@nzikas Yes, this has been a thorn in my side for years. The only reason I wrote questions like 12.1 is to model Jorion's approach in Chapter 17. This is not the intuitive approach to SaR; the intuitive approach provides assets and liabilities as separate variables, such that S = A - L, ΔS = ΔA - ΔL, and variance of the change in surplus is a natural variance of the difference between to variables, σ^2(ΔA - ΔL). Jorion's approach, it seems, has the advantage of only specifying a single volatility (rather then two volatilties); instead of volatility of asset and volatility of liabilities, he only provides "volatility of the surplus." The problem with this is surplus tends to operate near the zero axis which makes it unwieldy (impossible actually) such that he necessarily "normalizes" the initial value of the assets. Here is the key assumption:

The relevant variable is the surplus S, defined as the difference between the value of assets A and liabilities L.

**The change then is ΔA - ΔL. Normalizing by the initial value of assets, we have Return(surplus) = ΔS/A = ΔS/A - ΔL/L*L/A = R(asset) - R(liabilities)*L/A [Jorion's formula 17.3] **
In this way, my question 12.1 mimics Jorion's example, see quoted below. Effectively, the return on surplus is a return on assets; i.e., ROA of 5% --> 5% * $800 assets = +$40.0 mm, and whereas Jorion writes "volatility of the surplus is 9.4 percent" and then scales the assets, I edited my question reflect that 18.0% is actually applied to the assets! But, let's see, if I could re-write it to give the

*same exact answer* but

**without any regard to the assigned reading**, I might write it something like the following:

- "12.1. Public Employee Retirement Fund (PERF) has $800 million in assets and $720 million in liabilities, for a current surplus (S) of $80 million. The expected annual return on assets, ROA, is 5.0%. The volatility of assets 18.0%, with a normal distribution. Over the next year, neither will the liabilities change nor will they experience any volatility. At the end of the year, there is a 1.0% probability that the pension will have a deficit of what amount?"

Notice how my slight re-write basically eliminates liabilities as a random variables and makes this a VaR question about the assets. I showed you this to highlight how Jorion's approach only needs one volatility, but by "normalizing by assets" both assets and liabilities are considered

I've asked GARP many times to replace this reading....

The better approach is something like this, but this gives a different answer, I am not trying to match outcomes here

- "12.1. Public Employee Retirement Fund (PERF) has $800 million in assets and $720 million in liabilities, for a current surplus (S) of $80 million. The expected annual return on assets, ROA, is 9.0%. The expected annual return on liabilities is 5.0%. [note this gives the same expected growth in surplus of +40.0 mm]. The volatility of assets 18.0%, with a normal distribution. The volatility of liabilities is [some value]%. The correlation between the return on assets and liabilities is [some value]. Over the next year, neither will the liabilities change nor will they experience any volatility. At the end of the year, there is a 1.0% probability that the pension will have a deficit of what amount?"
- Now we can use σ^2(S-L) to directly retrieve the variance of the surplus. I hope that's helpful!

Here is Jorion's SaR example in Chapter 17, which is the role model for my question 12.1:

"As an example, consider a hypothetical pension plan. Call it Public Employee Retirement Fund (PERF). PERF has $1000 in assets and $900 in liabilities, for a surplus S of $100 million. The duration of liabilities is 15 years; this high number is typical for pension funds. Assume that the expected return on the surplus, scaled by assets, is 5 percent. Using Equation (17.3), this translates into an expected growth of $50 million over 1 year, creating an expected surplus of $150 million. For Canadian pension funds, the typical volatility of the surplus is 9.4 percent, leading to an annual VAR of 22 percent, or $220 million at the 99 percent confidence dence level. Taking the deviation between the expected surplus and VAR, we find that there is a 1 percent probability that over the next year the surplus will turn into a deficit of $70 million or more. The tradeoff between tween this number and an expected surplus growth of $50 million defines the risk profile of the fund. If acceptable, risk budgeting then allocates the SAR of $220 million to different aspects of the investment process." -- Philippe Jorion. Value at Risk, 3rd Ed.: The New Benchmark for Managing Financial Risk (p. 433). Kindle Edition.

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