document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166502>Question 1166502</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An investor purchased a 91-day, $100,000 T-bill on its issue data for $99,326.85. After holding it for 42 days, she sold the T-bill for a yield of 2.72%.<BR>\n" );
document.write( "A. What was the original yield of the T-bill?<BR>\n" );
document.write( "B. For what price was T-bill sold?<BR>\n" );
document.write( "C. What rate of return (per annum) did the investor realize while holding this T-bill? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852749 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852749\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a multi-part investment problem involving the calculation of yields, selling price, and realized return for a U.S. Treasury Bill (T-bill). T-bills are zero-coupon bonds, and their yields are typically quoted on a **discount basis** or a **bond-equivalent yield (BEY)** basis, which we will assume here means the standard annualized simple interest rate.\r<BR>\n" );
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document.write( "## A. What was the original yield of the T-bill?\r<BR>\n" );
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document.write( "The original yield is the simple annual return the investor would have received had they held the T-bill for the full **91 days**.\r<BR>\n" );
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document.write( "* **Face Value ($FV$):** \$100,000<BR>\n" );
document.write( "* **Original Purchase Price ($P_{orig}$):** \$99,326.85<BR>\n" );
document.write( "* **Interest Earned ($I$):** $FV - P_{orig} = 100,000 - 99,326.85 = \$673.15$<BR>\n" );
document.write( "* **Original Holding Period ($t$):** 91 days\r<BR>\n" );
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document.write( "The yield ($Y$) is calculated as:<BR>\n" );
document.write( "$$Y = \frac{\text{Interest Earned}}{\text{Purchase Price}} \times \frac{\text{Days in Year}}{\text{Days to Maturity}}$$\r<BR>\n" );
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document.write( "Assuming a 365-day year (standard for BEY):<BR>\n" );
document.write( "$$Y = \frac{673.15}{99,326.85} \times \frac{365}{91}$$<BR>\n" );
document.write( "$$Y \approx 0.006777 \times 4.01099$$<BR>\n" );
document.write( "$$Y \approx 0.027178$$\r<BR>\n" );
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document.write( "The original yield of the T-bill was approximately **2.72%**.\r<BR>\n" );
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document.write( "## B. For what price was T-bill sold?\r<BR>\n" );
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document.write( "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 $2.72\%$.\r<BR>\n" );
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document.write( "The selling price ($P_{\text{sell}}$) is the present value of the face value discounted at the buyer's yield over the remaining term.\r<BR>\n" );
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document.write( "$$\text{Buyer's Yield} = \frac{\text{Interest Paid by Seller}}{\text{Selling Price}} \times \frac{365}{\text{Days Remaining}}$$\r<BR>\n" );
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document.write( "Here, the interest paid by the seller is the Face Value less the Selling Price: $I = FV - P_{\text{sell}}$.\r<BR>\n" );
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document.write( "$$0.0272 = \frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} \times \frac{365}{49}$$\r<BR>\n" );
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document.write( "Rearrange the equation to solve for $P_{\text{sell}}$:\r<BR>\n" );
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document.write( "1.  Isolate the price difference ratio:<BR>\n" );
document.write( "    $$\frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} = 0.0272 \times \frac{49}{365}$$<BR>\n" );
document.write( "    $$\frac{100,000 - P_{\text{sell}}}{P_{\text{sell}}} \approx 0.0272 \times 0.13425 \approx 0.003651$$\r<BR>\n" );
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document.write( "2.  Let $R$ be the ratio $0.003651$:<BR>\n" );
document.write( "    $$100,000 - P_{\text{sell}} = R \times P_{\text{sell}}$$<BR>\n" );
document.write( "    $$100,000 = P_{\text{sell}} (1 + R)$$<BR>\n" );
document.write( "    $$P_{\text{sell}} = \frac{100,000}{1 + 0.003651}$$<BR>\n" );
document.write( "    $$P_{\text{sell}} = \frac{100,000}{1.003651} \approx 99,636.23$$\r<BR>\n" );
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document.write( "The T-bill was sold for **\$99,636.23**.\r<BR>\n" );
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document.write( "## C. What rate of return (per annum) did the investor realize while holding this T-bill?\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "* **Holding Period ($t$):** 42 days<BR>\n" );
document.write( "* **Original Purchase Price ($P_{orig}$):** \$99,326.85<BR>\n" );
document.write( "* **Selling Price ($P_{\text{sell}}$):** \$99,636.23<BR>\n" );
document.write( "* **Gain ($G$):** $P_{\text{sell}} - P_{orig} = 99,636.23 - 99,326.85 = \$309.38$\r<BR>\n" );
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document.write( "The realized rate of return ($R_{real}$) is calculated as:<BR>\n" );
document.write( "$$R_{\text{real}} = \frac{\text{Gain}}{\text{Purchase Price}} \times \frac{\text{Days in Year}}{\text{Holding Days}}$$\r<BR>\n" );
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document.write( "$$R_{\text{real}} = \frac{309.38}{99,326.85} \times \frac{365}{42}$$<BR>\n" );
document.write( "$$R_{\text{real}} \approx 0.003115 \times 8.69048$$<BR>\n" );
document.write( "$$R_{\text{real}} \approx 0.027071$$\r<BR>\n" );
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document.write( "The investor realized an annual rate of return of approximately **2.71%**. </SPAN>\n" );
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