i) Nominal, Effective and Equivalent RateTwo annual rates of interest with different conversion periods are said to be equivalent if they earn the same compound amount for the same time.

For instance, in one year, the compound amount of ₱10 invested a) at 12% compounded semi-annually is ₱10 = = ₱11.236b) At 12.36% compounded annually is ₱10 (1+ 0.1236) = ₱11.236

Notice that 12% compounded semi-annually is equivalent to 12.36% compounded annually

When interest is compounded more than once a year, the given rate is called thenominal rate. The effective rate is the rate that, when compounded annually, produces the same compound amount each year as the nominal rate j compoundedm times a year. In example above, 12% is a nominal rate while 12.36% is an effectiverate.

Derivation of formula:

Let the effective rate be denoted by w. To compute w, let a principal P be investedat these two investment rates. The amount of P at the effective rate w at the end of one year is The amount of P in one year, at the nominal rate j compounded m times a year is

Since the two compound amount are equal (= ), then by substitution, =

= , solving for w = - 1

and j can be expressed in terms of w as

Note that if the nominal rate j is compounded annually, then w = j.

Example 1: Find the effective rate equivalent to 12% compounded quarterly.

Example 2: What nominal rate compounded semi-annually is equivalent to investing at 8% effective rate?

Example 3: What rate compounded quarterly is equivalent to 14% compounded semi-annually?

Example 4: Comparison of Two Rates Which is better, to invest money at 6 % compounded monthly or 6.5% compounded semi-annually? (Hint: To compare two rates of interest is to compare their effective rates)

Example 1: Find the effective rate equivalent to 12% compounded quarterly.

Solution:Given: j = 12% = 0.12

m = 4w = ?

= - 1

= - 1 = - 1 = 0.1255088 = 12.55%

Thus, investing at 12.55% compounded annually is equivalent to investing at 12%compounded quarterly.

Example 2: What nominal rate compounded semi-annually is equivalent to investing at 8% effective rate?

Solution:Given: w = 8% = 0.08

m = 2j = ?

=

1 + 0.08 = , take square root both sides or power of ½. 1.0392305 = 0.0392305 = j = 2(0.0392305)

j = 0.078461 j = 7.85%

Example 3: What rate compounded quarterly is equivalent to 14% compounded semi-annually?

Solution:Given: = 14% = 0.14

= 2 = ?= 4

=

= 4 = 4 = 4 = 4(0.034408) = 0.1376321 = 13.76%

Thus, investing at 13.76% compounded quarterly is equivalent to investing at 14%compounded semi-annually.

Derivation:

= = = =

Example 4: Comparison of Two Rates Which is better, to invest money at 6 % compounded monthly or 6.5% compounded semi-annually? (Hint: To compare two rates of interest is to compare their effective rates)

Solution:Given: = 6% = 0.06 = 6.5% = 0.065

= 12 = 2 =? =?

= - 1 = - 1 = - 1 = - 1 = 0.0616778 = 0.0660563

Since is greater than , then it is better to invest at 6.5% compounded Semi-annually than to invest at 6% compounded monthly.

Problem Set:1. Find the effective rate equivalent to 12% compounded

a) annuallyb) semi-annuallyc) quarterlyd) monthly

2. What nominal rate, converted semi-annually, is equivalent to 6.5% effective rate?3. What rate compounded monthly will yield the effective rate 7%?4. Find the rate compounded semi-annually that is equivalent to 15% compounded monthly.5. What simple interest rate is equivalent to 12% compounded quarterly in a 3-year transaction?6. What nominal rate converted semi-annually is equivalent to 9% simple interest in a 5-year transaction?7. Which investment is better: 9% (m=2) or 9% (m=4)?8. BDO bank offers 6% (m=4) on savings account while BPI bank offers 6.5% (m=2). If you are a depositor, in which bank would you prefer to put your money?9. Which investment yields a higher interest: 8% effective rate or 7.5% compounded semi-annually?

Reference Book:1) Mathematics of Investment by Victoria C. Naval, N.B. Gorospe, et., al. 2) Fundamentals of Investment Mathematics by Presentacion Gabriel and Anita C. Ong

Assignment: Pages: 36Items: 2, 11, 12, and 14Book: Mathematics of Investment by Victoria C. Naval, N.B. Gorospe, et., al.

ii) Continuous Compounding

Interest may also be converted very frequently; weekly, daily, or hourly. What happens when interest is converted very frequently, or the frequency of conversionbecomes infinite? The table below gives the amounts for converted interest at a nominal rate of 7%, at different frequencies in one year.

The value of ₱100 at the End of One Year as m Becomes Infinite

Nominal Rate (j):7% converted

Frequency ofConversion (m)

Amount of ₱100 at the end of 1 year

annuallysemi-annuallyquarterlymonthlyweeklydaily

1241252365

₱107₱107.1225₱107.1859₱107.22901₱107.24576₱107.25006

Notice in table that as m increases, the compounded amount increases marginally.Frequency compounding thus will increase the interest, but only slightly. For example, ₱100 earns ₱7.24 when converted weekly, and ₱7.25, or a centavo more,when converted daily. For this reason, depositors need not be attracted to banksadvertising interest compounded daily unless they have considered other factors.

Interest converted very frequently is referred to as being converted continuously.

At the nominal rate j compounded continuously, the amount of a principal P invested for t years is Solving for P, the present value of F due at the end of t years at the nominal rate jcompounded continuously is

PThe base e is a constant where e = 2.71828… The value of and can be obtained using a scientific calculator. (see the derivation of these formulas in the book).

Example 1: Find the amount of ₱200 at the end two years if the interest rate is 9% compounded: a) semi-annually b) quarterly c) monthly, and d) continuously.

Example 2: Find the present value of ₱5,000 due in 5 years at 7% converted continuously.

Example 3: Find the present value of ₱750, due in 4 years, at the interest rate of 8% converted a) annually, b) quarterly, and c) continuously.

Example 4: Find the accumulated value of ₱9,000 at the end of 5 years if it is invested at 9% converted continuously.

Example 5: Find the accumulated value of ₱8,700 at the end of 10 years if it earns effective interest of 7% in the first 3 years and 10% converted continuously in the remaining years.

Example 1: Find the amount of ₱200 at the end two years if the interest rate is 9% compounded: a) semi-annually b) quarterly c) monthly, and d) continuously.

Solution:The amount at the end of 2 years at a) 9%, m = 2 is = 200 = 200 = ₱ 238.50b) 9%, m = 4 is F = ₱200 = ₱238.97c) 9%, m = 12 is F = ₱200 = ₱239.28d) 9% continuously is = ₱200 = ₱239.44

Example 2: Find the present value of ₱5,000 due in 5 years at 7% converted continuously.

Solution:

P = ₱5,000 = ₱5,000 (0.704688) = ₱3,523.44