Analyse balances and graphs of an affine cash-flow model

Analyse a periodic cash-flow model

An affine cash-flow recurrence gives the next balance from the current balance, a multiplier and a fixed cash flow. Calculate indexed values in order and represent a discrete model with separate points

1. Start from V_0 Use V_0 as the balance before the first transition and use the index to count periods \(\displaystyle V_0=2000,\quad V_1=1.05V_0+300\)
2. Apply interest before cash flow Each transition first multiplies the previous balance by the period multiplier and then applies the end-of-period cash flow \(\displaystyle V_{n+1}=1.05V_n+300\)
3. Use the previous balance each period After finding one balance, substitute it into the next transition rather than returning to the initial balance \(\displaystyle V_2=1.05V_1+300\)
4. Plot indexed balances as separate points A discrete recurrence gives one point for each integer index, so the points are not joined unless a continuous model is specified \(\displaystyle \left(0,V_0\right),\quad\left(1,V_1\right),\quad\left(2,V_2\right)\)

Worked example

A balance starts at $1500. It earns 8% nominal interest per year, compounded half-yearly, and receives a $120 addition at the end of each half-year. Find the balance after two half-year periods and give the points for a discrete graph

1. \(\displaystyle i=\frac{8}{100\times2}=0.04,\quad R=1.04\)The half-yearly rate is 0.04, so the period multiplier is 1.04
2. \(\displaystyle V_1=1.04\times1500+120=1680\)The first transition applies interest to $1500 before adding $120
3. \(\displaystyle V_2=1.04\times1680+120=1867.2\)The second transition uses the first balance of $1680 as its starting value
4. \(\displaystyle \left(0,1500\right),\quad\left(1,1680\right),\quad\left(2,1867.2\right)\)Plot one separate point for each integer index in the discrete model

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