document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162507>Question 1162507</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A corporation issues a RM2, 500, 000 piece of commercial paper that comes due in 2 years. If it sets up a sinking fund with an institution that pays 8%, what will its quarterly payment be? Set up a schedule showing the accumulation of the assets in this sinking fund. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854401 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854401\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import pandas as pd\r<BR>\n" );
document.write( "\n" );
document.write( "# Given values<BR>\n" );
document.write( "fv = 2500000<BR>\n" );
document.write( "annual_rate = 0.08<BR>\n" );
document.write( "compounding_frequency = 4<BR>\n" );
document.write( "years = 2<BR>\n" );
document.write( "n = years * compounding_frequency<BR>\n" );
document.write( "i = annual_rate / compounding_frequency\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate Payment (PMT)<BR>\n" );
document.write( "# FV = PMT * ((1 + i)**n - 1) / i<BR>\n" );
document.write( "pmt = (fv * i) / ((1 + i)**n - 1)\r<BR>\n" );
document.write( "\n" );
document.write( "# Sinking Fund Schedule<BR>\n" );
document.write( "schedule = []<BR>\n" );
document.write( "balance = 0\r<BR>\n" );
document.write( "\n" );
document.write( "for period in range(1, n + 1):<BR>\n" );
document.write( "    interest = balance * i<BR>\n" );
document.write( "    increase = pmt + interest<BR>\n" );
document.write( "    balance += increase<BR>\n" );
document.write( "    schedule.append({<BR>\n" );
document.write( "        \"Quarter\": period,<BR>\n" );
document.write( "        \"Payment\": pmt,<BR>\n" );
document.write( "        \"Interest Earned\": interest,<BR>\n" );
document.write( "        \"Increase in Fund\": increase,<BR>\n" );
document.write( "        \"Fund Balance\": balance<BR>\n" );
document.write( "    })\r<BR>\n" );
document.write( "\n" );
document.write( "df_schedule = pd.DataFrame(schedule)\r<BR>\n" );
document.write( "\n" );
document.write( "# Format for output<BR>\n" );
document.write( "print(f\"Quarterly Payment: {pmt}\")<BR>\n" );
document.write( "print(df_schedule)\r<BR>\n" );
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document.write( "# Write to CSV<BR>\n" );
document.write( "df_schedule.to_csv(\"sinking_fund_schedule.csv\", index=False)\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Quarterly Payment: 291274.49783440644<BR>\n" );
document.write( "   Quarter        Payment  Interest Earned  Increase in Fund  Fund Balance<BR>\n" );
document.write( "0        1  291274.497834         0.000000     291274.497834  2.912745e+05<BR>\n" );
document.write( "1        2  291274.497834      5825.489957     297099.987791  5.883745e+05<BR>\n" );
document.write( "2        3  291274.497834     11767.489713     303041.987547  8.914165e+05<BR>\n" );
document.write( "3        4  291274.497834     17828.329463     309102.827298  1.200519e+06<BR>\n" );
document.write( "4        5  291274.497834     24010.386009     315284.883844  1.515804e+06<BR>\n" );
document.write( "5        6  291274.497834     30316.083686     321590.581521  1.837395e+06<BR>\n" );
document.write( "6        7  291274.497834     36747.895317     328022.393151  2.165417e+06<BR>\n" );
document.write( "7        8  291274.497834     43308.343180     334582.841014  2.500000e+06\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "[file-tag: code-generated-file-0-1776522428876279223]\r<BR>\n" );
document.write( "\n" );
document.write( "To calculate the quarterly payment for the sinking fund, we use the formula for the future value of an ordinary annuity:\r<BR>\n" );
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document.write( "$$FV = PMT \times \frac{(1 + i)^n - 1}{i}$$\r<BR>\n" );
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document.write( "Where:<BR>\n" );
document.write( "* $FV = \text{RM } 2,500,000$ (The future amount needed)<BR>\n" );
document.write( "* $r = 0.08$ (Annual interest rate)<BR>\n" );
document.write( "* $m = 4$ (Quarterly compounding)<BR>\n" );
document.write( "* $i = \frac{r}{m} = \frac{0.08}{4} = 0.02$ (Quarterly interest rate)<BR>\n" );
document.write( "* $t = 2 \text{ years}$<BR>\n" );
document.write( "* $n = t \times m = 2 \times 4 = 8$ (Total number of quarterly payments)\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. Quarterly Payment Calculation<BR>\n" );
document.write( "Rearranging the formula to solve for $PMT$:\r<BR>\n" );
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document.write( "$$PMT = \frac{FV \times i}{(1 + i)^n - 1}$$<BR>\n" );
document.write( "$$PMT = \frac{2,500,000 \times 0.02}{(1 + 0.02)^8 - 1}$$<BR>\n" );
document.write( "$$PMT = \frac{50,000}{(1.02)^8 - 1}$$<BR>\n" );
document.write( "$$PMT \approx \text{RM } 291,274.50$$\r<BR>\n" );
document.write( "\n" );
document.write( "The quarterly payment required is **RM 291,274.50**.\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Sinking Fund Schedule<BR>\n" );
document.write( "The following schedule shows the accumulation of assets in the sinking fund over the 2-year period.\r<BR>\n" );
document.write( "\n" );
document.write( "| Quarter | Payment (RM) | Interest Earned (RM) | Increase in Fund (RM) | Fund Balance (RM) |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| 1 | 291,274.50 | 0.00 | 291,274.50 | 291,274.50 |<BR>\n" );
document.write( "| 2 | 291,274.50 | 5,825.49 | 297,099.99 | 588,374.49 |<BR>\n" );
document.write( "| 3 | 291,274.50 | 11,767.49 | 303,041.99 | 891,416.48 |<BR>\n" );
document.write( "| 4 | 291,274.50 | 17,828.33 | 309,102.83 | 1,200,519.31 |<BR>\n" );
document.write( "| 5 | 291,274.50 | 24,010.39 | 315,284.89 | 1,515,804.20 |<BR>\n" );
document.write( "| 6 | 291,274.50 | 30,316.08 | 321,590.58 | 1,837,394.78 |<BR>\n" );
document.write( "| 7 | 291,274.50 | 36,747.90 | 328,022.40 | 2,165,417.18 |<BR>\n" );
document.write( "| 8 | 291,274.50 | 43,308.34 | 334,582.84 | 2,500,000.02* |\r<BR>\n" );
document.write( "\n" );
document.write( "*\*Minor difference due to rounding.*\r<BR>\n" );
document.write( "\n" );
document.write( "The complete schedule has been saved to `sinking_fund_schedule.csv`. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );