A physicist view on Finance (IV) — ETF, pension and risk management?

ETF, pension and risk management

Not long ago I was discussing with a FinTech’ friend about pension planning and wealth management. The topic we were discussing was

If you were to retire tomorrow at the age of 60 with a million euros, how would you invest it on ETF markets?

I was immediately hooked on the question. How would I? As simple and straightforward as the question may be, I did not have a straight answer on the top of my head. Of course, it is obvious that investing on riskier markets than state protected French ‘livret A’-type accounts would increase the possible financial outcomes, thus increasing the pension I could receive.

But what would be a sound pension planning strategy on the ETF markets?

On the figure below, three examples of how things could unfold until my death if I invested in a risky asset while paying myself a pension coupon with regular installments.

• I choose a coupon that is too high (bottom curve) I end up with no money before the end of my life.
• I choose a coupon that reaches zero the day of my death (middle curve — this would probably be extremely lucky, but for the sake of presentation, let’s assume that I can do that!), I have been able to live off my pension plan but I am not able to leave something to my children.
• I choose an even smaller coupon (top curve) which leaves a positive wealth at the end of my life, however, I possibly have restrained myself too much with a too low-value coupon on my pension plan.

The topic is fascinating from a modelling perspective as it requires adequate statistical description of financial assets (ETFs here) to run as trustworthy as possible Monte Carlo simulations and it requires to adapt usual portfolio management theories because of the pension paid on a periodic basis.

Here are my thoughts…

For those in a hurry…

The short answer is that I would decide to get monthly pension payments and although I would monitor regularly, I would update my strategy after each coupon payment, if market conditions had changed drastically. For example, a drastic change would modify the value of statistical parameters like trend and volatility, on which I based my initial decision.

The strategy would be an to invest in ETFs while assessing the risk of performing worse than a state protected account.

I would decide on the coupon value trading off two parameters: life expectancy and risk aversion.

• Life expectancy is usually known for any age and I would choose the average value at my age of retirement.
• Risk aversion is a trade off parameter between potential gains (left figure below) and potential losses (right figure below) over the span of my pension plan.

The Greek letter in x-coordinate is the dimensionless risk aversion parameter (defined in the mathematical section). The y-coordinate is a ratio to my initial wealth.

For those curious to know more about the Maths…

If you want even more mathematical insights, I redacted a more detailed note here. All python scripts are also available on Github.

Wealth dynamics

The cash flow diagram below presents the variables and parameters involved in the retirement plan investment strategy. It shows what the investor would do between two periods of investment k and k+1.

• µ is the return of the risky asset and r is the return of the risk-less asset.
• c is the monthly pension, or ‘coupon’.
• At the beginning of each investment period, the coupon is paid and the rest is allocated between a risky asset and the risk-less asset.
• If there is not enough capital for the full amount of the coupon, then the remaining wealth is paid.

• At the beginning of the next period and before the payment of the coupon, the investor balances out their positions.

As often in financial modelling, simplifications arise from the introduction of discounted variables

With these discounted variables, the wealth variation is equal to the value variations in the risky asset minus the paid coupon. As the value variation in the risky asset occur at k+1 and the coupon is paid in k, an additional discount is given to the first term as shown below.

Investment strategy

As a retired-from-work-life investor planning my pension, I would be rather wary of the risks I take. I would particularly make sure at every steps that I am comfortable with the risk-return trade off. Taking risk to increase my gains is an interesting thought that becomes very unattractive if I perform far less than the risk-less investment.

In mathematical terms, the below risk aversion condition is a good translation of my wariness.

For a level α of my choosing (5% is the most common choice), I ensure with the available data that at every next step, I will not under-perform the risk-less asset by a factor 1-ρ.

ρ is my risk aversion parameter. The closer ρ gets to one, the greater fraction of my capital I accept to loose with a probability lower or equal to α.

Now if we assume that the returns of the risky asset are Gaussian¹ with an average m and a standard deviation σ, the previous condition may be restated as

Another aspect is that getting loans to invest in the risky asset is out of the question as these types of leverage generally increase my risk. As a result, the amount available for the risky asset is limited to the available capital. The amount invested in the risky asset has thus to be constrained by the condition

Now that I know the upper boundaries of capital to be invested in the risky asset and as I also want to maximize the expected returns, the optimal strategy is to allocate it all to the risky asset. Mathematically, this means that

Complete model

By simply adding the two previous modelling bricks together, the full dynamics of the wealth is given by the set of equations below.

Now that we have a fully stated model for the wealth dynamics, all there is left to do is to find a methodology to search for the appropriate coupon.

Matching the life expectancy with the maturity of the investment

The stochastic nature of the risky asset’s returns makes it impossible to know for sure if I will be able to maintain my portfolio until the end of my life. Using Monte-Carlo simulations, I can however calculate the average life expectancy of an investment of coupon c and risk aversion parameter ρ.

In other words I can solve the problem

where κ(c(ρ), ρ) is the Monte Carlo computed average of the investment’s life span for a given risk aversion parameter ρ.

Below are some results I had using the following parameters

• the risk free rate is set at r = 0.005/12 which is the livret-A rate,
• we consider the maturity K = 24 which is equivalent to two years,
• the number of samples in the Monte-Carlo simulations is S = 100000, and finally
• the risky asset is the Lyxor ETF CAC 40 for which monthly data gives m = 0.02592 and σ = 0.06164 from 1st April 2020 to 31st March 2021.

Also for the sake of simplicity in the implementation, I introduced the variable

which spans from 0 to 0.77 if α=0.05.

The coupon curve (top left) describes the coupon value c(ξ) all values of ξ. Note that there are some very little numerical instabilities due to the root finding algorithm. This should not stop our understanding of the results. The dotted line the risk-less benchmark coupon obtained if no money was invested in the risky asset. Quite clearly, as the investor increase their risk-taking sensitivity, they can increase the coupon they pay themselves.

The average life span (top right) is unsurprisingly constant. The purpose of this figure is to show explicitly that as the investor increases their risk, the uncertainty of the investment duration increases too. The dashed lines are one standard deviation away from the average and it would not be uncommon for unlucky investors to reach zero wealth 4 months before the planned maturity.

Additional wealth (bottom left) is the capital surplus from the risky positions. The average increases with with the risk and so does the possibility of negative outcomes as shown by the lower dashed line which gets quite close to zero, indicating probable losses.

The 5%-Expected Shortfall (bottom right) confirms this last statement. This last statistics compares the final total paid coupons to the final total paid risk-less benchmark coupon. The riskier the investor is, the higher is the average of the 5% worst losses which can easily be up to 10% of the benchmark. So any risk-taking pension-planning investor needs to ask themselves, are those 10% worth the risk?

Wrapping up

This short document demonstrate that with a precise wealth management strategy, an investor can determine how they can gain from investing in ETFs. This study assumes rather simple market dynamics hypotheses, however, the general framework provides an accurate methodology to determine optimal pension management.

Potential improvements are

• control of the target final average wealth at maturity to open the door to inheritance and transmissions,
• risky returns modeled with a stochastic volatility model,
• adding the random nature of life expectancy,
• investing in multiple ETFs,
• computing wealth dynamic with undiscounted values to avoid underflows,
• optimizing the root finding algorithm to avoid numerical inaccuracies.

1. This is a HUGE assumption. As shown in a previous post, returns are usually well fitted with a Student distribution which should have implications on the risk measures. To start with, it is however a sound choice to keep the model complexity at the lowest, refinements can be done later on aspects regarding the stochastic modelling of returns.

The above references an opinion and is for information purposes only. It is not intended to be investment advice. Seek a duly licensed professional for investment advice.