4. Statistical Risk Management

To get us started, I have re-generated the GLIVE and GDEAD matrix, but this time starting with GL0=5000 investors (instead of the 1000 used in the prior chapter). For now, I have maintained the same age x = 65, and Gompertz parameters (m = 90, b = 10), although I will modify those as we progress thru the chapter. For now, I have also assumed the same r = 4%, and σ = 3%, which means that as far as the R-script is concerned EXR=0.04 and SDR=0.03. As noted in Chap. 3, this leaves a 10,000 by 30 matrix TONDV and DETFV, representing the dividends and fund value.

My first objective in this chapter is to extract some summary metrics from these large and extensive simulation results (besides pretty pictures), which can give users a quick sense of the stability of the projected dividends and fund, but in a manner that is completely objective and statistically sound. Every time the user modifies any of the investment, mortality or tontine dividend payout rule parameters, the entire TONDV matrix will change, and I would like to create some summary metrics. First up is the mean value, standard deviation and relative variability of the tontine dividends:

                  
                    mean(TONDV)
                  
                  
                    > 7.710858
                  
                  
                    sd(TONDV)
                  
                  
                    > 1.002761
                  
                  
                    sd(TONDV)/mean(TONDV)
                  
                  
                    > 0.1300454
                  
                

The average dividend is $7.71 thousand dollars per year, per initial $100, 000 investment, which is very close to the κ ₁ = 7.67%. The standard deviation divided by the mean, or the tontines dividend variability is approximately 13%, which is across 10,000 scenarios and 30 years. While this number doesn’t account for differences in time or the time value of money, it is a baseline of sorts, one that I will return to later.

The next issue I would like to examine is the existence of any possible trend of the tontine dividends over time: which is something we don’t want, and shouldn’t happen if the payout rules are being followed. I define a new vector mtd as the median tontine dividend every year, from year i = 1 to the final i = 30. The value mtd[j] is assigned the 50th percentile value of the TONDV matrix, to remind users that the median is the 50% mark. I could also use the built-in command median as well, and will do so interchangeably. While on the topic of choices, I could have also used the mean value or perhaps even the modal value of the N = 10, 000 scenario dividends in each of the 30 years as the metric on which I focus on for stability; they are all debatable.

                  
                    mtd<-c(); t<-1:TH
                  
                  
                    for (j in 1:TH){
                  
                  
                      mtd[j]<-as.numeric(quantile(TONDV[,j],0.50))
                  
                  
                    }
                  
                  
                    fit<-lm(mtd~t)
                  
                

The last line in the above script generates a simple linear regression using the lm() command, in which time is the independent variable and the median tontine dividend is the dependent variable. This is obviously not the only way to examine the pattern of (median) dividends over time, and an alternative is to fit some sort of time series model, test for unit roots, etc. The key though is to test whether or not there is any trend in the median dividends, which is reflected in the coefficient term.

                  
                    summary(fit)
                  
                 
                  
                    Call:
                  
                  
                    lm(formula = mtd ~ t)
                  
                 
                  
                    Residuals:
                  
                  
                           Min         1Q     Median         3Q        Max
                  
                  
                    -0.0167151 -0.0046365  0.0000555  0.0036623  0.0166466
                  
                 
                  
                    Coefficients:
                  
                  
                                 Estimate Std. Error t value Pr(>|t|)
                  
                  
                    (Intercept) 7.669e+00  2.953e-03 2596.97   <2e-16 ***
                  
                  
                    t           8.656e-05  1.663e-04    0.52    0.607
                  
                  
                    ---
                  
                  
                    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
                  
                 
                  
                    Residual standard error: 0.007885 on 28 degrees of freedom
                  
                  
                    Multiple R-squared:  0.009578, Adjusted R-squared:  -0.02579
                  
                  
                    F-statistic: 0.2708 on 1 and 28 DF,  p-value: 0.6069
                  
                

So, although the variability in the tontine dividends do increase and grow over time as evidenced by Fig. 4.1, the median tontine payout is rather stable. I will return to this basic tontine dividend stability test and use it to detect (unwanted) trends in payouts over time as well as to examine and summarize worst-case scenarios.

For example, if I generate similar regression results but with a dependent variable (i.e. on the left) that is defined as the natural logarithm of the lowest percentile dividend, log(quantile(TONDV[,j],0.01)) for example, the resulting coefficient on the regressor is approximately negative 1% and is statistically significant at the highest levels of confidence. In English: The worst case scenario gets worse over time. But you already knew that from Fig. 4.1. Later on in this chapter I will examine a few of the (10,000) individual sample paths of the tontine dividends, versus the confidence intervals, to develop yet another angle and perspective on their behaviour over time.

Along the path to a deeper understanding of the simulation, R-script and results, the next parameter I would like to focus on is the standard deviation of returns, occasionally denoted by σ, or SDR in the R-script. How does it affect the TONDV matrix? Figures 4.2 and 4.3 offer a perspective on the answer.

Note the only difference is the variability in the portfolio investment return driving PORET. The GLIVE and GDEAD matrix are exactly the same, and I simply generated TONDV values using the different portfolio returns. Figure 4.2 has σ = 0.1% and Fig. 4.3 assumed σ = 4%. Although the median path for both figures are constant, and both pass the stability regression coefficient test I explained earlier, the 10,000 scenarios reflected in Fig. 4.3 are much riskier than those underlying Fig. 4.2. Readers and users can confirm that the ratio: sd(TONDV)/mean(TONDV) is a mere 1.5% when σ = 0.1%, but jumps to 17% when σ = 4%. This tontine dividend dispersion ratio is computed numerically, but in fact can also be derived via analytic methods in continuous time assuming Gompertz mortality and LogNormal returns, but taking us beyond the mandate of this book.

Nevertheless, the wide bands for the 98% confidence interval are not driven by the shrinking and small pool of survivors, although there is a trace of that in Fig. 4.2. Rather, the over $10,000 range in annual tontine dividends (from $4 to $14 thousand) for investors who survive to their mid-90s is entirely driven by investment returns. This means two things. First, we might want to think more carefully about how to reduce and control σ, to reduce the variability of payouts. Second, and just as importantly, the tontine sponsor must ensure they model future investment return volatility properly, or the tontine dividends themselves might fluctuate more than anticipated. I’ll return to both of these matters later on in Chap. 6 when I discuss alternative investment model-assumptions and derivative-based strategies for controlling volatility. For now, let’s examine a related question.

For the sake of completeness, I will stitch the various pieces of R-script I displayed and used in the prior chapter and place them in the next page as one (long) sequence of commands. The code begins with the Gompertz survival curve and the temporary life annuity function, then proceeds to define the system parameters and the various matrices I have already discussed at length, it then populates those matrices. The very last line performs my quick-and-dirty regression test for the stability of the tontine dividends over time. Users can simply cut-and-paste the code (which I am calling version 1.0), run it in R and generate a full set of results. In later chapters, when I augment the basic code, I will use different version numbers and will focus on the parts of the R-script that have to be changed.

I now return to the basic core simulation results in which the initial size of the pool is GL0=1000 investors, the assumed return is r = 4%, and the simulated investment returns are: EXR=0.04 with SDR=0.03, with the usual age and Gompertz parameters x=65 with m=90 and b=10. I generated N=10000 scenarios in which an initial investment f0=100 thousand dollars is tracked over a time horizon of TH=30 years. And, in this particular “run” the expected tontine dividend over the 30 years and 10,000 scenarios is: mean(TONDV)=7.73 thousand dollars per year, with a variability of sd(TONDV)/mean(TONDV)=0.134, or approximately 13%. The average tontine fund value (at year end) over those 30 years and 10,000 scenarios is: mean(DETFV)/1000=43.47 million dollars.

                  
                    # Modern Tontine (MoTo) Fund Version 1.0
                  
                  
                    # No Death Benefits Paid or Reserved
                  
                  
                    TPXG<-function(x,t,m,b){exp(exp((x-m)/b)*(1-exp(t/b)))}
                  
                  
                    TLIA<-function(x,y,r,m,b){
                  
                  
                      APV<-function(t){exp(-r*t)*exp(exp((x-m)/b)*(1-exp(t/b)))}
                  
                  
                      sum(APV(1:(y-x)))}
                  
                  
                    x<-65; m<-90; b<-10; GL0<-1000; TH<-30; N<-10000
                  
                  
                    EXR<-0.04; SDR<-0.03; r<-0.04; f0<-100
                  
                  
                    kappa<-c()
                  
                  
                    GLIVE<-matrix(nrow=N,ncol=TH)
                  
                  
                    GDEAD<-matrix(nrow=N,ncol=TH)
                  
                  
                    TCPAY<-matrix(nrow=N,ncol=TH)
                  
                  
                    PORET<-matrix(nrow=N,ncol=TH)
                  
                  
                    DETFV<-matrix(nrow=N,ncol=TH)
                  
                  
                    TONDV<-matrix(nrow=N,ncol=TH)
                  
                  
                    STPRV<-matrix(nrow=N,ncol=TH)
                  
                  
                    for (i in 1:N){
                  
                  
                      GDEAD[i,1]<-rbinom(1,GL0,1-TPXG(x,1,m,b))
                  
                  
                      GLIVE[i,1]<-GL0-GDEAD[i,1]
                  
                  
                      for (j in 2:TH){
                  
                  
                        GDEAD[i,j]<-rbinom(1,GLIVE[i,j-1],1-TPXG(x+j-1,1,m,b))
                  
                  
                        GLIVE[i,j]<-GLIVE[i,j-1]-GDEAD[i,j]
                  
                  
                      }}
                  
                  
                    for (i in 1:N){
                  
                  
                      PORET[i,]<-exp(rnorm(TH,EXR,SDR))-1}
                  
                  
                    for (j in 1:TH){
                  
                  
                      for (i in 1:N){
                  
                  
                        STPRV[i,j]<-prod(PORET[i,1:j]+1)^(-1)
                  
                  
                      }}
                  
                  
                    for (i in 1:TH){kappa[i]<-1/TLIA(x+i-1,x+TH,r,m,b)}
                  
                  
                    for (i in 1:N){
                  
                  
                      TONDV[i,1]<-kappa[1]*f0
                  
                  
                      DETFV[i,1]<-f0*GL0*(1+PORET[i,1])-TONDV[i,1]*GLIVE[i,1]
                  
                  
                       for (j in 2:TH){
                  
                  
                        TONDV[i,j]<-kappa[j]*DETFV[i,j-1]/GLIVE[i,j-1]
                  
                  
                    DETFV[i,j]<-DETFV[i,j-1]*(1+PORET[i,j])-TONDV[i,j]*GLIVE[i,j]}}
                  
                  
                    for (i in 1:N){TCPAY[i,]<-cumsum(TONDV[i,])}
                  
                  
                    mtd<-c(); t<-1:TH
                  
                  
                    for (j in 1:TH){mtd[j]<-median(TONDV[,j])}
                  
                  
                    fit<-lm(mtd~t)
                  
                  
                    summary(fit)
                  
                

What I would like to do next is (cherry) pick a few choice years and carefully examine the distribution of tontine dividends in those years, and the various factors that are correlated (or not) with those dividends. In particular, I will define three new vectors, td5<-TONDV[,5], as well as td15<-TONDV[,15], and td25<-TONDV[,25], which is obviously the entire array (and range) of payouts to survivors in the 5th, the 15th and the 25th year of the modern tontine fund. I get my simulation results, their mean values are as follows (and your numbers should be very close to):

                  
                    mean(td5)
                  
                  
                    > 7.686171
                  
                  
                    mean(td15)
                  
                  
                    > 7.722712
                  
                  
                    mean(td25)
                  
                  
                    > 7.771013
                  
                

First things first, let’s examine (and interpret) the correlation of these three vectors (of three numbers). I will create a matrix via cbind() and then use the cor() command to generate a 3x3 matrix of correlation values. I could obviously have done this for all 30 years and urge users to do that at their leisure, but the story this matrix tells is the same.

                  
                    dv<-cbind(td5,td15,td25)
                  
                  
                    > cor(dv)
                  
                  
                               td5      td15      td25
                  
                  
                    td5  1.0000000 0.5210515 0.3752384
                  
                  
                    td15 0.5210515 1.0000000 0.7206721
                  
                  
                    td25 0.3752384 0.7206721 1.0000000
                  
                

Notice how the numbers are all positive and decay over time but are rather high. Here is how to interpret them. The tontine dividend survivors receive at the age of 90 (which is 25 years after they invested at age 65) assuming they are alive, exhibits a correlation of 38% with the tontine dividend received two decades prior at the age of 70. If the dividend was above average in year 5, it will be above average in year 25. There is a clear persistence in the dividend. The converse is disappointing. If the year-5 tontine dividend didn’t reach expectations then there is a good chance that 20 years later the tontine dividends will be under-average as well. If the first few years or periods are disappointing the rest are likely to disappoint as well.

Some readers will recognize this as the sequence of returns effect which is a well-used phrase in the retirement income arena. It should be intuitive. The reason tontine dividends are likely to be under-average in the first few years is because investment returns were under-average in those years. Yes, it could be that “too many” investors survived and that’s why the tontine dividends were low, but in all likelihood the investment returns dominated. This is yet another reason to implement volatility controls in the first few years of the fund, perhaps using put and call options. I’ll get back to this later.

Now, in the spirit of stress-testing and enterprise risk management let’s assume a hypothetical under which deaths were expected to occur based on Gompertz mortality with (m = 90, b = 10), but the reality was different. Using those Gompertz parameters the initial tontine dividend was set at κ ₁ = 7.67% at age x = 65, using a valuation rate of r = 4%. However, assume that in fact nobody actually died during the first 10 years of the fund. All 1000 investors survived to age 75. What happens to the fund and tontine dynamics in that case? Intuitively, the lack of deaths implies that individual tontine dividends will be forced downwards during the first decade (and perhaps beyond) because of the larger-than-expected number of survivors. Eventually though, as investors do begin to die at a normal rate (if such a thing exists) the tontine dividends will stabilize with the thermostat effect, but that will take some time. To get a more precise sense of how this will all affect the modern tontine fund, I will now generate a revised GLIVE and GDEAD matrix, one in which the first 10 columns (and years) are rather boring. Namely, the GLIVE values are all set equal to GL0 and the GDEAD values are set to zero. This can be easily and quickly done using the following modified script. Nothing else is changed anywhere in the master script.

                  
                    # Simulating Modified Gompertz Values
                  
                  
                    # Realizing No Deaths in First 10 Years
                  
                  
                    x<-65; m<-90; b<-10; GL0<-1000
                  
                  
                    for (i in 1:N){
                  
                  
                      GDEAD[i,1:10]<-0
                  
                  
                      GLIVE[i,1:10]<-GL0
                  
                  
                      for (j in 11:TH){
                  
                  
                        GDEAD[i,j]<-rbinom(1,GLIVE[i,j-1],1-TPXG(x+j-1,1,m,b))
                  
                  
                        GLIVE[i,j]<-GLIVE[i,j-1]-GDEAD[i,j]
                  
                  
                      }}
                  
                

Once you have generated both matrices, be sure to use summary(GLIVE[,1:10]) to confirm that during the first decade under all N = 10, 000 scenarios there are no deaths. Then, in year #11, dying begins based on (Binomially simulated) Gompertz mortality, a.k.a. q ₇₅, for a 75-year-old. I should note that this (simple and unrealistic) modelling assumption also means that nature is not catching-up and killing more investors after age 75, to account for the missing deaths in the first decade. Rather, the investors who were supposed to die in the first decade escape death altogether, as if the tontine fund started at age 75 with 1000 investors. In reality, if indeed none of the GL0=1000 investors died during the first decade, I would anticipate seeing a slightly elevated mortality rate (relative to the Gompertzian value) at age 75 and beyond. But I leave this rather advanced discussion of more realistic models of life and death to a later stage, and at this point focus exclusively on the behaviour of tontine dividends during and after that peculiar decade.

Figure 4.4 displays the usual 98% confidence intervals, with the obvious flat line for the first decade and then a sudden (again, unnatural) jump in deaths starting in year #11. Notice how after the first decade the distribution of deaths resembles the earlier figures, which is exactly what one would expect under Gompertz mortality.

More importantly, moving on to the actual tontine dividends we can now use the latter part of the master script after the PORET segment and matrix should remain unchanged. Why? Recall that the stochastic model generating market portfolio returns is independent of the stochastic model generating life and death. Bottom line, after running the entire script the following script provides a statistical summary of the tontine dividends—using the revised GLIVE and GDEAD matrix—in the first few years of the modern tontine fund’s life. Can you spot the pattern?

                  
                    > summary(TONDV[,1])
                  
                  
                       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
                  
                  
                      7.671   7.671   7.671   7.671   7.671   7.671
                  
                  
                    > summary(TONDV[,3])
                  
                  
                       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
                  
                  
                      6.383   7.290   7.517   7.527   7.753   8.937
                  
                  
                    > summary(TONDV[,5])
                  
                  
                       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
                  
                  
                      5.752   7.032   7.342   7.358   7.659   9.491
                  
                  
                    > summary(TONDV[,7])
                  
                  
                       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
                  
                  
                      5.291   6.755   7.125   7.148   7.518   9.870
                  
                  
                    > summary(TONDV[,9])
                  
                  
                       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
                  
                  
                      4.822   6.447   6.868   6.900   7.311   9.694
                  
                

Indeed, let’s go thru the first 10 years of the life of the modern tontine fund in this (rather bizarre) universe during which nobody dies. At the end of the first year the tontine dividend is $7.67 thousand per survivor, a number that was set and fixed at time zero. But, with all those extra survivors (that weren’t expected) the tontine fund value at the end of year #1 will be lower than originally anticipated, leading to a pre-programmed and necessary reduction in the tontine dividend for year #2. In fact, this process continues during the entire decade and the tontine dividends must constantly be revised downwards albeit based on a well-defined and transparent algorithm. Notice how the median tontine dividend in year #3 is reduced to $7.52 thousand, and then $7.34 thousand and then $7.12 thousand and then $6.87 thousand. The 1st and 3rd quartile of the entire distribution of tontine dividends exhibits the same shrinking pattern, which is the thermostat effect. But then, starting in year #10 the tontine dividends stabilize as people (finally) start dying at the (originally) anticipated Gompertzian rate. Figure 4.5 displays that stabilization process.

To wrap up this particular discussion one could summarize the last few pages by loosely declaring that the absence of any deaths during the first decade will reduce the tontine payout rate by about 110 or so basis points, from the initially promised 7.67 to the sustainable 6.55, albeit slowly and over the course of a decade. Needless to stay, this doesn’t account for the standard deviation or dispersion in payouts, but it does enable users and readers to intuit the impact of misestimating mortality.

In some sense, my job is done. With this chapter I have delivered the core of the simulation and discussed a number of risk management aspects. In theory, you can use Chap. 3 and this chapter to design and manage a very simple and crude modern tontine. The next few chapters are really about additions, augmentations and alternative formulations.