# 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.

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.