Speaker 1: High and welcome to another edition of the Odd Thoughts podcast.
Speaker 1: I'm Tracy Alloway and I'm Joe. Wasn't all Joe, I
Speaker 1: got to ask when you were at school, were you
Speaker 1: good at math? So it's really funny that you should
Speaker 1: ask that, um, Because so my dad is a physics professor,
Speaker 1: and he started me on math training when I was
Speaker 1: very young. And I was really good at math and
Speaker 1: really good at mental math and super good at multiplications
Speaker 1: up until like fourth grade. And then as soon as
Speaker 1: it like hit the level of like where I actually
Speaker 1: had to do work and couldn't just do stuff in
Speaker 1: my head, I just like I just totally became very average.
Speaker 1: So I went from being really good at math to
Speaker 1: really mediocre very fast. But I do really love math,
Speaker 1: and I do really like doing a sort of math
Speaker 1: in my head and think about math and stuff like that.
Speaker 1: So I think I have a love hate relationship with math,
Speaker 1: Like I find it very very difficult to do and
Speaker 1: is probably my most hated subject, but conceptually I think
Speaker 1: it's really interesting and statistics. I was actually quite good
Speaker 1: at UM, so I like thinking about math ideas I
Speaker 1: hate actually doing the math. I mean, I think I
Speaker 1: think we're probably in the same boat on this one. Okay, well,
Speaker 1: we're going to talk about maths, and yeah, we're not
Speaker 1: going to do do math because that would be the
Speaker 1: world's most boring podcast ever. But we are going to
Speaker 1: talk about mathematical ideas and specifically how they applied to
Speaker 1: investment and markets and finance. And we have a very
Speaker 1: cool guest who is probably better better able to talk
Speaker 1: about math and investments than just about anyone else. Yeah,
Speaker 1: that's right. So, anyone who's ever heard of long term
Speaker 1: capital Management, you know that there was a quant there
Speaker 1: co founder called Victor Hagani and basically was hugely instrumental
Speaker 1: in the founding of that firm and is a mathematical
Speaker 1: expert of the highest order, I suppose you would say, right.
Speaker 1: And so these days there's so much interest in like
Speaker 1: algorithms and computers and quantitative finance and stuff, and there
Speaker 1: of course really ahead of the curve on a lot
Speaker 1: of these ideas, and now there's much more interest. So
Speaker 1: we're going to talk about the connection between math and
Speaker 1: finance and particularly some important mathematical concepts that investors should understand.
Speaker 1: Maybe we'll get an LTCM questioned in there too. Who knows.
Speaker 1: Let's bring in Victor Hagani. Like I said, he was
Speaker 1: at l T c M, but he is now the
Speaker 1: CEO of ELM Partners, which is basically a portfolio of
Speaker 1: low cost index and exchange traded funds. Victor, thanks so
Speaker 1: much for joining us. Thank you very much for having
Speaker 1: me so. Victor. We actually brought you on after reading
Speaker 1: a paper that you did, UH basically about what coin
Speaker 1: tossing and the probabilities involved in coin tossing? Can each
Speaker 1: us about investment? Can you tell us about that paper?
Speaker 1: So it came out of an experiment that that I
Speaker 1: did with the colleague of mine UM who I worked
Speaker 1: with at Ellen Partners, Rich Dewey, and we had heard
Speaker 1: about some research that was that had been done involving
Speaker 1: coin flipping and how people UH managed situations where they
Speaker 1: were given a favorable odds kind of investment opportunity. And
Speaker 1: I know we can't remember it with with these things,
Speaker 1: sometimes you can't quite remember where the ideas come from.
Speaker 1: But we decided to do this experiment where we would
Speaker 1: UM give gives depends some some real money and allow
Speaker 1: them to flip a coin that was biased to come
Speaker 1: up six likely to come up heads tails, and we
Speaker 1: told them that to begin with, and we gave them
Speaker 1: half an hour to flip to bed as much of
Speaker 1: their starting as they wanted. And at the end, however,
Speaker 1: were much money they had left in their bank, we
Speaker 1: would pay them up to a maximum amount of two
Speaker 1: and fifty dollars. And what we found was that UM
Speaker 1: our participants, who were pretty quantitatively quantitatively trained UM young
Speaker 1: men and women, didn't do very well and they didn't
Speaker 1: kind of get some of the basic concepts of UM
Speaker 1: decision making under uncertainty or the they didn't quite get
Speaker 1: the independent nature of the flips and the fact that
Speaker 1: it just made sense to keep betting on heads to
Speaker 1: debt you know, some modest constant proportion of how much
Speaker 1: they had in their bank at any point in time
Speaker 1: on heads and so UM. Yeah, it was, it was.
Speaker 1: It was really interesting to think about, you know, how
Speaker 1: people were we're having trouble with that, and you know,
Speaker 1: to give us some ideas for trying to help with
Speaker 1: UM with education as well. On on that, on that topic,
Speaker 1: explain real quickly the exact mechanics they were supposed to play.
Speaker 1: They had twenty five dollars and they were supposed to bet,
Speaker 1: what explained to us what the nature of the bet is,
Speaker 1: and then what did the lessons show about mistakes that
Speaker 1: people might or might not make when they invest. Sure,
Speaker 1: so the you know, the exact mechanics of it were that,
Speaker 1: you know, we told that the people to come for
Speaker 1: a lecture and uh, and then we asked them to
Speaker 1: get out their laptops and to play this game. So
Speaker 1: we gave them twenty five dollars that turned up on
Speaker 1: their screen and their banks and their bank accounts or
Speaker 1: their bank roll, and then they could bet up to
Speaker 1: the on the flip of a coin, and they could
Speaker 1: do it repeatedly. Some people flipped the coin three hundred
Speaker 1: times in the thirty minutes that they had and if
Speaker 1: they won the slip, then their bank roll would go
Speaker 1: up and and and vice versa. And however much they
Speaker 1: were left with at the end, we actually told them
Speaker 1: and we did pay them as a check or cash,
Speaker 1: which was you know, especially for a bunch of college students,
Speaker 1: which were the majority of our subject. You know, it
Speaker 1: was very welcome. And the two d and fifty dollar
Speaker 1: maximum that we were going to pay them, uh, we
Speaker 1: only told them that if they got close to it,
Speaker 1: so we told them that there was a maximum payout
Speaker 1: to begin with, but it was only when they got
Speaker 1: to a point where they could reach the two hundred
Speaker 1: and fifties. So if they had two d and twenty
Speaker 1: five dollars in their bank account and they were betting
Speaker 1: thirty dollars on heads, we would say, by the way,
Speaker 1: the most will pay you is two fifties. So you
Speaker 1: might want to do see your best from thirty dollars
Speaker 1: to twenty five dollars, because there's no point in winning
Speaker 1: two hundred and fifty five dollars. We won't pay you that.
Speaker 1: The most surprising thing, uh, in a way, was the
Speaker 1: fact that people would relatively frequently bet on tails, um,
Speaker 1: you know, even though we told him it was likely
Speaker 1: to be heads. You know, even though in general, you know,
Speaker 1: heads was coming up more frequently for most people, you know,
Speaker 1: after they have flipped a number of times, they still
Speaker 1: felled them, particularly after a string of head So like
Speaker 1: if they got four heads in a row, they were
Speaker 1: then more likely to bet on tails. Not everybody. That
Speaker 1: seems like a deep failure of numerousy to ever bet
Speaker 1: on tails, even and to think that something like the
Speaker 1: past streak of flips. Is any bearing on the next flip, Yeah,
Speaker 1: it is, But it's just that it's like the deep
Speaker 1: seated need that we have, you know, to sort of
Speaker 1: see a story and random thing. It's very you know
Speaker 1: that given that like half of the people did you know,
Speaker 1: half of these subjects at some point bet on tails,
Speaker 1: and like thirty percent of them bet on tails fair
Speaker 1: amount of the time. So there's something kind of deep
Speaker 1: seated and there my mom. I had my mom through
Speaker 1: the experiment and we talked about it afterwards, and she
Speaker 1: said to me, I know that I should never bet
Speaker 1: on tails, but I just couldn't resist. So she knew it,
Speaker 1: she knew it didn't make any sense, but she just
Speaker 1: couldn't resist. And it was interesting. We did another experiment
Speaker 1: and following up on this, this the same as interview
Speaker 1: question about the family planning that if you uh, you
Speaker 1: know that if if in a if you're going to
Speaker 1: in a society, if if everybody wants to have a
Speaker 1: girl and so they keep having children, each family has
Speaker 1: children until they have a girl, does that change the
Speaker 1: expected number of boys and girls? And most people feel
Speaker 1: that it does, even though when father as a coin flip,
Speaker 1: you can kind of see that they're independent, and you
Speaker 1: know that there's nothing there's really nothing much you can
Speaker 1: do to change the expect number of boys being equal
Speaker 1: to the expect number of girls to any finite horizon.
Speaker 1: So the point of those types of experiments is essentially
Speaker 1: that the optimum investment strategy is dictated by maths, right,
Speaker 1: and yet we choose to ignore it for whatever reason
Speaker 1: because we instinctively don't understand probabilities, or there's some emotional
Speaker 1: thing going on. Yeah, I mean I think people, you know,
Speaker 1: understand it. I mean like our subjects were really quantitatively trained.
Speaker 1: I mean they understood all of this, you know, there
Speaker 1: they were some of them were even mathematicians that at
Speaker 1: one of the universities where we did it, and and
Speaker 1: U some of and some of the subjects were also
Speaker 1: professional investment investment professionals that had both know a lot
Speaker 1: of maths and econ and finance training. So they understand it.
Speaker 1: But I think there is this sort of there's something
Speaker 1: deep seated that that sort of comes up and steers
Speaker 1: us off the path. And so you know, it's kind
Speaker 1: of like quite a lot of specific training is probably
Speaker 1: what's needed to get people to be disciplined, and you know,
Speaker 1: to be disciplined, it's not a lot of fun. I
Speaker 1: mean thinking about you're sitting there flipping a coin for
Speaker 1: half an hour and you're just trying to get of
Speaker 1: your bank roll on it and keep betting on head.
Speaker 1: It reminds me of reading about professional poker players who
Speaker 1: know that they can make a steady profit playing limit poker,
Speaker 1: which is a very mathematical, no, very little bluffing version
Speaker 1: of the game of poker, but they're just bored out
Speaker 1: of their minds when they play it. So they no
Speaker 1: limit is more fun and more exciting. There's it's a
Speaker 1: little less mathematical and more sort of based on emotion. Uh,
Speaker 1: they are more inclined to lose, Like you know, these games,
Speaker 1: these sort of sure things are not very enjoyable practices. Yeah. Yeah, Well,
Speaker 1: and think about index investing, right, I mean kind of
Speaker 1: the most you know, the most boring thing you could
Speaker 1: do is to take all of your savings and to
Speaker 1: put it into two index funds. You know, very few
Speaker 1: people really do that, and very you know, very few
Speaker 1: people do that and stick to it. I mean, people
Speaker 1: will do it and then they sort of will come
Speaker 1: back and feel that they need to change it because
Speaker 1: you know, there was an election, or there was a
Speaker 1: change in interest rates or something. So it's sort of
Speaker 1: fighting that that urged to us. That fighting the urge
Speaker 1: to be active is difficult in a lot of different contexts.
Speaker 1: We're all suckers for a sense of control. Let's talk
Speaker 1: about a different mathematical concept that's incredibly important to investing
Speaker 1: and that is compounding. This sort of I forget who
Speaker 1: said it. Maybe it's like Einstein, someone famous said something
Speaker 1: about compounding or being one of the most powerful forces
Speaker 1: on earth. Yeah, I think that they say Einstein may
Speaker 1: said something like that. As strange as it me. Yeah,
Speaker 1: I don't know why he would have been talking about it,
Speaker 1: but I think he did say something about it for
Speaker 1: whatever reason. Um, what don't people understand? What is it?
Speaker 1: Why is compounding such an important concept to understand? And
Speaker 1: what do people what do people get wrong about this? Well?
Speaker 1: You know, I think that you know that in these
Speaker 1: sort of investing things or mass things in general. You know,
Speaker 1: one of the things that really gets us is nonlinearities,
Speaker 1: you know, as things that are not proportional, and compounding
Speaker 1: is one of those things. So, um that the growth
Speaker 1: of your money doesn't kind of go up in a
Speaker 1: straight line. It goes up in this exponential line. It
Speaker 1: starts off growing slowly, and then as it gets bigger,
Speaker 1: it's growing faster in terms of the amount of money
Speaker 1: by which it's growing, I mean, the rate of growth,
Speaker 1: let's say, stays the same. And and so you know,
Speaker 1: when you start to look at relatively long periods of time,
Speaker 1: which are the kinds of periods of time that are
Speaker 1: relevant to us in terms of building savings for retirement
Speaker 1: or or uh, you know, our our our sort of
Speaker 1: personal security longer term, or for our family or our kids. Um,
Speaker 1: you know, those long term horizons are important, and and
Speaker 1: compounding and small effects really magnify out there. So you know,
Speaker 1: the one that we always that we can hear a
Speaker 1: lot about, right, is sort of the effect of fees.
Speaker 1: You know that, Um, you know that if you're compounding
Speaker 1: at a five percent return because you're paying two percent fees,
Speaker 1: or if your account compounding at a seven percent return,
Speaker 1: that what you wind up with at the end is
Speaker 1: not proportional to seven over five, right, that that seven
Speaker 1: winds up giving you a lot more um at the
Speaker 1: end because it's it's you know, it's it's one point
Speaker 1: oh seven being raised to a power divided by one
Speaker 1: point oh five being raised to a power. So you know,
Speaker 1: everything kind of gets magnified by by compounding, and so yeah,
Speaker 1: you get a lot of you know, like another thing
Speaker 1: that you know, sort of similar to sees as taxes.
Speaker 1: So if we can invest in a way where we
Speaker 1: don't pay tax until the end of our investment horizon, um,
Speaker 1: you know, we wind up with a lot more money
Speaker 1: than if we're paying the same rate of tax on
Speaker 1: our growth every year that we go along. So um,
Speaker 1: like you know, an example of that would be let's
Speaker 1: say that you have a uh an investment that has
Speaker 1: an eight percent rate of return and let's say a
Speaker 1: tax rates or fifty percent. Just to make the mass simple, Um, Well,
Speaker 1: after thirty years, if you were well, if you're paying
Speaker 1: tax every year, then your eight percent return duringly like
Speaker 1: a four percent after tax return. So if you have
Speaker 1: a if you have a hundred thousand dollars and you're
Speaker 1: investing it, then after tax, that hundred thousand dollars has
Speaker 1: grown to three four thousand dollars after thirty years at
Speaker 1: this four percent rate of growth. Half of the five
Speaker 1: but it's instead you're deferring your tax to the end.
Speaker 1: Then you're growing at eight percent right, um, because you're
Speaker 1: not paying any tax on it. But at the end
Speaker 1: you have to pay tax on all your games. And
Speaker 1: when you do that, you wind up with clothes to
Speaker 1: double the money. After thirty years, you wind up with
Speaker 1: like five fifty thou dollars and almost a six percent
Speaker 1: instead of a four percent rate of return. So that
Speaker 1: stuff really kicks in over these long horizons and is important.
Speaker 1: You know, small differences wind up being big differences because
Speaker 1: it's compounding. What's your favorite um financial formula for investing? Like,
Speaker 1: if you had to choose one, UM, well, I don't know.
Speaker 1: I guess, uh, you know, one of the simplest ones
Speaker 1: one that there's been on my mind lately. I don't
Speaker 1: know if it's and I think if I had more
Speaker 1: time to think of it, I find a better one,
Speaker 1: but it's been on my mind. Of that is is
Speaker 1: what's known as Sharp equality, from a paper that William Sharp,
Speaker 1: the Nobel Prize winner, wrote, um in the early it
Speaker 1: was I think the paper was called like the Aristhetic
Speaker 1: of Active Investing, And in that he just made the
Speaker 1: very simple, uh statement, that the return on the average
Speaker 1: actively managed dollar has to equal the return of market
Speaker 1: minus minus fees on the active stuff, and that comes
Speaker 1: about the market return is UM must equal a weighted
Speaker 1: average of the reach end of the passive and active
Speaker 1: segments of the market. So if the uh, if the
Speaker 1: total market return is the same as the index thing
Speaker 1: return of the passive part, then you know, it's sort
Speaker 1: of like, you know, if if two equals one plus one,
Speaker 1: then two minus one equals one is kind of um.
Speaker 1: You know, I guess I suppose the way of seeing it.
Speaker 1: So it's a very simple. It's kind of like in physics,
Speaker 1: the the idea of the conservation of energy UM and
Speaker 1: you know, so what are the practical ramifications of that?
Speaker 1: From an investor standpoint? This sort of I get it
Speaker 1: sounds like an identity essentially, what do the how does
Speaker 1: that manifest itself practically in terms of making investing decisions? Well,
Speaker 1: it just helps us a lot in terms of thinking
Speaker 1: about what we're doing when we choose active strategies. That
Speaker 1: for an active strategy to be working for us UM
Speaker 1: that we have to believe that there's some other active
Speaker 1: strategy that's losing money and we have to be able
Speaker 1: to identify, um, you know why and who that's likely
Speaker 1: to be. You know that if we're that that, if
Speaker 1: we're if we think that we're making you're kind to
Speaker 1: make money, we really be sure of who we're making
Speaker 1: the money from. And it's not really it's some game essentially, yes,
Speaker 1: you know, within that space. I mean, at least too,
Speaker 1: you know, I think that at least to a first approximation,
Speaker 1: it's the it's a valid identity. I mean, there's some
Speaker 1: caveats and so on that people would bring into it,
Speaker 1: but I kind of like that. It's simple. Um. It
Speaker 1: reminds us of Bill Sharp, who is a really cool guy.
Speaker 1: I think it's I think it's a really useful It's
Speaker 1: really it's a really useful one. Um to to remember.
Speaker 1: Oh um, I promised a potential l t c M question. Um, so,
Speaker 1: I guess like one of the other things we've observed
Speaker 1: in markets recently is the rise of smart beta, but
Speaker 1: also risk parity strategies, and some people have likened risk
Speaker 1: parity to the old black shoal portfolio insurance of THEES,
Speaker 1: and some people have connected l t c MS collapse
Speaker 1: with black shoals. So I guess I'm just curious how
Speaker 1: you feel about risk parity and how you feel about
Speaker 1: the downsides of math addicts in finance. For me, the
Speaker 1: really short answer is that the the the leverage, you know,
Speaker 1: my ltc M experience has just made me not want
Speaker 1: to use leverage, um explicitly in any sort of investment strategy. Uh,
Speaker 1: you know, for myself, for anybody that I would be
Speaker 1: trying to help. Um, you know it. Leverage has its
Speaker 1: place in our financial system. It has its place perhaps
Speaker 1: within the investment community, but personally, um, you know, it
Speaker 1: was that that was you know, that's that was the
Speaker 1: primary cause of the problems at ltc M. And so
Speaker 1: for me anyway, I mean, I know the arguments for
Speaker 1: risk parity, UM, you know, it may well be that
Speaker 1: the aversion to leverage by people like me is what
Speaker 1: makes using a moderate amount of leverage a good idea.
Speaker 1: You know, That's what some people that are proponents of
Speaker 1: risk parity would argue that it's an inefficiency that a
Speaker 1: bunch of people like me now are averse using leverage.
Speaker 1: But I'm averse using it. I don't. So I'm not
Speaker 1: a fan of risk parity because I'm sort of you know,
Speaker 1: I just don't want to. I don't feel that I
Speaker 1: need to use leverage to get better quality returns. I
Speaker 1: think that the returns afforded by the marketplace without using leverage,
Speaker 1: and the risk attached there too, is all sufficient for me.
Speaker 1: And then I can go to sleep and not worry
Speaker 1: about having to reduce exposures because my leverage is causing
Speaker 1: me to do that. What about financial formulas in general
Speaker 1: and maths in investing, what are the downsides? Well, you know,
Speaker 1: models used in investing are are very useful. That they're
Speaker 1: they're a way of us um you know, thinking that
Speaker 1: that if we in one one of my colleagues one
Speaker 1: said that, uh, think about just the yield, yield to
Speaker 1: maturity of a bond. Think about that as a model. So,
Speaker 1: you know, for a while, you know, at some point
Speaker 1: in time, yield to maturities didn't wasn't really used. So
Speaker 1: people used to talk about the price of a bond.
Speaker 1: They talked about the current yield the coupon divided by
Speaker 1: the price, and then somebody and then people started to
Speaker 1: use yield to maturity or yield to worse more, Well,
Speaker 1: yield is just a much more useful thing to use
Speaker 1: and thinking about comparing different bonds with each other, implies
Speaker 1: volatility is a more useful way of thinking about comparing
Speaker 1: stock options to each other. There's nothing kind of magical,
Speaker 1: It doesn't tell you what to do, but it's just
Speaker 1: a more useful that that these models are a useful
Speaker 1: way of decomposing things into more intuitive quantities that we
Speaker 1: can that we can use in our decision makings. So
Speaker 1: I think that um, you know, math in finance is
Speaker 1: is useful, for sure, there's no doubt about that. But
Speaker 1: you know, but when we start to try to optimize
Speaker 1: things too much using math, when we when we try
Speaker 1: to get um, you know, trying trying to become too
Speaker 1: optimal and following you know, sort of narrow mathematical rigor
Speaker 1: too far, is extremely dangerous. Right. So it's it's sort
Speaker 1: of the you know, you you come up with a
Speaker 1: whole portfolio of different investments and you look at an
Speaker 1: optimization of that, and it tells you to do things
Speaker 1: that that common sense would tell you probably don't make
Speaker 1: sense to do. So taken to an extreme, I think
Speaker 1: that that math, that sort of mathematical outcome can lead
Speaker 1: us to uh, the dangerous places sometimes. But that's that's
Speaker 1: a great question. I wish I had more time to
Speaker 1: think about it and give you a better answer to it.
Speaker 1: That's a great answer, And Victor Hagani of ELM Funds
Speaker 1: really appreciate you coming on. Fascinating conversation, looking forward to
Speaker 1: reading and learning more about some of these concepts, and
Speaker 1: I think uh listeners will have learned a lot from this. Well,
Speaker 1: thank you very much as a pleasure, Joe, was that
Speaker 1: mathematical enough for you? I think that was just like
Speaker 1: the sort of a perfect level of mathematical sophistication while
Speaker 1: being able to understand the concepts without actually having to
Speaker 1: attempt to do math over the over audio, which I
Speaker 1: think would be tough. I mean, I sympathize with the
Speaker 1: coin tossers because if you think that like a coin
Speaker 1: toss has a fifty chance of coming up heads or tails,
Speaker 1: then if you got five in a row, well I
Speaker 1: suck at probabilities. I mean I get like, like you
Speaker 1: know there is something in your gut, like like you're
Speaker 1: something like that's exactly right, Like you really have to
Speaker 1: sort of sublimate your intuition and your feelings about how
Speaker 1: things work. Although then the question is like if you
Speaker 1: had a coin and say it came up twenty times
Speaker 1: in a row, you might think that it's going to
Speaker 1: be heads forever because then it's like broken other way. Um,
Speaker 1: but then it's really fascinating and like you know, like
Speaker 1: I said that the poker comparison, it's like it's not fun.
Speaker 1: Like if you're sticking to rules and it's like, yeah,
Speaker 1: everybody knows we should just put our money in a
Speaker 1: bond index fund, in a stock index fund or and
Speaker 1: leave it there. But it's really tough to be disciplined
Speaker 1: about these sort of rules and investing. Yeah. But conversely,
Speaker 1: you know, as LTCM to some extent demonstrated, it can't
Speaker 1: all be maths, right, Like the models sometimes need to
Speaker 1: be used with human judgment, even though they're useful in
Speaker 1: many ways. If if something big is happening, or if
Speaker 1: the model doesn't seem to be performing, you kind of
Speaker 1: have to step back and go way to second what's
Speaker 1: going on, or just the intuition that a model you're
Speaker 1: taking a huge risk, even though if you're leveraging thirty
Speaker 1: to one and obviously, as Victor pointed, or much much
Speaker 1: bigger and some at some points and as Victor pointed
Speaker 1: out at this point in his career, he doesn't have
Speaker 1: any interest after that experience in sort of applying leverage
Speaker 1: to finance at this point connotative finance. On his point
Speaker 1: about models, I did think that was really interesting, which
Speaker 1: is that you don't necessarily want to over determine what
Speaker 1: markets are going to do for models, but that models
Speaker 1: can provide a lot of insight just in sort of like, uh,
Speaker 1: sort of assessing where things are, and the idea of
Speaker 1: like volatile implied volatility being a sort of yeah, like
Speaker 1: the fact that all these things are in fact models
Speaker 1: that help you sort of compare one thing to another.
Speaker 1: I never thought of that because it's everyone uses it, right,
Speaker 1: we don't even think of them as models. Yeah, all right,
Speaker 1: Well that was a fun discussion. That was great. Let's
Speaker 1: say goodbye goodbye everyone. Thank you very much for listening.
Speaker 1: I'm Joe wisn't Thal. You can follow me on Twitter
Speaker 1: at the Stalwart and I'm Tracy Alloway. I'm on Twitter
Speaker 1: at Tracy Alloway. Thanks for listening. Year to E