# Testability of TSM (Term Structure Models)

#### nicholasjalonso

##### New Member
Good morning,

In all of the term structure models, we have "dw" serving as a normal random variable, in which David recommends to use Norm.S.InV on that uniform random variable. I'm curious if anyone could provide how to prepare for this, given that there is a "randomization" factor in the model that's been shown on BT to be a formula in excel. How would we be expected to execute one of these models on the P2 exam?

Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nicholasjalonso Please see the 5th bullet in my recent WIFE blog entry at https://forum.bionicturtle.com/threads/week-in-financial-education-2021-03-29.23750/post-87816 . GARP provides a z-lookup table on the exam (see example below, from my linked-to thread). This table enables a candidate to perform the inverse standard normal CDF, =NORM.S.INV(p), by "inverting" the table: that is, finding the probability in the body and identifying its associated quantile (Z value).

See below. So for example if you wanted to retrieve N^(-1)(0.20%), then you can see on the table that z = -2.88 (which is rounded of course) which is next to Pr(X<0.0020).
Hi @julienfrancaoui Yea this is a basic skill that is already much-discussed in the forum; e.g., see https://forum.bionicturtle.com/threads/probability-function.23487/post-83885

See below. The Z-lookup table gives you the relationship between N(z) = p; for example (see red below) N(-2.33) = 0.0099 ≅1.0%. We need to be able to use this lookup table "interactively" by which I mean we need to be able to invert to get z = N^-1(p) as in -2.33 = N^-1(1.0%). Further, we need to be facile with the fact that all of these values are one-tailed CDFs, by definition. If we want a two-sided 99.0% confidence interval, then we want 2.58 at 99.0% (because if we want 1.0% to be shared between each tail, then it's the same as the one-tailed quantile at 0.50%, see purple below). VaR is always one-sided, but CIs are typically two-sided. Okay, I've now re-explained this for literally the one hundreth time on the forum! I hope it's helpful and feel free to use search in the future because it saves us all a lot of time when you look to see if a question has been previously asked/answered....

#### nicholasjalonso

##### New Member
Thank you for your response. My question is more in line with the Term Structure Models incorporating Drift (i.e., Cox-Ingersoll, Model 1, Model 2, etc.) where we say that "dw" is a random variable. How is the randomization performed?

Taken from the Study Notes on the TSM: "Assume our uniform random variable happens to be 0.40 such that the random standard normal = -0.2533 = NORM.S.INV(40%). Each step accepts a different random normal."

Where does the different random normal variable come from?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nicholasjalonso Oh, okay well the randomization is the essence of a simulation. In Excel, it is achieved with RAND() a function so common that it takes no arguments: it generates a (pseudo-) random number between 0 and 1, which you will notice also bounds the CDF probability. A probability must also lie between 0 and 1.0 (100%). So the 40% to which you refer is just an illustrative example of a continuous random number, [0,1], which becomes the probability input into the inverse standard normal CDF N^(-1)(p) = q, in this case N^(-1)(40%) = -0.2533. Not just in those term structure (TS) models you mention, but for pretty much every Tuckman's TS model, "dw" is a random normal which time-scales the random standard normal: dw = sqrt(Δt)*NORM.S.INV(RAND()),

In terms of the exam, you would need to be provided a random value, again which in this case is merely a random CDF probability, from 0 to 100%. Also, in this case, it happens to inform an inverse normal, but inverse transformation can take any (analytical) distribution and translate the probability, p, into its associated quantile. If you were given a random p, it would correspond to a probability in the lookup table.

At the same time, GARP would be at least as likely to go ahead and give you the already-executed N(40%) = -2533 value, or the already-executive dw = N(40%)*sqrt(Δt) = whatever. Similar to how, in the BSM option pricing model, they are at least as likely to provide N(d1) and N(d2) values fully executed, so to speak. In summary, the exam can't ask you to generate your own random probability, p, by definition! (everyone would get a different result!). Instead, if applicable, it would need to provide (eg), random p = 0.40 or p = 0.83 or whatever. Thanks,