Why would a life insurance company agree to pay out a nominal amount that is higher than the principal given to them?
As mentioned in other answers, the insurance company is assuming that they will earn a return on the principal they receive from the annuitant which is greater than they are giving the annuitant.
The basic principle behind this is called the time value of money. Broadly speaking (assuming positive interest rates) money now is worth more than money in the future, assuming you can earn the risk free rate of interest on the money.
As mentioned in the other answers, given the amount they will be paying out just on the guaranteed portion, they need to earn a relatively hefty return on the $100,000 of ~5.2% just to break even. So clearly they have calculated that for that portion they will earn more than this amount.
Is it that simple?
It's not quite as simple as assuming a 5.2% rate of return for 20 years, as some 80 year olds will live to an age greater than 100, and the insurance company will want to cover those payments as well (since it's a lifetime annuity). If it was a 20 year guaranteed term annuity, then yes, it would be that straightforward, but this is a lifetime annuity, as a result survival has to be taken into account.
It's also important to understand that it's not as simple as taking $100,000, investing it for 20 years and ending up with $160,000 at the end of it. The reason why it's not that simple is that we need to pay the annuitant $664 per month, every month, for the full 20 year term. As a result, we don't gain the returns on these payments over the full 20 year period. This has the effect of us needing a significantly higher interest rate to fund the annuity than we would if we were just sticking the $100,000 in an investment and taking it all out at the end of the 20 year period as a lump sum.
How do they calculate what it is worth? or How do they know what they need to earn to make a profit?
The formula for the Expected Present Value of a lifetime annuity with a guarantee period is denoted1:
EPV = P * (agt + vt * tpx * a(x+t) )
- EPV is the Expected Present Value. In this case, assuming no profit, you want the EPV to be equal to $100,000, the money the annuitant used to buy the annuity)
- t is the time, in years, that you want your guarantee period to last for (in this case 20 years)
- P is the nominal value of the annuity payments over a single year (in this case $664 * 12 = $7,968)
- v is the discount factor for a single year at interest rate i (this is how we equate the value of money today with the value of money tomorrow)
- agt is a guaranteed annuity with a term of t years (this is the guaranteed portion of the lifetime annuity)
- a(x+t) is the value of a lifetime annuity given to a person aged x+t with no guarantee period where the principal is 1 of your favourite currency ($/£/€). This is the remainder of the lifetime annuity which is not guaranteed.
- tpx is the probability that a person aged x will live to age x+t (ie the probability that the insurance company will have to pay that payment)
This formula can be used to calculate any of the three common unknowns for an annuity (the single premium, the interest rate required, or the annual/monthly payments to the annuitant).
They also all assume that the annuity is being paid in arrears (ie I pay you $100,000 now and you pay me my first payment after one period has elapsed [1 month]). There are additional modifications you can make to calculate an annuity-immediate (one that pays you in advance of each interest period), but they aren't relevant for the purposes of this question.
But, you may ask, how do you calculate these different formulae?
Three of them are pretty straightforward (math wise)
- v = 1/(1+i) where i is the interest rate you are expecting to earn
- agt = ( 1 - vt )/i
- a(x+t) = infsumt=0(vt * tpx+t) where infsumt=0 indicates you are taking a sum from t=0 to t=infinity.
- Since we know that humans don't live forever, most insurance companies will have an assumed "omega age" at which all policyholders are assumed to die. This is typically either 120 or 125. 120 is the more commonly used assumption. As a result this formula reduces to:
- a(x+t) = omega_age - (x+t)sumt=0(vt * tpx+t)
Note: the formulae presented above assume an annual payment. If it were to be a monthly payment further adjustments would need to be applied to both the guaranteed annuity formula and the lifetime annuity formula to take account of this.
The final term (tpx) doesn't have a nice formula, and the calculation of this value is a large part of the reason you have actuaries.
tpx is calculated using historical data, to figure out the likelihood of a person aged x dying for each period of time after that age. Studies like the Continuous Mortality Investigation in the UK devote significant resources to figuring out what these values are, and producing models to predict changes to this value in the future. these sort of investigations produce a table of expected survival probabilities for a whole country (eg the UK).
These country wide tables are adjusted by individual insurers for factors like:
- Healthcare improvements (and thus life expectancy improvements) over time
- The fact that insured policyholders tend to be healthier than the average member of a population of a country (due to the fact they are thinking about their health and spending money on it)
- Differences in mortality experience by area and earnings (some areas, like coal mining towns, tend to have a significantly worse mortality experience due to exposure to coal dust and other carcinogens which shorten lifespan)
- Whether or not the person smokes
As a result there is no simple formula for this, and in general actuaries will take this from a set of precalculated tables, that are updated at least yearly, sometimes more frequently. These tables will, in general, be produced by the Actuarial Research team within each company. The same team will also determine a view as to what future potential mortality improvements will be, which are then applied on top of the tabulated base table values.
How does an insurance company make a profit on this arrangement?
It is important to remember that they are offering the annuitant an annuity with a minimum return of 5.2% return baked into the annuity, for the life of the policyholder. In order to make a profit on selling this annuity, the insurance company needs to add some margin onto that amount, and that margin comes from what they expect they can earn on the money markets with the money.
To allow for some measure of profit, and for the policyholders who live beyond the age of 100 being paid, insurance companies will require an investment return that is higher than that calculated purely on the guaranteed portion of the annuity. So in this case it's likely to be somewhere in the region of 6 - 16% (the exact % will depend on the specific market the annuity is being sold in, the regulatory environment the insurance company is regulated in, and the level of profit the insurance company is looking to achieve).
Where does the 6 - 16% figure come from?
It comes from my experience working in the actuarial side of insurance companies in Europe for almost a decade. Due to the competitive nature of the insurance market, most insurers will (on average) look for a profit margin of 5 - 10% on their book of policies, with 5% being typical (in my experience).
In addition, they will need to earn some additional investment income on the principal to cover the period after the guarantee term. In the specific case of an 80 year old annuitant this additional interest requirement will be a relatively small compared to their whole book of policies, due to the minute amount of policyholders who live beyond age 100. If they were offering a similar return for a policyholder aged 60, then the additional interest required would be more significant as a much larger portion of the population live beyond age 80.
Where does the 5.2% figure come from?
The 5.2% figure comes from calculating the interest rate that would be required to support the annuity-certain portion of the annuity (ie the 20 year guaranteed portion). To do this we take our annuity formula from above:
and multiply it by the 1 year annuity payments (ie P), $664 * 12 = $7,968.
Using these two pieces of information and the fact we know what payment the annuitant will be making ($100,000) we setup the equation:
- $100,000 = $7,968 * ( 1 - (1/(1+i) )20 )/i
and solve for i. The fastest way to do this is to setup the formula in excel and goal-seek for the right interest rate. Note: If you know the interest rate this formula is much easier to use to find one of the other two values.
If we calculate that interest rate we get get 4.92%. As you may notice, this is not 5.2%, and I promised 5.2%! The difference here comes from the fact that this formula uses annually compounded interest as opposed to monthly compounded interest. If instead we use monthly compounded interest the formula we need to use is:
- $100,000 = $7,968 * ( 1 - (1/(1+i) )20 )/( 12 * ( (1+i)(1/m) - 1 ) )
The modification we've done is a standard modification. To translate a figure compounded annually to one compounded m times per year you multiply the figure by i/r(m), where r(m) is the annual rate of interest compounded m times per year.
In our case, this means simply replacing the annual interest rate (compounded annually), i, in the denominator of our formula with the annual interest rate (compounded monthly), r(12) = (12 * ( (1+i)(1/m) - 1 )).
When you do this you obtain an interest rate of 5.19%, or 5.2% if you round it.
This answer ignores all of the costs that the insurer needs to cover above and beyond the policy benefits. These costs would be covered as part of the "6%-16%" figure I mentioned earlier, but profit would also need to come out of that figure.
What sort of costs am I ignoring?
- Salaries (typically ~30% of an organisations cost base)
- Buildings maintenance and insurance costs (yes insurers need to insure their buildings)
- Transactional costs for operating on the financial markets
- Fraud investigations (some people are happy to submit fraudlent claims because they view it as a "victimless crime", but it's not, it is accounted for in the premiums of everyone insured by that company)
- Future regulatory changes (insurance companies are highly regulated businesses and changes, like Solvency 2, can cost significant amounts of money to become compliant with)
- Counterparty Risk (what happens if one of their providors goes bust, they still need to provide whatever service that provider was providing)
Obviously what effect all of these things have on the interest rate depends heavily on how large the book of policies the insurance company has is (annuities are policies too!). Larger books can make use of economies of scale to require smaller interest rate returns on their policies, and thus offer lower premiums (and attract more policyholders).
1: There is a more formalised notation for this, to differentiate guaranteed annuities from a life annuity, but the stackexchange site doesn't allow for formatting that notation outside of using pictures, which aren't friendly to screen readers.