Compound Interest and Effective Rates

EXERCISE 01

Deposit €2,000 in an account quoting a nominal annual rate of 7.2%, compounded monthly. No further deposits or withdrawals are made.

A.Calculate the balance after four years and after four and a half years.
B.Calculate the effective annual rate.

A.

n_{4y}=4(12)=48

FV_{48}=2{,}000(1.006)^{48}\approx 2{,}665.22

n_{4.5y}=4.5(12)=54

FV_{54}=2{,}000(1.006)^{54}\approx 2{,}762.62

FV_n=2{,}000\left(1+\frac{0.072}{12}\right)^n

Horizon	Months	Balance (€)
4 years	48	2,665.22
4.5 years	54	2,762.62

B.

EAR=\left(1+\frac{0.072}{12}\right)^{12}-1

7.4424% per year.

Match the number of periods to the periodic rate: 54 months requires a monthly rate, not an annual rate.

Compound Interest and Effective Rates

EXERCISE 02

Use compound-interest equivalence. A stated nominal annual rate is divided by its conversion frequency before compounding.

A.Convert a 3% effective monthly rate into an effective annual rate.
B.Convert a 12% effective annual rate into an effective four-month rate.
C.Find the monthly rate from an 8% nominal annual rate convertible monthly.
D.Convert a 12% nominal annual rate convertible quarterly into an effective two-month rate.

A.

i_{annual}=(1+i_{period})^m-1

i_a=(1.03)^{12}-1

42.576089% per year.

B.

i_{4m}=1.12^{1/3}-1

3.849882% per four months.

C.

i_{period}=\frac{j^{(m)}}{m}

i_m=\frac{0.08}{12}

0.666667% per month.

D.

i_q=\frac{0.12}{4}=0.03

i_{2m}=(1.03)^{2/3}-1

1.990131% per two months.

Compound Interest and Effective Rates

EXERCISE 03

Invest €2,000 for four months at an effective annual rate of 10%. Use compound interest and a fractional year of 4/12.

A.Calculate the accumulated amount.

A.

FV=C(1+i)^t

FV=2{,}000(1.10)^{4/12}

€2,064.56

Compound Interest and Effective Rates

EXERCISE 04

€4,000 earns 10.50% effective annual compound interest until it reaches €5,800.

A.Find the required time in years.

A.

t=\frac{\ln(M/C)}{\ln(1+i)}

t=\frac{\ln(5{,}800/4{,}000)}{\ln(1.105)}

3.721391 years.

Compound Interest and Effective Rates

EXERCISE 05

A capital triples in 7 years, 7 months and 15 days under compound interest. Use a 30/360 year.

A.Find the effective two-month interest rate.

A.

i=(M/C)^{1/n}-1

n=6\left(7+\frac7{12}+\frac{15}{360}\right)=45.75

i_{2m}=3^{1/45.75}-1

2.430403% per two months.

Compound Interest and Effective Rates

EXERCISE 06

Invest €10,000 for six years: 5% annually in year 1, 7% in years 2–3 and 4% in years 4–6. Under simple interest, all interest is calculated on the original principal; under compound interest, it is reinvested.

A.Find the final amount and equivalent constant annual rate under both regimes.

A.

M_s=10{,}000[1+0.05+2(0.07)+3(0.04)]=13{,}100

\bar i_s=\frac{0.05+2(0.07)+3(0.04)}6

5.166667% simple annual rate.

M_c=10{,}000(1.05)(1.07)^2(1.04)^3

Compound amount: €13,522.50.

\bar i_c=[1.05(1.07)^2(1.04)^3]^{1/6}-1

5.158120% effective annual rate.