# 3.4 Treasury Bills and Commercial Papers

## LEARNING OBJECTIVES

- Use simple interest in solving problems involving treasury bills and commercial papers.

## Treasury Bills

**Treasury bills,** also known as T-bills, are short-term financial instruments that both federal and provincial governments issue with maturities no longer than one year. Approximately 27% of the national debt is borrowed through T-bills.

Here are some of the basics about T-bills:

- The Government of Canada regularly places T-bills up for auction every second Tuesday. Provincial governments issue them at irregular intervals.
- The most common terms for federal and provincial T-bills are 30 days, 60 days, 90 days, 182 days, and 364 days.
- T-bills do not earn interest. Instead, they are sold at a
**discount**and redeemed at full value. This follows the principle of “buy low, sell high.” The percentage by which the value of the T-bill grows from sale to redemption is called the**yield**or rate of return. From a mathematical perspective, the yield is calculated in the exact same way as an interest rate is calculated, and therefore the yield is mathematically substituted as the discount rate in all simple interest formulas. - The
**face value****of a T-bill**(also called*par value*) is the maturity value, payable at the end of the term. It includes both the principal and yield together. - T-bills do not have to be retained by the initial investor throughout their entire term. At any point during a T-bill’s term, an investor is able to sell it to another investor through secondary financial markets. Prevailing yields on T-bills at the time of sale are used to calculate the price.

## Commercial Papers

A **commercial paper** (or paper for short) is the same as a T-bill except that it is issued by a large corporation instead of a government. It is an alternative to short-term bank borrowing for large corporations. Most of these large companies have solid credit ratings, meaning that investors bear very little risk that the face value will not be repaid upon maturity.

Commercial papers carry the same properties as T-bills. The only fundamental differences lie in the term and the yield:

- The terms are usually less than 270 days but can range from 30 days to 364 days. The most typical terms are 30 days, 60 days, and 90 days.
- The yield on commercial papers tends to be slightly higher than on T-bills because corporations do carry a higher risk of default than governments.

## How Treasury Bills and Commercial Papers Work

Mathematically, T-bills and commercial papers operate in the exact same way. The future value for both of these investment instruments is always known because it is the face value. Commonly, the two calculated variables are either the present value (the price) or the yield (the interest rate).

#### Calculating the Price

- The face value, yield, and time before maturity must be known. Draw a timeline if necessary, as illustrated below, and identify the following:
- The face value ([latex]S[/latex]).
- The yield ([latex]r[/latex]) on the date of the sale, which is always expressed annually. Remember that mathematically the yield is the same as the discount rate.
- The number of days remaining between the date of sale and the maturity date. Count the first day, but not the last day.

- Solve for the present value using [latex]\displaystyle{P=\frac{S}{1+r\times t}}[/latex], which is the price of the T-bill or commercial paper. This price is always less than the face value.

#### Calculating the Yield Rate

- The face value, price, and time before maturity must be known. Draw a timeline if necessary, as illustrated below, and identify:
- The face value ([latex]S[/latex]).
- The price on the date of the sale ([latex]P[/latex]).
- The number of days remaining between the date of sale and the maturity date. Count the first day, but not the last day.

- Apply formula [latex]I = S − P[/latex], to calculate the interest earned during the investment.
- Apply simple interest formula, [latex]I=P \times r \times t[/latex], rearranging for [latex]r[/latex] to solve for the interest rate (or yield or rate of return).

## EXAMPLE

A Government of Canada 182-day T-bill has a face value of $100,000. Market yields on these T-bills are 1.5%. Calculate the price of the T-bill on its issue date.

**Solution:**

**Step 1:** The given information is

[latex]\begin{eqnarray*} S & = & \$100,000 \\ r & = & 1.5\% \mbox{ (per year)} \\ & = & 0.015 \\ t & = & \frac{182}{365}\end{eqnarray*}[/latex]

**Step 2:** Solve for the present value, [latex]P[/latex].

[latex]\begin{eqnarray*} P & = & \frac{S}{1+ r \times t} \\ & = & \frac{100,000}{1+ 0.015 \times \frac{182}{365} }\\ & = & \$99,257.61 \end{eqnarray*}[/latex]

An investor will pay $99,257.61 for the T-bill. If the investor holds onto the T-bill until maturity, the investor realizes a yield of 1.5% and receives $100,000.

## EXAMPLE

Pfizer Inc. issued a 90-day, $250,000 commercial paper on April 18 when the market rate of return was 3.1%. The paper was sold 49 days later when the market rate of return was 3.63%. Calculate the price of the commercial paper on its date of sale.

**Solution:**

Note that the historical rate of return of 3.1% is irrelevant to the price of the commercial paper today. The number of days elapsed since the date of issue is also unimportant. The number of days before maturity is the key piece of information.

The original term of the commercial paper was 90 days and 49 days have passed. So, there are 41 (90-49) days remaining until maturity.

**Step 1:** The given information is

[latex]\begin{eqnarray*} S & = & \$250,000 \\ r & = & 3.63\% \mbox{ (per year)} \\ & = & 0.0363 \\ t & = & \frac{41}{365}\end{eqnarray*}[/latex]

**Step 2:** Solve for the present value, [latex]P[/latex].

[latex]\begin{eqnarray*} P & = & \frac{S}{1+ r \times t} \\ & = & \frac{250,000}{1+ 0.0363 \times \frac{41}{365}} \\ & = & \$248,984.76 \end{eqnarray*}[/latex]

An investor pays $248,984.76 for the commercial paper on the date of sale. If the investor holds onto the commercial paper for 41 more days (until maturity), the investor realizes a yield of 3.63% and receives $250,000.

## TRY IT

A 60-day, $90,000 face value commercial paper was issued when yields were 2.09%. What was its purchase price?

**Click to see Solution**

[latex]\begin{eqnarray*} P & = & \frac{S}{1+ r \times t} \\ & = & \frac{90,000}{1+ 0.0209 \times \frac{60}{365}} \\ & = & \$89,691.85 \end{eqnarray*}[/latex]

## EXAMPLE

Marlie paid $489,027.04 on the date of issue for a $500,000 face value T-bill with a 364-day term. Marlie received $496,302.21 when he sold it to Josephine 217 days after the date of issue. Josephine held the T-bill until maturity. Determine the following:

- Marlie’s actual rate of return.
- Josephine’s actual rate of return.
- If Marlie held onto the T-bill for the entire 364 days instead of selling it to Josephine, what would his rate of return have been?

**Solution:**

Calculate three yields or rates of return ([latex]r[/latex]) involving Marlie and the sale to Josephine, Josephine herself, and Marlie without the sale to Josephine.

**Marlie’s rate of return for the sale.**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$489,027.04 \\ S & = & \$496,302.21 \\ t & = & \frac{217}{365} \end{eqnarray*}[/latex]

**Step 2**: Calculate the interest, [latex]I[/latex].

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 496,302.21-489,027.04 \\ & = & \$7,275.71 \end{eqnarray*}[/latex]

**Step 3:** Solve for the interest rate, [latex]r[/latex].

[latex]\begin{eqnarray*} r & = & \frac{I}{P \times r \times t} \\ & = & \frac{7,275.17}{489,027.04 \times \frac{217}{365}} \\ & = & 0.025 \\ & \rightarrow & 2.5\% \end{eqnarray*}[/latex]

When Marlie sold the T-bill after holding it for 217 days, he realized a 2.50% rate of return.

**Josephine’s rate of return.**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$496,302.21 \\ S & = & \$500,000 \\ t & = & \frac{147}{365} \end{eqnarray*}[/latex]

**Step 2**: Calculate the interest, [latex]I[/latex].

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 500,000-496,302.21 \\ & = & \$3,697.79 \end{eqnarray*}[/latex]

**Step 3:** Solve for the interest rate, [latex]r[/latex].

[latex]\begin{eqnarray*} r & = & \frac{I}{P \times r \times t} \\ & = & \frac{3,697.79}{496,302.21 \times \frac{147}{365}} \\ & = & 0.0185 \\ & \rightarrow & 1.85\% \end{eqnarray*}[/latex]

When Josephine held the T-bill for 148 days, she realized a 1.85% rate of return.

**Marlie’s rate of return without the sale.**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$489,027.04 \\ S & = & \$500,000 \\ t & = & \frac{364}{365} \end{eqnarray*}[/latex]

**Step 2**: Calculate the interest, [latex]I[/latex].

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 500,000-489,027.04 \\ & = & \$10,972.96 \end{eqnarray*}[/latex]

**Step 3:** Solve for the interest rate, [latex]r[/latex].

[latex]\begin{eqnarray*} r & = & \frac{I}{P \times r \times t} \\ & = & \frac{10,972.96}{500,000 \times \frac{364}{365}} \\ & = & 0.0225 \\ & \rightarrow & 2.25\% \end{eqnarray*}[/latex]

If Marlie had not sold the note to Josephine and instead held it for the entire 364 days, he would have realized a 2.25% rate of return.

## TRY IT

A 90-day Province of Ontario T-bill with a $35,000 face value matures on December 11, 2022. Farrah works for Hearthplace Industries and notices that the company temporarily has some extra cash available. If she invests the money on October 28, 2022, when the yield is 4.94%, and sells the T-bill on November 25, 2022, when the yield is 4.83%, calculate how much money Farrah earned and the rate of return she realized.

**Click to see Solution**

Farrah’s purchase price on October 28.

[latex]\begin{eqnarray*} P & = & \frac{S}{1+r \times t} \\ & = & \frac{35,000}{1+0.0494 \times \frac{44}{365}}\\ & = & \$34,792.81\end{eqnarray*}[/latex]

Farrah’s selling price on November 25.

[latex]\begin{eqnarray*} P & = & \frac{S}{1+r \times t} \\ & = & \frac{35,000}{1+0.0483 \times \frac{16}{365}}\\ & = & \$34,926.05\end{eqnarray*}[/latex]

Farrah’s interest.

[latex]\begin{eqnarray*} I & = & 34,926.05-34,792.81 \\ & = & \$133.24\end{eqnarray*}[/latex]

Farrah’s rate of return.

[latex]\begin{eqnarray*} r & = & \frac{I}{P \times t} \\ & = & \frac{133.24}{34,792.81 \times \frac{28}{365}} \\ & = & 0.0499 \\ & \rightarrow & 4.99\% \end{eqnarray*}[/latex]

## TRY IT

Philippe purchased a $100,000 Citicorp Financial 220-day commercial paper for $96,453.93. He sold it 110 days later to Damien for $98,414.58, who then held onto the commercial paper until its maturity date.

- What is Philippe’s actual rate of return?
- What is Damien’s actual rate of return?
- What is the rate of return Philippe would have realized if he had held onto the note instead of selling it to Damien?

**Click to see Solution**

Philippe’s rate of return for the sale.

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 98,414.58-96,453.93 \\ & = & \$1,960.65 \\ \\ r & = & \frac{I}{P \times r \times t} \\ & = & \frac{1,960.65}{96,453.93 \times \frac{110}{365}} \\ & = & 0.0674 \\ & \rightarrow & 6.74\% \end{eqnarray*}[/latex]

Damien’s rate of return.

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 100,000-98,414.58 \\ & = & \$1,585.42 \\ \\ r & = & \frac{I}{P \times r \times t} \\ & = & \frac{1,585.42}{98,414.58 \times \frac{110}{365}} \\ & = & 0.0535 \\ & \rightarrow & 5.35\% \end{eqnarray*}[/latex]

Philippe’s rate of return without the sale.

[latex]\begin{eqnarray*} I & = & S-P \\ & = & 100,000-96,453.93 \\ & = & \$3,546.07 \\ \\ r & = & \frac{I}{P \times r \times t} \\ & = & \frac{3,546.07}{96,453.93 \times \frac{220}{365}} \\ & = & 0.061 \\ & \rightarrow & 6.1\% \end{eqnarray*}[/latex]

**Exercises**

- A 60-day, $90,000 face value commercial paper was issued when yields were 2.09%. What was its purchase price?

**Click to see Answer**$89,691.85

- A Government of Canada V39065 issue 90-day T-bill achieved its highest rate of return on May 24, 2000, with a yield of 5.74%. It realized its lowest rate of return on February 26, 2010, with a yield of 0.16%. Calculate the purchase price of a $100,000 T-bill on each of these dates. In dollars, how much more yield did an investor realize in 2000 than in 2010?

**Click to see Answer**$1356.15

- A Government of Canada V121780 issue 364-day T-bill achieved its highest rate of return on May 17, 2000, with a yield of 6.31%. It realized its lowest rate of return on May 6, 2009, with a yield of 0.42%. Calculate the purchase price of a $55,000 T-bill on each of these dates. In dollars, how much more yield did an investor realize in 2000 than in 2009?

**Click to see Answer**$3026.69

- Pawan is the marketing manager for Cyanamid Canada. His company will execute a marketing program half a year from now that requires a $500,000 investment. If his finance department had $485,000 to invest today into a 182-day $500,000 face value commercial paper yielding 4.55%, would it have enough money to purchase the commercial paper? Show calculations to support your answer.

**Click to see Answer**$488,907.82; cannot purchase for $485,000; $3907.82 more is required

- A 182-day Province of Manitoba T-bill with a face value of $250,000 was issued 102 days ago, when the yield was 4.3%. What is its purchase price today if the current rate of return is 4.53%?

**Click to see Answer**$247,542.21

- William purchased a $110,000 270-day commercial paper on its date of issue, when the yield was 3.89%. He sold it 178 days later when yields had increased to 4.03%. How much money did William earn on his investment?

**Click to see Answer**$1970.63

- Dollar Thrifty Automotive Group issued a $1,000,000, 180-day commercial paper. A bank purchased the paper for $975,560.21 on the issue date. What was the yield for the commercial paper on its issue date?

**Click to see Answer**5.08%

- A 90-day Province of Ontario T-bill with a $35,000 face value matures on December 11, 2022. Farrah works for Hearthplace Industries and notices that the company temporarily has some extra cash available. If she invests the money on October 28, 2022, when the yield is 4.94%, and sells the T-bill on November 25, 2022, when the yield is 4.83%, calculate how much money Farrah earned and the rate of return she realized.

**Click to see Answer**$133.24, 4.99%

- On August 8, 2022, Harriet invested in a $150,000 Canadian Wheat Board commercial paper on its date of issue with a 220-day term at a yield of 5.98%. On October 15, 2022, she sold the commercial paper to another investor when the current market rate was 5.75%. Calculate the amount of money earned and the rate of return realized.

**Click to see Answer**$1710.69, 6.34%

- Philippe purchased a $100,000 Citicorp Financial 220-day commercial paper for $96,453.93. He sold it 110 days later to Damien for $98,414.58, who then held onto the commercial paper until its maturity date.
- What is Philippe’s actual rate of return?
- What is Damien’s actual rate of return?
- What is the rate of return Philippe would have realized if he had held onto the note instead of selling it to Damien?

**Click to see Answer**a. 6.74%, b. 5.35%, c. 6.10%

#### Attribution

“8.6: Application: Treasury Bills & Commercial Papers” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“8.6: Application: Treasury Bills & Commercial Papers” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.