Comprehensive Notes on Interest Rates

Interest Rates

Simple Interest

• Definition of Interest: Payment by a borrower for using a lender's capital.

• Principal: The amount borrowed or invested (original principal).

• Simple Interest (S.I.): Computed on the principal only, not reinvested, used for short-term loans (less than a year).

Elements of Simple Interest

• $s$: Principal, present value, initial value (in monetary units).

• $t$: Time period (length of the loan or investment) in years.

• $I$: Amount paid for the use of money (interest) in monetary units.

• $S$: Accumulated value, future value, or final value; $S = s + I$ (in monetary units).

• $i$: Interest rate per year (interest earned over a year when 1 m.u. is invested).

Example:

If $s = 100$, $t = \frac{2}{12} = \frac{1}{6}$ (2 months), and $i = 7\%$, then:

$I = s \cdot i \cdot t = 100 \cdot 0.07 \cdot \frac{1}{6} = \frac{7}{6} \approx 1.1667$
$S = s + I = 100 + \frac{7}{6} = \frac{607}{6} \approx 101.1667$

Simple Interest Formulas

• Interest: $I = s \cdot i \cdot t$

• Accumulated Value: $S = s + I = s + s \cdot i \cdot t = s(1 + i \cdot t)$

• Accumulation Formula: $S = s(1 + it)$

• Discount Formula: $s = \frac{S}{1 + it}$

*Time value of money: money received now is worth more than in the future

Accumulation and Discount Factors

• Accumulation factor: $u = 1 + i$ (value over 1 year of 1 m.u. of today).

• Present value (discount) factor: $v = \frac{1}{1 + i}$ (present value of 1 m.u. a year from now).

Basic Formulas Summary

• $I = s \cdot i \cdot t$

• $S = s(1 + i \cdot t)$ (accumulation relation; given $s, i, t \implies S$)

• $s = \frac{S}{1 + i \cdot t}$ (present value/discount relation; given $S, i, t \implies s$)

Time period Considerations

• The interest rate ($i$) and time period ($t$) must be consistent.

• If the time is given in months or days, convert it to years.

• Interest can be calculated using the formulas:

  • $I = s \cdot i \cdot \frac{\text{no. of days}}{360 (\text{ordinary interest}) \text{ or } 365}$

  • $I = s \cdot i \cdot \frac{\text{no. of months}}{12}$

• Splitting the year into m equal periods.

  • $t_m = t \cdot m$ is the time measured in the number of periods.

  • $I = s \cdot i<em>m \cdot t</em>m$

  • $S = s(1 + i<em>m \cdot t</em>m)$

Equivalent interest rates

• Two interest rates are said to be equivalent if they produce the same interest over the same period of time starting from the same principal

• $i = m \cdot i_m$

Compound Interest (C.I.)

• Interest due is added to the principal at the end of each interest period and earns interest in the next period.

• The time between successive interest computations is the compounding or conversion period.

• C.I. is used for loans longer than one year.

Compound Interest Example

Find the accumulated value of $1,000 after three years at an interest rate of 24% per year convertible annually:

• After 1 year: $1000 \cdot (1 + 0.24 \cdot 1) = 1000 \cdot 1.24 = 1240$

• After 2 years: $1240 \cdot (1 + 0.24 \cdot 1) = 1240 \cdot 1.24 = 1537.6$

• After 3 years: $1537.6 \cdot (1 + 0.24 \cdot 1) = 1537.6 \cdot 1.24 = 1906.624$

• Alternatively:$S=1000 \cdot 1.24^3 = 1906.624$

Compound Interest Formulas

• $S = s(1 + i)^t$

• Where:

  • $i$ = interest rate/year

  • $t$ = time in years

• Interest earned: $I = S - s = s[(1 + i)^t - 1]$

• Accumulation factor: $u = 1 + i$

• $S = s \cdot u^t$

• Discount factor: $v = \frac{1}{1 + i}$

• $s = S \cdot v^t = \frac{S}{(1 + i)^t}$

Basic Formulas (Conversion period = 1 year)

• $S = s(1 + i)^t = s \cdot u^t$ (accumulation relation; $S$ = final/future value).

• $s = \frac{S}{(1 + i)^t} = S \cdot v^t$ (present value/discount relation; $s$ = principal/present value).

• $I = S - s = s(u^t - 1)$

• $i = \sqrt[t]{\frac{S}{s}} - 1$

Example: Doubling Capital

How long will it take to double a capital attracting annual interest of 6%?

• $i = 6\% = 0.06$

• $S = 2s$

• $S = s(1 + i)^t$

• $2 = (1.06)^t$

• $ln(2) = t \cdot ln(1.06)$

• $t = \frac{ln(2)}{ln(1.06)} \approx 11.895 \text{ years}$

Basic Formulas (m compounding periods per year)

• $S = s(1 + i<em>m)^{t \cdot m} = s \cdot u</em>m^{t \cdot m}$

Compound vs. Simple Interest Example

• Compute the compound interest earned on $1000 for one year at 1%/day. Compare with simple interest.

• Simple interest: $I = s \cdot i \cdot t = 1000 \cdot 0.01 \cdot 365 = 3650$

• Compound interest: $S = 1000(1 + 0.01)^{365} = 1000 \cdot (1.01)^{365} \approx 37,783.43$

• $I_c=S-s=37,783.43-1000=36,783.43$

• Remark: $i = (1+\frac{i_m}{m})^m -1$

Nominal Interest Rate (N.I.R.)

• An interest rate is called nominal if the period of time on which the interest rate is announced (usually one year) is different from the compounding period.

• Notation: $p_m$, where $m$ is the number of compounding periods.

• $i<em>m = \frac{p</em>m}{m}$

N.I.R. Basic Formulas

• $p<em>m$ = N.I.R./year; $m$ = compounding periods/year; im = interest rate/period; t = term in years; $t</em>m = t \cdot m$ = term in no. of periods.

• $S = s(1 + i<em>m)^{t \cdot m} = s(1 + \frac{p</em>m}{m})^{t \cdot m}$

Effective Interest Rate

• $i_{eff}$ is the interest earned over a year when 1 m.u. is invested.

• Remark: $i<em>{eff} = (1+\frac{p</em>m}{m})^m -1$

The Time Value of Money

• Receiving $100 today is not the same as receiving $100 one year ago or one year from now.

• Cannot add, subtract, or compare payments made at different times directly.

• Need to determine the value of payments at the same moment by accumulating or discounting.

• $(n, t<em>1)$ (n due at time $t</em>1$) is equivalent to $(r, t)$ at a given interest rate $i$:

  • $r = n(1 + i)^{t - t_1}$

• If t > t1, move money forward in time (accumulate): $r = n(1 + i)^{t - t</em>1}$
  Time Diagram = Accumulation

• If t < t1, move money backward in time (discount): $r = n(1 + i)^{t - t</em>1}$
  Time Diagram = Discount