Model and interpret a periodic cash-flow recurrence

Define the balance at a consistent instant

Let Bₙ be the balance immediately after the nth end-of-period payment or contribution, with B₀ the starting balance. If interest proportion i is applied first, followed by signed cash flow C, then Bₙ₊₁=(1+i)Bₙ+C. Deposits are positive; repayments or withdrawals reduce the balance. This is a mathematical model with constant stated conditions

A model starts at B₀=800, adds 1% interest each month, then a deposit of 40. Find B₁

A balance earns 1% interest each month. A deposit of 40 is made before interest. Which monthly update is correct?

Match nominal rates to the period

If a stated nominal annual percentage j is convertible m equal times per year, the period proportion is found by dividing the nominal percentage by 100, then by the number of periods per year. An effective annual rate is a different quantity and cannot simply be divided by m. State the time step, frequency and rate convention before forming the recurrence

A nominal annual rate of 6% is convertible monthly. What is the exact monthly multiplier?

Use the finite geometric sum

With end-of-period signed cash flow C and multiplier R≠1, the displayed explicit model applies. The display also gives the case when R=1. Keep full precision through the calculation and round only the final requested money value. A model with actual per-period rounding is a different recurrence

Build and assess a loan model, ungraded

A mathematical loan model starts at 3000, adds 2% interest per month and then removes a payment of 400. Define the balance timing, derive Bₙ and calculate B₂. Explain why the same fixed payment cannot be applied indefinitely once the balance is repaid. Independent model formulation and evaluation are ungraded

Model: a finite valid financial horizon

Let the balance be recorded just after each monthly payment while the loan remains outstanding. The displayed working gives the recurrence, explicit formula and first two balances. The equilibrium 20000 is repelling, so it is not this loan’s limiting debt. At the final repayment, reduce the scheduled amount to the balance plus that period’s interest; continuing to a negative balance would change the meaning of the model. Fees, changing rates and payment interruptions are excluded assumptions