Analyse a cash-flow model with changed conditions

A model changes its end-of-period addition after two periods. Which starting value should the new stage use?

Reuse the Year 10 frequency experiment with $2000, a fixed nominal annual rate of 6%, two years and no other transactions. Predict how both the period rate and number of updates change. The main investigation below adds cash flows, so this explorer alone cannot calculate it

When is one ordinary TVM calculation an appropriate tool for a stage?

Handle changed cash-flow conditions

A changed-condition model has one recurrence before a specified point and another recurrence after it. Calculate the first stage, carry its change-point balance into the second stage, and replace only the stated condition

1. 1Split at the change point Apply the first set of conditions up to the change point, then use the resulting balance as the starting value for the second stage \(\displaystyle V_0\longrightarrow V_1\longrightarrow\cdots\longrightarrow V_c\)
2. 2Carry the exact seam balance Keep the balance at the change point exact when it becomes the starting value for the new conditions \(\displaystyle V_c=6537.5\quad\Rightarrow\quad V_{c+1}=1.05V_c+700\)
3. 3Change only the named condition If only the cash flow changes, keep the period multiplier and the financial role unchanged \(\displaystyle R_2=R_1=1.05,\quad D_1=500,\quad D_2=700\)
4. 4Round at the requested boundary Round the final displayed balance after the stage-two calculations rather than feeding a rounded seam into the next stage \(\displaystyle V_4=8642.59375\rightarrow8642.59\)

Start with a goal and choose an adequate model

A fictional community programme starts with $8000 and needs at least $9500 exactly one year later. The model assumes 4% nominal annual interest compounded quarterly, no fees or taxes, and $200 contributed at the end of each quarter after interest. Carry exact balances internally and round only displayed results. The starting amount is quarter 0.

1. 1Choose, construct and check Explain why a no-contribution compound-interest model is inadequate once contributions are included. Select a spreadsheet, recurrence table or TVM solver and justify it. Construct the complete quarter 0 to quarter 4 table, give the recurrence, and independently verify the final value by a geometric-sum expression. Calculate the goal shortfall using exact values
2. 2Record actual technology settings Use the account contributor’s viewpoint: initial and regular deposits are paid out, while the final balance is received. Record N, annual percentage, PV, PMT, FV, payments per year, compounding periods per year and END/BEGIN. Run the chosen tool yourself, save its actual fields/formulas and verify that the first-row output agrees with the recurrence
3. 3Keep an independent investigation record Choose and justify the model and technology yourself. Save all complete tables, entered fields or spreadsheet formulas, checks, comparisons and written conclusions before opening the model response. Complete construction, written judgement and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Start with a goal and choose an adequate model

Compare your saved work with this conditional model. A different suitable tool or method is acceptable if its units, viewpoint, timing, precision and constraints agree

1. 1Model and complete table The no-contribution model is 8000(1.01)^n and would omit the deposits. The adequate baseline is B(0)=8000 and B(n+1)=1.01B(n)+200 for n completed quarters. All balances are 8000,8280,8562.8,8848.428,9136.91228. Independently, B(4)=8000(1.01)^4+200[1+1.01+(1.01)^2+(1.01)^3]. It displays as $9136.91; the exact shortfall is $363.08772
2. 2Tool record and viewpoint A TVM record is N=4, I%=4, PV=−8000, PMT=−200, FV unknown positive, P/Y=4, C/Y=4 and END. Here I% means the nominal annual percentage, not the quarterly percentage. A spreadsheet can use B2=8000 and B3=1.01*B2+200, copying the formula through four updates; apply display formatting without ROUND in the recurrence. Verify B(1)=8280 before trusting the last result. Opposite viewpoint is valid only if all money signs reverse consistently

Given information

\(\displaystyle p=4,\quad t=1\text{ year}\)

How many quarterly updates occur in the one-year plan?

Given information

\(\displaystyle B_0=8000,\quad B_{n+1}=1.01B_n+200\)

Display the exact balance after quarter 2 to two decimal places

Given information

\(\displaystyle B_0=8000,\quad B_{n+1}=1.01B_n+200,\quad n=4\)

Display the baseline year-end balance to two decimal places

Which signs describe the contributor’s viewpoint in the baseline TVM record?

Revise the same plan at its exact change point

At the end of quarter 2, after its $200 contribution, the programme considers three changes: increase the last two contributions to $300; add a one-off $500 immediately at that change point while keeping regular contributions $200; or do both. No extra time elapses at the lump-sum event. Then impose a total additional-cash limit of $400 beyond the original four $200 contributions

1. 1Construct competing interventions Build all four complete branch records, including the baseline and the balance immediately before and after any lump sum. Compare at the same year-end date. Name exactly which settings differ in each counterfactual, rather than calling all changes one effect. Record separate TVM stages or explicit spreadsheet updates
2. 2Choose and prove a feasible revision Determine which listed plans meet the exact target and which satisfy the extra-cash limit. Keeping the last two contributions at $300, find the least one-off addition in whole cents at quarter 2 that meets the target. Prove the cent below fails using unrounded balances, even if it displays as $9500.00. State the total extra cash and the size of the remaining margin
3. 3Keep an independent investigation record Choose and justify the model and technology yourself. Save all complete tables, entered fields or spreadsheet formulas, checks, comparisons and written conclusions before opening the model response. Complete construction, written judgement and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Revise the same plan at its exact change point

Compare your saved work with this conditional model. A different suitable tool or method is acceptable if its units, viewpoint, timing, precision and constraints agree

1. 1Complete branches Every branch begins 8000,8280,8562.8 at quarters 0,1,2 before an extra payment. With larger final deposits, quarters 3 and 4 are 8948.428,9337.91228. With a $500 lump sum only, the immediate new starting balance is 9062.8 and later values are 9353.428,9646.96228. With both, later values are 9453.428,9847.96228. Increasing the last deposits adds $201 to the final balance; the lump sum adds $510.05. Those are different interventions and their effects add in this linear recurrence
2. 2Budget and minimum cent proof Larger deposits alone use $200 extra but miss the target. A $500 lump sum, with or without larger deposits, exceeds the $400 extra-cash limit. With larger deposits and a variable lump L, the final balance is 9337.91228+1.0201L. The exact least L is 8104386/51005, approximately 158.8939516, so the least whole-cent addition is $158.90. At $158.89 the exact final is 9499.995969, below the target despite displaying $9500.00. At $158.90 it is 9500.00617, above the target by $0.00617. Total extra cash is $200+$158.90=$358.90
3. 3Two-stage tool record Stage one uses N=2, I%=4, PV=−8000, PMT=−200, P/Y=C/Y=4, END, giving positive FV=8562.8. After the immediate lump sum, stage two uses N=2, I%=4, PV=−8721.7, PMT=−300, P/Y=C/Y=4, END, giving positive FV=9500.00617. Do not restart at −8000, treat the lump as a third quarter, add it twice, or round the seam before entering the next stage

Given information

\(\displaystyle B_2=8562.8,\quad B_{n+1}=1.01B_n+300\)

Display the year-end balance if only the last two contributions rise to $300

Given information

\(\displaystyle B_2=8562.8,\quad L=500,\quad R=1.01,\quad D=200,\quad N=2\)

Add the $500 immediately after quarter 2, then add $200 at the end of each of the two remaining quarters after interest. Display the year-end balance

Given information

\(\displaystyle F=9337.91228+1.0201L,\quad F\ge9500\)

Find the minimum whole-cent lump sum with the last two contributions at $300

Given information

\(\displaystyle L=158.90,\quad \Delta D=100,\quad n=2\)

Find the total extra cash used by that minimum-lump plan

Does the $158.89 lump-sum plan meet the exact at-least target because its final display is $9500.00?

Revise the purpose: finite payments or a lasting reserve

The programme now compares different uses of exactly $9500 at the start of a new payout year. This is a separately stated starting value, not an unnoticed rounding of the preceding model. Keep a fixed 1% quarterly rate and end-quarter cash flows. Consider four $2400 grants from a fund, borrowing $9500 to be repaid over four quarters, or retaining the $9500 nominal principal while paying a regular grant indefinitely

1. 1Choose the role and technology Explain which recurrence and viewpoint fits each objective. Construct all four annuity rows and record its TVM fields. For a four-quarter zero-target model, solve the exact equal payment, then find the greatest whole-cent annuity withdrawal leaving a nonnegative final balance and the least whole-cent loan payment giving zero or below. Prove each adjacent-cent boundary. Explain why these inequalities require opposite rounding directions
2. 2Constant balance is an identity Derive the quarterly grant that preserves the nominal principal indefinitely under fixed assumptions. Verify it by substitution into the recurrence and by a finite four-period TVM check. Explain why a large finite N alone is not a perpetuity proof, and why an interest-only loan with the same balance still owes its principal. State a settlement convention before treating a negative model balance as an actual final loan payment
3. 3Keep an independent investigation record Choose and justify the model and technology yourself. Save all complete tables, entered fields or spreadsheet formulas, checks, comparisons and written conclusions before opening the model response. Complete construction, written judgement and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Revise the purpose: finite payments or a lasting reserve

Compare your saved work with this conditional model. A different suitable tool or method is acceptable if its units, viewpoint, timing, precision and constraints agree

1. 1Finite annuity and its fields For the fund, A(0)=9500 and A(n+1)=1.01A(n)−2400. Rows 0 through 4 are 9500,7195,4866.95,2515.6195,140.775695, so the displayed remainder is $140.78. From the investor’s viewpoint use N=4, I%=4, PV=−9500, PMT=+2400, FV unknown positive, P/Y=C/Y=4 and END. For the borrower, the same balance recurrence has PV=+9500, PMT=−2400 and FV negative: the remaining amount is still owed
2. 2Inverse boundary and cash constraints The exact zero-target payment is 9500(1.01)^4/[1+1.01+(1.01)^2+(1.01)^3]=9885738095/4060401, approximately 2434.670392. A fixed $2434.67 payment leaves +0.00159233; a fixed $2434.68 payment leaves −0.03901168. Thus $2434.67 is the greatest whole-cent annuity withdrawal that does not exhaust the fund before the boundary, while $2434.68 is the least fixed whole-cent loan payment that meets a zero-or-below screening target. For an actual final settlement, adjust the last payment under a stated posting/rounding rule rather than paying a negative balance. The earlier posted-cents lesson shows why exact-carry and actual cent settlement are distinct
3. 3Perpetuity and interest-only comparison A constant fund requires D=0.01×9500=$95 each quarter because 1.01×9500−95=9500. Repeated substitution proves the same balance at every whole quarter under unchanged assumptions. A finite TVM check uses N=4, I%=4, PV=−9500, PMT=+95, FV=+9500, P/Y=C/Y=4, END. An interest-only borrower reverses all money signs and still owes $9500 after the four payments. A $100 perpetual quarterly grant instead requires a nominal principal of $10000 at this rate. Preserving nominal principal does not guarantee purchasing power, and fixed rates/no fees are substantive assumptions
4. 4Conditional choice The purpose determines the model: finite larger grants draw down principal, a perpetuity preserves nominal principal but provides a much smaller grant, and borrowing creates future repayment obligations. A calculated balance does not establish affordability, real product suitability or future rates. State the chosen objective and budget rather than recommending whichever model displays the largest balance

Given information

\(\displaystyle A_0=9500,\quad A_{n+1}=1.01A_n-2400\)

Display the fund remainder after four $2400 grants to two decimal places

Given information

\(\displaystyle A_0=9500,\quad R=1.01,\quad N=4\)

Withdrawals occur at the end of each quarter, after interest. Find the greatest fixed whole-cent quarterly withdrawal that leaves a nonnegative balance after four quarters

Given information

\(\displaystyle L_0=9500,\quad R=1.01,\quad N=4\)

Payments occur at the end of each quarter, after interest. Find the least fixed whole-cent quarterly loan payment satisfying the zero-or-below four-quarter target

Given information

\(\displaystyle P=9500,\quad i=0.01\)

Find the quarterly payment that preserves the nominal $9500 balance

What establishes the modelled perpetual balance, rather than only a finite trial?

Test a rate change and report a justified revision

Return to the one-year saving goal and the exact quarter-two balance $8562.80. Suppose the nominal annual rate falls to 2% for the last two quarters. The original $200 end-quarter contributions remain available, and extra contributions can be placed only at or after the quarter-two change point. The extra-cash budget is still $400. Compare the previously selected plan with every possible timing of that bounded extra budget

1. 1Sensitivity and a feasibility bound Recalculate the $158.90 lump plus two $300 contributions under the new rate. Then find an upper bound for the final value of any plan using at most $400 extra: justify why placing all extra cash at the earliest permitted time is best when the multiplier exceeds one. Determine whether merely reallocating the $400 can still meet the target. If not, find the minimum whole-cent immediate extra amount when later contributions remain $200
2. 2Final investigation report Write the goal, successive model choices, tool rationale and actual settings/formulas, complete branch evidence, independent checks and a conditional recommendation. Distinguish the original feasible plan from the changed-rate infeasibility result. State which constraint would need revision and what further information could invalidate the model. Do not treat simulated returns as guaranteed or choose a real product from these numbers
3. 3Keep an independent investigation record Choose and justify the model and technology yourself. Save all complete tables, entered fields or spreadsheet formulas, checks, comparisons and written conclusions before opening the model response. Complete construction, written judgement and actual external-tool operation are not automatically marked; review them yourself or with a teacher

Model response: Test a rate change and report a justified revision

Compare your saved work with this conditional model. A different suitable tool or method is acceptable if its units, viewpoint, timing, precision and constraints agree

1. 1Sensitivity results The new quarterly multiplier is 1.005. With $158.90 added at quarter two and two $300 contributions, the exact next values are 9065.3085 and 9410.6350425, so the goal is missed. Moving all $400 extra to the change point, while retaining the original later $200 contributions, gives 9453.65207 at year end. This is an upper bound because moving any nonnegative extra payment earlier multiplies it for at least as many periods when R>1. It is below 9500, so no rearrangement of that $400 budget can work under these constraints
2. 2Boundary and revised condition With the original later $200 contributions, the exact required immediate extra is 10007954/22445, approximately 445.8879038. The least whole-cent amount is $445.89, which gives 9500.00211725. The adjacent $445.88 gives 9499.992017. Thus the budget would need at least $45.89 more under this immediate-payment plan, or the goal, deadline or another substantive assumption must change. This is a mathematical constraint report, not a request to spend money
3. 3Example conditional conclusion At the original fixed nominal 4%, the $158.90 immediate extra plus two $100 contribution increases reaches the exact goal within the $400 extra budget, but its margin is only $0.00617. Under the stated rate cut, the original proposal fails and even the best timing of the $400 cannot reach the goal. Record both models rather than silently using the old rate. A real decision would also need fees, tax, timing, access and uncertainty information; the fixed-rate, no-fee outputs are not guarantees. Preserve the independent formulas, complete tables and adjacent-cent checks with the report

Given information

\(\displaystyle B_2=8562.8,\quad L=158.90,\quad R_2=1.005,\quad D_2=300,\quad N=2\)

Add the lump sum immediately after quarter 2, then add $300 at the end of each of the two remaining quarters after interest. Display the final value after the rate cut to two decimal places

Given information

\(\displaystyle B_2=8562.8,\quad L=400,\quad R_2=1.005,\quad D_2=200\)

Two quarters remain. Add the regular $200 at each quarter end after interest; extra cash may be added from immediately after quarter 2 onward. Display the best possible year-end balance from at most $400 extra under the cut-rate assumptions

Given information

\(\displaystyle (8562.8+L)(1.005)^2+200(1+1.005)\ge9500\)

Find the minimum immediate whole-cent extra amount under the cut rate with later $200 contributions

Which conclusion follows from the model and feasibility bound?