February 6, 2023 – Welcome to another installment of my Safe Withdrawal Rate Series. See the landing page of this series here for an intro and a summary of all the posts I’ve written so far. On the menu today is an issue that will impact most retirees: we all likely receive supplemental cash flows in retirement, such as corporate or government pensions, Social Security, etc. Some retirees opt for an annuity, i.e., transform part of their assets into a guaranteed, lifelong cash flow.
Of course, if you are a long-time reader of my blog and my SWR series you may wonder why I would write a new post about this. In my SWR simulation toolkit (see Part 28), there is a feature that allows you to model those supplemental cash flows and study how they would impact your safe withdrawal rate calculations. True, but there are still plenty of unanswered questions. For example, how do I evaluate and weigh the pros and cons of different options, like starting Social Security at age 62 vs. 67 vs. 70 or receiving a pension vs. a lump sum?
Also, you might want to perform those calculations separately from the safe withdrawal rate analysis, from a purely actuarial point of view. For example, we may want to calculate net present values (NPVs) and/or internal rates of returns (IRRs) of the different options before us. Clearly, NPV and IRR calculations are relatively simple, especially with the help of Excel and its built-in functions (NPV, PV, RATE, IRR, XIRR etc.). However, the uncertain lifespan over which you will receive benefits complicates the NPV and IRR calculations. How do we factor an uncertain lifespan into the NPV calculations? Should I just calculate the NPV of the cash flows up to an estimate of my life expectancy? Unfortunately, the actuarially correct way is more complicated. But Big ERN to the rescue, I have another Google Sheet to help with that, and I share that free tool with you.
Let’s take a look…
An Actuarial NPV/IRR tool
Let’s start with a simple example: A 55-year-old male early retiree in average health has access to a life-long corporate pension worth $600 a month. Alternatively, he can cash out the pension and receive $100,000. What’s the best choice for this retiree?
There isn’t one unique answer, but we can address several questions to evaluate this lump-sum vs. lifetime annuity tradeoff:
- Considering the average death probabilities from the Life Table used at the U.S. Social Security Administration (SSA) – latest release here – what is the expected internal rate of return of the net cash flows?
- Assuming a fixed target rate of return, what is the net present value (NPV) of this cash flow stream assuming the SSA life tables?
- We can also ignore the life table probabilities and calculate an IRR for an assumed specific time of death. For example, if this retiree believes that he gets benefits exactly until, say, age 84, what would be the IRR in that case?
- And likewise, for that lifespan up to age 84, what would be the NPV under the target IRR?
- Assuming a certain target rate of return, how long does the retiree have to survive to break even from a financial perspective?
Lots of questions. Let’s look for the answers. Here’s the link to the Google Sheet:
Notice that you need to save your own sheet in your own Google account first. I can’t give you permission to edit my clean Google Sheet because you’d likely mess it up for everyone else.
In the Google Sheet, the cells in dark orange are the user-provided inputs. The main outputs are in green and the other cells are used for computing. Please change only orange cell inputs! The main inputs are pretty straightforward:
- Enter the age in years and months.
- From the pulldown menu, select either SSA-Male or SSA-Female to use the respective Social Security Administration life table assumptions.
- Since we’re not all average Americans we can also adjust the death probabilities to model longer or shorter life expectancies. For the baseline case, I leave this parameter at 1.00, because the baseline model assumed an average life expectancy. But later we can play with that parameter and see how sensitive our results are.
- We set the Target IRR at 5.00% (nominal), so roughly in line with an investment-grade (BAA) corporate bond yield in early 2023.
- The age for the NPV Calculation. Consider this a case study for when this retiree dies at exactly age 84.
- The cash flows: I assume that the retiree takes the annuity and receives $600 per month starting in month 0. Sometimes your benefits would start a month later, but that wouldn’t make a huge difference here.
- What’s the deal with the month 0 negative cash flow? The $100,000 that you forego is an opportunity cost. If you net that with the first monthly pension income you get -$99,400.
And we can now read off the results:
- The IRR is 4.915% when using the SSA Life Tables.
- If you require a 5% target IRR, then the pension is worth -$872.85, so the future pension payments don’t entirely cover your initial $100,000 outlay.
- If you were to survive until age 84 the pension looks much more attractive. That would generate a 6.112% internal rate of return!
- And likewise, at your target IRR of 5%, your annuity generates a positive NPV of $12k+. Conditional on surviving that long, the future payments more than compensate for the initial $100k opportunity cost.
- At the 5% target IRR, you’ll need to survive up to age 78 years and 1 month to cross over into “positive financial territory,” i.e., recover the initial $100,000 opportunity cost. Again, this uses the 5% annualized discount rate.
Ignoring the Opportunity Cost
Alternatively, we could have calculated the NPV of the pension itself, ignoring the opportunity cost. That’s what I did in the calculation below: evaluate the $600 monthly cash flows only. Notice that three features in my Google Sheet are no longer usable: The two IRR calculations and the crossover calculation, because we only consider the positive cash flows.
These calculations will be helpful in a scenario where the retiree doesn’t have a cash-out option and/or wants to assign a value to his or her future cash flows. In the base case example, we can still read off the two NPVs: $99,127 and $112,092 for the probability-weighted and the Death at age 84 scenarios, respectively. Notice that those values differ by precisely the $100,000 opportunity cost in the above calculation because that value has a discount factor of 1.0000.
Side Note: How much of an error do we make when we ignore the survival probabilities?
Since I made such a big issue out of the difference between the actuarially correct way – discounting cash flows with survival probabilities – and the incorrect way of a cash flow from the pension up to the life expectancy, how much of a difference would that be? That’s easy to answer. In the Google Sheet, I set the age for the NPV calculations equal to the life expectancy, 80.69 years in this case, and compared the two different NPV estimates. With a certain death at age 80.69, you get an NPV of $5,773, but with uncertainty around the exact age at death, you get a -$872.85 NPV. That’s a $6,646 difference, which is quite substantial for a pension with a $100,000 cashout value.
Why the significant difference? Very simple, if there is uncertainty around the age at which the retiree dies then you certainly benefit from living longer and you lose from dying earlier. But since the later cash flows are more heavily discounted, the gains will not sufficiently compensate you for the losses from dying earlier, because those cash flows are not as heavily discounted. So, this numerical example shows very nicely how important it is to get the math and the actuarial assumptions right. A $6,000+ difference in the NPV can often make a difference between an attractive and unattractive annuity or pension arrangement!
Adjusting the life expectancy
What if you’re not the average person? I’m a generally healthy person, with a healthy body mass index, with some of my ancestors living into their late 80s or even 90s. I’m a non-smoker and I don’t do any “stupid things in stupid places with stupid people.” Thus, my life expectancy should be a bit longer than the SSA average. How do I account for that? Glad you asked because I devised a way to scale the death probabilities to generate more realistic life expectancy estimates. So, imagine that our 55-year-old retiree believes that he has a life expectancy of roughly three years longer than the average American male. We can play with the parameter “Death Prob scaling” to accomplish exactly that. For example, if we set this parameter to 0.7 we raise the base life expectancy from 80.69 years to 83.98 years, see the screenshot below. The way I model this is to assume that over the entire life span, this individual has a 30% reduced death probability every single month. So, if your baseline death probability in month zero at age 55 was 0.0612%, your scaled death probability is only 0.0428% or 0.7×0.0612%. Is this a good assumption? I’m sure actuaries have more sophisticated models that let you input a ton of extra demographic information and then would custom-tailor your death vs. survival probabilities. But with my limited time and resources, this is what I ran with. It’s certainly better than working with only the SSA assumptions! If you don’t like my assumptions, please come up with a better model. It takes a model to beat a model!
So, how much of a difference would that make in my calculations? Please see the screenshot below. Notice that the life expectancy is now 3.29 years longer, which raises the expected horizon to just under 29 years. The Survival-probability-weighted numbers are now greatly improved. You raise your IRR to about 5.5% and the NPV to about $5,725 when using a fixed target return rate of 5%. With the improved life expectancy, this pension starts to look quite attractive. Of course, 5.5% is still way behind an expected equity return, but considering the pension as a safe bucket, fixed-income asset, the return looks quite attractive under my assumptions.
Do I get an 8% return per year for delaying my Social Security?
If you delay Social Security from your normal retirement age of 67 (for most people in or close to my age cohort) to age 70 you raise your benefits by 24%. With compounding, that’s 7.4% p.a.; not quite but pretty close to 8%. Likewise, if you planned to take benefits early, at age 62, but you waited five more years, you would get a 100/70-1 or roughly 43% increase over five years. That’s again a 7.4% compounded annualized increase.
But the 8% return claim is not just wrong due to some bad rounding and confusing arithmetic vs. geometric returns. The 8% figure is nonsensical because by waiting one year you may get 7.4% more benefits but you also lose one year of benefits. To study the tradeoff between claiming at different ages we need to do a lot more than this back-of-the-envelope calculation.
Let’s look at an example where a retiree is 67 years old and could claim Social Security immediately and receive $2,000 a month or wait 36 months and receive 1.24x$2,000 = $2,480 a month. The differential cash flow for delaying benefits by three years is -$2,000 for the first three years and +$480 for all subsequent months. Notice: it’s not +$2,480 but +$480 at age 70+! In other words, we take the $2,480 cash flow starting in month 36 but we also subtract the opportunity cost of not claiming at age 67. Let’s plug that into the toolkit and see what happens, please see the screenshot below. The first observation: your IRRs are much smaller. All the returns here should be considered real, inflation-adjusted returns because the cash flows are all inflation adjusted. So, curb your enthusiasm and accept leaner returns. In the case of this sample retiree, the IRR of delaying benefits is only 1.2%. If you make it to age 87, it’s still “only” 3.286%. A far cry from the 8% estimate floating around on the web. In fact, even if you survived all the way to age 119.9, your IRR wouldn’t get much above a 7% internal rate of return. And finally, to reach a 2.5% real internal rate of return you’d have to survive until at least age 85 plus 7 months!
Why is the implicit return so low? The benefit increases or reductions from delaying or filing early are roughly actuarially fair. They are supposed to factor in a very modest real rate of return, maybe about in line with the long-term average real U.S. Treasury rate. Let’s be real, friends, our federal government wouldn’t shower us regular slobs with an 8% annualized real return. The real generous gifts go to the defense or pharmaceutical industries, but I digress.
Higher Life Expectancy
What about someone with a higher life expectancy? Let’s go back to a male retiree, age 67, but with a 0.7 death probability scaling. Now we’re making progress. By increasing the life expectancy by 2.7 years, we also increase the IRR to above 2.6%. This, in turn, means that at a target 2.5% discount rate we’re now at a positive $838.75 NPV. Yeah, you get a little bit extra, but in the big scheme, that’s not a large amount. The NPV of Social Security at age 67 (only counting the +$2,000 cash flows) is $355,611.95 when using a 2.5% target IRR, so the improvement in the NPV from delaying benefits of $838.75 is really a drop in the bucket.
Female retirees with a higher life expectancy
You can get noticeably better outcomes when looking at a female retiree. Assuming again the 0.7 scaling applied to the already lower death probabilities of a female retiree, we now get an IRR of 3.59% and an NPV advantage of almost $9,000 when using a 2.5% annual discount rate. Please see the screenshot below. And again, some people will complain that 3.59% is much lower than they can make with their VTSAX. I know, but keep in mind that these are real, inflation-adjusted returns, and they are perfectly safe without any equity volatility. So, for a safe, fixed-income bucket investment, any real return north of 2.5% and certainly 3.5% is a great return. If you have a better-than-average life expectancy, you should undoubtedly delay your Social Security benefits.
Future research, extensions
Notice that in this simple toolkit, I’ve abstracted from a few other potential benefits of Social Security. First and foremost is joint spousal retirement planning. For example, when the older spouse with a shorter life expectancy has higher benefits, it’s often beneficial to claim benefits at age 70. When that older spouse dies, the surviving spouse can then take over the higher benefits. In today’s post, I have ignored the joint spousal benefit calculations, but I may add that all at a later point. There is an excellent tool at opensocialsecurity.com already, so I’m not rushing to add that feature now.
Another benefit of Social Security is the advantageous tax treatment. Only up to 85% of the benefits are taxable. So, maximizing lifetime benefits is essential!
And talking about taxes, here’s another reason to use a very sharp pencil and craft a careful personalized analysis: taking a large lumpsum today might push you into a higher tax bracket, while small future pension payments may not. It’s possible that such tax considerations might make the pension even more attractive than it already is.
Side note: beware of the “Worst of the Web”
People shall be forgiven when they miss some of the subtleties of actuarial calculations, like calculating IRRs up to the life expectancy vs. using survival-probability-weighted cash flows. Sometimes the differences are not that great, so as a quick-and-dirty first estimate we can certainly just look at the IRR and NPV up to the life expectancy. But I’ve seen much worse out there; mind-blowing examples of financial and mathematical illiteracy that I just wanted to feature here as a warning about how we should take everything floating on the internet with a grain of salt.
The first common mistake is to ignore the time value of money altogether, effectively setting the discount rate to 0%. So in other words, in this context people will often argue that since the $100,000 cash-out value is only worth about 167 monthly premiums, the crossover point occurs before age 69. Compare that to your life expectancy of 80+ and you’re good to go with this pension. Uhm, wrong!
The same funny math is common when gauging the pros and cons of Social Security timing. You’ll be surprised how widespread this error is. I’ve seen this on financial adviser pages. Even reputable major brokerage houses, like Fidelity, publish this nonsense on their website as one of their “Viewpoints.” See this link, comparing lifetime Social Security benefits when claiming benefits at different ages. Note that the future benefits are not discounted but just added up to one fixed age of 95. I want to avoid beating up Fidelity too much because it’s my preferred broker. I’m sure other large brokerages also publish this rubbish written by people with a similar disregard for elementary accounting and finance principles, like the time value of money. But this is the financial misinformation we’re often dealing with out there!
Another error is messing up the time value of money calculations. One fellow FIRE blogger produced results even worse than if he had simply set the discount rate to 0%. Specifically, he makes these two insane assumptions:
- Instead of (slightly incorrectly) discounting cash flows up to a specific life expectancy or (correctly) discounting with survival probabilities, this brainiac blogger discounts the future cash flows – and I am not making this up – up to INFINITY! As in “forever.” Probably because the geometric sum formula is much simpler when discounting to infinity rather than a set future date. But it’s also wrong and it doesn’t even pass the smell test because a pension or annuity NPV must be different at age 50 vs. 90.
- He also discounts future benefits of a pension by the (intermediate-term) U.S. Treasury Rate (e.g., 10-year Treasury, currently at 3.532% as of February 3, 2023). (granted, for corporate pensions, this blogger indeed uses an adjustment factor of 0.95, but that only raises the effective discount rate to 3.72%, still too low compared to IG corporate yields). Not a good idea – actuaries typically prefer an IG corporate bond yield, e.g., somewhere between the AAA at 4.28% and the BAA yield at 5.28% as of early February.
Now take the annual cash flow, say $7,200 in the base case scenario, and divide that by the 10-year Treasury rate (3.532%), and you get $7,200/0.03532=$203,8416. Or $193,658 when applying the 0.95 risk factor. In either case, that’s even “wronger” than just using the life expectancy times benefits: $7,200×25.69=$184,968. Note that the correct NPV was only $99,127, less than half of the infinite-horizon value. I’m going to spare the fellow the embarrassment and not mention him here. But you might already guess who he is; I’ve had a run-in with him before on another issue.
So, I hope that with my little toolkit here I’ve taken away some of the excuses for spreading bad math on the internet. If people bother to read my post and use it…
Wow, I was able to write an entire blog post without any safe withdrawal rate simulations. I wanted to offer this Google Sheet because sometimes folks ask me about my views on annuities and pensions and I like to be able to refer people to a simple tool where they can punch in their numbers and play around themselves. Saves me a lot of time!
I am also planning to write a separate post about how annuities, pensions, and Social Security timing work in the context of my safe withdrawal rate toolkit (see Part 28 for a guide and the link to that Google Sheet). I didn’t want to squeeze those two major topics into one blog post because I’m already past 3,500 words.
Obviously, the guaranteed payments likely look even more attractive in a withdrawal rate analysis because longevity is correlated with running out of money in retirement. Anything that hedges this longevity risk, like an annuity, pension, or Socal Security, will look good when optimizing a failsafe withdrawal rate. But then again, that’s not 100% guaranteed, either. Annuities and pensions are often just nominal, i.e., not CPI-adjusted, so they would not do well in today’s high-inflation environment. Critics could also argue that it’s most important to hedge against sequence risk during the first 5-10 years of retirement, so a safe asset phased out over that short-to-medium term will likely hedge better against Sequence Risk than an annuity that runs your entire life. The annuity payments are stretched too thin over the whole retirement horizon. And they might be too low during the first ten years of retirement when Sequence Risk is a concern and too high later in retirement when you don’t need a Sequence Risk hedge. All interesting issues to be talked about in a future post. Stay tuned!
You might have noticed that the SSA life table uses annual data, but I prefer monthly simulations. How did I go from annual to monthly numbers? Simple. I assumed that the SSA annual numbers refer to the survivors on their birthdays at that age. Then I interpolated the monthly numbers in between with a cubic spline interpolation (interpolate.splev, using the scipy package). I then noticed that the interpolation was whacky for young and very old cohorts. So I transform the annual survival rates to (geometric) monthly rates for ages 0-20 and 100-119. But I kept the cubic spline interpolation for ages 20-100. See the interpolated survivors and death probabilities below:
Thanks for stopping by today! Please leave your comments and suggestions below! Also, make sure you check out the other parts of the series, see here for a guide to the different parts so far!
All the usual disclaimers apply!
Picture Credit: Pixabay.com