This document describe a toy example for the use of the package systemicrisk.

```
library(systemicrisk)
```

Suppose we observe the following vector of total liabilities and todal assets.

```
l <- c(714,745,246, 51,847)
a <- c(872, 412, 65, 46,1208)
```

The following sets up a model for 5 banks:

```
mod <- Model.additivelink.exponential.fitness(n=5,alpha=-2.5,beta=0.3,gamma=1.0,
lambdaprior=Model.fitness.genlambdaparprior(ratescale=500))
```

Choosing thinning to ensure sample is equivalent to number of

```
thin <- choosethin(l=l,a=a,model=mod,silent=TRUE)
```

```
## Warning in findFeasibleMatrix_targetmean(l, a, p = u$p, targetmean =
## mean(genL(model)$L > : Desired mean degree is less than minimal degree that
## is necessary.
## Warning in findFeasibleMatrix_targetmean(l, a, p = u$p, targetmean =
## mean(genL(model)$L > : Desired mean degree is less than minimal degree that
## is necessary.
```

```
thin
```

```
## [1] 60
```

Running the sampler to produce 1000 samples.

```
res <- sample_HierarchicalModel(l=l,a=a,model=mod,nsamples=1e3,thin=thin,silent=TRUE)
```

```
## Warning in findFeasibleMatrix_targetmean(l, a, p = u$p, targetmean =
## mean(genL(model)$L > : Desired mean degree is less than minimal degree that
## is necessary.
```

Some examples of the matrics generated are below.

```
res$L[[1]]
```

```
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 0 0 714
## [2,] 497 0 0 0 248
## [3,] 0 0 0 0 246
## [4,] 0 51 0 0 0
## [5,] 375 361 65 46 0
```

```
res$L[[2]]
```

```
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0000 36.763750 3.163915 26.61477 647.45756
## [2,] 169.2638 0.000000 61.836085 0.00000 513.90011
## [3,] 0.0000 226.614770 0.000000 19.38523 0.00000
## [4,] 0.0000 4.357677 0.000000 0.00000 46.64232
## [5,] 702.7362 144.263803 0.000000 0.00000 0.00000
```

The sampler produces samples from the conditional distribution of matrix and parameter values given the observed data. To see the posterior distribution of the liabilities of Bank 1 towards Bank 2:

```
plot(ecdf(sapply(res$L,function(x)x[1,2])))
```

All the caveats of MCMC algorithms apply. In particular the samples are dependent.

Some automatic diagnostic can be generated via the function diagnose.

```
diagnose(res)
```

```
## Analysis does not consider 5 entries of matrix
## that are deterministic (diagonal elements, row/column sum=0 or forced result).
## All remaining elements of the liabilities matrix have moved during sample run.
## ESS in matrix:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 600.8 693.3 1000.0 880.0 1000.0 1000.0
## ESS in theta:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 447.2 717.7 751.6 766.3 864.9 1000.0
```

Trace plots of individual liabilities also shoud show rapid mixing - as seems to be the case for the liabilities of Bank 1 towards Bank 2.

```
plot(sapply(res$L,function(x)x[1,2]),type="b")
```

Trace plot of the fitness of bank 1.

```
plot(res$theta[1,],type="b")
```

Also, the autocorrelation function should decline quickly. Again, considering the liabilities between bank 1 and bank 2:

```
acf(sapply(res$L,function(x)x[1,2]))
```

In this case it decays quickly below the white-noise threshold (the horizontal dashed lines).