Continuous compounding for T.Bill

[RaamZen]

Hello,

My question is regarding below:
The Quoted Price of a 180-day Treasury bill is 9.00. What is the True Yield (a.k.a., effective return) to the investor under an actual/365 basis (ACT/365) with continuous compounding?

Solution:
Cash Price (Y) = 100 - P * n/360 = 100 - 9.00 * 180/360 = 95.50; Continuously compounded true yield on ACT/365 basis is given by: 365/180 * LN(1 + 4.5/95.50) = 9.3367%

Can you explain how 4.5/95.50, as I suppose, quoted price is 9.00 and cash price is 95.50. I dont understand this part ?

Thank you.

 

[RaamZen]

I suppose, I managed to get the reasoning (was bit too fast to post to forum)

That is due to the price delta change from face value. 100 - 95.50 / 95.50 which gives the discrete frequency. Converting back to continuos by log and number of days, it gives the right one.

 

[David Harper CFA FRM]

Hi @RaamZen This post might be helpful: https://forum.bionicturtle.com/threads/what-is-true-yield.9052/#post-37787

In your example, 9.0 is the quoted price because the "discount rate" is 9.0% per annum: $100 * 9.0% * 180/360 = $4.50 interest and therefore a cash price of $100.00 - $4.50 = $95.50; i.e., the buyer pays $95.50 (cash price) and receives back $100.00 (face value) in 180 days.

The simple (periodic) interest is therefore $4.50/$95.50 = 4.7120%.

We can unpack the logic of 365/180 * LN(1 + 4.5/95.50) = 9.3367% this way just to observe:

• The simple six-month rate of 4.7120% is annualized with 2*4.7120% = 9.4241%; this is a semi-annual (discrete) rate with ACT/360 day count.
  
• We can convert this into continuous per the usual: m*LN(1+y/m) = 2*LN(1 + 9.4241%/2) = 9.20879% but this is act/360
  
• So we translate this into act/365: 9.20879%*365/360 = 9.3367%. I hope that helps!

 

[Dr. Jayanthi Sankaran]

Hi David,

Just a small typo above (copied below):

In your example, 9.0 is the quoted price because the "discount rate" is 9.0% per annum: $100 * 9.0% * 180/360 = $4.50 interest and therefore a cash price of $100.00 - $4.50 = $95.50; i.e., the buyer pays $95.50 (cash price) and receives back $100.00 (face value) in 180 days. The simple (periodic) interest is therefore $4.50/$95.50 = 4.7120%.

We can unpack the logic of 365/180 * LN(1 + 4.5/95.50) = 9.3367% this way just to observe: the 4.7120% is annualized with 2*4.7120% = 9.3367% (should be 2*41720% = 9.424%) which is then a semi-annual (discrete) rate under an ACT/360 day count basis. We can convert this into continuous per the usual: m*LN(1+y/m) = 2*LN(1 + 9.3367%/2) = 9.20879% (should be 2*LN(1 + 9.424%/2) = 9.20879% but this is act/360 so we translate this into act/365: 9.20879%*365/360 = 9.3367%.

Thanks a lot
Jayanthi

 

[David Harper CFA FRM]

Yes, thank you @Jayanthi Sankaran, sharp eye! corrected above ...