Binomial Limit for a Portfolio Loss Distribution

Summary

The document explains a limit used in a Vasicek portfolio loss distribution: a binomial cumulative probability with a threshold proportional to the number of exposures converges to an indicator at the binomial success probability. The sum is recognized as the probability that a binomial random variable with parameters n and s is at most the integer part of nx.

Dividing that variable by n gives the sample proportion of successes. By the law of large numbers, this proportion converges to s as n grows, so the cumulative probability tends to one when the threshold x is at least s and to zero when it is below s. The response gives the core probabilistic argument, though its displayed final probability contains a parameter typo: the binomial variable should retain success probability s, not x. The result also relies on interpreting the limit at the threshold appropriately.

Key ideas

• The binomial sum is the cumulative probability that a binomial count does not exceed the threshold.
• Dividing the count by n converts it to a sample proportion.
• The law of large numbers makes that proportion converge to the success probability s.
• The limiting cumulative probability is an indicator depending on whether x is at least s.
• The response’s final display appears to contain a parameter typo, using x where s is intended.

Tags

# What is the limiting distribution of loss portfolio?


# What is the limiting distribution of loss portfolio?












I am working through this paper on Vasicek's portfolio loss distribution.

On page 3 he mentions that by the law of large numbers,

$$\lim_{n\to\infty}\sum_{k=0}^{\lfloor nx \rfloor} \binom{n}{k}s^k(1-s)^{n-k} = 1_{\{x \ge s \}}$$ where $s \in (0,1)$,

Here $x$ is some real number, $k$ is an integer and $\binom{n}{k}$ is the binomial coefficient.

but I can't quite work out why this is the case.

I have tried some sort of binomial distribution argument but haven't been very successful!

Any help would be greatly appreciated.

## Answer by M. Jeunesse (score 1, accepted)

https://quant.stackexchange.com/a/26091

$$\begin{split}\sum_{k=0}^{[nx]}\binom{n}{k}s^k(1-s)^{n-k}& =\sum_{k=0}^n \mathbb{1}_{k\leq [nx]} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \sum_{k=0}^n \mathbb{1}_{k\leq nx} \binom{n}{k}s^k(1-s)^{n-k} \\ & = \mathbb{P}(\mathcal{B}(n,s)\leq nx)\\ & = \mathbb{P}\left(\frac{\mathcal{B}(n,x)}{n}\leq x\right)\end{split}$$

where $\mathcal{B}(n,s)$ is a binomial of parameters $(n,s)$

using the law of large number you get $\frac{\mathcal{B}(n,s)}{n}\to_{n\to\infty} s$ and you can conclude.

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