Model and interpret a periodic cash-flow recurrence

Periodic cash-flow recurrences

A periodic cash-flow model updates a balance by applying interest and then adding or subtracting a fixed amount. The recurrence records both the period multiplier and the direction of the cash flow

1. Name the balance at each period V_n is the balance after n periods, and V_0 is the balance before the first period's interest \(\displaystyle V_0=\text{initial balance},\quad V_{n+1}=\text{balance after the next period}\)
2. Convert the nominal rate to a period rate For a nominal annual rate r% with p periods per year, divide the percentage by p before forming the multiplier \(\displaystyle r=6\%,\ p=2\quad\Rightarrow\quad i=\frac{6}{100\times2}=0.03\)
3. Apply interest before cash flow Multiply the previous balance by the period multiplier, then add the end-of-period cash flow \(\displaystyle V_{n+1}=RV_n+sD\)
4. Choose the cash-flow sign Use +D for an addition to the balance and -D for a repayment or withdrawal \(\displaystyle V_{n+1}=1.03V_n+200\quad\text{or}\quad V_{n+1}=1.03V_n-200\)

Worked example

A savings balance starts at $2500. It earns 8% nominal interest per year, compounded quarterly, and a $150 deposit is made at the end of each quarter. Find the balance after three quarterly periods

1. \(\displaystyle i=\frac{8}{100\times4}=0.02,\quad R=1.02\)The nominal annual rate is divided by four to obtain the quarterly rate and multiplier
2. \(\displaystyle V_1=1.02\times2500+150=2700\)The first quarter applies interest to the initial balance before the deposit
3. \(\displaystyle V_2=1.02\times2700+150=2904\)The second quarter uses the previous balance of $2700 as its starting value
4. \(\displaystyle V_3=1.02\times2904+150=3112.08\)The balance after three quarterly periods is $3112.08

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