Question

Periodic interest rates.You have a savings account in which you leave the funds for one year without adding to or withdrawing from the account. Which would you rather​ have: a daily compounded rate of 0.040​%, a weekly compounded rate of 0.285​%, a monthly compounded rate of .35%, a quarterly compounded rater of 4.00​%, a semiannually compounded rate of 7.5%, or an annually compounded rate of 14%? What is the effective annual rate​ (EAR) of a daily compounded rate of 0.040​%?

​(Round to two decimal​ places.)

Answer 1

Given the following information,

Daily Compounded rate	0.040%
weekly Compounded rate	0.285%
Monthly Compounded rate	0.35%
Quarterly Compounded rate	4.00%
Semiannually Compounded rate	7.50%
Annually Compounded rate	14%

Part 1: Daily compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 365

APR = Annual Percentage Rate

Since,

daily compounded rate = APR/365

So,

APR = 365*Daily compounded rate

APR=365*0.04 = 14.6%= 14.6/100 = 0.146

Substituting these in the EAR formula, we get

EAR = (1+ (0.146/365))^365 - 1

EAR = (1+ (0.0004))^365 - 1

EAR = (1.0004)^365 - 1

EAR = 1.1572 - 1

EAR = 0.1572 = 15.72%

Part 2: Weekly compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 52

APR = Annual Percentage Rate

Since,

Weekly compounded rate = APR/52

So,

APR = 52*Weekly compounded rate

APR = 52*0.285 = 14.82%= 14.82/100 = 0.1482

Substituting these in the EAR formula, we get

EAR = (1+ (0.1482/52))^52 - 1

EAR = (1+ (0.0028))^52 - 1

EAR = (1.0028)^52 - 1

EAR = 1.1595 - 1

EAR = 0.1595 = 15.95%

Part 3: Monthly compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 12

APR = Annual Percentage Rate

Since,

Monthly compounded rate = APR/12

So,

APR = 12*Monthly compounded rate

APR = 12*0.35 = 4.2%= 4.2/100 = 0.042

Substituting these in the EAR formula, we get

EAR = (1+ (0.042/12))^12 - 1

EAR = (1+ (0.0035))^12 - 1

EAR = (1.0035)^12 - 1

EAR = 1.0428 - 1

EAR = 0.0428 = 4.28%

Part 4: Quarterly compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 4

APR = Annual Percentage Rate

Since,

Quarterly compounded rate = APR/4

So,

APR = 4*Quarterly compounded rate

APR = 4*4 = 16%= 16/100 = 0.16

Substituting these in the EAR formula, we get

EAR = (1+ (0.16/4))^4 - 1

EAR = (1+ (0.04))^4 - 1

EAR = (1.04)^4 - 1

EAR = 1.1699 - 1

EAR = 0.1699 = 16.99%

Part 5: Semi-annually compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 2

APR = Annual Percentage Rate

Since,

semiannually compounded rate = APR/2

So,

APR = 2*Semi-annually compounded rate

APR = 2*7.5 = 15%= 15/100 = 0.15

Substituting these in the EAR formula, we get

EAR = (1+ (0.15/2))^2 - 1

EAR = (1+ (0.075))^2 - 1

EAR = (1.075)^2 - 1

EAR = 1.1556 - 1

EAR = 0.1556 = 15.56%

Part 6: Annually compounding rate

We know that effective annual rate formula is given by,

EAR = (1+ (APR/n))^n - 1

Where EAR = Effective Annual Rate

n = number of compounding periods = 1

APR = Annual Percentage Rate

Since,

annually compounded rate = APR/1

So,

APR = 1*annually compounded rate

APR = 1*14 = 14%= 14/100 = 0.14

Substituting these in the EAR formula, we get

EAR = (1+ (0.14/1))^1 - 1

EAR = (1+ (0.14))^1 - 1

EAR = (1.14)^1 - 1

EAR = 1.14 - 1

EAR = 0.14 = 14%

In this case since the compounding period is one year APR = EAR = 14%

Therefore,

Type of interest rate	Calculated EAR(%)
Daily Compounded rate	15.72
weekly Compounded rate	15.95
Monthly Compounded rate	4.28
Quarterly Compounded rate	16.99
Semiannually Compounded rate	15.56
Annually Compounded rate	14

From the above table it is easy to understand that the quarterly compounded rate gives the gives EAR. Hence I invest my funds in savings account which gives quarterly Compounded rate of 16.99%.