Converting Discount Curves Across Rate Conventions
Summary
The document explains that a discount curve does not need to use one universal compounding or day-count convention. For no-arbitrage pricing, the underlying discount factor for a given cash flow date is the key quantity; different conventions can express that same factor using different quoted rates and year fractions. This is relevant when valuing cash flows, including fair-value measurement under IFRS 13.
It illustrates the method by equating discount-factor formulas under two day-count conventions, then solving for the corresponding rates. It also compares continuous compounding with periodic compounding and derives conversions by setting the two discount-factor expressions equal. The general procedure is to write each convention’s discount factor, equate them, and solve for the rate in the target convention. The answer is conceptual and assumes consistent dates and discount factors; it does not discuss curve-building details, market-specific conventions, or practical implementation issues.
Key ideas
- A discount factor is the invariant quantity when expressing the same cash flow discounting under different conventions.
- Day-count changes require adjusting the quoted rate so the discount factor remains equal.
- Compounding changes can be handled by equating the discount-factor formulas and solving for the new rate.
- The conversion method depends on correctly applying each convention’s year fraction and compounding rules.
Tags
Full text
# Characteristics of a Discount Curve
# Characteristics of a Discount Curve
Does the discount curve used for discounting cash flows have to be a zero coupon, annual compounding, actual by actual day basis curve? In practice, does a curve used for discounting necessarily have certain attributes, if so what are those?
Also, how can we convert a curve from one compounding and day count to another compounding and day count?
Am working on IFRS 13 where I need to compute discounted cash flows to measure fair value of an asset/liability. I have tried to find answers to these questions but have not found anything conclusive. There is mention on how to convert a discount rate from one day count to another but not a discount curve and also not while taking into consideration the compounding change as well.
## Answer by Chris Taylor (score 2)
https://quant.stackexchange.com/a/35808
By no arbitrage, market participants need to agree on the values of the discount factor, even if they are using different conventions (day count, compounding period) to convert the discount factor into a rate.
For example, consider two discount factors computed using continuous compounding, where one is computed using the 30/360 day count (year fraction $t_{30/360}$) and the other is using ACT/365 (year fraction $t_{ACT/365}$). Then
$$ d_{30/360} = e^{-r_{30/360} t_{30/360}} \\ d_{ACT/365} = e^{-r_{ACT/365} t_{ACT/365}} $$
and by no arbitrage, you must have $d_{30/360} = d_{ACT/365}$ and hence
$$ e^{-r_{30/360} t_{30/360}} = e^{-r_{ACT/365} t_{ACT/365}} $$
which implies that
$$ r_{30/360} = \frac{t_{ACT/365}}{t_{30/360}} r_{ACT/365} $$
Similarly, consider a continuous rate $r_c$ and a rate compounded $k$ times per year, $r_k$ where the year fraction is $t$ and the number of periods is $n=kt$. Then
$$ d_c = e^{-r_ct}\\ d_k = \left( 1 + \frac{r_k}{k} \right)^{-kt} $$
and by no arbitrage, $d_c=d_k$, so
$$ e^{-r_ct} = \left( 1 + \frac{r_k}{k} \right)^{-kt} $$
which implies that
$$ r_c = k\log\left( 1 + \frac{r_k}{k} \right) $$
and
$$ r_k = k \left( e^{r_c/k} - 1 \right) $$
In general, to convert from one rate convention to another, write the expression for the discount factor in terms of the rate in each convention, equate the two expressions, and solve for one rate in terms of the other.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.