Interest-only loans and perpetuities

Constant-balance cash-flow models

A constant-balance model applies interest and then removes the periodic cash flow. The balance is unchanged when the cash flow exactly equals the interest for one period

1. Convert the annual rate Divide the nominal annual percentage rate by the number of periods per year to obtain the rate for one period \(\displaystyle i=\frac{r}{100p}\)
2. Calculate one period's interest Multiply the starting balance by the period rate \(\displaystyle P=24000,\quad i=0.015\quad\Rightarrow\quad I=iP=0.015\times24000=360\)
3. Match the payment to interest A constant balance is obtained when the periodic cash flow equals the interest for one period \(\displaystyle D=I=(R-1)P=iP\)
4. Compare amounts for direction A cash flow below one period's interest makes the balance grow, while a cash flow above it makes the balance fall \(\displaystyle D<I\Rightarrow V_{n+1}>V_n,\quad D>I\Rightarrow V_{n+1}<V_n\)

Worked example

A community fund has a balance of $18,000 and earns 6% nominal annual interest compounded quarterly. It pays an equal amount at the end of each quarter. Find the payment that keeps the fund balance constant and write the recurrence

1. \(\displaystyle i=\frac{6}{100\times4}=0.015\)The nominal annual rate is divided by four to obtain the quarterly rate
2. \(\displaystyle I=iP=0.015\times18000=270\)One quarter's interest is 270 dollars on the starting balance
3. \(\displaystyle D=I=270\)The payment must equal one quarter's interest to leave the balance unchanged
4. \(\displaystyle R=1+i=1.015,\quad V_{n+1}=1.015V_n-270\)The fund receives interest first and then pays 270 dollars at the end of each quarter

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