Its not possible to directly compare a yield rate and discount rate due to them being fundamentally different calculations. How should we derive a formula that will enable comparison?
Let’s start with a future value calculation of a yield instrument:
$ \text{FV} = \text{PV} \times \left( 1 + \text{y} \times \frac{\text{day count}}{\text{Annual basis}} \right)$
Restating this formula to isolate the Present Value:
$ \text{PV} = \frac{\text{FV}}{\left( 1 + \text{y} \times \frac{\text{day count}}{\text{Annual basis}} \right)}$
By contrast, let’s use the definitional formula of a discount instrument:
$ \text{PV} = \text{FV} \times \left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)$
Since we have both formulae equating to the PV, we can set these equations equal to one another:
$ \frac{\text{FV}}{\left( 1 + \text{y} \times \frac{\text{day count}}{\text{Annual basis}} \right)} = \text{FV} \times \left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)$
Hold onto your hat, we’ll now solve for y in a step-by-step fashion:
$ \frac{\text{FV}}{\text{FV}} = \left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right) \times \left( 1 + \text{y} \times \frac{\text{day count}}{\text{Annual basis}} \right) $
$ \left( 1 + \text{y} \times \frac{\text{day count}}{\text{Annual basis}} \right) = \frac{1}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
$ \text{y} \times \frac{\text{day count}}{\text{Annual basis}} = \frac{1}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} – 1 $
$ \text{y} \times \frac{\text{day count}}{\text{Annual basis}} = \frac{1 – \left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
$ \text{y} \times \frac{\text{day count}}{\text{Annual basis}} = \frac{1 – 1 + \text{d} \times \frac{\text{day count}}{\text{Annual basis}}}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
$ \text{y} \times \frac{\text{day count}}{\text{Annual basis}} = \frac{\text{d} \times \frac{\text{day count}}{\text{Annual basis}}}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
$ \text{y} = \frac{\text{d} \times \frac{\text{day count}}{\text{Annual basis}}}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} \times \frac{\text{Annual basis}}{\text{day count}} $
$ \text{y} = \frac{\text{d} \times \frac{\text{day count}}{\text{Annual basis}}\times \frac{\text{Annual basis}}{\text{day count}}}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
$ \text{y} = \frac{\text{d}}{\left( 1 – \text{d} \times \frac{\text{day count}}{\text{Annual basis}}\right)} $
And that’s the formula for converting a discount rate to a yield.