Module-5-Annuties
Introduction
Annuity: A series of payments made at
equal intervals of time.
Examples: House rents, mortgage
payments, instalment payments on
automobiles and interest payments on
money invested.
.
R- The periodic payment of the annuity
n- The numer of payments made
i- the interest rate per payment interval
S- The accumulated value of the annuity; the amount of money you will
have at the end of the annuity’s term.
A- The present value of the annuity or discounted value, the amount of
money which must be set aside today to allow a specified payment for a
predetermined period of time.
Terms used in Annuity
Annuity-certain: An annuity such that payments
are certain to be made for a fixed period of
time.
Example: A 30 year mortgage, because there
are precisely 360 monthly payments
made during the duration of the loan
Contingent annuity:
An annuity under which the payments are not
certain to be made.
A common type of contingent annuity is one in
which payments are made only if a person is alive
(Life Annuity).
Example:A monthly retirement payment paid
during the life of a retiree.
Term: The fixed period of time for which
payments are made.
Payment interval or conversion period:The
time between payments is called the
payment interval.
Simple Annuity:When the payment period
coinsides with the conversion period, the
annuity is said to e simple.
Annuity-immediate:
An annuity under which payments of 1 are
made at the end of each period for n
periods.
Annuity-Due:
The payments are made at the beginning of
the period
General Annuity:Annuity is not simple is
called general annuity.
Ordinary Annuity:The payments are made
at the end of each payment interval the
annuity is called ordinar annuity.
Accumulated value
The accumulated value of an ordinary
annuity is computed by
S=RX accumulation factor.
Accumulated value of an annuity due is
computed by
S=Rxaccumulation factorX(1+i)
Present Value
Present value of an ordinary annuity is
computed by
A=RX discount factor.
Accumulated value of an annuity due is
computed by
A=Rxdiscount factorX(1+i)
Example-1
Find the accumulated value of an annuity of $700
invested at the end of each quarter for 5 years at
an annual rate of 8% compounded quarterly.
Solution:
Accumulation factor for n=20(4X5) and
i=0.08/4=0.02 is=24.29737
S=RX accumulation
factor=$700X24.29737=17008.159
Example-2
Find the accumulated value of an annuity of $700
invested at the beginning of each quarter for 5 years
at an annual rate of 8% compounded quarterly.
Solution:
Accumulation factor for n=20(4X5) and i=0.08/4=0.02
is=24.29737
S=RX accumulation
factorX(1+i)=$700X24.29737X1.02=17348.322
Example-3
Find the present value of an annuity of $300 at the end of
each month for 5 years at 9% compounded monthly.
Solution:
Discount factor for n=60(12X5) and i=0.009/12=0.0075 is
48.17337.
A=RX discount factor=$300X48.17337=14452.011
Example-4
Find the present value of an annuity of $300 at the
begining of each month for 5 years at 9%
compounded monthly.
Solution:
Discount factor for n=60(12X5) and
i=0.009/12=0.0075 is 48.17337.
A=RX discount X(1+i)
factor=$300X48.17337X1.0075=14560.4011
Sinking Fund Method:
In Sinking Fund Method will provide us with an
amount of depreciation as well as provide funds for
the replacement of this asset when an asset need
replacement like the end of life of an asset. Under
this method, we charged depreciation on the value
of asset but will not be credited to the asset
account instead we will credit to sinking fund
account.
At the end of each accounting year, the total
amount of sinking fund credited in a year will
be invested in the outside marketable
security to provide cash for the replacement
of an asset when needed.
What is the Sinking Fund Method?
The sinking fund method is a technique
for depreciating an asset while generating
enough money to replace it at the end of
its useful life. As depreciation charges are
incurred to reflect the asset's falling value, a
matching amount of cash is invested. These
funds sit in a sinking fund account and
generate interest.
The sinking fund method is a depreciation
technique used to finance the replacement
of an asset at the end of its useful life.
Companies rarely use the sinking fund
method of depreciation because of its
complexity.The sinking fund method is
mainly used by large-scale industries, such
as utility companies, that require expensive,
long-term assets to function.
Example
The accumulation of sinking fund in which $10000 is to accumulated in
1 year if payments are made Quarterly into an account which pays 5%
compounded quarterly, for the above case Construct a sinking fund
schedule.
Solution:
𝑆
10000
𝑅 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 = 4.07563=2453.60
Perio
d
Initial
Interest
amoun at 1.25%
t
Quarterl Increase Final
y
in fund mount
paymen
t
1
0
0
2453.60
2453.60
2453.60
2
2453.6
0
30.67
2453.60
2483.67
4953.67
3
4953.6
7
55.50
2453.60
2509.10
7462.77
4
7462.7
7
93.28
2453.60
2546.88
10009.65
Amortization
Amortization typically refers to the process
of writing down the value of either a loan or
an intangible asset. It also refers to
the repayment of loan principal over time.
Amortization schedules are used by lenders,
such as financial institutions, to present a
loan repayment schedule based on a
specific maturity date
In business, amortization refers to spreading
payments over multiple periods. The term is
used for two separate processes:
amortization of loans and amortization of
assets
Amortization is a method of spreading the
cost of an intangible asset over a specific
period of time, which is usually the course of
its useful life. Intangible assets are nonphysical assets that are nonetheless
essential to a company, such as patents,
trademarks, and copyrights. The goal in
amortizing an asset is to match the expense
of acquiring it with the revenue it generates.
Example
Construct an amortization schedule for a one and half year loan
of $5000 at 5% interest which is to repaid in quarterly
installments over one and half years.
Solution:
𝐴
5000
𝑅 = 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 = 5.74601=870.1690
Paymen Paymen Interest
t
t
Number amount
Reducti Principa
on to
l
principa balance
l
$5000
1
870.169
0
62.5
807.669
4192.33
1
2
870.169
0
52.40
817.769
3374.56
2
3
870.169
0
42.18
827.989
2546.57
3
4
870.169
0
31.83
838.339
1708.23
4
Difference between depreciation and
Amortization
• The key difference between amortization and depreciation is that amortization
is used for intangible assets, while depreciation is used for tangible assets.
• Another major difference is that amortization is almost always implemented
using the straight-line method, whereas depreciation can be implemented
using either the straight-line or accelerated method.
• Finally, because they are intangible, amortized assets do not have a salvage
value, which is the estimated resale value of an asset at the end of its useful
life. Depreciated assets, by contrast, often have a salvage value. An asset's
salvage value must be subtracted from its cost to determine the amount in
which it can be depreciated.
Capital Budgeting
Capital budgeting is the process a business
undertakes to evaluate potential major
projects or investments. Construction of a
new plant or a big investment in an outside
venture are examples of projects that would
require capital budgeting before they are
approved or rejected.
As part of capital budgeting, a company
might assess a prospective project's lifetime
cash inflows and outflows to determine
whether the potential returns that would be
generated meet a sufficient target
benchmark. The process is also known as
investment appraisal.
Capital budgeting is a process of evaluating
investments and huge expenses in order to
obtain the best returns on investment.
Capital budgeting is a predominant function
of management. Right decisions taken can
lead the business to great heights. However,
a single wrong decision can inch the
business closer to shut down due to the
number of funds involved and the tenure of
these projects.
FEATURES OF CAPITAL BUDGETING
• It involves high risk
• Large profits are estimated
• Long time period between the initial
investments and estimated returns
Example
Which would you prefer to receive(i) $12000 at the end of 3 years plus
$25000 at the end of 5 years, or (ii) $2250 at the end of each quarter
for the next 5 years.Assume that money is worth 10% annually and is
compounded quarterly.
Solution:
The present value of $ 12000 , 3 years from now is
𝑆
12000
12000
𝐴 = (1:𝑖)𝑛=(1.025)12 = 1.34489=8922.66(n=3X4=12; i=0.10/4=0.025)
The present value of $25000, 5 years from now is
𝑆
25000
25000
𝐴 = (1:𝑖)𝑛=(1.025)20 = 1.63862=15256.74(n=3X4=12; i=0.10/4=0.025)
The total present value under option(i) is $24179.4004
Under option (ii) the present value of $2250 per quarter for 5 years is
A=Rxdiscount factor=$2250X15.58916=$35075.61
Option (ii) is preferale, since the present value of the money is larger.
Thank You