(a) For an interest rate of 18% per year, compounded continuously, calculate the effective monthly and annual interest rates.
(b) An investor requires an effective return of at least 15%. What is the minimum annual nominal rate that is acceptable for continuous compounding?
(a) The nominal monthly rate is r = 18%/12 = 1.5%, or 0.015 per month. By Equation [4.11], the effective monthly rate is
\quad \quad \quad i % per month = { e }^{ r } − 1 = { e }^{ 0.015 } – 1 = 1.511%
Similarly, the effective annual rate using r = 0.18 per year is
\quad \quad \quad i % per year = { e }^{ r } − 1 = { e }^{ 0.018 } − 1 = 19.722%
(b) Solve Equation [4.11] for r by taking the natural logarithm.
\quad \quad \quad { e }^{ r } − 1 = 0.15\quad \quad\quad \quad { e }^{ r } = 1.15
\quad \quad \quad \ln { { e }^{ r } } = \ln { 1.15 }
\quad \quad\quad \quad r = 0.13976
Therefore, a rate of 13.976% per year, compounded continuously, will generate an effective
15% per year return. The general formula to find the nominal rate, given the effective
continuous rate i , is r = \ln { (1+i) }.
Solution by Spreadsheet:
(a) Use the EFFECT function with the nominal monthly rate r = 1.5% and annual rate r = 18% with a large m to display effective i values. The functions to enter on a spreadsheet
and the responses are as follows:
Monthly: = EFFECT(1.5%,10000) effective i = 1.511% per month
Annual: = EFFECT(18%,10000) effective i = 19.722% per year
(b) Use the function in Equation [4.6] in the format = NOMINAL(15%,10000) to display the nominal rate of 13.976% per year, compounded continuously.
\quad \quad \quad i = { e }^{ r } – 1 \quad \quad\quad \quad\quad\quad [4.11]
The NOMINAL spreadsheet function finds the nominal annual rate r. The format is
=NOMINAL(effective_rate,compounding_frequency_per_year)
= NOMINAL( { i }_{ a }%, m ) \quad \quad \quad [4.6]