Market -Consistent and risk neutral valuation

This is revision of Subject CT8 which shows that we price assets (particularly derivatives) by constructing a replicating portfolio that has a value equal to the expected value of the payoffs using a risk-free discount rate and risk-neutral probabilities.

Best wishes

Mark

 

The proof is a long one, eg two chapters of CT8 are devoted to the five step method which demonstrates that we discount at the risk-free rate. So I'm afraid I can't summarise it in the space we have here. However, for ST2 you only need to know that it is true. If you would like to refresh your knowledge then I suggest you find a book on derivatives in the library.

Best wishes

Mark

 

It may be easiest to explain with an example.

Imagine we have a ten year pure endowment with sum assured of 100. The policyholder wants to change the sum assured to 150 after three years.

By equating policy values on the premium basis the insurer will take the accrued profit from the old policy, ie three years of profit. The insurer will expect to get the profit from a new seven year contract. This feels right as the insurer gets 10 years of profit in total.

That's fine as long as the premium basis hasn't changed. However, now imagine that interest rates have gone down since the contract was sold and so now the contract is more expensive. Using the original premium basis for the new seven year contract will now be loss making. Any profits made on the old contract will be outweighed by the losses on the new contract as the new contract has a larger sum assured and longer term (7 instead of 3).

This problem can be solved by equating policy values using the new updated premium basis, so that the insurer makes the correct profit on the new contract.

Best wishes

Mark