# What's the interest rate in this example of irregular payments?

Loaned out \$10K, with the following payments (like, these are all the payments there will ever be)

``````Period 1     2     3     4     5     6
\$0    \$0    \$0    \$0    \$26K  \$4K
``````

How would I calculate the interest rate?

• Just to be clear, you lent \$10,000 and will be repaid a total of \$30,000? – quid Mar 21 at 1:35
• yes that's right – james Mar 21 at 1:36

Noting the year-by-year balances ...

``````y0 = 10000
y1 = y0 (1 + r)
y2 = y1 (1 + r)
y3 = y2 (1 + r)
y4 = y3 (1 + r)
y5 = y4 (1 + r) - 26000
y6 = y5 (1 + r) - 4000

∴ y6 = -4000 + (1 + r) (-26000 + 10000 (1 + r)^5)
``````

Solving for the balance in year 6 being equal to zero, i.e. fully repaid

``````∴ r = 23.925%
`````` One method: set up a spreadsheet with a cell labelled "rate", and columns labelled "balance" and "payments". Enter the payments, initialize the first balance as 10,000, then set each subsequent balance as the previous balance plus the previous balance times the rate minus the payment. Now go to "goal seek"/"solver", and find the rate that sets the final balance to zero. Depending on the timing of the payments, you may need to add in another period. For instance, if the nth payment is made after n years, then you should start you sheet at period 0 so that interest has accrued by period 1.

Let's call the interest rate `r`.

After one year, you owe `10*(1+r)`, after two years `10*(1+r)^2`, etc, then after 5 years, you owe `10*(1+r)^5-26`, and finally after 6 years `(10*(1+r)^5-26)*(1+r)-4` - which is `=0`.

You can now solve the equation for `r`.

• 2 follow ups: 1) where does the `-26` come from in "after 5 years, you owe `10*(1+r)^5-26`" 2) where does the `-4` come from in the second equation – james Mar 21 at 1:50
• @james Each payment reduces the amount owed. Aganju is calculating in thousands of dollars. Also, note that this calculation corresponds to the internal rate of return. – nanoman Mar 21 at 2:41
• ok but i'm still not sure where the `-26` and `-4` come from I'm afraid.. – james Mar 21 at 2:44
• @james After 5 years, the original 10K balance has grown to `10K * (1+r)^5`, and then 26K is subtracted for the payment made that year (making a payment reduces the balance). Once all payments have been made, the balance should be zero. – nanoman Mar 21 at 3:12
• Put another way, the \$26K payment covers the interest that accrued over the previous 5 years, plus some (but not all) of the original \$10k principal, leaving some principal to accrue more interest in the last year. – chepner Mar 22 at 16:52