I struggle with the following exercise, where the correct answer is supposed to be "no":

A riskless security with cash flow $C_1, C_2, \dots, C_n$ has a market price of $\sum_{i=1}^n C_i\,d(i)$. The discount factor $d(i)$ denotes the present value of $\$1$ at time $i$ from now. Is the formula still valid if the cash flow depends on interest rates?

I don't even know another way to valuate a security with known future cash flows other than with the usual formula $\mathrm{PV} =\sum_{i=1} C_i\,d(i)$.


1 Answer 1


If the cash flows depend on the (random) interest rates, then the $C_i$ are random variables and so would be the sum $\sum\limits_{i=1}^n C_id(i)$. However, initial market prices need to be constants, they cannot be random (because you need to know how much this claim is worth right now). The $d(i)$ are real numbers though since they are discount factors (typically prices of default-free zero-coupon bonds, which we can observe in the marketplace). So, what you need to do is to specify a model for the interest rate (say a short rate model) and then, you can compute the the (conditional) expectation of the cash flows $C_i$. It basically boils down to pricing a bond with variable coupons.

Recall that in general the no-arbitrage price of any claim paying $\xi$ is given by $\mathbb{E}^\mathbb{Q}\left[\frac{B_t}{B_T}\xi\bigg|\mathcal{F}_t\right]$.


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