Fractional-Year Discounting and Compounding Assumptions
Summary
The document examines how to interpret annual effective-rate discounting when the cash-flow horizon is a fraction of a year. It starts from the present-value relation that divides a future value by one plus the annual rate raised to the time in years. The author is unsure whether a fractional exponent, such as one and a half years, implies meaningful half-year compounding, and compares it with simple prorated interest for a half-year.
It also questions whether annual compounding is the right convention when a security pays interest more frequently, using a semiannual coupon instrument as an example. The note does not resolve these questions or specify a market convention, day-count basis, or rate definition. The key modeling distinction to clarify is whether the quoted rate is an effective annual rate, a nominal rate with a stated compounding frequency, or a continuously compounded rate; those conventions imply different discount factors. Coupon payment frequency alone does not determine the appropriate curve discounting convention.
Key ideas
- Fractional-year discounting depends on how the annual rate is defined.
- An annual effective rate raised to a fractional time represents a mathematical compounding convention.
- Nominal rates with specified frequencies lead to different discount factors from effective annual rates.
- A bond’s coupon payment schedule does not by itself determine the discounting convention.
- The document poses the convention questions but does not provide a resolution.
Tags
Full text
# Understanding risk-free discounting with fractional powers
# Understanding risk-free discounting with fractional powers
#### Introduction
Discounting a value $X_{T}$ is premised through the presence of a risk-free security with a rate of $r$ per annum that yields the same value at time $T$, in years. That is
\begin{align*} P(1+r)^{T} &= X_{T} \\ \\ \implies P &= \frac{X_{T}}{(1+r)^{T}} \end{align*}
$P$ is the present value of $X_{T}$ and is the convention for discounting cash flows.
#### Understood Examples
Let $T=1$: Then
\begin{align} P(1+r)^1 &= X_{1} \end{align}
represents investing $P$ in a risk-free security compounding interest over a single year, yielding $X_{1}$
Let $T=2$: Then
\begin{align} P(1+r)^2 &= X_{2} \end{align}
represents investing $P$ in a risk-free security compounding interest every year for 2 years, yielding $X_{2}$
#### Misunderstood Examples
Now let $T=1.5$, or a year and 1-half: Then
\begin{align} P(1+r)^{1.5} &= X_{1.5} \\ \\ P(1+r)(1+r)^{0.5} &= X_{1.5} \end{align}
represents investing $P$ into a risk-free security for a year, compounding the interest - though then compounding $P(1+r)$ over a half-year. This doesn't make sense to me as one cannot expand into $(1+r)^{0.5}$. What is the assumption here? In practice, if I wanted to exit out of the security - then wouldn't the final value be
\begin{align} P(1+r)\left(1+\frac{180r}{360}\right), \end{align} where half a year worth of interest is accumulated?
Another point of confusion is the compounding period? Why is it assumed that the risk-free security compounds interest yearly? Why doesn't it compound half-yearly or quarterly? Surely we'd follow the guidelines of say U.S. treasuries, which pays every 6 months. Therefore, I'd re-invest 6-months worth of interest into U.S. treasuries and continue the the process. Though then the discounting equation should be
\begin{align} P\left(1+\frac{r}{2}\right)^{2T} &= X_{T} \\ \\ \implies P &= \frac{X_{T}}{\left(1+\frac{r}{2}\right)^{2T}} \end{align}
#### Conclusion
I'm looking for an understanding into my misunderstood examples, and the precise assumptions behind risk-free discounting.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.