Simple Interest
- Simple interest is calculated using the
principal only, ignoring any interest that had
been accrued in preceding periods. In
practice, sample interest is paid on short-term
loans in which the time of the loan is
measured in days.
I = Pni
F = P + I = P + Pni
F = P(1 + ni)
where:
I = interest
P = principal or present worth
n = number of interest periods
i = rate of interest per interest period
F = accumulated amount or future worth
(a) Ordinary simple interest is computed on the basis
of 12 months of 30 days each or 360 days a year.
1 interest period = 360 days
(b) Exact simple interest is based on the exact number
of days in a year, 365 days for an ordinary year
and 366 days for a leap year
1 interest period = 365 or 366 days
Cash-Flow Diagrams
- A cash-flow diagram is simply a graphical
representation of cash flows drawn on a time
scale. Cash-flow diagram for economic
analysis problems is analogous to that of free
body diagram for mechanics problems
- receipt (positive cash flow or cash inflow)
- disbursement (negative cash flow or cash
outflow)
Compound Interest
- In calculations of compound interest, the
interest for an interest period is calculated on
the principal plus total amount of interest
accumulated in previous periods. Thus,
compound interest means "interest on top of
interest”
𝐅 = 𝐏 (𝟏 + 𝐢)𝐧
The quantity (1 + i)^n is commonly called the "single
payment compound amount factor and is designated
by the functional symbol P/Pn. Thus
𝐅 = 𝐏 (𝐅/𝐏. 𝐢%. 𝐧 )
The symbol F/P. 1%, n is read as "F given P at i per
cent in a interest periods:
𝐏 = 𝐅 (𝟏 + 𝐢)−𝐧
Rates of Interest
(a) Nominal rate of interest
- The nominal rate of interest specifies the rate
of interest and a number of interest periods in
one year
𝒓
𝒊=
𝒎
where: I = rate of interest per interest period
r = nominal interest rate
m = number of compounding periods per
year
If the nominal rate of interest is 10% compounded
quarterly then i=10%/4 = 2.5%, the rate of interest
per interest period
(b) Effective rate of interest
- Effective rate of interest is the actual or exact
rate of interest on the principal during one
year.
- If P1.00 is invested at a nominal rate of 15%
compounded quarterly, after one year this
will become
𝒊𝒆 = (𝟏 +
𝒓 𝒎
)
𝒎
Equation of Value
- An equation of value is obtained by setting
the sum of the values on a certain comparison
or focal date of one set of obligations equal to
the sum of the values on the same date of
another set of obligations
Discount
- Discount on a negotiable paper is the
difference between the present worth (the
amount received for the paper in cash) and
the worth of the paper at some time in the
future (the face value of the paper or
principal). Discount is interest paid in
advance
𝒅 = 𝟏 − (𝟏 + 𝒊)−𝟏
𝒅
𝒊=
𝟏−𝒅
where: d = rate of discount for the period involved
i = rate of interest for the same period
Continuous Compounding and Discrete Payments
In discrete compounding the interest is compounded
at the end of each finite-length period, such as a
month, a quarter or a year. In continuous
compounding, it is assumed that cash payments
occur once per year. but the compounding is
continuous throughout the year,
𝑭 = 𝑷(𝟏 +
𝒓 𝒎𝒏
)
𝒎
Where: r = nominal rate of interest per year
r/m = rate of interest per period
m = number of interest periods per year
mn = number of interest periods in n years
ANNUITIES
An annuity is a series of equal payments occurring at
equal periods of time
P = value or sum of money at present
F = value or sum of money at some future time
A = a series of periodic, equal amounts of
money
n = number of interest periods
i = interest rate per interest period
Ordinary Annuity
- An ordinary annuity is one where the
payments are made at the end of each period
(1 + 𝑖)𝑛 − 1
𝐹 = 𝐴[
]
𝑖
1 − (1 + 𝑖)−𝑛
𝑃 = 𝐴[
]
𝑖
Deferred Annuity
- A deferred annuity is one where the first
payment is made several periods after the
beginning of the annuity.
(1 + 𝑖)𝑛 − 1
𝐹 = 𝐴[
] (1 + 𝑖)−𝑚
𝑖
1 − (1 + 𝑖)−𝑛
𝑃 = 𝐴[
] (1 + 𝑖)𝑚
𝑖