motivated by a question of Wonko and "the math" by Hymas ... on Webring
Suppose you can buy a 30day 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 30day 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 Nday 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?
Right.
Example: For our 30day bill, above, the purchase price is 1/(1+ 30(0.0273)/365) = $0.99776.
That means the 30day gain factor is 1/0.99776 and ...
>Huh? 30day 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 60day 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 90day gain equal to (1/0.99776)(1+ 60Y/365).
>Uh ... that's (30day gain)*(60day gain), but what's your point?
Okay, here's the question. Would it be better to buy a 90day bill rather than 30day then 60day notes?
>Definitely, it the 60day pays 50%, 30 days from now.
Well, yes ... so it'd depend upon that Yvalue. If we expect it to be large, we'd buy a 30day then a 60day.
Suppose you could buy a 90day 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 90day gain factor is then 1/0.99290.
Remember the gain factor when you bought a 30day then a 60day with a yield Y?
That total 90day gain factor was (1/0.99776)(1+ 60Y/365).
In order for this to be a better strategy than a 90day bill, we'd want the 60day yield (after the first 30 days)
such that the total 30then60day gain is greater than a single 90day bill. That is, we'd want the 60day 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
>90day wins!
Yes, but if the 60day yield satisfied [2], then we'd switch to 60day 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 60day 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 Y_{1} is the yield for an N_{1}day bill,
and Y_{2} is the yield for an N_{2}day bill,
then Y =

>I assume that N_{2} is bigger than N_{1}.
Uh ... yes, else all bets are off
>It's still too complicated.
That's where Hymas comes in
If'n you put N_{1} = N_{2} , 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 ...
