Interpreting the Stochastic Interest-Rate Discount Factor

Summary

The document explains why the process defined as the exponential of the negative accumulated short rate is called a discount factor. For a cash flow paid at time t, multiplying by this factor converts the future amount into its present value at time zero. When the interest rate varies stochastically, writing the discount factor as a process packages the accumulated-rate calculation into a reusable quantity.

The explanation is conceptual and gives the present-value relationship directly; it offers no numerical example or discussion of alternative valuation conventions. Its interpretation assumes that the stated money-market rate is the rate used to discount the cash flow over the interval. The document does not address risk-neutral pricing, uncertainty about cash-flow amounts, or how the discount factor is estimated in practice.

Key ideas

• The discount factor is the exponential of the negative integral of the short-rate process.
• Multiplying a cash flow at time t by the discount factor gives its time-zero present value under the stated setup.
• Writing the factor as a process avoids repeatedly expressing the accumulated interest-rate integral.
• The interpretation depends on using the specified money-market rate for discounting.

Tags

# What is the meaning of the discounted process defined from the interest rate process?


# What is the meaning of the discounted process defined from the interest rate process?












Assume a money market has interest rate process $R(t)$. In Shreve's Stochastic Calculus for Finance II, formula (5.2.17) on page 215 defines the discounted process as $$ D(t) = e^{-\int_0^t R(s) ds}. $$

Why is $D(t)$ called "discounted" process?

Does it mean that any value at time t times $D(t)$ will give its present value at time $0$?

## Answer by SRKX (score 5, accepted)

https://quant.stackexchange.com/a/7624

Indeed, $D(t)$ is the discount factor used to compute the present value of a cash flow at time $t$:

$$PV = D(t) \cdot CF_t$$

It is more convenient to write it that way when you assume stochastic interest rate because you don't have to write the integral all the time.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.