Maximizing the Long-Run Returns of Retirement Savings

[FCCC]

Superannuation funds are important and poorly implemented

Australia prepares its people for retirement by requiring them to save a portion of their income in “superannuation funds”, which currently hold about $3 trillion dollars’ worth of citizen wealth. In 2025, the required proportion of income saved will increase to 12 percent. Considering the enormous wealth this system controls, it’s important for the system to work well. Unfortunately, a recent government report pointed out significant failures:

If there were no unintended multiple accounts (and the duplicate insurance that goes with them), members would have been collectively better off by about $2.6 billion a year. If members in bottom-quartile of … products had instead been in the median of the top-quartile … they would collectively have gained an additional $1.2 billion a year.

In fact, the last time I checked the data, every single large retirement fund was outperformed by Vanguard’s global index (over a 10-year period). Most workers know nothing about investing, nor should they need to—the system should do the hard work of getting them the best deal. To do this, the government report recommends the creation of a committee that would help people choose the best 10 (or fewer) funds to invest in. In other words, the report’s solution is essentially to hire people who will solve the problem.

A fundamental overhaul of the system might be a better fix. The goal of my proposed system is to maximize the long-run returns. First, we will build a basic model, and then fix the primary issue. Still, there are obvious issues with it, but I think it could be modified into a system that would work well. If you want to point these issues out, please recommend a fix as well.

The proposed system

Each dollar of forced savings goes up for auction. The highest-bidding fund gets the dollar immediately after the auction (i.e. the present value of the funding is $1):

$$PV\left(Funding\right)\equiv 1.$$

Then, after a previously agreed-upon period of time, the fund pays the government the second-highest bid (i.e. the future value of the price paid is equal to the second-highest bid):

$$FV\left(Price\right)\equiv Bi{d}_{2nd}.$$

All of these bets go into one big collective fund, and each citizen’s return is the average return of the fund.

Profit of the fund

The future value of the funding is equal to the present value of the funding, i.e. $1, plus the net return it can generate from that dollar:

$$\begin{array}{cc}FV(Funding{)}_i& =PV\left(Funding\right)(1+NetRetur{n}_i)\\ & =1+NetRetur{n}_i.\end{array}$$

The future value of the fund’s profit is equal to future value of the funding it receives minus the future value of the price it pays:

$$\begin{array}{cc}FV(Profit{)}_i& =FV\left(Funding\right)-FV\left(Price\right)\\ & =1+NetRetur{n}_i-Bi{d}_{2nd}.\end{array}$$

The fund’s profit-maximizing bid

Suppose that funds maximize profits. Thus, the $i$th fund bids its profit-maximizing bid ($Bi{d}_i^{\ast }$), which, in a second-price auction with non-colluding bidders, is the bid ($Bi{d}_i$) that would make its profit zero if it paid its bid:

$$Bi{d}_i=Bi{d}_i^{\ast }⟺FV(Profit{)}_i=0\text{and}FV(Price)=Bi{d}_i.$$

The profit-maximizing bid can be found by making the relevant substitutions:

$$\begin{array}{cc}Bi{d}_i=Bi{d}_i^{\ast }⟺0& =1+NetRetur{n}_i-Bi{d}_i^{\ast }\\ Bi{d}_i^{\ast }& =1+NetRetur{n}_i.\end{array}$$

Note that each fund’s profit-maximizing bid is independent of the other funds’ bids.

Sufficient criteria to maximize returns

The future value of citizens’ savings is maximized if

1. the fund that can produce the highest net return gets the dollar, and

2. all funds maximize their net return, and

3. the winning fund’s profit approaches zero.

The proposed system satisfies the sufficient criteria

Criterion 1: Best fund gets the money

Fund $i$ gets the dollar only if its bid is the highest. We substitute in its profit-maximizing bid, and find that the winning bidder must have the highest net return:

$$\begin{array}{cc}\text{Fund}i\text{gets the dollar}⟺Bi{d}_i& >Bi{d}_j\text{for all}j\ne i\\ ⟺1+NetRetur{n}_i& >1+NetRetur{n}_j\text{for all}j\ne i\\ ⟺NetRetur{n}_i& >NetRetur{n}_j\text{for all}j\ne i.\end{array}$$

I.e. criterion 1 is satisfied.

Criterion 2: The winner maximizes their net return

Recall that the winning bidder pays the second-highest bid. Recall that, barring collusion, a profit-maximizing fund will bid independently of the bids of any other funds—the winning fund cannot affect the second-highest bid. So, to maximize its profits, the winning fund will maximize the remaining factors, which happen to equal its net return:

$$\begin{array}{cc}FV\left(Profit\right)& =max(1+NetRetur{n}_i-Bi{d}_{2nd})\\ & =max\left(NetRetur{n}_i\right)+1-Bi{d}_{2nd}\end{array}$$

I.e. criterion 2 is satisfied.

Criterion 3: The fund’s profit is zero

The winning fund’s profit equals the difference between the total benefit it would provide and that of the second-best fund:

$$\begin{array}{cc}FV\left(Profit\right)& =1+NetRetur{n}_i-Bi{d}_{2nd}\\ & =1+NetRetur{n}_i-(1+NetRetur{n}_j)\\ & =NetRetur{n}_i-NetRetur{n}_j.\end{array}$$

I.e. if the best two funds are equally capable, the profit is zero, and criterion 3 is satisfied.

EDIT: Addressing the main weaknesses of the model

The definition for the future value of profit assumes no bankruptcy

A fund can’t make unbounded negative profits: They’ll go bankrupt once their fund’s value goes below zero. So funds could profit by taking high-volatility bets even if the expected value of the bet is below zero. To prevent this, we probably want capital-rich intermediates to underwrite the funds. Or you could have a secondary market that effectively acts as an underwriter.

The problem for underwriters is that the market is risky. To make their optimal bids, funds need to estimate their exact return at an exact time. And that’s not possible.

Making the bidding easier

To make it easier for the funds to estimate their optimal bid, the bids can be stated as whatever the return of the agreed-upon index is (probably a global index) plus some constant.

$$\begin{array}{cc}Bi{d}_i^{\ast }& =1+Retur{n}_{Index}+C.\end{array}$$

In other words, when the fund makes its bid, it only gives the value $C$. Recall that the optimal bid also equals $1+NetRetur{n}_i$. Solving for the constant, we get

$$\begin{array}{cc}Bi{d}_i^{\ast }& =Bi{d}_i^{\ast }\\ ⟺1+NetRetur{n}_i& =1+Retur{n}_{Index}+C\\ ⟺C& =NetRetur{n}_i-Retur{n}_{Index}\\ & =GrossRetur{n}_i-InvestmentCost{s}_i-Retur{n}_{Index}\\ & =ExcessRetur{n}_i-InvestmentCost{s}_i\end{array}$$

Therefore

$$\begin{array}{cc}Bi{d}_i^{\ast }& =(1+Retur{n}_{Index})+(ExcessRetur{n}_i-InvestmentCost{s}_i).\end{array}$$

This allows bidding from index funds (whose excess return over the index is equal to their tracking error, which should be close to zero) to bid the following

$$\begin{array}{cc}Bi{d}_i^{\ast }& =1+Retur{n}_{Index}-InvestmentCost{s}_i.\end{array}$$

So, the only variable they need to estimate is their cost of investment.

Underwriters will only support funds who can estimate their costs well (I’m not sure if this is difficult or not) and have low tracking error. Forcing all bids to be made as an estimate of $C$ also leaves the door open for funds that can reliably beat the market (they would make bids with positive values for $C$).

Why use a global index as a reference point?

The argument for using a global index as a reference point is a bit hand-wavy, but it’s close enough to being true. In the long run, funds can’t exceed the global index by significant amounts. If they did, they’d eventually end up with 100% of market, which would make the index return be whatever the fund’s return is. That fund can’t outperform the market if it is the market. Therefore, matching the return of the global index should be equivalent to maximizing long-run returns. And using that return as a reference point for bidders reduces the cost to investors.

So why have a bidding process at all? What not just put everyone in Vanguard or BlackRock index funds? If, at some point in the future, these funds get worse at matching the index, or they have higher costs, or they’re outperformed by a new competitor, there needs to be a mechanism to switch to the new best fund. Having auctions ensures this transition happens quickly, keeping the market competitive.

[Ramiro]

I think there’s a small typo, probably from your previous post on prisons:

Note that each prison’s profit-maximizing bid is independent of the other prisons’ bids

I like this idea; we should have many more second-price auctions out there. Do you have any further references about it?

[FCCC]

Caught red handed. I’d been thinking about this idea for a while and was trying to get the maths to work last night, so I had my prison/​immigration idea next to me for reference.

I like this idea; we should have many more second-price auctions out there. Do you have any further references about it?

Thanks. I’m not the best person to ask about auctions. For people looking for an introduction, this video is pretty good. If anyone’s got a good textbook, I’d be interested.

[Larks]

I think this is a neat idea.

However, I think it might have problem with the fiscal limits of the asset managers; there is a reason that even hedge funds with sophisticated clients do not structure themselves this way. At the moment most asset managers do not have very large balance sheets—even Blackrock, the largest in the world, only has $5bn of cash, and a market cap of 85bn. If the winning fund under-performed the second bid by 0.5%, they would face a 15bn loss—or, more likely, they would go bankrupt and the pensioners would bear the loss. Even if you divided the funds up between multiple managers the total capitalisation of the industry is not that large, and the winning bids would disproportionately be submitted by low-capitalisation funds that wanted a free call-option. This gives managers an asymmetric payoff curve that encourages them to take a lot of risk.

To solve this you could try regulating the asset managers, but at that point you have basically re-invented insurance companies, and they would not be able to take much risk.

Another possible solution would be to implement very harsh penalties for the individual managers. But I think it would be difficult to calibrate these penalties well, and might make it hard to attract talent.

[FCCC]

Yeah, it’s definitely flawed. I was more thinking that the bids could be made as a difference between an index (probably a global one). So the profit-maximizing bids for the funds would be the index return (whatever it happens to be) minus their expected costs. And then you have large underwriters of the firms, who make sure that the fund’s processes are sound. What I’d like is everyone to be in Vanguard/​Blackrock, but there should be some mechanism for others to overthrow them if someone can match the index at a lower cost.

[Larks]

Ahhh, so basically the idea is that no underwriter would be willing to vouch for anything but a credible index shop. Seems plausible.

[FCCC]

Yeah, that’s right. The problem with my toy model is that it assumes that funds can actually estimate their optimal bid, which would need to be an exact prediction of the their future returns at an exact time, which is not possible. Allowing bids to reference a single, agreed-upon global index reduces the problem to a prediction of costs, which is much easier for the funds. And in the long run, returns can’t be higher the return of the global index, so it should maximize long-run returns.

However, most (?) indices are made by committees, which I don’t like, so I wanted to see other people’s ideas for making this workable. (But index committees are established and seem to work well, so relying on them is less risky than setting up a brand-new committee as proposed in that government report.)

[Larks]

My understanding is the committees generally make rules for the indices, and then apply them relatively mechanistically, though they do occasionally change the rules. I think it is hard to totally get rid of this. You need some way to judge that a company’s market cap is actually representative of market trading, as opposed to being manipulated by insiders (like LFIN was). Presumably if the index committee changed it to something absurd the regulator could change their index provider for the next year’s bidding, though you are at risk of small changes that do not meet the threshold for firing.

As a minor technical note gross returns often are (very slightly) higher than the index’s, because the managers can profit from stock lending. This is what allows zero-fee ETFs (though they are also somewhat a marketing ploy).