ML Aggarwal Class 12 Maths Solutions Section C Chapter 02 Annuities

2. Annuities

2.1 Annuity

An annuity refers to a series of payments of the same amount, made at regular time intervals. Common examples include weekly salaries, monthly mortgage payments, recurring deposit contributions, quarterly dividend payouts, and yearly life insurance policy premiums.

The interval between one payment and the next is known as the payment period or payment interval. This could be weekly, monthly, quarterly, yearly, or any other fixed duration. The span from the start of the first interval to the finish of the final interval is called the term or duration of the annuity. Each payment amount in an annuity is referred to as the periodic payment of the annuity. An annuitant is the individual who gets paid. The total of all payments within a single year is known as the annual rent. For instance, if Rs. 2000 is paid every quarter, the annual rent amounts to Rs. 8000.

The amount, or future value, of an annuity represents the total sum owed at the conclusion of the annuity's term. It consists of principal and interest combined. In other words, the amount equals the combined total of each payment, compounded at interest up to the end of the term. As an example: if Rs. 100 is set aside monthly over 7 years, and after 7 years the depositor obtains Rs. 10000, then Rs. 10000 is the amount of the annuity, Rs. 100 is the periodic payment, 7 years is the term or duration, and one month is the payment period or payment interval.

Present value of an annuity means the value today of a stream of equal regular payments over a specified time span. Consider this situation: you take out a vehicle loan of Rs. 2 lacs and repay it through 24 equal monthly instalments of Rs. 10000 each across 2 years. In this case, Rs. 2 lacs is the present value of this annuity.

Annuities fall into three main categories:

(i) In annuities certain, the count of payments is predetermined - that is, the payments have fixed start and end dates. An illustration is borrowing money from a bank and paying it back in, say, 10 equal yearly instalments, with the first payment due 1 year from the borrowing date.

(ii) A contingent annuity occurs when the duration relies on a particular event that is uncertain. A typical case is the yearly payments of life insurance premiums, which cease when the individual passes away.

(iii) A perpetual annuity or perpetuity is an annuity that never terminates - it goes on without end. Because of this, there is no final payment; they keep happening forever. Owning freehold land, where you can collect rent indefinitely, is one example.

Annuities receive further classification into three groups by payment timing:

(i) An ordinary annuity or immediate annuity is one in which payments happen at the conclusion of each period. That is, the initial payment occurs at the conclusion of the first period, and continues the same way. Common instances are auto loan repayment and house mortgage payments.

(ii) An annuity due occurs when payments are made at the start of each period. Examples are life insurance premium payments, recurring deposit payments, and house rent.

(iii) A deferred annuity happens when payments begin after a given number of periods have passed. Insurance companies typically offer pension schemes of this type. When an annuity sits unpaid over several years (or payment intervals), it is referred to as unpaid or fore-borne during that time. The complete amount for these periods, together with the interest earned, is the amount of deferred annuity.

2.2 Amount and Present Value of Ordinary Annuities

When discussing annuities certain, we typically refer to ordinary (or immediate) annuity, in which payment happens at the end of each payment interval. The following notations will be applied throughout this chapter:

A = amount of each instalment
V = present value of annuity
M = (future) amount of annuity
r = rate of interest (on one rupee per payment interval)
n = number of instalments (in case of annuity certain)

Note: if payments occur semi-annually and the interest rate is 6% p.a., then r = 0.03.

When solving annuity questions, drawing a line diagram is often quite helpful.

Consider an ordinary annuity of Rs. 10000 per year over four years with money valued at 5%. (Think about a scenario in which you purchase a computer and pay four yearly instalments of Rs. 10000 each.)

Here A = Rs. 10000, r = 0.05, n = 4.

To calculate the present value, V, of all four instalments taken together, consider the present value of each separately. The value of Rs. 10000 payable in one year equals Rs. 10000/(1.05) paid today. The value of Rs. 10000 payable in two years equals Rs. 10000/(1.05)² paid today, and so forth.

Therefore,

V = Rs. 10000/(1.05) + Rs. 10000/(1.05)² + Rs. 10000/(1.05)³ + Rs. 10000/(1.05)⁴

= Rs. 10000/(1.05) [1 + 1/(1.05) + 1/(1.05)² + 1/(1.05)³]

= Rs. 10000/(1.05) [1 - (1/(1.05))⁴] / [1 - 1/(1.05)]

= Rs. 10000/0.05 [1 - 0.8227] = Rs. 35460

To find the future amount M, we note that Rs. 10000 paid at the conclusion of the first year is equivalent to paying Rs. 10000 (1.05)³ at the end of the fourth year, and so forth.

Therefore,

M = Rs. 10000 (1.05)³ + Rs. 10000 (1.05)² + Rs. 10000 (1.05) + Rs. 10000

= Rs. 10000 [1 + 1.05 + (1.05)² + (1.05)³]

= Rs. 10000 [(1.05)⁴ - 1] / [1.05 - 1]

= Rs. 10000/0.05 [1.2155 - 1] = Rs. 43100

These four instalments of Rs. 10000 each equal Rs. 35460 paid immediately or Rs. 43100 paid four years hence. We can verify: Rs. 35460 paid now would equal Rs. 35460 (1.05)⁴ = Rs. 43100 approximately.

Now let us work out a generalised formula.

Present value, V = A/(1 + r) + A/(1 + r)² + ... + A/(1 + r)ⁿ

= A/[(1 + r)ⁿ] [1 + (1 + r) + ... + (1 + r)ⁿ - ¹]

= A/[(1 + r)ⁿ] [(1 + r)ⁿ - 1] / [(1 + r) - 1]

= (A/r) [1 - (1/(1 + r))ⁿ]

V = (A/r) [1 - (1 + r)⁻ⁿ]

Remember that r is the interest rate per payment period and n is the number of interest periods. If payment is not annual, r and n values must be recorded correctly.

Similarly, the future amount M of an annuity is

M = A(1 + r)ⁿ⁻¹ + A(1 + r)ⁿ⁻² + ... + A(1 + r) + A

= A[1 + (1 + r) + ... + (1 + r)ⁿ⁻¹]

= A [(1 + r)ⁿ - 1] / [(1 + r) - 1]

M = (A/r) [(1 + r)ⁿ - 1]

Observe that M = V(1 + r)ⁿ, which is logical.

2.3 Amount of Annuity Left Unpaid

When an annuity is left unpaid over n interest periods, the amount that could be paid at the conclusion of the nth interest period is simply the amount of ordinary annuity certain.

Therefore,

M = (A/r) [(1 + r)ⁿ - 1]

An unpaid annuity is also referred to as a foreborne annuity for that time span.

Illustrative Examples

Example 1. Find the present value and amount of an ordinary annuity of 8 quarterly payments of Rs. 500 each, the rate of interest being 8% per annum compounded quarterly.
Answer: Here periodic instalment, A = Rs. 500, number of periods n = 8, and rate of interest r = 8% p.a. = 0.02 (per quarter).

Exam Tip: Always convert the given annual rate to the applicable period rate before using any formula - this ensures your calculation remains accurate.

Example 1 (continued). Calculate Present value and Amount.
Answer: Present value, V = (A/r) [1 - (1 + r)⁻ⁿ] = (500/0.02) [1 - (1.02)⁻⁸]

Let x = (1.02)⁻⁸ ⇒ log x = -8 log 1.02 = -8(0.0086) = -0.0688 = 1.9312 ⇒ x = 0.8535

V = (500/0.02) [1 - 0.8535] = 500 × 50 × 0.1465 = Rs. 3662.50

Now, amount of annuity, M = (A/r) [(1 + r)ⁿ - 1] = (500/0.02) [(1.02)⁸ - 1]

Let x = (1.02)⁸ ⇒ log x = 8 log 1.02 = 8(0.0086) = 0.0688 ⇒ x = 1.171

M = (500/0.02) [1.171 - 1] = 500 × 50 × 0.171 = Rs. 4275

Thus, the present value of annuity is Rs. 3662.50 and amount is Rs. 4275.
In simple words: We find how much Rs. 500 is worth today for each of 8 payments, adding them all together. Then we figure out how much all these payments grow to with interest by the end.

Exam Tip: Modern calculators give V = Rs. 3662.74, M = Rs. 4291.48 - verify your method matches the computing tool allowed in your exam.

Example 2. Find the amount of an annuity of Rs. 2000 payable at the end of every month for 5 years if money is worth 6% per annum compounded monthly. (I.S.C. 2004)
Answer: Here, monthly instalment = A = Rs. 2000, number of periods = n = 5 × 12 = 60, and rate of interest r = 6% p.a. = (6/12)% per month = 0.005 per period per rupee.

Amount of annuity = M = (A/r) [(1 + r)ⁿ - 1]

= (2000/0.005) [(1 + 0.005)⁶⁰ - 1]

= (2000/0.005) [(1.005)⁶⁰ - 1]

= Rs. 139540.06
In simple words: You put in Rs. 2000 every month for 60 months. Each deposit earns interest at a monthly rate. By the end, all your deposits plus all the interest add up to Rs. 139540.06.

Exam Tip: When dealing with monthly/quarterly payments, always divide the annual rate by the number of periods per year - a common error is forgetting this step.

Example 3. What amount should be set aside at the end of each year to amount to Rs. 10 lacs at the end of 15 years at 6% per annum compound interest?
Answer: Let A be the amount set aside at the end of each year. Here number of periods, n = 15, rate of interest per period = 0.06, amount of annuity, M = Rs. 1000000.

Using M = (A/r) [(1 + r)ⁿ - 1], we get

Rs. 1000000 = (A/0.06) [(1.06)¹⁵ - 1]

⇒ A = (60000/[(1.06)¹⁵ - 1]) = Rs. 42962.76

∴ A = Rs. 42960 approx.
In simple words: To gather Rs. 10 lacs in 15 years at 6% interest, you need to save about Rs. 42960 every year.

Exam Tip: This is a reverse calculation - work backwards from the desired future amount to find the periodic payment needed.

Example 4. A man borrowed some money and returned it in 3 equal quarterly instalments of Rs. 4630.50 each. What sum did he borrow if the rate of interest was 20% per annum compounded quarterly? Find also the interest charged.
Answer: We have to find present value of an ordinary annuity certain. Here, periodic instalment A = Rs. 4630.50, number of periods, n = 3, rate of interest = 20% p.a. i.e. 5% per quarter i.e. 0.05 per period

∴ Present value V = (A/r) [1 - (1 + r)⁻ⁿ] = (4630.50/0.05) [1 - (1.05)⁻³] = Rs. 12610

Thus, the sum borrowed was Rs. 12610

Now, total money repaid = (3 × 4630.50) = Rs. 13891.50

∴ Interest paid = Rs. 13891.50 - Rs. 12610 = Rs. 1281.50
In simple words: The person borrowed Rs. 12610. They paid back Rs. 13891.50 in three equal parts. The extra amount they paid, Rs. 1281.50, is the interest charged.

Exam Tip: Always identify whether you are finding present value (what was borrowed) or future amount (what will be owed) - the formula changes.

Example 5. An iPod is purchased on instalment basis, such that Rs. 8000 is to be paid on the signing of the contract and four yearly instalments of Rs. 3000 each, payable at the end of the first, second, third and fourth years. If compound interest is charged at 5% per annum, what would be the cash price of the iPod? (Take 1.05⁻⁴ = 0.8227). (I.S.C. 2009)
Answer: Here, cash down payment = Rs. 8000, yearly instalment = Rs. 3000, number of periods = n = 4, and rate of interest = 5% p.a. = 0.05 per period per rupee.

Present value of annuity = V = (A/r) [1 - (1 + r)⁻ⁿ]

= (3000/0.05) [1 - (1 + 0.05)⁻⁴]

= (3000/0.05) [1 - (1.05)⁻⁴]

= (3000/0.05) (1 - 0.8227)

= (3000 × 0.1773)/0.05 = Rs. 10638

∴ Cash price of the iPod = Rs. 8000 + Rs. 10638 = Rs. 18638
In simple words: The buyer pays Rs. 8000 upfront and then Rs. 3000 yearly for 4 years. When we calculate what all those future yearly payments are worth in today's money and add the upfront payment, we get the total cash price.

Exam Tip: Cash down payments at the beginning are already in today's value - never discount them again.

Example 6. Mr. Aggarwal buys a house at Rs. 3000000 for which he agrees to make payments at the end of each year for 10 years. If the money is worth 10% p.a., find the amount of each instalment. [Take (1.1)⁻¹⁰ = 0.3855] (I.S.C. 2008)
Answer: Here, present value V = Rs. 3000000, n = 10, r = 10% = 0.1

We wish to find the amount of each instalment, A.

Using V = (A/r) [1 - (1 + r)⁻ⁿ], we get

Rs. 3000000 = (A/0.1) [1 - (1.1)⁻¹⁰]

⇒ A = (3000000 × 0.1) / (1 - (1.1)⁻¹⁰)

= (300000) / (1 - 0.3855)

= (300000) / (0.6145)

= Rs. 488201.79

Hence the value of each instalment is Rs. 488202 approximately.
In simple words: The house costs Rs. 30 lacs today. To repay this with interest at 10% over 10 years in equal yearly payments, each payment must be about Rs. 488202.

Exam Tip: When solving for A (the periodic payment), rearrange the formula carefully and use the given values to substitute - precision matters.

Example 7. Amit borrows Rs. 60000 at 6% effective and promises to repay the loan in 20 equal instalments beginning at the end of first year. Find the value of each instalment.
Answer: Here, present value V = Rs. 60000, n = 20, r = 6% = 0.06.

We wish to find the amount of each instalment, A.

Using V = (A/r) [1 - (1 + r)⁻ⁿ], we get

Rs. 60000 = (A/0.06) [1 - (1.06)⁻²⁰]

⇒ A = (60000 × 0.06) / (1 - (1.06)⁻²⁰) = Rs. 5231.07

Hence, the value of each instalment is Rs. 5231 approximately.
In simple words: Amit borrowed Rs. 60000 and must repay it through 20 equal yearly payments. Each payment works out to about Rs. 5231.

Exam Tip: Always verify your answer by checking that the total of all payments plus any interest paid exceeds the original loan amount.

Example 8. A person takes a loan on compound interest and returns it in 2 equal annual instalments. If the rate of interest is 16% p.a. and the yearly instalment is Rs. 1682, find the principal and the interest charged with each instalment.
Answer: Here, A = Rs. 1682, r = 0.16, n = 2

So, principal V = (A/r) [1 - (1 + r)⁻ⁿ]

= (1682/0.16) [1 - (1.16)⁻²]

= 2700

Hence, principal is Rs. 2700

Now, the first instalment of Rs. 1682 includes interest on Rs. 2700 for 1 year i.e. (2700 × 0.16 × 1) = Rs. 432, and principal repayment of (1682 - 432) i.e. Rs. 1250.

Hence, principal remaining after first payment = (2700 - 1250) = Rs. 1450

Now, second instalment of Rs. 1682 includes interest on the remaining principal i.e. (1450 × 0.16 × 1) = Rs. 232, and principal repayment of (1682 - 232) = Rs. 1450.
In simple words: The person borrowed Rs. 2700. The first payment of Rs. 1682 consists of Rs. 432 interest plus Rs. 1250 towards the main loan. The second payment covers the remaining Rs. 1450 plus interest on it.

Exam Tip: In each instalment, interest is charged only on the remaining unpaid balance, not on the original principal.

Example 9. Sanjay borrows a loan of Rs. 400950 on condition to repay it with compound interest at 6% p.a. by annual instalments of Rs. 150000 each. In how many years will the debt be paid off?
Answer: Here, present value, V = Rs. 400950, each instalment A = Rs. 150000, r = 6% p.a. = 0.06

We wish to calculate the number of instalments, n.

Using V = (A/r) [1 - (1 + r)⁻ⁿ]

Rs. 400950 = (150000/0.06) [1 - (1.06)⁻ⁿ]

⇒ 1 - (1.06)⁻ⁿ = (400950 × 0.06) / (150000) = 0.1604

(1.06)⁻ⁿ = 1 - 0.1604 = 0.8396

Taking logs, -n log 1.06 = log 0.8396

⇒ -n × 0.0253 = -0.759

⇒ n = 3

Hence, it will take 3 years to pay the debt off.
In simple words: Sanjay borrowed Rs. 400950 and pays Rs. 150000 every year. After working through the maths, we find he will completely repay the debt in exactly 3 years.

Exam Tip: When solving for n, use logarithms - this allows you to work with exponents and find how many periods are needed.

Example 10. You have taken a loan and have to repay it in 10 annual instalments of Rs. 1000 each. The rate of interest is 10% p.a. If you want to pay off the loan in annual instalments of Rs. 2000 each, how many instalments will be required to be paid? What would be the amount of last instalment?
Answer: Amount of loan borrowed is the present value of annuity of Rs. 1000. Here A = 1000, n = 10, r = 0.1

∴ Loan borrowed, V = (A/r) [1 - (1 + r)⁻ⁿ]

= (1000/0.1) [1 - (1.1)⁻¹⁰]

= Rs. 6145 (approx.)

Now if n instalments of Rs. 2000 will pay off this loan, then

Rs. 6145 = (2000/0.1) [1 - (1.1)⁻ⁿ]

⇒ 1 - (1.1)⁻ⁿ = (6145 × 0.1) / (2000) = 0.3072

⇒ (1.1)⁻ⁿ = 0.6928

⇒ -n log 1.1 = log 0.6928

⇒ -n (0.0414) = -0.1594

⇒ n = (0.1594) / (0.0414) = 3.85

Thus, four instalments will be required, first three of Rs. 2000 each and the fourth less than Rs. 2000.

Now present value of first 3 instalments of Rs. 2000 each

= (2000/0.1) [1 - (1.1)⁻³] = Rs. 4974

∴ Present value of fourth instalment = (6145 - 4974) = Rs. 1171

Hence, amount of fourth instalment = Rs. 1171 × (1.1)⁴ = Rs. 1714 (approx.)
In simple words: You originally owed enough to need Rs. 1000 yearly for 10 years. If you want to pay faster with Rs. 2000 yearly, you need only 4 payments - but the last one is smaller, around Rs. 1714.

Exam Tip: When the number of periods comes out to a decimal (like 3.85), round up to the next whole number and calculate the adjusted final payment.

Example 11. A man borrows Rs. 37500 and agrees to repay in semi-annual instalments of Rs. 2250 each, the first due in 6 months. How many payments must he make if rate of interest is 6% compounded semi-annually?
Answer: Here we have an ordinary annuity certain of present value V = Rs. 37500, periodic payment A = Rs. 2250, rate of interest per period, r = 0.03 and we have to find the number of payments, n.

V = (A/r) [1 - (1 + r)⁻ⁿ] ⇒ Rs. 37500 = (2250/0.03) [1 - (1.03)⁻ⁿ]

⇒ 1 - (1.03)⁻ⁿ = (37500 × 0.03) / (2250) = 0.5

⇒ (1.03)⁻ⁿ = 0.5

⇒ -n log 1.03 = log 0.5

⇒ -n (0.0128) = -0.3010

⇒ n = (-0.3010) / (-0.0128) = 23.51

Thus, the money may be repaid in 23 instalments with 23rd instalment slightly more than Rs. 2250, or the money may be repaid in 24 instalments, the 24th instalment slightly less than Rs. 2250. Following the procedure of previous example, the amount of 24th instalment

= Rs. {37500 - (2250/0.03) [1 - (1.03)⁻²³]} × (1.03)²⁴

= Rs. 1020.22

Thus, there will be 23 instalments of Rs. 2250 each and 24th instalment of Rs. 1020.22.
In simple words: The person can either pay 23 times (with the last one slightly higher) or pay 24 times (with the last one slightly lower, around Rs. 1020). The choice depends on preference.

Exam Tip: When the calculated number of periods is not a whole number, provide both options: exact period count with an adjusted final payment, or an extra period with a smaller final payment.

Example 12. Anshul buys a flat for Rs. 800000 on the following conditions: 25% cash down payment and balance in 10 equal semi-annual instalments, the first to be paid six months after the date of purchase. Calculate the amount of each instalment, if the rate of interest is 10% per annum compounded half-yearly. Also calculate the total amount of interest paid by Anshul.
Answer: Cash down payment = 25% of Rs. 800000 = Rs. 200000

Balance = Rs. 800000 - Rs. 200000 = Rs. 600000

This is to be paid in 10 equal semi-annual instalments, say A.

Here V = Rs. 600000, n = 10, r = 0.05

V = (A/r) [1 - (1 + r)⁻ⁿ]

⇒ Rs. 600000 = (A/0.05) [1 - (1.05)⁻¹⁰]

⇒ A = (600000 × 0.05) / (1 - (1.05)⁻¹⁰) = Rs. 77702.75

Hence, the amount of each instalment is Rs. 77703 approximately.

Now to calculate the interest paid, we note that in lieu of Rs. 600000 cash payment, 10 instalments of Rs. 77703 each were paid.

Hence, total interest paid = (77703 × 10 - 600000) = Rs. 177030.
In simple words: Anshul paid Rs. 200000 upfront. The remaining Rs. 600000 is divided into 10 equal payments of about Rs. 77703 each. In total, by paying in instalments instead of cash, Anshul paid about Rs. 177030 as interest.

Exam Tip: Total interest paid = (number of instalments × each instalment) - actual balance owed.

Example 13. Mr. Mehta purchased a house, paying Rs. 50000 down and promising to pay Rs. 2000 every 3 months for the next 10 years. The interest is 6% p.a. compounded quarterly. (i) What is the cash value of the house? Round off your answer to nearest Rs. 100. (ii) If Mr. Mehta misses first 6 payments, how much should he pay at the time of 7th payment to bring himself upto date? (iii) If at the end of 5th year, he wants to finish his liability by a single payment, how much should he pay?
Answer: (i) We have an ordinary annuity certain, where each payment A = Rs. 2000, number of payments = 40, rate of interest per period = 6% p.a. = 1.5% per quarter = 0.015 per quarter

Present value of annuity, V = (A/r) [1 - (1 + r)⁻ⁿ]

⇒ V = (2000/0.015) [1 - (1.015)⁻⁴⁰]

= Rs. 59831.69

Hence, V = Rs. 59832 approximately.

Thus, total cash value of house = Rs. 50000 + Rs. 59832 = Rs. 109832 approx = Rs. 109800 (correct to nearest Rs. 100)

(ii) At the time of seventh payment, equivalent of first 6 missed payments has also to be paid. Thus, total payment required at end of 7th period is the amount of annuity of 7 terms; hence, amount required to be paid

= (A/r) [(1 + r)ⁿ - 1] = (2000/0.015) [(1.015)⁷ - 1]

= Rs. 14645.99

i.e. Rs. 14646 approximately.

(iii) If at the end of 5th year i.e. at the time of 20th payment, he wants to finish off the liability, then lump sum payment required

= Rs. 2000 + (2000/0.015) [1 - (1.015)⁻²⁰]

= Rs. 2000 + Rs. 34337.28

= Rs. 36337 approximately.
In simple words: (i) The house is worth about Rs. 109800 in cash value today. (ii) If Mr. Mehta misses 6 payments, he must pay Rs. 14646 at the 7th payment to catch up. (iii) If he wants to pay everything off at year 5, one lump payment of Rs. 36337 clears his debt.

Exam Tip: For missed payments, find the amount (future value) of those payments at the catch-up date. For early payoff, calculate the present value of remaining payments plus any immediate payment due.

Exercise 2.1

1. Mr. X purchased an annuity of Rs. 2500 per year for 15 years from an insurance company which reckons the interest at 3% compounded annually. If the first payment is due in one year, what did the annuity cost X?

2. Find the amount of an ordinary annuity if payment of Rs. 600.00 is made at the end of every quarter for 10 years at the rate of 4% per year compounded quarterly. (I.S.C. 2005)

3. A man borrowed some money and paid back in 3 equal annual instalments of Rs. 2160 each. What sum did he borrow if the rate of interest charged was 20% p.a. compounded annually? Find also the total interest charged. (I.S.C. 2007)

4. Anshul has been depositing Rs. 1000 at the end of every year out of his pocket money in a savings account which pays 3 1/2 % effective. What is the amount in his credit just after the tenth deposit?

5. Find the amount and the present value of an ordinary annuity of Rs. 150 a month for 6 years 3 months at 6% compounded monthly.

6. Calculate the amount of ordinary annuity of Rs. 7000 at the rate of 10% per annum compounded annually for 10 years.

7. Find the present value of an annuity of Rs. 1200 payable at the end of each 6 months for 3 years when the interest is earned at 8% per year compounded semi-annually.

8. Find the future amount of an ordinary annuity of 12 monthly payments of Rs. 1000 that earn an interest at 12% per year compounded monthly.

9. The price of a tape recorder is Rs. 1661. A person purchased it by making a cash payment of Rs. 400 and agrees to pay the balance with due interest in 3 half-yearly equal instalments. If the dealer charged interest at the rate of 10% per annum compounded half-yearly, find the value of the instalment. (I.S.C. 2011)

10. Mr A buys a television for Rs. 6000 down and Rs. 500 per month for the next 12 months. If interest is charged at 9% compounded monthly, find the equivalent cash value.

11. A washing machine was on sale for Rs. 13500 cash or Rs. 6000 as initial payment and the rest paid in equal monthly instalments over a period of 10 months at 20% interest per annum compound interest. Find the equal monthly instalments to be paid, giving your answer to nearest rupee, if not exact.

12. At 6 month intervals Mr. Gupta deposited Rs. 5000 in a savings account paying 6% compounded semi-annually. The first deposit was made when his son was 6 months old and the last deposit was made when the son was 21 years old. The money remained in account and was presented to the son on his 25th birthday. How much did he receive?

2.4 Amount and Present Value of Annuity Due

The earlier section discussed ordinary or immediate annuity, in which payment takes place at the end of each period. With annuity due, payment happens at the start of each period. For example, monthly house rent paid upfront each month is annuity due.

Note: first payment is made at the start of the first period i.e. time zero; last payment happens at the start of the nth period i.e. time (n - 1).

Here, present value

V = A + A/(1 + r) + A/(1 + r)² + ... + A/(1 + r)ⁿ⁻¹ = (A/r) (1 + r) [1 - (1 + r)⁻ⁿ]

and future amount

M = A(1 + r)ⁿ + A(1 + r)ⁿ⁻¹ + ... + A(1 + r) = (A/r) (1 + r) [(1 + r)ⁿ - 1]

Note: this can easily be derived from the formulae of ordinary annuity by multiplying by the factor (1 + r). This makes sense if you picture that all instalments move from the end of each period to the beginning, and thus amount to (1 + r) times more.

Illustrative Examples

Example 1. Find the amount and present value of an annuity due of Rs. 500 per quarter for 8 years and 9 months at 6% compounded quarterly.
Answer: Here, rate of interest r = 1.5% per interest period = 0.015, number of interest periods, n = 4 × 8 + 3 = 35, and each instalment, A = Rs. 500

Present value of annuity due,

V = (A/r) (1 + r) [1 - (1 + r)⁻ⁿ]

= (500/0.015) (1.015) [1 - (1.015)⁻³⁵]

= Rs. 13740.86

Amount of annuity due,

M = (A/r) (1 + r) [(1 + r)ⁿ - 1]

= (500/0.015) (1.015) [(1.015)³⁵ - 1]

= Rs. 23137.98
In simple words: The annuity due means payments start right away (at the beginning). We calculate what those payments are worth today and what they grow to, accounting for the fact that each payment starts earning interest one period earlier than in an ordinary annuity.

Exam Tip: Always multiply the ordinary annuity formula by (1 + r) to convert it to annuity due - this accounts for the extra period of interest each payment earns.

Example 2. What equal payments made at the beginning of each year for 10 years will pay for a piece of land priced at Rs. 400000, if money is worth 7% per annum compounded annually?
Answer: Here we have an annuity due with rate of interest, r = 7% p.a. = 0.07, number of interest periods, n = 10, present value, V = Rs. 400000

We wish to calculate the amount of each instalment, A

Now, for annuity due, present value

V = (A/r) (1 + r) [1 - (1 + r)⁻ⁿ]
In simple words: To find how much to pay yearly at the start of each year to purchase land worth Rs. 400000, we use the annuity due formula and solve for the payment amount.

Exam Tip: For annuity due problems, confirm whether payments begin immediately (at time zero) or after a delay - this determines your formula choice.