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# Risk Neutral Interest Rate Process

#### LuEs

##### New Member
Hello Everybody,

I have a question concerning the chapter 8 of Fixed Income Securities. At the end of the chapter, Tuckman shows two different trees, one depicting the true interest rate process an one the risk-neutral interest rate process. It is right to use "risk-neutral" for defining that process even if there are no risk neutral probabilities ?

Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @LuEs I think that's a great observation! Tuckman's usage of the term "risk-neutral" has presented challenges over the years (and some debates on this forum). I happen to think that he is correct, but I would like to note that he variously refers to "risk neutral probabilities," "risk-neutral rates," "risk-neutral tree" and even "risk-neutral process." I think we can start with, and agree on, the idea that a risk-neutral measures (e.g., risk neutral probability) align the expected discounted value with the market price (e.g., https://en.wikipedia.org/wiki/Risk-neutral_measure, or as Tuckman writes in Chapter 7 w.r.t. risk neutral probabilities: "The probabilities of for the up and down states are the assumed true or real-world probabilities. But there are other probabilities, called risk-neutral probabilities, that do cause the expected discounted value to equal the market price.). In really casual terms, the risk-neutral measure incorporate risks (e.g., lowers the price of an asset to reflect natural risk aversion due to outcome uncertainty).

Chapter 8 introduces this way, as a transition from arbitrage-free to equilibrium models:
From assumptions about the interest rate process for the short-term rate and from an initial term structure implied by market prices, Chapter 7 showed how to derive a risk-neutral process that can be used to price all fixed income securities by arbitrage. Models that follow this approach, i.e., models that take the initial term structure as given, are called arbitrage-free models. A different approach, however, is to start with assumptions about the interest rate process and about the risk premium demanded by the market for bearing interest rate risk and then derive the risk-neutral process. Models of this sort do not necessarily match the initial term structure and are called equilibrium models. -- Tuckman, Bruce; Serrat, Angel. Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (p. 229). Wiley. Kindle Edition.

Consequently, my interpretation is that Chapter 7 concerns risk-neutral probabilities which are adjusted to match the traded price, which is implicitly an arbitrage-free type approach. Compared to your example at the end of Chapter 8, which is an equilibrium-type approach, such that the same risk-neutral concept is applied but rather than adjusting probabilities, the rates are adjusted (both have the effect of discounting further the present value, so they incorporate risk aversion. I get confused semantically because a risk-neutral measure is actually incorporating risk-aversion). My evidence includes this, I hope this helps!
"Continuing with the assumption that investors require 20 basis points for each year of duration risk, the three-year zero, with its approximately two years of duration risk,6 needs to offer an expected return of 40 basis points. The next section shows that this return can be effected by pricing the three-year zero as if rates next year are 20 basis points above their true values and as if rates the year after next are 40 basis points above their true values. To summarize, consider trees (a) and (b) below. If tree (a) depicts the actual or true interest rate process, then pricing with tree (b) provides investors with a risk premium of 20 basis points for each year of duration risk. If this risk premium is, in fact, embedded in market prices, then, by definition, tree (b) is the risk-neutral interest rate process." -- Tuckman, Bruce; Serrat, Angel. Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (pp. 237-238). Wiley. Kindle Edition.

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