Episode 399: James Choi - Portfolio Theory in a Spreadsheet

20:41Benjamin Felix:That could exert a force, but I think that's more of a second order force, not kind of a first order force.

20:47James Choi:Yeah, interesting. And the last one is labor income characteristics, but you kind of touched on that earlier. If you're more stock-like, it's like you've got a smaller bond position.

20:55Benjamin Felix:Yeah, but there's an additional twist to that, which is there are two components of labor income risk. There is transitory labor income risk, and then there's permanent labor income risk. So transitory labor income risk is like, you know, I get laid off, I'm out of work for six months, and then I find another job and I'm back to normal. So these are risks that don't, in expectation, persist over time. And then there is that permanent risk, which we might call career risk. Do you make partner at your law firm? Do you get that promotion? do you get demoted and there's a stigma on you for the rest of your life, that sort of thing.

21:28Benjamin Felix:Now, it turns out that the amount of permanent versus transitory income risk that you have is different by education level. So if you are a high school dropout, then you have relatively high transitory labor income risk. You're kind of getting laid off a lot more than a college graduate would. But you don't have a lot of career risk because when you're employed, you're kind of doing the same job to the first approximation. Whereas a college graduate doesn't have that much transitory labor income risk, not zero, but not relatively low. But they have a lot of career risk, a lot of permanent income risk.

22:05Benjamin Felix:And it turns out, and this is something that we discovered in our paper, by the way, I should mention that this is a paper that was co-authored with Sanya Liu and Peng Chong Liu, both finance PhD students. Peng Chong almost graduated, Sanya now graduated and working on Wall Street. So what we found was that the transitory labor income risk actually does very little to affect your optimal stock allocation. So that kind of rolls off the back of the agent that we're modeling. It's a permanent income risk that is really scary and causes you to choke up on the bat, essentially, and become more conservative in your portfolio.

22:41Benjamin Felix:Maybe ironically, and people might have had a different instinct, it's the high school dropout, the all else equal would be more aggressive in their portfolio allocation than the college graduate.

22:53James Choi:That's counterintuitive. I think we have questions later about how people should think about estimating their future wages, which I think would be related to that, but we'll get there in a bit. For this paper, can you talk about how you and your co-authors set up and solved the portfolio choice problem?

23:07Benjamin Felix:It was very simple on one dimension, which is we just took the Kaka Gomes and Manhout model and we solved it for thousands of different parameter sets that were within kind of a realistic range of what we thought would be relevant for people who would want to use the model. What turned out to be difficult, and this is why the paper kind of took six years to come into fruition, is when you're doing these numerical optimizations, it's a little bit of dark art to get these things to work. It's not like, you know, you just put it to solver and excel and it gets you the solution and you're done. And then strange solutions come out and you're like, I can't possibly be right.

23:46Benjamin Felix:And so you kind of need to constrain the search in certain ways and extrapolate off the grid in certain ways. And these things that you think shouldn't matter actually end up mattering. Now, when you're solving the model for one parameter set, you kind of look at it and you're like, OK, that's funny. OK, we'll tweak it in this way and that way. Now that the solution is working out just fine and then you're done. When you're solving for thousands of different parameter sets, you need a really general way to make the solution robust. And so that's what took up a lot of our time.

24:16James Choi:How did you approximate your numerical solutions with a simplified model?

24:20Benjamin Felix:Well, so we knew from the Kaka Gomes and Mannhout paper that the solutions that came out looked a lot like the old Merton 1971 solutions, where you have risk-free human capital, just the value of the bond appeared to be different than if you were discounting future labor income at the risk-free interest rate. That hint was there in the 2005 paper, but they didn't run with that very much. But we said, well, if it looks like human capital when it's risky but uncorrelated with stock market return behaves like a bond, well, how do we value bonds? We apply discount rates to future cash flows or future expected cash flows.

25:02Benjamin Felix:So what if we did the same thing with human capital? So what is the discount rate that I should apply to my labor income at age 60 or my social security benefit at age 75 and so on? So we just kind of went about trying to find discount rates for future human capital. And there was no guarantee at the end of the day that we would be able to find discount rates that would result in present discounted values of human capital that then would closely match the solutions that were coming out of the numerical optimization of the CockoGums and ManHout model. It just so happened that working off of this hunch and kind of finding these discount rates, we were able to actually get a pretty close fit.

25:46Benjamin Felix:So that was a happy result, but not one that was a complete shock because we were able to eyeball the graph and say, gee, it really looks like these things are behaving like bonds that are discounted at higher rates than the risk-free interest rate. So that was kind of step one. You got kind of a series of discount rates for different ages and for each parameter set. So you have like many, many thousands of discount rates. Well, next step is to provide an easy way for people like you and me, who are casual observers and users of this research, to calculate a discount rate for themselves. And so then we just provided kind of these arithmetic approximations as a function of the model parameters.

26:28Benjamin Felix:How much do you multiply risk aversion by? How much do you multiply the risk premium by? And then you kind of add up these products. So basically we ran a regression to approximate the discount rates as a function of the model parameters. And turned out the very simple functions of the model parameters provided pretty good fits to these discount rates that are popping out of our own optimization. So happily, and again, there was no guarantee that this was going to happen, relatively simple calculations allow you to come to pretty close approximations of the optimal solutions.

27:05James Choi:You say pretty close. They're like really close. Can you talk about how close the approximate solutions are to the precise numerical solutions?

27:12Benjamin Felix:I'm talking a little bit of generalities because there are, of course, thousands of different parameter sets. And within those thousands of different parameter sets, there are 80 different ages. There are also different asset allocations that are associated with different levels of financial wealth that has been accumulated up to each age. And so basically, if you were taking goodness of fit over the entire range of solutions that we computed for the entire range of parameter sets, talking about an average deviation of about three percentage points, four percentage points from what the actual numerical optimization would suggest.

27:52Benjamin Felix:So that's kind of one way to look at it, just how much scatter there is. another way to look at it is if you put best fit line where I put on the vertical axis, the allocation that was coming out of the numerical optimization. And then on the horizontal axis, you put on the approximate solution that we were providing. And you look at what's the slope of that relationship. It's very, very close to one. We think the fit there is pretty good. The last thing that we did was we kind of ran this exercise where we said, take a 22-year-old. If that 22-year-old followed the actual optimal asset allocation strategy over their entire lives, what is the expected discounted stream of lifetime utility that they would get?

28:33Benjamin Felix:Now, instead, let's have this 22-year-old follow our approximate strategy instead. And then we can measure how bad is the welfare loss from following our approximate strategy instead of the actually optimal strategy. And it turned out that this was kind of less than a 0.1 % welfare loss over the course of the lifetime.

28:50James Choi:That's one of the most exciting parts of the paper. It was thrilling. I'm not even kidding. That was such a fun part to read.

28:56Benjamin Felix:You guys are such nerds. My gosh, need to get out more.

29:00James Choi:Oh, it was so good. Were there cases of parameter sets or whatever where the approximation was not good? Are there cases where we can say, well, it doesn't work as well for these parameters?

29:10Benjamin Felix:That's a good question. I don't think that we drilled down to that level, but maybe that's something we should take a look at after this interview.

29:18James Choi:That'd be interesting. Were you guys surprised when you saw how close it was, especially with the welfare loss analysis?

29:24Benjamin Felix:You have to be cautiously optimistic when you do these research projects, because if you're pessimistic, then you never kind of finish or even start these things. And so there's a positive illusion maybe that's associated with any research project. And so it worked out. It's like, of course it worked out. But I'm sure that somebody who was not involved in the research project may have been more surprised because they would have been more pessimistic.

29:50James Choi:So I have to ask, James, how does your approximately optimal solution compare to rules of thumb like the classic 100 minus your age in stocks?

29:59Benjamin Felix:So the 100 minus age or 60-40, they're actually not bad. And I think that's why these evolutionarily, these rules have had their persistence. You do worse with these rules, but it's not like awful. I have this table in front of me. So if you used 100 minus age in your percent of your portfolio in equities, then what we're calculating is across all the parameter sets we consider that you would lose 2 % of lifetime welfare as a 22-year-old. If you were just 60 % equity for your entire life, then you would lose 3.75 % of your lifetime utility across all of our parameter sets. And that's in contrast to our rule where you're losing 0.06 % of your lifetime utility.

30:46Benjamin Felix:I think you follow our rules instead of the optimal. Now, there are some things that you can do that are really quite terrible. So if you are 0 % equity for your entire life, then you're losing 7.9 % of lifetime welfare. It starts feeling quite big. If you're 100 % equities for your entire life, then you lose 11.8 % of your lifetime welfare. There's a lot of heterogeneity, actually, in that particular figure. If your relative risk aversion has value of four, we talked about relative risk aversion as being kind of how afraid are you of risks. We think the four is a pretty reasonable number that a lot of people probably have.

31:26Benjamin Felix:And 100 % equities for your entire life is only going to result in a 0.56 % welfare loss. So actually, where we think a lot of people are, 100 % equities for your entire life is actually pretty good. It's just that if your relative risk aversion is 10, which we think is kind of on the border of pathologically risk averse, then you're losing like 30 % of your lifetime welfare by being 100 % equities. But I think realistically, there's not a lot of people there. And so I'd say that 100 % equities, probably for a lot of people is not a terrible strategy. And I do have a faculty colleague here at Yale who is 100 % equities, not a particularly young guy at this point, but he says he's just betting on the equity premium and he's done pretty well for himself.

32:11James Choi:That part of the paper was super interesting where it's like, I read the welfare loss from being 100 % equities and it was worse on average across all the parameter sets than being just in cash. And I was like, whoa, what is going on there? But then when you talk about the heterogeneity for a normal level of risk aversion, that welfare loss is actually really small. It's just for a really risk averse person, you're losing a ton of welfare by being in stocks.

32:34Benjamin Felix:As a footnote to that, the 0 % equity is really terrible across all the different parameter sets. 0 % equities is where a lot of people are. One of the lessons that comes out of this paper, which is an old lesson, but this is just bringing it back to the fore, is that when you have human capital, equities are kind of awesome for you because you have this huge, huge bond-like asset in your implicit portfolio. And so you should not be afraid of putting that money into stocks because if you do experience a loss, well, you're cushioned by that enormous stock of human capital you have. And so you can afford to take a 20 % loss in your financial portfolio if you have a human capital stock that is like three, four, five, six times the amount of money that you've already saved to date.

33:22James Choi:For someone with normal-ish risk aversion, 100 % equities actually looks okay over the life cycle, but your optimal allocation is still better. If we think about that, what is the optimal portfolio look like over the life cycle of, I don't know, like a typical household?

33:37Benjamin Felix:Well, it's just going to be very heavily stocks, like 100 % stocks. And we do cap the formula so that you can't have more than 100 % stocks in your portfolio, and you'd be 100 % stocks for a very long time during working life. And then eventually, you would start de-risking your portfolio, but that might not be until your 40s or your 50s. The reason that you de-risk the portfolio is that your human capital stock is getting run down. So the depressing truth is every paycheck I get, in some sense, I'm not getting any richer or poorer. It's just my human capital is sublimated into financial capital, but it's the same number of dollars.

34:11Benjamin Felix:And now I need to kind of make an allocation decision with the new dollars that has just been transferred from my human capital to my financial capital. But as I get older, I have fewer paychecks coming to me in the future. And so my human capital stock is getting run down over time. More of my lifetime wealth is held in my financial portfolio rather than my human capital. So I need to de-risk my financial portfolio as I get older, because I have less and less of this bond-like human capital in my total wealth portfolio as time goes on.

34:40James Choi:I just want to put a finer point on that. Why does your model often recommending higher equity allocations than we see like in popular personal finance advice or even past academic papers?

34:53Benjamin Felix:It really is about the human capital angle and the fact that most of this advice is ignoring human capital for understandable reasons. Human capital is hard to deal with. It's hard to calculate. We don't see human capital trading in markets. We don't have a valuation for it. It's just a hard mathematical problem. It's a hard conceptual problem. And so we do what we do with a lot of hard problems, which is we ignore the hard part and solve the part that is easy. And so if you have nothing but this pot of money that you saved up to now, then gosh, yeah, like maybe we should be a little more conservative with that pot of money because it's all we got.

35:28Benjamin Felix:But if I have like two,$3 million of labor income coming to me in the future, then if I lose like$20 ,000 in my financial portfolio, like not that big of a deal.

35:38James Choi:You mentioned one of your Yale colleagues who is 100 % equities. Did the results of this paper have an effect on how you and your co-authors or anybody else, maybe in academia that read it, how they think about their personal asset allocations?

35:50Benjamin Felix:I don't know if it's changed anybody's allocations at the moment. So at first, it's a relatively new paper that's been released. That said, the insights of the paper, or the qualitative insights of the paper are not new. This has kind of been known for at least 20 years. I have talked to faculty colleagues, not in finance, who say that they are not anywhere close to 100 % stocks. And say, oh, like, you know, there's this kind of this thing, Cuban Capital, Merton, Cocker Gomes, Manhouse. Some non-finance colleagues know my views on this. I don't know if they change their portfolios. They're in those conversations.

36:25And then for my own personal asset allocation,

36:27Benjamin Felix:I've been 100 % stocks for a long time until very recently. That's because actually for me, I felt like the problem was much easier because I'm a tenured professor at a pretty creditworthy institution, very hard for them to fire me. So if I discounted my future expected wages at the risk-free discount rate, I'm not going to be so far off. So that's what I had been doing for a long time. Even if you futz around the discount rate a little bit, it would just be pretty hard to get myself away from the 100 % equities boundary for the last couple of decades.

37:01James Choi:Makes sense. How did you decide which variables and specifications to include in your approximately optimal model?

37:08Benjamin Felix:It was not that difficult. Cocker Gomes and Manhout did a lot of the hard work for us. They had all of these parameters that they had put into their model saying that these are the variables that we think are the most important for determining what somebody's portfolio allocation would be over the life cycle. And so we just took those variables. And for the most part, we just varied those variables over what we thought of as realistic ranges. The one thing that we did not vary among their parameters was just the time discount rate, kind of the generic rate of impatience. That was basically because I don't think that people have a good sense of what their discount rate actually is.

37:46Benjamin Felix:There's a great survey article that was published like a couple decades ago, where it outlined the attempts of economists over the decades to estimate what time discount rates are for people. So you plotted each estimate as a point on this graph, and on the horizontal axis was the year in which the estimate was published. So you see this big cloud of points, and as time goes on, there's no narrowing of the cloud. You measure discount rates from just all over the place, no sign of converging to a consensus. And I think just the way that you ask these questions greatly determine the parameter that you get.

38:24Benjamin Felix:And so I think the introspection is a very hard task for people to be able to introspect and figure out what their discount rate is. So we ended up not messing with that because we just thought that if I ask you, is your discount rate 0.97, 0.98, 0.99, you're not going to know. And so just not that useful to vary that.

38:42James Choi:In the abstract of the paper, you talk about practical finance, how you basically want to make this stuff solvable in a spreadsheet for a normal person. Can you take us through an example of calculating the optimal equity allocation using your model in a spreadsheet?

38:59Benjamin Felix:I will do a screen share. I did post a Google Doc where actually your listeners can go to the Google Doc. I'm sure you'll put it in some link on your website. It's also on my Yale faculty website right underneath the PDF for this paper, which is called Practical Finance, you can kind of go to a Google Doc. So what the link in my website will go to is instructions on how to use the spreadsheet. And then you click on the link in the instructions, and it'll make a copy of the spreadsheet into your own Google Drive. And what you'll get is a spreadsheet with two different tabs. One is full inputs, one is with weight imputed.

39:37Benjamin Felix:I'll explain the difference in a minute, but there are instructions up at top here. And then it's going to ask you for certain parameters to be entered. The important one is how risk averse are you? One to 10. And in the instructions that accompany the spreadsheet, there's kind of this thought experiment that you can go through to figure out what is my level of risk aversion. Here I put in five. Just ask you, you know, how old is the first adult in the household. Spreadsheet does accommodate two adults, up to two adults in the household. How much have you saved to date? So this would be your investable net worth.

40:14Benjamin Felix:What do you think the stock market's real return is going to be going forward? And 5 % is kind of a reasonable value given valuation ratios, but obviously opinions will vary greatly on that. Now, what is the real risk-free interest rate that you're facing right now? For a US user, I would say this would be like a 30-year tips real interest rate because you kind of want a long-term interest rate that's going to apply for most of your life. So in the blue is the inputs that it wants. And then in the green and the purple, you would just enter what you forecast your future labor income to be. And so here, I've just kept things simple.

40:52Benjamin Felix:And you're going to earn $100 ,000 in today's dollars. So these are all in kind of inflation-adjusted terms. Up to age 64, and then I say at age 65, I'm going to stop working. I'm going to start collecting social security benefits. So these would be any retirement benefits that are risk-free at the point they start being collected. And so then I forecasted$40 ,000 of social security benefits for myself up through age 100. And a crucial thing to kind of know when you're using the spreadsheet is that you are supposed to put a forecast for each age up through age 100. Even if you think you're not going to make it to age 100, all that mortality risk is baked into the approximation.

41:33Benjamin Felix:As long as you don't know that you are for sure not going to make it to age 100, you should put in a value for each age through 100. So that's for the first adult. And then I have those equivalent values for the second adult in the household. It's the same pattern,$100 ,000 during working life, all the way through age 100, I start collecting those$40 ,000 per year social security benefits. So what the spreadsheet does is just uses the approximations that we came up with to compute what is the value of your human capital. So here it's$2.2 million. For reference, I do have in the bottom here, if you had no human capital, what fraction of your portfolio should be allocated to equities?

42:18Benjamin Felix:And here it's 17%. But then given that you have the$2.2 million of human capital and that you've saved$500 ,000 to date, what fraction of your financial portfolio should be in equities and it's 91%. So that's using the full range of inputs to the spreadsheet in order to get a recommendation. For a lot of us, it's going to be hard to forecast our wages over the course of an entire career and then to figure out what our social security benefits going to be. So in the second tab, I have this wage imputed methodology where you just need to enter, in addition to everything we'd entered before, what is your current wage?

43:01Benjamin Felix:And what is your current retirement benefit? So for somebody in working life, the current retirement benefit would be zero because they're not collecting anything right now. And then we will grow automatically your current wage at kind of the average growth rate over the life cycle for a college graduate. And so that reduces the data entry burden where the only thing you need to enter is the stuff here in the blue. There is a little extra input here where in the US, there's this thing called the social security spousal benefit, where if one member of the couple has a much shorter earnings history or much lower earnings history than the primary earner, then they can get a bigger benefit by claiming the spousal benefit.

43:42Benjamin Felix:So this is just asking, when we do this imputation of your future social security benefit, should we consider one of the members of the household to be claiming the Social Security spousal benefit at retirement. And then you get to a similar output where you have what is the value of your human capital, our estimate of that, what fraction of your portfolio should be in equities. If you had no human capital, and then given that you have the human capital that you do, how much of your financial portfolio should be in equities? And here it's 89%.

44:17James Choi:How sensitive is the equity portfolio you a share to risk aversion? Like it's at five now, if you put it to four in the spreadsheet, what do we see?

44:24Benjamin Felix:Now you're up against 100 % versus if I went up to six, then you dropped to 70%. Not surprisingly, your risk tolerance does matter for what you should be doing.

44:36James Choi:You did mention that there's a kind of a thought experiment. Can you talk more about how people should approach figuring out what their risk aversion number is? The thought experiment that we use is suppose that you are facing this lottery.

44:50Benjamin Felix:You flip a coin. The coin comes up heads. Then you need to live on$100 ,000 for the next year. You have to spend it. You can't save it. You can't borrow to spend more. You're going to actually live on$100 ,000 for the next year. Coin comes to tails. You have to live on$50 ,000 for the next year. So you have this 50-50 gamble. and then suddenly a genie appears and says, I can take this gamble away from you. And in return, I will give you a for sure amount of money that you will have to live on for the next year. And so the thought experiment is, what is the for sure amount of money that would make you exactly indifferent between keeping the gamble and trading it in for that for sure amount?

45:31Benjamin Felix:And so if you are perfectly risk neutral, if your risk aversion is zero, then your answer would be$75 ,000. I'm exactly indifferent between 50-50,$100 ,000,$50 ,000 versus a for sure$75 ,000 amount. The more risk averse I am, the lower the for sure amount is going to be that's going to make me exactly indifferent between keeping the gamble and having the for sure amount. And so we have a table that we provide where we say, you know, if your for sure amount is X, that corresponds to risk aversion of Y.

46:05James Choi:I know you gave the easy option for this, but how should people approach forecasting their labor income?

46:11Benjamin Felix:Most people will start with the easy option. Another way to go is there are some data out there on income trajectories for certain careers. So that's imperfect, of course, but you could see if I'm in this industry for some person that is 20 years older than me, what are they making right now on average for someone 40 years ahead of me? What are they making on average? That's the way to go. Now, that obviously excludes the effect of just generic economic growth. If economic growth is like 2 % per year from now until 20 years from now, then we should expect that to kind of lift all the boats. Just taking the average income of people 20 years ahead of me in my career would be an underestimate of what I would probably actually earn.

46:58Benjamin Felix:But then there's the old problem of like, everything's changing anyway. And maybe, you know, we will all be AI slaves in the future and be depending upon the largest of our AI overlords. The future is hard to predict, right?

47:12James Choi:The thing that's cool about the spreadsheet though, is that if you just took your current wage and projected forward, say it's constant in real terms, you can see the effect that different assumptions would have on your optimal equity allocation. So if you're like, oh, well, what would happen if I do think I have an above inflation career trajectory for my income? You can see how that would affect it. I think that's, even if we can't predict the future, we can see how different futures should affect our current asset allocation, which I think is pretty useful for people. Yeah.

47:38Benjamin Felix:And I think that what people would also see is that for, especially young people, even under some pessimistic views of what their income trajectory is, the formula would probably still recommend a pretty aggressive, probably 100 % equity allocation for them because even under pretty negative scenarios, they probably have a lot of labor income coming to them in the future relative to the amount of money that they've already saved.

48:01James Choi:One of the other inputs on the sheet is investable net worth. What does that include? Is your house in there? What's not in there? What is?

48:08Benjamin Felix:So this is the pain point, actually. Turns out that housing is extraordinarily hard to model. It's illiquid. It has high transactions costs. The risks of housing are poorly understood. If my house value goes down by 10%, but I'm living in it in the neighborhood just fine and I never plan to move, then no big deal. I haven't realized the loss at all. Versus my house value went down by 10 % and it's because my neighborhood went to pot. That's terrible. It depends a lot. And when I plan to move, when I move, am I moving to a neighborhood whose house price growth is pretty correlated with mine or not?

48:45Benjamin Felix:So if my house price falls by 10%, that neighborhood that I'm going to move to has house prices that have also fallen by 10%. Again, no big deal. I was kind of hedged. So those kinds of considerations. And then you have mortgages that are amortizing over time. And so that's another thing that you need to kind of keep track of. So housing is a huge, huge headache for all models like this. What do we do in our case? Well, we don't have housing in our model. How do we deal with problems that are really, really hard? We ignore them for now. And so the model, very strictly speaking, is for a renter for life.

49:20Benjamin Felix:And now the question is, well, like a lot of people have housing. How do we deal with that? the off-label recommendation that I give is ignore housing. So kind of take out home equity, ignore the mortgage in your investment net worth calculation. So make the investment net worth kind of all of your non-housing assets minus all of your non-housing debts. And that I would call that investment net worth. Now, surely that can't be the actual right answer, but it's kind of the best that we have right now. I was having a conversation with this very distinguished real estate professor who's at Wharton at UPenn.

49:57Benjamin Felix:And I asked him, you're an expert in real estate, world-renowned scholar. How is your financial portfolio allocation affected by the fact that you own a house in Philadelphia? And he said, not at all. I basically ignore the house and just allocate my portfolio as if the house didn't exist. So yeah, at least I'm not alone here.

50:17James Choi:That is so funny. You're right. It's like if you said, okay, should a renter owner have a riskier financial asset portfolio? You'd have to ask what neighborhoods do they live in? Where do they plan on living in the future? Do they have a mortgage? It's not a straightforward question to answer. No, not at all. How frequently should people be updating their assumptions and their portfolios?

50:38Benjamin Felix:If you updated once a year, that will be perfectly fine. These things are not quickly moving. And so So I don't think that your optimal portfolio allocation quickly moves. One time, I was personally more enamored with volatility timing. So this is this strategy that observes that the volatility of the market going forward is highly predictable, but the average return on the market is no different whether you're in a really volatile time or not volatile time. And so voila, the obvious implication is pull back on the market when things are really volatile, be aggressive in the market when things are less volatile.

51:19Benjamin Felix:It turns out that when you backtest the strategy, it's not like fantastic. A lot of the bloom on that has come off for me. And so that was kind of my one little carve out when you might want to maybe pay attention to your portfolio, adjust the portfolio more often. Absent that, I don't think there's a lot of justification for fiddling with your portfolio too often.

51:41James Choi:Just comes back to Mertenshare, I guess. If markets are more volatile and volatility predicts future volatility, you should be taking, well, your equity share should be lower when markets have been volatile.

51:50Benjamin Felix:And that was how I did my personal portfolio. You had the Merton share and you have the variance of the market return in the denominator, just sticking the VIX squared into the Merton share, and then kind of take my Merton share, adjust it for my human capital. There was an equity allocation for me. For the vast majority of the time, even when things are a little scary, you are at 100 % equity share just because human capital is pretty huge for people that aren't very close to retirement. But every once in a while, things pull off the 100 % boundary, even for somebody middle age. Like, you know, during COVID, VIX went to like 83 or whatever it was.

52:25Benjamin Felix:That was a scary time. And I did pull back at that time. Liberation Day, Trump tariffs, scary time and pulled back there. So during COVID, volatility timing worked quite well for me. The Trump tariffs, they did not work so well for me because the markets came back so quickly. So do you still do it? No. As I said, the bloom has come off for me. And then, you know, since the Liberation Day episode, there hasn't been a huge VIX spike since then. So there hasn't been an opportunity to, but I think that I'm more reluctant to do it now than I would have been in the past.

52:57James Choi:That's cool. You mentioned earlier about how there's no leverage. We're constrained at 100 % equity. Do you have a sense of what would happen to the model's advice for a typical household over the life cycle if we relaxed the no leverage constraint?

53:09Benjamin Felix:In the model, strictly speaking, the agent never wants to lever because returns are log normally distributed in the model, which means that a negative 100 % return is, or very close to negative 100 % return has some tiny, tiny, tiny probability of happening. which means that if you lever up, you have some tiny, tiny probability of having negative wealth. You can't borrow in the model. And so that means that your consumption is like negative, then things blow up because that's just not allowed. You're kind of like infinitely unhappy if consumption is negative or zero. And so in the model, you never kind of want to do that.

53:49Benjamin Felix:Now, in real life, we don't think that the stock market has hardly any chance of going to negative 100%. Leverage, I think, can be advisable for somebody who is relatively young. And so I think that if you were to do that, probably the way to go is through one of these leveraged funds, kind of 2x, 3x. They're actually surprisingly reasonable investments. They get a bad rap in some circles because their leverage resets every day. I think that's a feature, not a bug. And that's a crucial feature for ensuring that these funds never return less than negative 100%. You need to kind of understand that you're not going to get 2x or 3x the market exactly over longer investment horizons.

54:31Benjamin Felix:There's a charge out there that these funds go to zero for sure, probability one, as the investment horizon goes. And that's just not true. I mean, you can run the simulation, see it's not true. And then you can actually see the experience of these funds themselves. They've been running for like 20 years now. And no, they have not gone to zero. In fact, they have vastly outperformed an unlevered investment in the S &P. Here, I'm talking about leveraged funds on the S &P 500. There's kind of crazy stuff like leveraged funds on NVIDIA stock, which I would not recommend. The other thing that gives me more confidence is that these leveraged funds, having been around for 20 years, they went through the 2008 financial crisis.

55:07Benjamin Felix:They went through COVID. They didn't blow up. There were kind of exotic products that did blow up under some of these circumstances. They didn't blow up. I really understand well how they're generating that leverage through equity swaps, very plain vanilla financial contracts. And we know that they borrow at very low rates, 70 basis points above treasuries is the estimate. So if you're going to do it, I think, you know, especially for somebody who's young, it's not unreasonable. You just have to be emotionally, relationally prepared where like, sometimes you might lose 20 % of your investment or 30 % of your investment in a day.

55:42Benjamin Felix:If you kind of have your why ready, I know why I'm doing this. I know the theory. I know I have this human capital buffer behind me, then I lose 20 % in a day. That's painful, but it's not like the end of the world. But if I need to sit at the dinner table with my spouse and try to think, am I ruining my family's future because I just lost 20 % of our wealth in a day, there are other considerations that you have to think about.

56:08James Choi:That makes me think of John Cochran's analogy about the inflation index perpetuity and how it's technically risk-free, but good luck explaining that to your spouse when it's mark-to-market value drops by 40 % or whatever. Absolutely. It's funny. Really interesting about leveraged ETFs. The episode that was released today, the day that we're recording, was with Hank Bessenbender. He's got a paper out on single stock constant leveraged ETFs. You mentioned those being not the ones you want to touch, but we did have quite a bit of discussion about index constant leveraged ETFs, like the 2X and the 3X and all that kind of stuff.

56:42James Choi:I'm hoping that he'll include analysis of those in a future update of his single stock ETF paper. Because one of the things he looks at is he calls them frictional costs. How much does it cost to use those products versus just borrowing and investing in something? And I think he does a lot to debunk the concept of volatility decay, which is I think what's been used historically to argue against these constant leverage funds. Super interesting. That's the second very smart person in the last little while that's told us that volatility decay is probably misunderstood and that the constant leverage funds, even though they're like the fund providers say that these are designed for daily replication.

57:18James Choi:But like you said, you shouldn't expect to get 2x the return of the index in the long run. You're going to get less than that. But the fact that they have the daily resets and the volatility decay is probably not as much of an issue as people have been historically told.

57:31Benjamin Felix:Well, if you're going to borrow on margin, then you're dealing with margin calls. Enormous headache. Do you really need that headache in your life, big virtue of these leveraged funds, you never get a margin call. There's kind of a separate thing of like, you know, I have student loans. I'm paying a 4 % interest rate in my student loans. Should I keep that alive so I can invest more in the stock market? Like that seems like quite reasonable to me. And there you don't face margin calls until your liquidity is preserved in those types of cases.

57:57James Choi:Access to leverage too. If a typical person wants to go and borrow 2X at a reasonable interest rate, like it's not that easy to do to invest in stocks.

58:07Benjamin Felix:these funds through equity swaps, highly collateralized. That's why they're borrowing a tragedy plus 70 basis points. Good luck getting that for yourself.

58:14James Choi:Yeah, yeah, exactly. Really interesting. I'm glad that you brought that up and that we had that part of this discussion. It's something I've been thinking about a lot. Leverage in general, but then using these constant leverage ETFs as a source of leverage for a typical person. Great perspective on that. Last couple of questions here. There's a paper that I know you've seen because I've seen you speak about it. Beyond the status quo, a critical assessment of lifecycle investment advice. We've had one of the co-authors, Scott Sederberg, on this podcast a few times to talk about that research. They find that a 100 % equity portfolio with one-third domestic and one-third international stocks is optimal over the full lifecycle for any level of risk aversion, I believe.

58:51James Choi:Can we recreate their result using your model and the inputs from their paper or the return assumptions from their paper?

58:58Benjamin Felix:You misspoke. It's one-thirds domestic, two-thirds international, not one-third, one-third. Oh, sorry.

59:03James Choi:Yeah, thank you.

59:05Benjamin Felix:I'll talk about that as kind of a separate issue, because in our model, we just have a single stock market that you're investing in. The virtue of the paper is that it brings to the fore the fact that for somebody with labor income, stocks are kind of awesome. And that's basically the message of the paper. Stocks are awesome, so awesome that you should be 100 % equities for your entire life. Now, there are a couple things that drive that result. One is just the generic stocks are awesome for people with labor income. The second is, at least for the draft that I saw, at least in the baseline calibration, they were using a fairly low level of risk aversion.

59:40Benjamin Felix:So it was like 3.8 or something like that. Not crazy low, but it's on the lower side. They say that the historical return on the stock market is what you can kind of expect going forward. We know that historically the equity premium has been enormous. And currently, valuation ratios have gone up a lot over the course of the last 50 years. So if you combine relatively low risk aversion, high equity premium, a lot of labor income, then you can kind of get pushed towards a lot of equity. And then the last thing I'll mention is that they, at least in the baseline calibration they have, is they're kind of assuming a 10 % constant savings rate over the course of the entire working life.

1:00:25Benjamin Felix:And I haven't checked this, and I'm not sure that they have exhibits in the paper that let you necessarily judge this. But it did seem like a kind of lowish savings rate relative to what an optimal model would suggest. And so now you have this agent in the model who is not accumulating like a ton of financial wealth. And if you have less financial wealth relative to your human capital, that's another force that pushes you towards more aggressive portfolio allocation. So you have kind of all these different forces in their setup that pushes you towards 100 % equities for your entire life. And indeed, that's what they end up finding.

1:00:59And so I think the virtue of the paper is that it does kind of make salient, again, that stocks are awesome and you should have more stocks probably than you do.

1:01:09Benjamin Felix:Now, on the one third domestic, two thirds international allocation, I think that is something that's generated by the way that they do their simulations. When I first saw that result, I was surprised because if they were relying upon historical performance for the forecasts, well, the U.S. has had an incredible century. Since the inception of the MSCI World Index, the U.S. stock market has had a higher average return and a lower variance. So kind of the international diversification just didn't pay off over the last half century. And so that's why I was surprised. How can you get this recommendation in the model to be two-thirds non-US, one-third US?

1:01:53Benjamin Felix:What turns out to happen is that they take the position that there is nothing special about the US during this time period. And so a US investor could have had a domestic stock market experience that matched Belgium's stock market experience or matched France's stock market experience or matched Sweden's stock market experience. So they're just like kind of randomly picking one of the countries in their data set and saying, this is going to be the domestic stock market return sequence you're going to get. And everything else kind of is the internationals. And so it turns out that when they're simulating the problem for the US investor, only like 5 % to 6 % of their simulations is the US stock market, actually the domestic stock market for the US investor.

1:02:40Benjamin Felix:Now the US had this extraordinary half-century run or century-long run. And so now what do you kind of want to do as an agent in this model? You want to maximize the chances that the US ends up having a big weight in your portfolio. And the way you do that is to put two-thirds of your portfolio into international stocks so that you catch that US rise with a big portion of your portfolio. So that's kind of what's happening. And so I think the critique out of that is the US is this enormous stock market, well over half global market cap at this point. And so do we really think that it's reasonable to think that the Belgian stock market historically is going to be a good proxy for the US's stock market return performance going forward?

1:03:25Benjamin Felix:This is kind of an untestable assumption and it really is almost philosophical in nature, but I guess that's what gives me pause is do I want to really deviate so much from market cap weights in my non-US, US portfolio by going only one third US, two thirds international when in fact the ratios by market cap weight are like kind of flipped.

1:03:47James Choi:They did find a pretty flat difference like the utility loss or however they measured it was pretty flat from kind of 5 % domestic to 50%, you probably could still be market cap weights and be pretty close. We loved the result because we and many Canadians have about a one-third home country allocation already. That's just for whatever reason, a very common home country bias for Canadian investors. So we saw the result and we're like, sweet, that's a great confirmation bias. Yeah.

1:04:14Benjamin Felix:I mean, I think it's just going to depend upon what country you're coming from as the investor. And not all home countries are created equal, To have 70 % of your portfolio in your home country as a US investor means a very different thing than if you're a Belgian investor.

1:04:29James Choi:For sure. Here's an interesting question for you. Given where US market valuations are, is the historical US experience a good proxy for the expected future US market experience?

1:04:41Benjamin Felix:Valuation ratios are very high right now. We just know that valuation ratios cannot keep on drifting upwards forever. From the old work of Bob Shiller or John Campbell, we would expect. returns going forward and stocks to be lower. Now, that being said, we also know that if you had tried to use price dividend ratios to form portfolios over the last 30, 40 years, you would have done pretty poorly. You and your fancy math, if you just guessed the historical average as your forecast, you would have done much better. This is kind of the Amit Goyal, evil Welch critique that none of these predictors do very well.

1:05:17Benjamin Felix:They certainly don't do better than just guessing the historical average to date.

1:05:21James Choi:In this paper we're talking about there, I'm just looking at it's in table seven panel D. They have risk aversion parameters from 0.5 up to 10. The optimal portfolio is pretty much the same. It's risk aversion 0.5. It's 32 % domestic and 68 % international. It's like a tiny, tiny difference compared to the kind of baseline result. Why do you think their result is so insensitive to risk aversion when yours is so sensitive to it.

1:05:49Benjamin Felix:In our results, if you have a very high equity premium and you haven't accumulated a lot of wealth, then you're up against the 100 % boundary basically for a pretty wide range. And so I think it's kind of a function of the fact that they're assuming a very high equity premium going forward and the fact that their agent, I think, is not accumulating a lot of wealth over time because the savings rate is relatively low.

1:06:13James Choi:One of the things that I thought was pretty interesting in that paper was that it kind of shows that, and they have some other older papers that look at this more specifically, but nominal bonds, especially when you look at all the countries in their sample have been pretty risky historically for long-term investors with real liabilities. And I thought maybe that was driving some of their result. Do you think there's anything to that idea?

1:06:32Benjamin Felix:There could be. I mean, I think from an investing perspective today, it's puzzling to me that any retail investor would hold anything but inflation index bonds, if they have that option within the US, it's kind of been shown that just regular treasuries have this convenience yield that's negative. And so basically, because treasuries are the grease that lubricates the wheels of the entire global financial system, there's a tremendous demand for collateral and all sorts of other things. And so the interest rate gets depressed for regular treasuries, and chips just don't serve that function. And so there was a paper a few years ago just showing that actually, you get a better deal on all-in basis by investing in tips.

1:07:12Benjamin Felix:And so treasuries, if you don't need the security as collateral or all this other stuff, other than just a source of return, for kind of a regular investor, there's not really any reason to hold anything but tips if you're gonna have fixed income exposure in the government bond market.

1:07:30James Choi:Yeah, that's really interesting. So the concept of making academic finance accessible to people with a spreadsheet is obviously brilliant. asset allocation is a natural place to start. What do you want to tackle next?

1:07:43Benjamin Felix:There are things that I would like to tackle that I don't quite have a handle on how to do it yet. I think housing is a huge black hole in our knowledge. It's only the biggest asset that most people have. How that should affect our asset allocation, things like I have a big mortgage. I have some money that I've saved up. Should I invest that money in the stock market or should I try to pay down my mortgage more quickly? You don't have all that much to say about it. And it's only like a major, major question that pretty much every homeowner faces for many, many years during their homeownership spell.

1:08:21Benjamin Felix:Housing is this huge black hole in our knowledge, but it's a huge black hole in our knowledge because it's really hard to figure out the solutions to these things. So that's what I would like to tackle, but I don't think that I'm going to tackle that soon because it's a really, really hard problem. There are other easier problems that one could think about tackling where I think there's not necessarily that much academic glory in tackling them. But an example would be like the 4 % withdrawal rule, which everybody knows and talks about. This is not the optimal withdrawal rule. It cannot be in the most obvious level.

1:08:57Benjamin Felix:If the level of interest rate falls, then the 4 % withdrawal rule becomes less sustainable. There are possibly approximations to what a sustainable withdrawal rate would be as a function of some market parameters that one could create. This is, in some sense, simple-minded, which is why there's no academic glory in creating these types of things. But that said, the 4 % withdrawal rule, I think, has been so influential. It has exerted such a big cultural footprint on our society. That could be worth kind of writing up a short little piece. Given this equity premium, given this interest rate, given this asset allocation, consumption growth rate that you want in retirement, and there are a lot of models that say that actually we should be expecting and desiring to consume less and less as we get older and older.

1:09:46Benjamin Felix:In that case, if you have preference parameters of such and such, that suggests a consumption decline rate of this amount. Now, how much should you be consuming in retirement in each year? So that could be an example of something that is, at least provide better guidance than the 4 % rule. It's not going to be perfect. I think that there are all sorts of other complications with spending and retirement, kind of medical expenses and government programs. And I think that your utility function just changes when you get older. My 80-year-old self, what does he want? I don't know. He's a stranger. He's kind of related to me, but does he want to travel the world?

1:10:23Benjamin Felix:Does he want to stay at home? I don't know this guy and I have to plan for him. That's kind of first order problem that is unsolved when it comes to like, how much should I be spending at age 72 when I'm early in retirement?

1:10:38James Choi:You call it simple minded, which is true. I think the 4 % rule has been so influential and popular because it is so simple. Like any person with even limited numerical ability and no financial knowledge can take that number and figure out how much they can spend or how much they need to save. But yeah, doing a better version of that that's similarly simple, I think would be incredible.

1:10:58Benjamin Felix:I know academic glory in it, but maybe big societal impact.

1:11:02James Choi:Practically useful, which has been your thing recently, I think, with the practical finance idea. Well, I said at the beginning that this paper was super cool, and I think you've proven that. Our listeners, I'm pretty sure are going to love this episode. I know I did. Oh my, they are going to lose it. This is pure gold, James. No, this is great. We really appreciate you coming to the podcast, James, and congratulations on another fantastic paper. Thank you. Pleasure to be here. Great to see you again.

1:11:30Cameron Passmore:Hey, everyone. It's producer Matt. Thank you so much for tuning in to this week's episode. Before we sign off, here's the disclaimer you've been waiting for. Portfolio management and brokerage services in Canada are offered exclusively by PwL Capital, which is regulated by the Canadian Investment Regulatory Organization and is a member of the Canadian Investor Protection Fund. Investment advisory services in the United States of America are offered exclusively by One Digital Investment Advisors, LLC. One Digital and PwL Capital are affiliated entities, and they mostly get on really well with each other.

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