motivated by a question of Wonko and "the math" by Hymas ... on Webring

Suppose you can buy a 30-day bill (it matures in 30 days and you get $1.00) with an annual yield of y.
Note: If the yield is 2.73%, we put y = 0.0273.

>It pays just $1.00?
If it pays $100, then just multiply the price by 100.

Okay. We write the daily yield as y/365 and the 30-day yield as 30y/365.
That means that if you pay P, then after 30 days it'd be worth P(1+ 30y/365) = $1.00, so the purchase price would be P = 1/(1+ 30y/365).

In general, an N-day bill with annual yield y can be bought for:

[1]     P = 1/(1+ Ny/365).

>And it'd be 100/(1+ Ny/365) if you get $100, after N days, right?

Example: For our 30-day bill, above, the purchase price is 1/(1+ 30(0.0273)/365) = $0.99776.
That means the 30-day gain factor is 1/0.99776 and ...

>Huh? 30-day gain factor?
Yes. That's the factor that turns your purchase price of $0.99776 into $1.00, in 30 days.

Okay, suppose, after 30 days, the yield on a 60-day bill is Y.
Then you'd pay ...

>You'd pay 1/(1+ NY/365), right?
Yes, just like it says in [1], but with N = 60.

Now your gain factor for the next 60 days would be: (1+ 60Y/365).
That'd make your total 90-day gain equal to (1/0.99776)(1+ 60Y/365).

>Uh ... that's (30-day gain)*(60-day gain), but what's your point?
Okay, here's the question. Would it be better to buy a 90-day bill rather than 30-day then 60-day notes?

>Definitely, it the 60-day pays 50%, 30 days from now.
Well, yes ... so it'd depend upon that Y-value. If we expect it to be large, we'd buy a 30-day then a 60-day.

Suppose you could buy a 90-day bill with an annual yield of 2.90%.
You'd pay what, for this bill? According to [1], above, you'd pay (1+90(0.0290/365)) = $0.99290.
Your 90-day gain factor is then 1/0.99290.

Remember the gain factor when you bought a 30-day then a 60-day with a yield Y?
That total 90-day gain factor was (1/0.99776)(1+ 60Y/365).
In order for this to be a better strategy than a 90-day bill, we'd want the 60-day yield (after the first 30 days) such that the total 30-then-60-day gain is greater than a single 90-day bill. That is, we'd want the 60-day yield Y to be such that

[2]     (1/0.99776)(1+ 60Y/365) > 1/0.99290.

>Wait! How do you know that 2.73% isn't already big enough?
Good question. Let's check:

    (1/0.99776)(1+ 60(0.0273)/365) = 1.006743
    1/0.99290 = 1.007151

>90-day wins!
Yes, but if the 60-day yield satisfied [2], then we'd switch to 60-day bills, after 30 days..

To satisfy [2], we'd need

[3]     Y > (0.99776/0.99290 - 1)*365/60 = 0.0298 and that means our 60-day yield must exceed 2.98% (30 days from now).

>Fat chance! So, whose gonna do that calculation?
Well, we can stick it all into a magic formula:
If Y1 is the yield for an N1-day bill,
and Y2 is the yield for an N2-day bill,
then Y =

>I assume that N2 is bigger than N1.
Uh ... yes, else all bets are off

>It's still too complicated.
That's where Hymas comes in
the Hymas Calculator

#1 Days: N1 =     #1 Yield: Y1 = %

#2 Days: N2 =     #2 Yield: Y2 = %

If you expect the yield of a (N2 - N1 ) day note to exceed %
when purchased N1 days in the future, then buy #1

If'n you put N1 = N2 , that %#@!$* division by (N2 - N1) (in the magic formula) gives infinity.
The calculator will swear, with NaN, which means No! and Numbskull.
>I thought it meant Not a Number.
You could be right ...