the major (and only) appeal.

### Make your own annuity

You *can* build your own annuity (BYOA)! In fact, that’s the way many people end up structuring their retirement. Basically it involves math. You need to predict what you think you can achieve with your investments (quite a feat in itself), then predict how long you’re going to need money for (good luck with that). Then you just plug in the numbers and BAM! You know exactly what you’ll make. Look, this is highly variable. That’s why it’s so expensive to buy an annuity. I’m not making a recommendation either way, but now you know the information you need to make an educated decision when the time comes. I’m going to talk about pensions shortly so make sure you understand the annuity before reading about pensions because they go hand in hand.

### Implementation and Taxes

From a tax perspective life annuity payments are regular income if they’re purchased with registered investments like RRSPs. This makes sense because the buyer never paid tax on the RRSP contributions so it’s all just income. Let’s face it, most of you who will buy annuities will be in the RRSP boat. Buying an annuity in a TFSA isn’t generally recommended since unregistered annuities get preferential tax benefits— you only pay tax on the ‘interest’. Some of you will have lots of unregistered money that you’ll used to buy an annuity. If you’re in this boat, start reading about the tax implications of unregistered annuities yourself. They are more complex and there are a large number of variables.

### Curve ball

Annuities work both ways. So far I’ve talked about using a large amount of money to purchase the right to get payments. But we’re all really familiar with mortgages, car loans and monthly payment plans. Those are annuities too! They just work the other way— the bank gives you a lump sum of money then you pay the bank back over a specified number of payments. When I said annuities are pretty much everywhere in finance I wasn’t kidding.

### So should I get a life annuity?

You have to decide whether or not it’s worth it to buy an annuity. Dependability has some serious advantages. That being said, it’s probably not a good idea to use all of your savings to buy an annuity. For example, say in retirement you need to make a big purchase. The working you would get a loan and then pay it off, with interest. The retired you who has a pile of cash can just pay cash for that big purchase and pay *no interest*! But if you have an annuity instead of a big pile of cash you’ll probably need a loan, with interest. There are literally hundreds of scenarios where an annuity could work against you if you don’t have another pool of cash in retirement. That is the cost of dependability.

## One thought on “The annuity: dependability has a price”