Lesson 3 of 9
Compounding conventions
One stated rate of 6 percent produces five different answers depending on how often it compounds. The effective rate is the only comparable form.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- A stated annual rate is not an answer until you know how many times a year it compounds.
- The effective annual rate is the one number that makes two differently quoted rates comparable.
- Beyond monthly compounding the gains are tiny, which is why the continuous form is used for convenience rather than accuracy.
Two instruments both quote 6 percent. One pays once a year, the other twelve times. They are not the same rate, and the difference is not a rounding question. It is the reason a quoted rate always needs a compounding frequency attached to it.
What 1,000 dollars becomes after one year at a stated 6 percent
- Once a year$1,060.00
- Twice a year$1,060.90
- Monthly$1,061.68
- Daily$1,061.83
- Continuous$1,061.84
The rule
Take the stated rate, divide it by the number of periods in a year, add one, and raise the whole thing to the number of periods.
At 6 percent compounded monthly, m is 12. So 1 plus 0.005, to the power 12, which is 1.061678. The effective rate is 6.1678 percent, not 6.
Why more frequent is more
Each sub-period's interest starts earning interest itself. Six percent paid once cannot do that. Six percent split into twelve payments of 0.5 percent means the first payment earns for eleven more months.
The gain shrinks fast. Going from annual to semi-annual adds 9 basis points. Going from monthly to daily adds 1.5 basis points. Going from daily to continuous adds less than a tenth of one basis point.
The continuous form
Push m towards infinity and the expression converges on e raised to the rate. At 6 percent that is 1.061837. This is why academic finance discounts with e: not because interest is paid every instant, but because a formula with no period count in it differentiates cleanly.
For an investor the practical point is narrower. Continuous and monthly compounding differ by less than two basis points a year at these levels, so the choice almost never changes a conclusion. The choice between annual and monthly can.
Where it bites
Three places, all of them ordinary. A bond quoting a semi-annual coupon yield against a loan quoting a monthly rate. A credit agreement stating a nominal rate where the effective cost is higher. And a spreadsheet that divides an annual discount rate by twelve to build monthly periods, which is exactly the error this lesson is about.
Check yourself
A stated 8 percent compounded quarterly has what effective annual rate?
- 8.00 percent
- 8.24 percent
- 32.00 percent
Show the answer
Answer: B.
1 plus 0.02, to the power 4, is 1.08243. Each quarter's interest earns interest for the rest of the year, which adds 24 basis points.Which two rates are closest to each other?
- Annual and semi-annual
- Semi-annual and monthly
- Daily and continuous
Show the answer
Answer: C.
The gain from compounding more often shrinks as the periods get shorter. Daily and continuous differ by less than a tenth of a basis point at 6 percent.You have an annual discount rate and you want monthly periods. What do you do to the rate?
- Divide it by 12
- Take the twelfth root of one plus the rate, then subtract 1
- Multiply it by 12
Show the answer
Answer: B.
Dividing by 12 produces a monthly rate that compounds back to more than the annual rate you started with. The twelfth root is the rate that compounds to exactly it.Sources
- Damodaran, Investment Valuation: the treatment of discrete against continuous compounding and when each is used.
- Williams, The Theory of Investment Value, 1938: discounting a dated stream, which is what a compounding convention makes precise.
What this leaves out
- The figures here are arithmetic on a stated rate. They say nothing about whether any rate is available to any investor.
- Day-count conventions differ between markets, so a real daily figure can vary slightly from the 365-period version used here.
- Nothing here is advice about any security.