The payout rate in the previous example wasn't 8%. It was 8.5%.ScubaHogg wrote: ↑Fri Sep 15, 2023 12:06 amYes I’m aware of the difference. But I’m also aware of the similarities, like how there is no maturity date or lump sum at the endpetulant wrote: ↑Thu Sep 14, 2023 7:42 pm
A perpetuity is a bond that pays coupons with no redemption date. It goes on forever unless affirmative action is taken by the issuer to buy the bond back. Hence, a perpetuity can be inherited by heirs. The most famous example is the British consol, a perpetual bond issued prominently in the late 18th and then the 19th century. Plenty of ink has been spilled on 19th century British social relations based on the inheritance of consols. On the other hand, a SPIA has an end date when the buyer passes away. For the average 70-year-old annuitant, heirs will receive nothing else. A SPIA is absolutely not a perpetuity.
Fair enoughMy "intuition" is not that a SPIA is a 17-year bond, i.e. a bond maturing in 17 years. I said clearly it should be compared at a first pass to a bond ladder. A bond ladder involves the purchase of bonds maturing each through a target year, with the maturing principal for that year plus interest from future years creating a cash flow. There are many threads on BH discussing bond ladders, most recently in the context of TIPS.
I’m still not convinced that using the duration formula for perpetuity makes more sense for a SPIA.I invited you to actually do the work of creating a spreadsheet that shows bond ladder cashflows to see how much principal has to be returned in the first few years to create the target cash flow. If you had done so, you would know that about half of the amount returned in the first few years is principal. The return of principal means that the duration--the sensitivity of the NPV of future payments to interest rate changes--of a SPIA is much lower than the expected payout period. A rough analogy is to mortgage-backed securities. Although they are issued for 30 years--much longer than the lifespan of almost all 70-year-old annuitants--they have a much shorter duration in part because all mortgage payments include a return of principal (in addition to prepayments such as paying off debts, refinancing, and moving). The average duration of MBB, the iShares MBS ETF, is 6.04 years per the iShares website. The average yield to maturity reported by iShares is 5.10%. Even without prepayments from early payments, refinancing, moving, and so on, the duration of a 30-year mortgage with an interest rate of 5.10% would be around 11-12, much less than the mortgage life of 30. The duration of a 15-year mortgage with an interest rate of 5.10% would be around 7, much less than the mortgage life of 15.
Let’s take this definition of duration
https://www.investopedia.com/terms/m/ma ... ration.aspUnderstanding the Macaulay Duration
The metric is named after its creator, Frederick Macaulay. Macaulay duration can be viewed as the economic balance point of a group of cash flows. Another way to interpret the statistic is that it is the weighted average number of years that an investor must maintain a position in the bond until the present value of the bond’s cash flows equals the amount paid for the bond.
That’s the big thing! When does the weighted average of cash flows received equal the amount paid for the bond. In your example the a SPIA had a duration of about 8 years. Well in the example I gave above a single male had a payout rate of about 8%. 8*8 is only 64% of the initial purchase price. Intuitively it makes no sense that 64% of 100 spread out over 8 years equals the purchase price made in the past.
The duration concept you're talking about isn't the future year at which the nominal dollar value of cash received equals the price of the bond. It's the future year at which the present value of future cash flows equals the purchase price. To calculate it, it's not 8.5% * n years = %. If you review the article you linked, it requires taking the cash flow for each year, multiplying it by a discount rate, and then multiplying each year's cash flow by the number of years into the future, like so:
Code: Select all
Discount Rate = 4.2% Life Expectancy = 17 years SPIA Premium = $100,000 Payout Rate = 8.5% Formula for Discounted = Payout/(1+Discount Rate)^Year Formula for Disc*Yr = Discounted * Year Year Payout Discounted Disc*Yr 1 8500 8157 8157 2 8500 7828 15657 3 8500 7513 22539 4 8500 7210 28840 5 8500 6919 34597 6 8500 6640 39844 7 8500 6373 44611 8 8500 6116 48929 9 8500 5869 52826 10 8500 5633 56330 11 8500 5405 59465 12 8500 5188 62256 13 8500 4978 64726 14 8500 4778 66895 15 8500 4585 68785 16 8500 4400 70413 17 8500 4223 71798 Sum of Disc*Yr = 816675 Sum of Disc*Yr / SPIA Premium = 8.16
Duration isn't a payback period used to evaluate investments. Duration is a measure of sensitivity to interest rate changes used to evaluate risk. The value of a perpetuity would be measured as the net present value of future coupon payments with no maturity--if it was sold at a later time, then the price at that point would be the net present value of future coupon payments at that time, meaning that all of the value always comes from future coupon payments.
To use the 8.5% SPIA example, in order for a perpetuity to produce $8500 in income per year when prevailing interest rates are 4.2%, then the market price would be 8500/.042 = $202,380. (You can also double check the perpetuity by using a formula like =PV(.042,10000,-8500,0).) The SPIA producing $8500 per year for a 70-year-old cost $100,000. The perpetuity is worth twice as much. If that isn't enough to prove to you that they're fundamentally different instruments, I've got a bridge in Alaska to sell you.
What duration is saying for a perpetuity is that, because it will receive payments indefinitely, the market value of the perpetuity is very sensitive to interest rates. One little flick of the rate, and the market value will swing around. If we can imagine a SPIA as having a market value after purchase, its duration must be shorter than the perpetuity because it has an end date. Its interest rate sensitivity must be less. How much less, of course, depends on the life expectancy of the annuitant.
It's possible that the person could collect for 70 years, but the expected period of collection for a 40-year-old buying a SPIA is more like 45 years. Undoubtedly the duration would be longer than for a 70-year-old purchasing a SPIA. One thing about NPV of future cash flow items is that as the period grows longer, the change in net present value is smaller since payments in the distant future are heavily discounted. For example, compare the NPV of a perpetuity paying $8500 per year when prevailing rates are 4.2%, which is about $202,380, to a 100-year mortgage with $8500 per year payments with a 4.2% interest rate. That mortgage's face value would be about $199,074. So, likewise, if a SPIA was going to last 50 years, its numbers would start to approach those of a perpetuity. But a 70-year-old's SPIA is not close to that yet.
But under the hood, so to speak, a SPIA is returning principal. For example, the Internal Revenue Code taxes nonqualified annuities as if they were returning principal and paying a bit of interest over the life expectancy of the annuitant. After life expectancy, all of the income is taxable. Did you ever make the Treasury ladder spreadsheet I asked you to make? Do you know what a Treasury ladder is? If you looked at the cashflows, I think you would agree that a mortgage is probably the other best comparison.ScubaHogg wrote: ↑Fri Sep 15, 2023 12:06 amAn MBS doesn’t make sense, since as the principal gets paid down, there is less ongoing interest more principal gets paid off each month. A SPIA isn’t returning principal, or paying interest. It’s just a payout rate. There is no remaining balance or residual claim.