Sep 12

Compound Frequencies (simple, semiannual, monthly, daily). 2007 FRM.

by David Harper, CFA, FRM, CIPM

FRM |

Learning Outcome

  • LO 21.6: Calculate simple, semiannual, monthly, daily, and continuously compounded rates given a discount factor or market interest rate for a specified interval.

Annual rate versus simple rate

If we say "the account will earn an annual interest rate of 10%," we are a bit unclear unless we also indicate the compounding frequency. If 10% is compounded annually, then the $1 grows to $1.10 at the end of one year:

compoundAnnual

 

Note that compounding annually is not the same thing as the simple interest rate. The simple interest rate is not compounded at all: a 10% simple interest rate means that one dollar borrowed becomes $1.10 at the end of some period, any period. To clarify:

  • If the annual rate is 10%, $1 compounded annually becomes $1.21 at the end of two years
  • If the simple rate is 10%, $1 becomes $1.10 at the end of whatever period (e.g., three months, two years)

 

Semi-annual compounding

Keep the interest rate at 10%, but now compound semiannually. If we let (m) equal the number of times we compound during the year, then semiannual compounding implies m=2. At the end of the year, $1 becomes a little more than $1.10 because ($1.05)($1.05)=$1.1025:

compoundSemiAnnual

 

Continuous compounding

If (m) is the number of times per year we compound, we can compound quarterly (m=4), monthly (m=12), daily (m=250 trading days or 360 calendar-like days). How far can we take this? Can we follow Buzz Lightyear 'to infinity and beyond?' Almost! We can compound continuously. This is money working at it hardest:

compoundContinuous

 

Continuous compounding is common in academic literature. The formula is elegance itself: Terminal value = (Initial value)exp[(rate)(number of years)]. So, above, $1.1052 = ($1)exp(10%). If we continuously compound at 10% over five years, the terminal value = $1.65 = ($1)exp[(10%)(5)].

 

Formulas and EditGrid spreadsheet

Given the following assumptions:

  • A = initial value
  • R = Annual interest rate
  • Rm = Interest rate compounded (m) times per year
  • Rc = Interest rate continuously compounded

If the initial value is (A) and the annual interest rate is (R), we compound (m) times per year over (n) number of years to get the terminal value:

compoundtotermina  

 

To solve for the continuous rate (Rc) given the periodic rate (Rm), use:

continuousreturnconversion

 

Conversely, to solve for the period rate (Rm) given the continuous rate (Rc), use:

periodicreturnconversion

 

The EditGrid spreadsheet below clarifies this is two brief columns. Note the assumptions: we invest $1 over 10 years at an annual rate of 10% (in yellow). The first column computes the terminal value under progressively more frequent compounding intervals. The second column computes the period returns that produce equivalent terminal values. In other words, the first column keeps the rate the same (10%) so the terminal values are higher with greater compounding frequency. The second column keeps the terminal value the same ($2.72) so the periodic returns change with the compounding frequency.

EditGrid Spreadsheet by bt/frm2007.

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