Pricing Securities with Interest-Rate-Dependent Cash Flows
Summary
The document explains why the usual present-value sum of known cash flows and discount factors cannot be applied directly when the cash flows depend on future interest rates. In that case, the cash flows are random variables, so their discounted sum is random as well, while the security's current market price must be a known value.
The answer proposes specifying an interest-rate model and valuing the claim using the conditional expectation of its future payments under an appropriate pricing measure. This frames the problem as pricing a security with variable coupons, using the broader no-arbitrage principle of discounting and taking a conditional expectation. The discussion is conceptual rather than a worked valuation: it does not specify a particular rate model, calibration procedure, or numerical example, and the result depends on the modeling assumptions.
Key ideas
- Interest-dependent future payments are random, so their discounted sum is not itself a current market price.
- A pricing model for interest rates is needed to value cash flows that vary with rates.
- The valuation uses a conditional expectation under a pricing measure.
- A bond with variable coupons is a useful analogy, but the document does not provide a model or worked example.
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Full text
# PV of security with interest-dependent cash flows
# PV of security with interest-dependent cash flows
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 $\mathrm{PV} =\sum_{i=1} C_i\,d(i)$.
## Answer by Kevin (score 3)
https://quant.stackexchange.com/a/48638
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 $\mathbb{E}^\mathbb{Q}\left[\frac{B_t}{B_T}\xi\bigg|\mathcal{F}_t\right]$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.