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9 MATHEMATICS OF FINANCEANNUITY
CHAPTER
A sequence of payments, generally equal in size, made at equal intervals of times is called an
Monthly Rent; premiums of LIC; deposit into a recurring account in a bank; equal monthly payments got by a retired government servant as pension and loan instalments to houses or automobiles etc.
The size of each payment of an annuity is called the periodic payment of the annuity.
The sum of all payments of an annuity made in one year is called its annual rent.
The duration between two successive payments of an annuity is called the payment period (or payment interval) of the annuity.
end of the last payment period is called the of the annuity.
The total Value of all the payments at the maturity time of an annuity is called the amount (or future value) of the annuity.
Sum of the present values of all the payments of an annuity is called the present value or capital value of the annuity.
TYPES OF ANNUITIES
Ordinary Annuity: If the payments of an annuity are made at the end of payment interval is called An Ordinary annuity or Regular annuity.
Annuity Due: If the payments of an annuity are made at the beginning of payment interval is called An Annuity Due or Annuity Immediate.
Perpetuity: A perpetuity is an annuity whose payments continue forever.
Note: In what is to follow, it is understood that the payment interval coincides with the interest period unless statement to the contrary is made.
ORDINARY ANNUITY OR ANNUITY REGULAR
Payments of an annuity are made at the end of payment interval.
i r m n () . 11 100
Where S = Amount of an Annuity
A= Value of each instalment
r = rate of interest
m = No. of conversion periods in a year
n = m.t = No. of instalments made in t years.
i = r m100 = Rate of interest of one conversion Period
Find (1 + i)n by calculator i.e. Type r 100 m + 1 Then push button then push = button (n - 1) times.
Then - 1 r 100m
Then A push = button (We get the required value of Amount)
Find the future value of an annuity of `500 is made annually for 7 years at interest rate of 14% compounded annually. [Given that (1.14)7 = 2.5023] (a) `5365.25 (b) `5265.25 (c) `5465.25 (d) None a SA i r m n () . 11 100536525 `
500 1 14 100 1 14 100536525 7 `
Find 14 100 1 7 As Type 14 100 + 1 Push = button 6 times.
Type - 1 14 then 100 (Because it is annually)
Then 500 = (we get the result)
`200 is invested at the end of each month in an account paying interest 6% per year compounded monthly. What is the future value of this annuity after 10th payment? Given that (1.005)10 =1.0511
(a) `2544 (b) `2144 (c) `2544 (d) None (a) is correct.
Here A = 200 ; r = 6% compounded monthly n = 10 = No. of payments.
FVSA i i n () 11 200 1 6 1200 1 6 12 10 0 00204560 `
Type 6 1200 + 1 Then push button then push = button 9 times.
Type - 1 Then 6 1200
Then Type 200 = buttons we get the required amount.
If (1 + i)n value is given in the question then use given value in the question otherwise answer may vary.
If a bank pays 6% interest compounded quarterly what equal deposit have to be made at the end of the each quarter for 3 years if you want to have `1500 at the end of 3 years?
(a) `117.86 (b) `115.01 (c) `150.50 (d) None of these
(b) is correct
`
Type 6
400 + 1 Then push button then push = buttons 11 times
Then push -1 6 400 buttons
Then push M+ button to save the typed value.
Then type 1500 then button then push “MRC” button 2 times then push = button.
[we get the required result]
PV = Present value = A i i n 11()
Type (1 + i) value then push ÷ button
Then push = buttons “n” times
Push GT button
Then type A (value) then push = button
Find the present value of an annuity which pays 200 at the end of each 3 months for 10 years assuming money to be worth 5% converted quarterly?
(a) `3473.86 (b) `3108.60 (c) `6265.38 (d) None of these Option (c) is correct
Here A = 200; m = 4; r = 5% 1/4 yearly.
t = 10 years n = mt = 4 × 10 = 40 year PV = ?
Type 5 400 + 1 then push button
Then push = buttons 40 times
Then Push GT button
Then type 200 = buttons
[We get the resulting value]
Mr. A borrows 5,00,000 to buy a house.
If he pays equal instalments for 20 years and 10% interest on outstanding balance what will be the equal annual instalment? (a) `58239.84 (b) `58729.84 (c) `68729.84 (d) None of these (b) is correct
Here PV = `5,00,000; r = 10% yearly
t = 20 years
n = 20; A = ?
Type 10 100 + 1 then push ÷ button
Push = buttons 20 times
Then Push GT button
Then M+ buttons to save the result.
Type 5,00,000 then push button then- MRC button 2 time and then = button.
(We get the required result)
Annuity Immediate/Due
Type r 100 m + 1 then push button
Push = buttons n + 1 - 1 = n times then push - 1 button then push button then push r value then push 100m value buttons.
Push - 1 button then button and then type A value & then push = button (we get the required result)
PAST EXAM QUESTIONS WITH SOLUTIONS (MEMORY BASED)
(a)83,042(b)90,100 (c)93,042(d)10,100 [Dec. 2015] (c) is correct
a)9517.56(b)9157.65
c)9715.56(d)9175.65 [June 2017]
Use Calculator tricks = ` 9157 option (b) is correct. `
(a)40,000(b)4,50,000
(c)4,80,000(d)50,000 [June 2017] R 796870 1 10 100 1 10 100 10
= ` 50,000 option (d) is correct.
(a) `20,456(b) `20,156
(c) `20,256(d) `20,356 [June 2018]
(a) is correct
FV = 2000 1 6 1200
= ` 20,456
(a) ` 45,00,000
(b) ` 50,00,000
(c) ` 55,00,000
(d) ` 60,00,000
(b) is correct i 12 1200 001 . Formula PV R i 50000 001 , . = ` 50,00,000 (b)is correct.
[June 2019]
(a)Favour for lessee
(b)Favour for lessor
(c)Not for both
(d)Can’t be determined [June 2019]
(a) is correct
Cost = ` 5,00,000.
So; GST = PV of Instalments made = PV = 51,272 11 10 100 10 i
* Type 121200111 times 1121200button. Then press (m+) button. * Type 55000 button then press MRC button then = button. We get ` 4337.
Type 101001 button 10 times then press GT button then 51272 , = button = ` 3,15,044.25. Which is less than ` 5,00,000.
So, Leasing is preferable. (a) is correct.
instalments
PV = R i 10 14 1200 10 14 1200 = ` 857.14 = ` 857.
(c) is correct.
FV = R 11 100 i r m n = 900 = 1 148 1200 1 148 1200 9 . . = ` 8511.31 = ` 8511 14.8 ÷ 1200 + 1 × = button 8 times -1 ÷ 14.8 × 1200 × 900 = button. We get FV ` 8511.
(a) F.V of ordinary annuity < F.V of annuity due
(b) F.V of ordinary annuity > F.V of annuity due
(c) P.V of ordinary annuity > P.V of annuity due
(d) None of these [Dec. 2020]
(a) is correct.
(a) ` 8,511 (b) ` 9,000 (c) ` 9,200 (d) ` 1,000 [Dec. 2020] (a) is correct
(a) is correct. ` (a) 13,040.27 (b) 15,847.90 (c) 14,674.21 (d) 16,345.11 [Dec. 2020] Loan amount = PV = R 11() i i n = 2500 11 4 100 10 i Type 14 ÷ 100 + 1 ÷ = button 10 times (Press) Then press GT button then × button. Type 2500 then = button. (Press) We get PV = ` 13,040.28 (a) is correct. ` (a) ` 2,500 (b) ` 5,000 (c) ` 7,500 (d) ` 10,000 [Jan. 2021]
(d) is correct
Discount rate = i = 7% = 7 100 = 0.07
Growing rate = g = 5% = 0.05
R = Value of each payment received = ` 200
PVA = R ig 200 007005 .. = ` 10,000 `
(a) ` 4,444(b) ` 8,756 (c) ` 3,491(d) ` 8,182 [Jan. 2021]
(d) is correct.
Monthly Instalment = A = ` 800 rate of interest = r = 6% p.a. compounded monthly
n = No. of Payments = 10
FV = A(n, i) = A 11 100 r r m n = 800 1 6 1200 1 6 1200 10 = ` 8182
[Calculator Tricks 6 ÷ 1200 + 1 × = 9 times –1 ÷ 6 × 1200 × 800 = button; we get ` 8182]
(a)Annuity regular for (n - 1) year plus the initial receipt in the beginning of the period
(b)Annuity regular for (n - 1) years
(c)Annuity regular for (n + 1) years
(d)Annuity regular for (n + 1) years plus the initial receipt in the beginning of the period [Jan. 2021] (a) is correct PV = R 11 1 1 i i n () = R 11 1 i i R n () = PV of Annuity Regular + Value of 1st instalment (a)is correct ` (a) ` 10,730.7 (b) ` 5,365.35 (c) ` 8,756(d) ` 9892.34 [Jan. 2021] (a) is correct
FV = A 111 100 n r m
Where m = No. of conversion periods in 1 year = 1
n = No. of payments made = mt = n = 1 × 7 = 7
