Percentages of older people are increasing in most countries. Moreover, in the United States and other countries there is a trend towards reliance on individuals' savings to supplement social retirement plans. This places a considerable burden on individual investors to choose appropriate strategies to use savings to efficiently provide income during their retirement years. This presentation will cover tools of economic analysis that can help inform such choices, including Monte Carlo analysis, equilibrium asset pricing theory, conditions for expected utility maximization and approaches for cost minimization. Their use will be illustrated in an analysis of a complex financial product in which an investment company and a life insurer jointly provide an investment fund with a guaranteed lifetime withdrawal benefit.

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Thank you. We’re running late - it’s been a long time. Let me first suggest that you stand up and stretch, please. I think it will make it a little less painful to listen to me. Thank you. Alright that’s it, you’re done. And second I’ll try to hit the highlights, because lunch awaits. You’ve heard from several speakers about the changing demographics in most countries. And I’m going to talk about a little bit of that problem. And where I think economics can help as we understand some aspects of it. Here’s a graph and if you don’t know this site, gapminder.com. It’s actually run by now a Swedish foundation in Stockholm. And it’s a fabulous site. You can get all this macro data and graph it in wonderful ways. So that may be the most important thing you’re going to hear from me. (Laughter) But these are fertility rates - I’ve done United States and Germany, since we’re in Germany. And what you see, and what you know of course, is fertility rates, which is children per woman roughly, have been falling at remarkable rates. And this is true, as you can find on the site in almost every country. Some countries are at higher level, but still the trend is down. And many countries are getting below 2.1 which is the magic number for stability in the population. And, of course, you can think about increased immigration, but nonetheless these are issues. The other piece of the puzzle is life expectancy. Again, I’ve shown Germany and the US. These are sort of squarely numbers you may know. These are computed each year by taking the mortality rates by age in that year and then chaining them. And so that’s why you get these huge tragic deviations for wars, for flu epidemics back in the early 1900s. But, again, life expectancy increasing - will it continue? That depends on the fight between the researchers who are working on cancer and the McDonald’s of the world. But somewhere - and here’s the result. These may be hard to see at your distance. But, again, we’ve alluded to this, others have in the talks. This is the support ratio which is 20 to 64 year olds divided by 65 plus. Sort of traditional workers per old folk, if you want to think of it that way. And the longer bars for 2008, the shorter projections for 2050. And what you see is alarming decreases in the worker types who could support traditionally retired people. And as was mentioned yesterday, Germany - the projection down towards the bottom: 1.6 and OECD 2.1. These are radical changes. And the implications for the economics and for the economies are, of course, profound. One last graph is a sort of a setup. The US is different from many countries from which you come in that, for better or for worse, we have changed the employer part of the savings for retirement to a system in which individuals make investments while they are working. And then have to figure out what to do with the money that they have from those investments when they retire. To a lesser or greater extent you see similar trends in the UK and Australia. And some trends, but nowhere near as big as thus far in the US, in other countries. And what you can see here for corporate retirement plans. The assets are the left bar. For plans within corporations in which individuals have individual savings accounts invested in mutual funds, whatever - the next bar. And then the third is a tax advantage savings vehicle, called an IRA, which may have come from one of the employer plans. But those last 2 bars indicate cases where the individual is making decisions as to how much to save, how to invest those savings and what to do upon retirement with those savings: Buy an annuity, put it in a mutual fund, go to Las Vegas - whatever it may be. And so we’re having this tectonic shift in which individuals in these countries, and perhaps in many of your countries, will all of a sudden have to make these terribly complex decisions which involve very, very substantial economic issues. And that’s what I want to talk a little bit about. And what I’m going to do is focus on a particular vehicle that is being offered for someone who has just retired. Which involves a kind of a combination of different financial products. And I want to show you how one can analyse it using pretty standard tools of economics. As a matter of fact, in this case extremely simple tools of economics. And again, think what’s going to happen here. We’ve got this money. I’m going to have 2 people and they are going to figure out: How to invest it. At what rate to spend it. To what extent to use an insurance company to help them deal with the uncertainty of mortality. And I want to project probabilistically the things that might happen. And try to understand some of the properties of this particular vehicle to deal with those issues. There are many, many, many, many possibilities. Because of the large amount of money that is coming to market from this sector, almost every part of the financial services industry is lusting after that money. And so there are insurance products, there are mutual fund products, there are financial advisors who have strategies. Many, many, many, many alternatives being offered for this very bewildered population. And it seems to me that it’s imperative that these folks get some help from disinterested third parties like you. So what do we need for this analysis? Well, at least the way I like to do it: we need some sort of stochastic models of mortality, actuarial tables and the like. We need to have some model, however crude, of the probability distributions of the returns on whatever vehicles might be used for investment. And in the particular product we’re looking at, generally the investments are a portfolio of stocks and bonds in roughly market proportions. And so I’m going to model that as the market portfolio which I’ll define in a minute. We need some sort of model about the stochastic behaviour of inflation. And we need some sort of valuation model of the sort Lars was talking about. His were far more sophisticated than the one you’re going to see here. But some notion of what you would pay today, or somebody would, for a dollar 5 years from now in a particular state of the world. So we need that sort of valuation function which I’ll talk about. And then we need to do a lot of scenarios in order to get any sense of the range of possibilities for what I’m going to use for Monte Carlo. I’m sorry, I’m racing fast here but I’m mindful of the time. And we need a good language in which to program this. I would argue that Matlab is as good as it gets. And there may be others. Except last night we were talking about programming languages. You might be able to do it faster - I’m sure you would in C++. But I for one have sworn never again to program in C++. Next, thank you. So here are the assumptions I’m going to make. The market portfolio which I want you to think of as all traded stocks and bonds in the world, held in market proportions. And this is important because there’s going to be a sense here of a market clearing equilibrium. So if this is the market which collectively everybody who holds traded securities holds. Or, if you will, the average investor in some sense holds this. And we want assumptions such that if these assumptions held, it would be consistent with at least some possible equilibrium in which this portfolio behaves as we assume. And I’m going to assume returns are independent year to year. In practice they use yearly differencing intervals with these products. I’m going to use yearly intervals for the simulation. And we’re going to assume independence. Which is a kind of a standard characteristic of a model of a market in which prices are anticipated. And random walks and those sorts of things, although many of the current models go well beyond that. This one doesn’t. I’m going to assume lognormal distributions which you can think of if, in fact, there were independence in say the monthly returns. Whatever their distributions might be when you compound, it will go towards the lognormal. So if you want to rationalise that in that manner, that works. But that’s what I’m going to assume. I’m going to assume an expected annual real return, real inflation adjusted, a 5% standard deviation of 12%. Remember these are stocks and bonds - for those of you who are used to thinking of stocks only. This is what I would call the true market portfolio theory. Inflation. This is a kind of a sideline. I’m going to assume it’s also independent and lognormally distributed. I’m going to assume, for convenience, it’s uncorrelated with the real returns of the market, which is certainly counterfactual. But probably not too unfortunate and has some attributes that are helpful in the simulation and the valuation. And I’m going to assume for this exercise expected inflation of 2.5% a year, and a standard deviation of 1%. Obviously, you change these parameters you’re going to change some of the values that you’re going to be seeing. And that’s an exercise that’s very helpful once you get the mechanism working. Present values. Lars talked about a stochastic discount factor. It’s basically the same concept. I like to think of them as present values or state prices following Arrow and Debreu. But the idea here is, what is the value today of a dollar to be paid 5 years from now if, and only if, the cumulative return on the market portfolio is 23%. So there’s this concept that the state prices are a function of the return on the market portfolio. Which is a surrogate for the broader concept of consumption that Lars was talking about. The idea here is - I’m going to use the term that I like, nobody else uses it: 'price per chance'. What is that? State price divided by the probability of that particular state. And that’s a kind of a normalisation that’s helpful as you think about some of these things. Now, here’s the key assumption. I’m assuming that only the market portfolio is priced for an investor with any given horizon. It’s a very, very strong assumption. But it is at least consistent with a possible multi-period equilibrium. And that’s about all I can say about it. It has the somewhat startling conclusion that that pricing kernel, that set of state prices, must look like the formula at the bottom. That is the price per chance must be a function of the cumulative return from today to that point in the future of the market in that state raised to a negative exponent. Another way of saying it is, this gives the implication that the market acts as if they were a single investor who had constant relative risk aversion. And that that coefficient is the same for every horizon, because you can’t really construct a pricing kernel that meets the condition of the market portfolio being efficient for every horizon. And you’ll see in a minute what I mean by 'efficient'. So here’s the pricing kernel in log-log scales, not surprisingly it’s a straight line. And that’s all I’ll say about that. So in practice, mechanically, what happens? You build a simulator. You create these humungous, at least by my standards, humungous matrices, where every row is a possible future scenario of the world and every column is a future year. And because the couple we’re going to look at are in their 60s, they might live for up to 50 more years. So you’ve got a matrix with as many scenarios as you feel you need times as many years as you need from the actuarial tables. And in practice, to keep the sampling error anywhere near under control, you’re going to need at least 100,000 scenarios. Each matrix is going to have about 5 million elements. And we’re going to have a bunch of them. You might say, well, how can we do that efficiently? And many of you already know. You can do it very efficiently, at least if you have Matlab and a machine with all-solid-state memory. Next time you buy a machine be sure to get - don’t have any discs going around; that’s not a good thing. So now let me - again, I know I’m racing. But you’re hungry, I’m hungry. I’ll be as quick as I can. So here’s the product we’re going to look at with this apparatus that I’ve described. And it’s an interesting product because it combines a number of features. People don’t like - we talked about this last night at dinner - people don’t like to buy annuities. Why should I give all my money to, you know, an insurance company that might go out of business. And if I die right away they’ve made all this money and they’ve got all my money and my kids don’t have anything. And on it goes. So they don’t like to buy annuities. But they face this grave uncertainty about their mortality. And so what do they do? They also don’t like to buy annuities because they can’t tap a fund in case there’s an emergency. Over the course of time people, those clever people in the financial services industry, came up with a number of products, including this one. And this is based on a product from a good investment company, Vanguard, which is itself a mutual and has low fees. So this is probably about as good as you’re going to get. And here’s the nature of the product. You have a million dollars. You go down and you put it in a Vanguard mutual fund. And then you take out 4.5% of that, $45,000, and you get to spend that in the first year. And, by the way, Vanguard takes out some money for their fees. And the insurance company that is involved in this strategy takes out some money for their fees. And I’ll tell you more about that. Then after a year you look at the value of the fund. If it’s below a million dollars then you get to take out $45,000 again. What you get to take out each year is 4.5% of something called the TWB, which is total withdrawal base. And that withdrawal base never diminishes. But if at any year anniversary the value is greater than the prior total withdrawal base, you kick that base up to equal the then value of the fund. So it’s a ratchet. And people love this. It never goes down, it can only go up. Now, remember, this is in nominal terms so your real income can fall. But your nominal income never can. So that’s basically how it works. And besides if you ever run out of money, if you’re still alive, the insurance company will keep cutting you the cheques for the $45,000, or whatever it may be at that time. So you have a life insurance or annuity component, mortality insurance, longevity insurance. And you have participation. And you can take money out of the fund, if there’s any left. More than the allowed amount at any time, although then the guarantees fall. So it’s an interesting combination. And here’s the Vanguard description: Steady income - that’s a good thing - with growth potential - that’s a great thing - and market protection - that’s wonderful. So it’s got it all. And these things are beginning to sell quite well in the US. We were talking last night. There are some available in Europe as well. So the question is, is this a good deal? And that depends on a number of things: What does it cost? What are the stochastic properties of the underlying drivers? And ultimately, what’s the utility function of the buyer, of the insured investors? I’m not going to talk about that third element today, I’ll just allude to it briefly. Come on, talk to me. Thank you. So here’s an example we’re going to look at. I’ve got a couple; they’re both 65, one is male, one is female. They’ve got a million dollars. The deal is they get to allow 4.5% of the total withdrawal base. Again, this is based on the actual product. And the fees are .57% from the current value of the account. That’s mostly for the investment management company. Although a little bit, I think, may go to the insurance company. And every year you pay 1.2% of that total withdrawal base. So that never goes down either. That’s the bad news. So these are the actual fees for the product that you could buy tomorrow. What do you need? You need mortality tables or some sort of a functional form. I use actuarial mortality tables of the probability that a 65 year old male will die each year in future year and these have progression. And some of the people I was talking to last night, know about these things in detail. So they’re in the simulation. And here’s an output. We’re looking only at 20 years from now - the year 20 years from now. And on the horizontal axis is the payment in real dollars, real dollars. And on the vertical axis is the probability that what you get will exceed that amount. So this is a cumulative probability distribution, but it’s drawn slightly differently than you’re used to. And I do that because I believe it has some behavioural advantages in terms of the way people think. They think, gee, I really need $40,000. What’s the chance I’ll have at least that? And it has other properties that seem to resonate with people. Here it is. This strategy, given all the assumptions and with the simulation, gives you the prospects viewed from today of getting this range of outcomes in real terms in year 20. And there is, of course, a corresponding figure for every year. Now, here is something interesting about it. Here I’ve plotted the state price per chance, that is the price, think of it just as price, on the vertical axis. And the amount of income you will get in year 20 for each of the 100,000 possible states of the world. And what you see is, it’s a mess. And that tells you this is not an efficient strategy in this sense. I could give you that same precise probability distribution with a strategy in which this graph would be just a nice curve. And I can do it in this case for 92% of the value that went into producing the graph we saw. That is there’s an 8% inefficiency in that it is not a least-cost strategy for producing the probability distribution. I’m not going to save any time at all. How do I do that? - That's another story; we’ll talk about that. But at least given the assumptions of the multi-period equilibrium, any strategy that does not give you a monotonic function in this diagram is inefficient. In that you can get the same probability distribution for that year at a lower cost. There are some complicating factors I’ll skip. Here is something else you can do and I’ll finish with this. You can figure out, given all your assumptions - caveat, caveat! - what the present value today is of the prospective outcomes. In other words, taking into account all the things that could happen, how likely they are and what the value today is for each of those contingent possible payments. And I would submit that’s an interesting thing for somebody contemplating this strategy versus some other to look at. And in this case you can see that about 72% of the million dollars is the present value of the income that those people could get from the investments directly. Another 15% is the present value of the income they could get from the insurance company after they run out of money. Another 3% is basically the insurance company’s profit margin. So the insurance product we talked last night about, where these annuities in general are, ‘overpriced’ – in this case you pay 3 to get 15; so the overhead is 3 over 15. Which is not outrageous; I’ve seen much worse. It basically goes to the investment company for management fees and whatever other little fees are in there. And then the present value of what the kids and the charities get is only 6% of your total initial million dollars. So this is a strategy, and not surprisingly, which is not particular generous ex ante to the estate. An annuity would give them value of nothing. And some strategies, in which you’d involve no insurance contracts and try to make the money last until you might die, generally will give the kids and the charities a lot of value. Because you’re probably going to die sooner and they’ll get what’s left. So strategies that people consider vary substantially in that respect. And believe me there are strategies in which the expenses to the financial services industry are much, much worse than this. And that’s something that, it seems to me, is important to try to evaluate as best one can. And hopefully guard against it. That’s it, thank you.

# William F. Sharpe (2014)

## Economic Analysis of Retirement Income Strategies

# William F. Sharpe (2014)

## Economic Analysis of Retirement Income Strategies

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