I haven’t written in a long time. On top of that my next entry is technical. It’s about the conversion between discount yield to money market yield. Reading one of Fabozzi’s book, I was puzzled about the equation. It was presented without the proper deduction or explanation.

The equation is:

m = (360 * Yd) / (360 – t * Yd)

where m is the Money Market equivalent yield, Yd is the discount yield, t is the number of days between the settlement and maturity.

This equation is required when you want to compare the return of a tresury bill (quoted in discount yield) and an equivalent CD (certificate of deposit). The day count convension for both is Act/360.

The discount yield for a t-bill is calculated with the following:

Yd = D * 360 / (t * F)

where D is the cash discount (actual amount of money that you get after t days per face value unit) and F is the face value.

The price of the t-bill is then P = F – D, assuming usually a face value of 100$.

The T-bill cash flow after t days is D (the discount). For a CD, the cash flow after t days is m * P * t / 360 where m is the money market yield and P is the price (or money invested).

To calculate equivalent yields, the cashflows must match. So, we have:

D = m * P * t / 360

Resolving for m, results into:

m = D * 360 / (t*P)

We extract D from the definition of the discount yield:

D = Yd * F * t / 360

which gives:

m = Yd * F / P

P = F – D = F – Yd * F * t / 360 = F ( 1 – Yd * t / 360) = F * ( 360 – Yd * t) / 360

We replace the price in the equation for m:

m = Yd * 360 / (360 – Yd * t)

which is the equation in Fabozzi.

This conversion is used to compare T-bill investments with money market investments because the discount yield cannot be directly compared with the money market yield.