1. Welcome To
A Session On
Annuity
©. S. B. Bhattacharjee

2. Business Mathematics (Eleventh Edition)
Your Textbook
Business Mathematics (Eleventh Edition) By
D.C. Sancheti  V.K. Kapoor
Chapter 8
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3. What is meant by Annuity?
Suppose, we want to make payment of Tk.5000 each year for 10 years to an account which earns 6 percent interest, compounded annually, and we want to calculate the total amount which will be in the account after 10 payments.
This series of periodic payments is called an annuity.
An ordinary annuity is a series of equal payments, each payment of which is made at the end of a period. Different annuities may be stated as under:
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4. i = Interest rate per period
M = Sum of the annuity
A = Payment per period
i = Interest rate per period
n = Number of periods
Continued…..
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5. What are various types of annuity?
Continued……
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6. Different types of annuities Annuity
Annuity Contingent
(Payments are to be made till the happening of some contingent event such as the death of a person, marriage of a girl ete.)
Annuity Certain
(Payments are to be made for a certain or fixed number of years)
Annuity due
(Where the first payment falls due at the beginning of the ist interval, and so on)
Annuity immediate
(Where the first payment falls due at the end of the first interval, and so on)
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7. Calculation of different types of annuities:
1. Formula for the present value of an Annuity due
Continued…..
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8. 2. Formula for the present value of an Immediate Annuity
3. Formula for the amount of an Immediate Annuity
4. Formula for the amount of Annuity due
5. Present Value of an Annuity
The present value of an Annuity is the sum of the present values of its installments.
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9. Example: 1
How much must be deposited each year at 5 percent compounded annually to accumulate Tk.1000 after 10 years from now?
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Continued…

10. Log (1.05)10
=10 log 1.05
= 10× =
Antilog = =
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11. A = Annuity (annual payment) i = Rate of interest n = Number of years
Example: 2
A man borrows Tk at 6% and promises to pay off the loan in 20 annual payments beginning at the end of the first year. What is the annual payment necessary?
, where
V = Present value
A = Annuity (annual payment)
i = Rate of interest
n = Number of years
Continued..
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12. Continued……
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13. Log (1.06)20 = 20 log 1.06 =20×0.0253 =0.506 Antilog 0.506 =3.206
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14. The sum will increase by 25%
Example: 3
How long will it take for a sum of money to increase by 25 percent at compound interest of 5 percent per year? [ Mathematics- Bowen ]
Answer :
The sum will increase by 25%
This means if the principal amount is Tk. 100, the increased compounded amount will be Tk (100+25) =Tk.125
A = Tk.125
P = Tk.100
Continued…
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16. Example: 4
How long will it take for sum of money to double itself at 6 percent, compounded annually? [Bowen]
Answer :
The compound amount will be double. Hence, if the principal amount is x, the compounded amount will be 2x.
We know,
A= P(1+r)n, where
A= Compounded amount
P = Principal amount
r = rate of interest
n = period
Here, A = 2x
Continued…
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18. b) What will be the simple interest in the above case?
Example: 5
Find the compound interest on Tk. 1,00,000 for 4 years at 5% per annum.
b) What will be the simple interest in the above case?
Answer :
We know,
A= P (1+r)n, where
A=compounded amount
P = Principal amount
r= Rate of interest
n = years
Continued…
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20. Now compound Interest = A - P =Tk.1,21,500 -1,00,000 = Tk.21,500/-
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Continued…

21. What will be the simple interest in the above case?
Simple Interest = P n r
= 1,00,000×4×0.05
= Tk. 20,000/-
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22. Find the compound interest on 25,800 for 5 years, if the
Example:6
Find the compound interest on 25,800 for 5 years, if the
rate of interest be 2% in the 1st year, interest in the second year,3% in the 3rd year and thereafter at 4% p.a.
We know,
A = P( 1+r)n, where
A = Compounded amount
r = Rate of interest per annum
n = Period
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Continued…

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Continued…

24. = 2.5% Continued… Rate of interest in the 2nd year =
Rate of interest in the 3rd year=3%
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Continued…

25. 0.04 log x = log (1.04)2 =2 log 1.04 =2×0.0170 = 0.034 Let x = (1.04)2
Rate of interest after 3rd year
0.04
log x = log (1.04)2
=2 log 1.04
=2×0.0170
= 0.034
Let x = (1.04)2
x = Antilog 0.034
=1.0816
(1.04)2=1.0816
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26. Example: 7
Find the compound interest on Tk. 6,950 for 3 years, if interest is payable half yearly, rate for the first two years being 6%, and for the third year 9% p.a.
P = 6950
n= 2,
.06
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Continued…

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28. Example: 8
Depreciation
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Continued…

29. We know, SV = PV (1-d)n, where SV = Scrap value PV = Purchase value
d = depreciation
n = period
Here, SV = Tk. 2250
PV = Tk. 5810
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Continued…

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31. Example: 9
A company buys a machine for TK 1,00,000. Its estimated life is 12 years and scrap value is TK What amount is to be retained every year from the profit and allowed to accumulate at 5 % C.I. for buying a new machine at the same price after 12 years. Or
The cost of a machine is TK. 1,00,000 and its effective life is 12 years. If the scrap realises only TK. 5,000, what amount should be retained out of profits at the end of each year to accumulate at 5 % C.I. p.a.
Continued………
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32. Amount of the annuity = Cost – Scrap value = TK 1,00,000 – TK 5,000
Answer.
Cost of the machine = TK. 1,00,000
Its scrap value = TK 5,000
Amount of the annuity = Cost – Scrap value
= TK 1,00,000 – TK 5,000
= TK.95,000
n = 12, i = 5%=
= 0.05
Annuity , A = ?
Continued……
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33. Let X = ( 1.05)12 log X = log (1.05)12 = 12 log 1.05 = 12×0.0212=
X = Antilog =
(1.05)12 =
Continued….
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35. Example: 10
A machine is depreciated in a such way that the value of the machine at the end of any year is 90 % of the value at the beginning of the year. The cost of the machine was TK. 12,000 and it was sold eventually as waste material for Tk 200. Find out the number of years during which the machine was in use.
We know
SV = PV (1- d )n, where
SV= Scrap value
PV= Purchase value
d = depreciation
n = period
Continued…
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36. Selling price i.e. Scrap value = TK. 200 PV = TK. 12,000 d =
Here,
Selling price i.e. Scrap value = TK. 200
PV = TK. 12,000
d =
Continued….
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37. the number of years during which the machine was in use = 39 years.
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38. Thank you
For
Attending
The Session
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