Risk neutral vs real world

[afterworkguinness]

Hi,
In Tuckman chapter 8 (risk premium subsection) he calculates the price of a two year zero using the up and down probabilities of 50% and says this is how the zero would be priced by risk neutral investors. Are these not the real world probabilities?
Thanks

 

[ShaktiRathore]

Hi
Yes you are right these 50% should be real world probabilities may be a typo. Other possibility is that tuckman assumes 50% as risk neutral probabilities while doing valuation.
Thanks

 

[afterworkguinness]

Thanks.

 

[David Harper CFA FRM]

That's a great observation @afterworkguinness. I actually think Tuckman does not have a typo here, but rather that this is due to the confusion between risk-neutral investors and risk-neutral pricing (which utilizes risk-neutral probabilities or risk-neutral trees). Here is how I would re-phrase his sentence, from "Risk-neutral investors would price a two-year zero by the following calculation:" to "If investors are not risk-averse (i.e., if investors are risk-neutral) then the market price of the two-year zero would equal its expected discounted values given by real-world (aka, true) probabilities."

He's introducing the risk premium. Although this section alters the interest rate tree rather than the probabilities, he's saying (effectively!) that if investors are neutral to risk, the real-world probabilities are equal to real-world probabilities; i.e., they can be used to price the bond. In any case, it is the introduction of the realistic assumption that investors are risk-averse. Here is where I think the confusing tends to lie. It is the introduction of the assumption that investors are risk-averse that creates a difference between risk-neutral and real-world probabilities. But the section goes on to alter the rate tree by increasing the discount rates (from 14.0% to 14.2%, eg) which has the effect of decreasing the discounted price of the bond. So, the key idea is unchanged: as risk aversion increases, the market price decreases and, if we want the discounted expected value to equal the (lower) market price, this implies either an increase in the discount rates or a change in the probabilities. I hope that helps reconcile.

 

[afterworkguinness]

Thanks for the detailed answer David. So when investors are risk neutral, risk neutral probabilities = real world probabilities and they only differ if the investor is risk averse?

I'm having difficulty understanding why, in general, real world probabilities are 50/50 but in a risk neutral world they are not.

 

[David Harper CFA FRM]

Sure thing @afterworkguinness

"When investors are risk neutral, risk neutral probabilities = real world probabilities and they only differ if the investor is risk averse?" Yes, that is correct.
"I'm having difficulty understanding why, in general, real world probabilities are 50/50 but in a risk neutral world they are not." Just borrowing from Tuckman, let's say the assumed real-world (aka, true) probabilities are 50% up and 50% down. Just to simplify, let's say this is a 50% chance that you (if up) will receive a payoff of $1.50 and a 50% chance that you will receive a $0.50 payoff. So the expected future payoff is the average of 1.50 and 0.50 or $1.00. As an alternative to this "instrument," you can simply invest at the risk free rate for a future payoff of $1.00. Consider these two choices:

• Invest today at price of $1.00*exp(-rT) for future riskless payoff of $1.00, or
• Invest today at X? for future expected payoff $1.00 due to 50% of $1.50 and 50% of $0.50

Both have expected future payoff of $1.00. If you are genuinely a risk-neutral investor, you are willing to pay $1.00*exp(-rT) for the second option; i.e., you don't care about the "volatility." In fact, if you are risk-seeking, you will pay more than $1.00*exp(-rT) for the 50% chance to make $1.50! But typically you are risk-averse such that you will pay less than $100*exp(-rT) due to the uncertainty of the latter alternative. If you are risk-averse, the price of the instrument must be less under uncertainty, and the probabilities must adjust to equate the discounted value with the traded price (i.e., the lower price you pay due to your risk aversion). I hope that helps,

 

[afterworkguinness]

Thanks David, I understand it now. Much appreciated.

 

[afterworkguinness]

David gave a great explanation above. After additional reading (and time for it to sink in), I thought I'd chip in the way I understand the difference in case it's helpful for anybody.

Assume in reality the probability of a zero coupon bond going up 50bps in 60 months is 50% and the probability of it going down 50bps in 6months is 1-50% = 50%. As Tuckman shows, we cannot calculate the market price of a risky security (one who's future value is not 100% certain) by simply calculating an expected discounted value using these real world probabilities. He shows the price calculated that way does not equal the market price and the reason is investors want compensation for the risk they are taking on (the risk being we are uncertain what the value is in the future).

Because of this, we use replicating portfolios to arrive at the market price. This is a bit more work than calculating an expected discounted value though, but we have to since the expected discount value gives us the wrong price. Specifically, expected discounted value gives up a price that is too high. As we established above, investors are not willing to pay that much for the security because of the risk.

There is an alternative to replicating portfolios though. We just need to estimate the probability of up and down jumps that gives us the correct market price. These new probabilities will consist of the real world probability + a premium for the risk, allowing us to arrive at the correct market price by calculating and expected discounted value. These are the risk neutral probabilities. The name is (in my opinion at least) confusing.

At this point, my explanation my sound contradictory to David's, but it's not. Confusion arises because they are termed 'risk neutral'.