Question 1166502: An investor purchased a 91-day, $100,000 T-bill on its issue data for $99,326.85. After holding it for 42 days, she sold the T-bill for a yield of 2.72%.
A. What was the original yield of the T-bill?
B. For what price was T-bill sold?
C. What rate of return (per annum) did the investor realize while holding this T-bill?
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! This is a multi-part investment problem involving the calculation of yields, selling price, and realized return for a U.S. Treasury Bill (T-bill). T-bills are zero-coupon bonds, and their yields are typically quoted on a **discount basis** or a **bond-equivalent yield (BEY)** basis, which we will assume here means the standard annualized simple interest rate.
## A. What was the original yield of the T-bill?
The original yield is the simple annual return the investor would have received had they held the T-bill for the full **91 days**.
* **Face Value ($FV$):** \$100,000
* **Original Purchase Price ($P_{orig}$):** \$99,326.85
* **Interest Earned ($I$):** $FV - P_{orig} = 100,000 - 99,326.85 = \$673.15$
* **Original Holding Period ($t$):** 91 days
The yield ($Y$) is calculated as:
$$Y = \frac{\text{Interest Earned}}{\text{Purchase Price}} \times \frac{\text{Days in Year}}{\text{Days to Maturity}}$$
Assuming a 365-day year (standard for BEY):
$$Y = \frac{673.15}{99,326.85} \times \frac{365}{91}$$
$$Y \approx 0.006777 \times 4.01099$$
$$Y \approx 0.027178$$
The original yield of the T-bill was approximately **2.72%**.
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## B. For what price was T-bill sold?
The T-bill was sold when it had **49 days remaining** to maturity ($91 - 42 = 49$ days). The selling price is determined by the **buyer's yield**, which was $2.72\%$.
The selling price ($P_{\text{sell}}$) is the present value of the face value discounted at the buyer's yield over the remaining term.
$$\text{Buyer's Yield} = \frac{\text{Interest Paid by Seller}}{\text{Selling Price}} \times \frac{365}{\text{Days Remaining}}$$
Here, the interest paid by the seller is the Face Value less the Selling Price: $I = FV - P_{\text{sell}}$.
$$0.0272 = \frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} \times \frac{365}{49}$$
Rearrange the equation to solve for $P_{\text{sell}}$:
1. Isolate the price difference ratio:
$$\frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} = 0.0272 \times \frac{49}{365}$$
$$\frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} \approx 0.0272 \times 0.13425 \approx 0.003651$$
2. Let $R$ be the ratio $0.003651$:
$$100,000 - P_{\text{sell}} = R \times P_{\text{sell}}$$
$$100,000 = P_{\text{sell}} (1 + R)$$
$$P_{\text{sell}} = \frac{100,000}{1 + 0.003651}$$
$$P_{\text{sell}} = \frac{100,000}{1.003651} \approx 99,636.23$$
The T-bill was sold for **\$99,636.23**.
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## C. What rate of return (per annum) did the investor realize while holding this T-bill?
The investor held the T-bill for **42 days**. The rate of return (annualized) is based on the gain realized over the purchase price during the holding period.
* **Holding Period ($t$):** 42 days
* **Original Purchase Price ($P_{orig}$):** \$99,326.85
* **Selling Price ($P_{\text{sell}}$):** \$99,636.23
* **Gain ($G$):** $P_{\text{sell}} - P_{orig} = 99,636.23 - 99,326.85 = \$309.38$
The realized rate of return ($R_{real}$) is calculated as:
$$R_{\text{real}} = \frac{\text{Gain}}{\text{Purchase Price}} \times \frac{\text{Days in Year}}{\text{Holding Days}}$$
$$R_{\text{real}} = \frac{309.38}{99,326.85} \times \frac{365}{42}$$
$$R_{\text{real}} \approx 0.003115 \times 8.69048$$
$$R_{\text{real}} \approx 0.027071$$
The investor realized an annual rate of return of approximately **2.71%**.
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