I think you have pointed out the main difference

For pricing, we use the information up to now while for risk management, what we what to do is to predict the future.

Thank you for replying.

For valuation, we also try to come up with an assumption about the underlying distribution, therefore, the payoff distribution. In this sense, we also "predict" the future by making assumptions about the future.
For risk purpose, we do similar things, making assumption about the risk factors, e.g., interest rates, computing the loss distribution, therefore, risk numbers.


Let's put my question differently: Is CAPM a pricing model or a risk model?
To me, the CAPM takes into account the risk factor and come up an expected return. At the same time, it is also a risk model. We just need to go one more step, compute the standard deviation of the return, then we can have the risk numbers,

Fair enough. I think the prediction we mean is slightly different. Actually, the risk model is trying to predict the future distribution of the price, while the pricing model is something like if we know the distribution of the underlyins's price, what is the derivative's price. Essentially, valuation do not need to predict something, the prediction you mean is just modeling the pattern of the price's behavior. Also, some derivatives such as forward contracts do not need to make the assumption on the underlings' distribution. Actually, for pricing issue, you know the underlying's drift will all be killed by the risk neutral measure, so that only the volatility matters. For risk model, such as the factor model, it try to predict the future's distribution namely the future situations depend on different factors. Its not a big question, so that we don't need to pay attention much. What you said is also reasonable.

In my opinion, CAPM actually is not a pricing model it is more like a risk model.
If you look carefully at CAPM, it does nothing on pricing but on allocation. So more precisely, it is a capital allocation model. CAPM is derived from Markowitz's  mean variance model. What CAPM says is that given the distribution (mean and variance) of different assets, if we solve the optimal problem (given fixed return, minimize the variance), the (excess) expected return between two assets (or portfolios) should have a linear relationship. In this sense, it did nothing but just find the optimal weight to solve the capital allocation problem and find that if we use these weights, the return has such a relationship. So what I mean is that CAPM, is very different from the "pricing" (no-arbitrage) model in modern finance. It is more like an equilibrium model in economics. For the "true" pricing model such as CIR's general equilibrium model, the price of the asset (or expected return) and the risk free rate together with the optimal weight, should both be given endogenously by the model not pre-specify return's mean and variance.

So this is my understanding.

Best,

Thank you very much for your input, even though I do not fully understand what you said for now.
Let me chew on it for a bit.

No problem, you are welcome