Construct and interpret an amortisation table

The table describes an investment addition. Which formula gives the principal increase in one row?

Reuse the Year 10 no-transaction experiment: $2,000, fixed 6% nominal annual rate, two years. Change annual, half-yearly, quarterly and monthly frequency. Predict the period rate and update count. This explorer does not post loan payments or implement the posted-cents table below

What must be converted before constructing a monthly amortisation table from a nominal annual rate?

Complete an amortisation table

An amortisation table records the interest, principal change and balance after each period. Use the posted balance from one row as the starting balance for the next row

1. 1Use the previous posted balance Interest for a row is calculated from the balance at the start of that row, which is the previous row's posted balance \(\displaystyle I_{n+1}=iB_n\)
2. 2Round row interest to cents Round the interest amount to cents before using it in the principal change and balance columns \(\displaystyle 0.04\times2456.88=98.2752\rightarrow98.28\)
3. 3Match the principal-change sign For an investment addition, the addition and interest both increase the balance; for a repayment, the payment reduces principal after interest \(\displaystyle C=A+I\quad\text{for an addition},\qquad C=A-I\quad\text{for a repayment}\)
4. 4Carry the new balance forward Use the new posted balance as the starting balance in the next row \(\displaystyle B_n=B_{n-1}+C_n\quad\text{or}\quad B_n=B_{n-1}-C_n\)

Construct and verify every loan row

A hypothetical loan starts at $1000.07 with 12% nominal interest per year compounded monthly. Interest is posted to the nearest cent each month, half away from zero. Payments occur at month end, after interest. Pay $260 normally, but use opening balance plus posted interest if that is less than $260, clearing the loan without an overpayment. There are no fees or other transactions

1. 1Build from index zero Define the month index and construct the complete table until the balance is zero. Include opening balance, posted interest, actual payment, principal reduction and closing balance. Choose a spreadsheet or recurrence table, justify the choice and record its actual formulas. Do not replace missing intermediate rows with only the final answer
2. 2Independent checks Verify each next opening equals the preceding closing. Sum all payments, interest and principal reductions. Check that total principal reductions equal the original loan and total payments minus original loan equals total posted interest. Explain why the final payment changes and why a negative computed balance is not an amount still owed
3. 3Save your work before comparison Save your complete tables, formulas, independently checked totals, assumptions and explanations before opening the model response. Full table construction, written reasoning and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Construct and verify every loan row

Compare each row and the independent checks, not only the final answer. Distinguish the explicitly named carry conventions

1. 1Rules and reproducible spreadsheet Let B(0)=1000.07, with n completed months. In each row calculate I=ROUND(0.01B,2), payment=min(260,B+I), reduction=payment−I and next balance=B+I−payment. For positive amounts here, the spreadsheet ROUND must implement the stated half-away rule; check it on 2.425 before relying on it. With opening in B2, use C2=ROUND(B2*0.01,2), D2=MIN(260,B2+C2), E2=D2-C2 and F2=B2+C2-D2, then B3=F2. Stop once the closing balance is zero
2. 2Every completed row Columns are month: opening, interest, payment, principal reduction, closing. Row 1: 1000.07,10.00,260.00,250.00,750.07. Row 2: 750.07,7.50,260.00,252.50,497.57. Row 3: 497.57,4.98,260.00,255.02,242.55. Row 4: 242.55,2.43,244.98,242.55,0.00. Index zero records the initial $1000.07 before any interest or payment
3. 3Totals and settlement Total payments are $1024.98, total posted interest is $24.91 and total principal reduction is $1000.07. Both 1024.98−1000.07=24.91 and 1000.07+24.91−1024.98=0 verify the ledger. A fourth full $260 payment would overpay by $15.02; the stipulated final-payment rule avoids this. A real contract might specify different settlement charges or dates

Given information

\(\displaystyle r=12\%\text{ nominal},\quad p=12\)

Give the monthly percentage number, without a percent sign

Given information

\(\displaystyle B_2=497.57,\quad i=0.01\)

Calculate posted interest for month 3, in dollars to two decimal places

Given information

\(\displaystyle B_3=242.55,\quad i=0.01,\quad D=260\)

Post interest to the nearest cent. If the opening balance plus posted interest is less than $260, pay that amount to clear the loan. Find the adjusted last payment, in dollars to two decimal places

Given information

\(\displaystyle P=1000.07,\quad D_1=D_2=D_3=260,\quad D_4=244.98\)

The fourth payment clears the loan. Find the total posted interest, in dollars to two decimal places

Which check tests a different relationship from merely copying the final row?

Compare a different precision contract

Keep the same initial $1000.07, monthly rate and first three $260 payments. Compare the posted-cents model with an explicitly exact-carry model that does not post rounded monthly interest. In the exact model, retain every internal fraction and at month 4 calculate the exact amount due before displaying a cents-rounded quote

1. 1Do not silently substitute models Construct the first three exact balances and the exact month-4 amount due. Compare the two final quotes to cents. Explain whether a payment rounded to cents would make an exact rational balance exactly zero. State what additional settlement convention would be needed for an actual cent-valued payment
2. 2Diagnose a rounding shortcut Check the month-4 posted interest from the unposted value 2.4255, and distinguish this from computing interest on the exact month-3 balance. Explain why simply rounding an exact-carry final result does not reconstruct the posted-cents ledger
3. 3Save your work before comparison Save your complete tables, formulas, independently checked totals, assumptions and explanations before opening the model response. Full table construction, written reasoning and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Compare a different precision contract

Compare each row and the independent checks, not only the final answer. Distinguish the explicitly named carry conventions

1. 1Exact balances and final quote The exact recurrence for the first three payments is E(0)=1000.07 and E(n+1)=1.01E(n)−260. It gives 750.0707,497.571407,242.54712107. The exact amount due at month 4 is 1.01×242.54712107=244.9725922807, displayed as $244.97. The posted-cents final payment is $244.98: the quotes differ by one cent because their contracts differ
2. 2Rounding and residual Posted month-4 interest is 0.01×242.55=2.4255, rounded to $2.43. The exact model instead uses the unrounded preceding balance. Paying $244.97 in the exact model leaves $0.0025922807, not exactly zero. A separate rounding or write-off settlement convention is needed to call a cent-valued payment final. An exact quote is not a posted ledger or an instruction about an actual lender

Given information

\(\displaystyle E_3=242.54712107,\quad R=1.01\)

Display the exact-carry month-4 amount due to two decimal places

A posted-cents model rounds each period's interest before carrying the balance forward; an exact-carry model retains fractions. Why can their final displayed quotes differ?

Check cash-flow direction and reverse inference

Use the retained investment example: initial $1800, 8% nominal interest compounded half-yearly, $250 added at each period end, with posted cents. Compare a separate annuity with the same initial amount and rate but $250 withdrawn at each period end. Then investigate the first interest observation $202.50 on a $9000 balance for a separate half-yearly account

1. 1Complete both roles Construct three complete rows for each role. Compare signs, total interest and final balances. Explain why principal increase includes both the addition and interest, while principal reduction is withdrawal minus interest. For the addition model, calculate the first-row result if the $250 instead arrives at the beginning of the period
2. 2Reverse information and uniqueness For the $9000 account, first assume $202.50 is only a rounded posted interest observation. Give an interval of possible period rates and exhibit two different rates producing that observation. Then add the condition that $202.50 was exact before rounding: determine the exact half-year rate and nominal annual rate. State why the extra condition matters. From an addition row with interest $72 and principal increase $322, infer the cash addition
3. 3Save your work before comparison Save your complete tables, formulas, independently checked totals, assumptions and explanations before opening the model response. Full table construction, written reasoning and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Check cash-flow direction and reverse inference

Compare each row and the independent checks, not only the final answer. Distinguish the explicitly named carry conventions

1. 1Addition and withdrawal tables Addition rows, listed as interest/principal increase/closing balance, are 72.00/322.00/2122.00,84.88/334.88/2456.88,98.28/348.28/2805.16. Total interest is $255.16, checked by 2805.16−1800−750. Withdrawal rows, listed as interest/principal reduction/closing balance, are 72.00/178.00/1622.00,64.88/185.12/1436.88,57.48/192.52/1244.36. Total interest is $194.36, checked by 1244.36−1800+750. Each next opening is the preceding closing
2. 2Timing and reverse addition A beginning-of-period addition gives 1.04(1800+250)=2132 in the first row, $10 more than the end-of-period result 2122. The extra deposit earns interest only under the beginning rule. The reverse addition is 322−72=$250, not 322+72.
3. 3Rate interval versus exact rate With positive half-away rounding, 202.495≤9000i<202.505 produces the same $202.50 posting. Thus 100(202.495/9000)≤period percentage<100(202.505/9000), approximately [2.24994444%,2.25005556%). Both 2.25% and 2.2500001% satisfy it. If first interest is explicitly exact before posting, i=202.50/9000=0.0225; the half-year rate is exactly 2.25% and the nominal annual rate is exactly 4.5%. Rounding alone does not identify that exact rate

Given information

\(\displaystyle B_0=1800,\quad i=0.04,\quad D=250\text{ withdrawn at period end}\)

Post each period's interest to the nearest cent. Find the three-period annuity balance, in dollars to two decimal places

Given information

\(\displaystyle C_1=322,\quad I_1=72\)

Find the equal cash addition in dollars

Given information

\(\displaystyle B_0=9000,\quad I_1=202.50,\quad p=2\)

Given that first-row interest is exact before posting, find the nominal annual percentage number

Which condition makes exact division valid for the rate-inference task?