T-bill price and yield

◈ 7 cards

Discount pricing on a 365-day year — price a Government of Canada T-bill from its quoted yield, and recover the yield from a price.

The convention

A T-bill is quoted by yield, not by price, and the Canadian convention turns one into the other with simple interest on an actual/365 basis: the yield is a rate per year, scaled by the actual number of days to maturity over 365. There is no compounding inside the term and there is no 360-day year — that is the US Treasury bill convention, and a formula copied from a US text gives a different number on this paper.

$Price per 100 = \frac{100}{1 + y \times \frac{d}{365}}$

where $y$ is the quoted yield as a decimal and $d$ the days to maturity. Inverting it recovers the yield from a price:

$y = (\frac{100}{P} - 1) \times \frac{365}{d}$

Worked example — Maritime Ferries buys a 91-day bill

The treasurer is quoted a 91-day Government of Canada T-bill at 4.20 %. Price per 100 of face:

$\frac{100}{1 + 0.042 \times \frac{91}{365}} = \frac{100}{1.010471} = 98.9637$

In Excel, =100/(1+0.042*91/365). For $250,000 face the cost is $98.9637 \div 100 \times 250{,}000 = \mathbf{\$247{,}409.32}$ — =price/100*face. At maturity the company receives $250,000; the $2,590.68 gap is its interest, earned as the discount.

Inverting. A 182-day bill is offered at a price of 98.75. Its yield:

$(\frac{100}{98.75} - 1) \times \frac{365}{182} = 0.012658 \times 2.005495 = 0.025386 \approx 2.54 %$

=(100/98.75-1)*365/182. The first factor is the return over the bill's life; the second annualises it on the 365-day year.

Price and yield move opposite ways

If the market yield on the 91-day bill falls to 3.80 %, the price rises to =100/(1+0.038*91/365) = 99.0615, and the same $250, 000 f a cecos t s$ 247,653.74. A lower yield is a smaller discount; a higher price is a smaller return. That inverse relation is the whole of bond mathematics in miniature, and Module 10 returns to it with coupons attached.

The 360-day trap

Price the same 4.20 % bill on a 360-day year and you get =100/(1+0.042*91/360) = 98.9495 — 1.4 cents per 100 lower, or about $35 on the $250,000 purchase. The exam's answer is marked on the value alone, so the divisor is not a detail: it is the mark.

Two more, cold

A 91-day bill priced at 99.20: =(100/99.20-1)*365/91 = 0.032347, about 3.23 %. A 182-day bill quoted at 3.00 %: =100/(1+0.03*182/365) = 98.5262 per 100.