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June 2026 · Question 5(b)

Time value of money: saving for a target sum

5marks
Priority
Tier A
Asked
6 of 10 sessions
Type
Numerical
Free sample

Question

Mr. X wants to get her daughter admitted into a medical college after 15 years from now. He Will require total Rs. 2,500,000 to get admission into the college. For this, he has identified a fund which pays interest at 9% p.a.

Required: Determine the amount to be invested:

  1. (i)If Mr. X decides to make annual payment into the fund at the end of each year.
  2. (ii)If Mr. X decides to invest a lump sum in the fund at the end of the year.
  3. (iii)If Mr. X decides to make annual payment into the fund at the beginning of each year.
Given
  • FVIF or CVF (n = 15, i = 9%) = 3.642
  • FVIFA or CVFA (n = 15, i = 9%) = 29.361

Answer

Mr. X must deposit Rs 85,146.96 at the end of each year, or a single sum of Rs 748,000 at the end of Year 1, or Rs 78,117.68 at the beginning of each year, to have Rs 2,500,000 after 15 years at 9% p.a.

Approach

The target is fixed: Rs 2,500,000 must be in the fund at the end of Year 15. The fund earns 9% a year. In each part we ask the same question: what deposit grows to Rs 2,500,000 by Year 15? Only the timing of the deposits changes, so only the factor changes.

  • Part i equal deposits at the end of each year. This is an ordinary annuity, so we divide by FVIFA.
  • Part ii one deposit at the end of the first year. It earns interest for 14 years only, so we use a single-sum factor for 14 years.
  • Part iii equal deposits at the beginning of each year. This is an annuity due, so we divide by .
Terms used
FVIF (future value interest factor)
The amount Re 1 grows to after n years at rate i. Here , so Re 1 today becomes Rs 3.642 in 15 years.
FVIFA (future value interest factor of an annuity)
The amount that Re 1 deposited at the END of every year grows to after n years. Here , so 15 deposits of Re 1 become Rs 29.361.
Ordinary annuity and annuity due
An ordinary annuity pays at the end of each period. An annuity due pays at the beginning of each period, so every payment earns one extra year of interest.
Future value
Rs 2,500,000 needed at the end of Year 15
FVIFA(n, i)
29.361 for n = 15 years and i = 9% (given)
Working
Part i annual payment at the end of each year
  1. Write the relation

    15 equal year-end deposits must accumulate to the target; FVIFA is given in the question.

  2. Solve for the payment
    =Rs 85,146.96 per year

    Deposits are made at the end of Year 1 to Year 15.

Time graph
Part i 15 year-end deposits compounded at 9% to Year 15
Part i 15 year-end deposits compounded at 9% to Year 15"> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Year 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 85,146.96 Net cash flow of each year: received ↑, paid ↓ every amount is carried forward to Year 15 at 9% →
YearCash flowAmountCompound factorFuture value
1–15Deposit at the end of each year85,146.9629.3612,499,999.89
Fund at end of Year 15 (Rs 2,500,000 after rounding)2,499,999.89

Part ii lump sum at the end of the year

The words "at the end of the year" are read as the end of Year 1. The college needs the money at the end of Year 15, so the deposit stays in the fund for years. We therefore discount Rs 2,500,000 back 14 years, not 15.

where
PVIF
present value interest factor: today's value of Re 1 received n years later
3.3417, so
Working
Part ii lump sum at the end of the year
  1. Years the money is invested
    =14 years

    Deposit is made at the end of Year 1; money is needed at the end of Year 15.

  2. Growth factor for 14 years
    =3.3417

    Not given in the question; compute it, or use (see the exam note).

  3. Discount factor for 14 years
    =0.2992

    PVIF is the reciprocal of FVIF.

  4. Lump sum required
    =Rs 748,000

    This amount, left for 14 years at 9%, grows to about Rs 2,500,000.

Time graph
Part ii Rs 2,500,000 at Year 15 discounted 14 years, back to the end of Year 1
Part ii Rs 2,500,000 at Year 15 discounted 14 years, back to the end of Year 1"> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Year 2,500,000 Net cash flow of each year: received ↑, paid ↓ ← every amount is brought back to Year 0 at 9%
YearCash flowAmountDiscount factorPresent value
15Amount needed for admission (value shown is at end of Year 1)2,500,0000.2992748,000
Lump sum to deposit at end of Year 1748,000
FVIFA(n, i)
29.361 (given)
(1 + i)
1.09: each deposit is made one year earlier, so it earns one more year of interest
Working
Part iii annual payment at the beginning of each year
  1. Factor for an annuity due
    =32.003

    Deposits are made at Year 0 to Year 14; each earns one extra year compared with part (i).

  2. Solve for the payment
    =Rs 78,117.68 per year

    Target divided by the annuity-due factor.

Time graph
Part iii 15 beginning-of-year deposits compounded at 9% to Year 15
Part iii 15 beginning-of-year deposits compounded at 9% to Year 15"> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Year 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 78,117.68 Net cash flow of each year: received ↑, paid ↓ every amount is carried forward to Year 14 at 9% →
YearCash flowAmountCompound factorFuture value
0–14Deposit at the beginning of each year78,117.6832.0032,500,000.11
Fund at end of Year 15 (Rs 2,500,000 after rounding)2,500,000.11
Table
Summary of the three options
PartOptionWhen paidFactor usedAmount (Rs)
Part iAnnual payment, ordinary annuityEnd of Year 1 to Year 1585,146.96
Part iiLump sumEnd of Year 1, once748,000
Part iiiAnnual payment, annuity dueBeginning of each year (Year 0 to Year 14)78,117.68
All three options produce Rs 2,500,000 at the end of Year 15 at 9% p.a.
Marks split
5 marks
  1. 1.5Part i Ordinary annuity relation and payment of Rs 85,146.96
  2. 1.5Part ii 14-year period identified and lump sum of Rs 748,000
  3. 1Part iii Annuity-due factor
  4. 1Part iii Payment of Rs 78,117.68
Our estimate of how a marker would divide the marks. ICAN does not publish a marking scheme.

Same kind of question

Question: ICAN CAP-II examination, June 2026, Paper 4. Open ICAN's Suggested Answers (PDF, page 14). The answer above is our own.

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