Discounting with Time-Varying Interest Rates and Dividends
Summary
This explanation describes how interest rates and dividend yields affect values over time. For a deterministic rate, continuously compounded growth is represented by integrating the rate across the time interval and exponentiating the result; discounting uses the corresponding negative exponent. If rates are constant, this reduces to the familiar rate-times-time expression. Discrete rate schedules can instead be accumulated through a product of period-by-period factors.
When rates are stochastic, the document frames bond valuation as a conditional expectation under a risk-neutral measure and notes that models such as Cox–Ingersoll–Ross, Ho–Lee, Hull–White, and Merton can represent rate dynamics. Proportional dividends are described as reducing future value, motivating valuation in terms of a forward price. The material is a conceptual answer rather than a worked market example: it gives no calibration, numerical comparison, or guidance for choosing a rate model, and its discrete compounding expression is only sketched.
Key ideas
- Integrating a deterministic rate over time gives the continuously compounded growth factor.
- Discounting reverses the growth exponent to obtain present value.
- A varying discrete rate schedule can be accumulated with period-specific factors.
- Stochastic rates require conditional valuation under a risk-neutral measure.
- Proportional dividends reduce forward value and need to be incorporated in pricing.
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# Integrating Interest and Dividend Functions
# Integrating Interest and Dividend Functions
How are interest rate and dividend functions integrated over time in practice?
For example, what does it mean in practice to discount a current price by $e^{\int_{t_m}^{T}r_s ds }$ where $r_s$ is the interest rate function or $e^{\int_{t_m}^{T}\delta_s ds }$ where $\delta_s$ is the dividend fuction.
## Answer by David Addison (score 1)
https://quant.stackexchange.com/a/33912
For a fixed interest rate or dividend payment/yield, integrating the function becomes:
$\large{P_T = } \Large{e^{\int_t^T r_s ds} = \Large{e^{r_s (T - t)}}}$
$P_T$ is usually understood to be the future value of a zero-coupon bond with a face value of \$1. The present value, $P_t$, is simply then:
$\large{P_t = } \Large{e^{\int_t^T -r_s ds} = \Large{e^{r_s (t - T)}}}$
If $r_s$ is a function, then that must be accounted for in the integration. For complex functions, it is common to estimate the quantity through discretization techniques. When discounting a discrete function, a product function may be used:
$\large{P_T=\prod_{t}^T(r_s + 1)}$;
for fixed $\large{r_s}$, $\large{P_T \to (r + 1)^{(T - t) + 1}}$
If $r_s$ is not a deterministic function (i.e., it is stochastic), we must now take the expectation under some risk-neutral measure, $\mathbb{Q}$, where the price of a zero-coupon bond at time, $T$, equals:
$\large{ \mathbb{E}[ P_T] = \mathbb{E^Q} [e^{\int _{t}^{T}r_{s}\,ds}|{\mathcal{F}}_{t}]}$
where $\mathcal{F_t}$ is a natural filtration process.
There exist many methods and models by which to express the expectation of a stochastic interest rate. E.g.: Cox-Ingersol-Ross; Ho-Lee; Hull-White; Merton; etc... The "best" model just depends on what you're trying to do.
When dividends are also present, it is common and mathematically convenient to assume they are a fixed proportion of $P_t$. The effect of proportional dividend is often thought of a exerting a "decay" on future value; it is thus common for pricing models to take the expectation of a "forward" value, $F_t$.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.