Unit study guide · NISM Series X-A
Time Value of Money
14 questions in this unit
What this unit covers
- A fixed positive amount will be received on a specified future date. Keeping that amount and date unchanged, what happens to its present value when the positive discount rate increases?
- ₹1,25,000 is invested for two years at an assumed return of 8% per year, compounded annually. No money is added or withdrawn. What is the value after two years?
- A payment of ₹1,33,100 is due exactly three years from today. At a discount rate of 10% per year, compounded annually, what amount today is equivalent to that payment?
- An investment grows from ₹2,50,000 to ₹3,02,500 in exactly two years, with no intermediate cash flows. What is its compound annual growth rate?
- An account credits interest monthly at 1% of the opening balance for that month. Interest remains in the account. What is the effective annual interest rate, rounded to two decimal places?
- A course currently costs ₹6,00,000. For this question only, assume its cost rises by 6% at the end of each year for three years. What is its expected cost at the end of year three, rounded to the nearest rupee?
- An investor deposits ₹10,000 at the end of each of years one, two and three. The account earns 10% per year, compounded annually. What is its value immediately after the third deposit?
- An investor deposits ₹10,000 immediately and makes two further ₹10,000 deposits exactly one and two years from today. At 10% per year, compounded annually, what is the account value exactly three years from today?
- A hypothetical investment pays ₹15,000 at the end of every year forever, starting one year from today. Payments do not grow. At a constant annual discount rate of 6%, what is its present value?
- A client will receive ₹10,000 at the end of each of the next three years. At a discount rate of 10% per year, what is the present value of these receipts, rounded to the nearest rupee?
- A client needs ₹1,57,625 immediately after making the third of three equal annual deposits. Deposits are made at the ends of years one, two and three. Assuming 5% annual compounding and no initial investment, how large must each deposit be?
- At 10% per year compounded annually, how many complete years are required for ₹2,00,000 to become ₹2,66,200, assuming no additional cash flows?
- CASE: A course costs ₹6,00,000 today and is payable in three years. Its cost rises 5% a year, compounded. What will it cost in three years?
- CASE: A course costing ₹6,00,000 today is payable in three years, and its cost rises 5% a year, compounded. A household earmarks ₹5,00,000 today in mutual funds assumed to earn 8% a year, compounded. What shortfall or excess is projected when payment is due?
Key ideas
From the practice bank · 1
A fixed positive amount will be received on a specified future date. Keeping that amount and date unchanged, what happens to its present value when the positive discount rate increases?
PV decreases because less money today is needed to reach the same future amount at the higher assumed rate. For annual compounding, PV = FV ÷ (1 + r)^n, where r is the annual rate in decimal form and n is the number of years. The future receipt and payment date are fixed. Increasing r makes the positive denominator larger, so PV becomes smaller. For example, ₹110 due in one year is worth ₹100 at 10%, but about ₹91.67 at 20%: ₹110 ÷ 1.20. Both present values remain positive. Remember: Higher discount rate, lower present value—when the future cash amount and timing remain unchanged.
Hint Common mistake: Confusing the effect of a higher return on future value with its opposite effect on the present value of a fixed future receipt.
X-A-005From the practice bank · 2
₹1,25,000 is invested for two years at an assumed return of 8% per year, compounded annually. No money is added or withdrawn. What is the value after two years?
₹1,45,800 correctly includes two complete years of annual compounding. Year 1: ₹1,25,000 × 1.08 = ₹1,35,000. The return is ₹10,000. Year 2: ₹1,35,000 × 1.08 = ₹1,45,800. The second-year return is ₹10,800. Equivalently, future value = ₹1,25,000 × (1.08)^2 = ₹1,45,800. The extra ₹800 is return on the first year’s ₹10,000. Remember: Apply the compound factor once for each complete period, or trace the accumulated balance period by period.
Hint Common mistake: Calculating every year’s return on the original principal when the question specifies compounding.
X-A-006From the practice bank · 3
A payment of ₹1,33,100 is due exactly three years from today. At a discount rate of 10% per year, compounded annually, what amount today is equivalent to that payment?
₹1,00,000 compounds to the stated payment after exactly three years. Use PV = FV ÷ (1 + r)^n with FV = ₹1,33,100, r = 0.10 and n = 3. The three-year factor is 1.10^3 = 1.331. Therefore PV = ₹1,33,100 ÷ 1.331 = ₹1,00,000. Check forwards: ₹1,00,000 becomes ₹1,10,000, then ₹1,21,000, then ₹1,33,100 after successive years. Remember: Check a present-value answer by compounding it back to the payment date.
Hint Common mistake: Subtracting n times the interest rate from future value instead of dividing by the compound growth factor.
X-A-007From the practice bank · 4
An investment grows from ₹2,50,000 to ₹3,02,500 in exactly two years, with no intermediate cash flows. What is its compound annual growth rate?
10% per year produces the exact stated ending value. The total growth factor is ₹3,02,500 ÷ ₹2,50,000 = 1.21. There are two years, so find the annual factor by taking the square root: 1.21^(1/2) = 1.10. Subtract 1 to obtain the rate: 1.10 − 1 = 0.10 = 10%. Check: ₹2,50,000 × 1.10 × 1.10 = ₹3,02,500. Remember: Annualising a multi-year return requires a root: = (ending value ÷ beginning value)^(1/years) − 1.
Hint Common mistake: Dividing a multi-year percentage gain by the number of years and labelling it CAGR.
X-A-008From the practice bank · 5
An account credits interest monthly at 1% of the opening balance for that month. Interest remains in the account. What is the effective annual interest rate, rounded to two decimal places?
12.68% is the effective annual growth rounded to two decimal places. A monthly rate of 1% means a monthly growth factor of 1.01. Twelve months require multiplying that factor twelve times. Starting with ₹100, the balance is ₹101 after month one and ₹102.01 after month two, because the second 1% applies to ₹101. After twelve months, ₹100 × 1.01^12 ≈ ₹112.6825. Effective annual rate = (1.01^12 − 1) × 100 ≈ 12.68%. Remember: Use the periodic rate and the actual number of compounding periods, then convert the final growth factor to a percentage.
Hint Common mistake: Treating a quoted monthly rate, its simple annual multiple and the effective annual rate as interchangeable.
X-A-009From the practice bank · 6
A course currently costs ₹6,00,000. For this question only, assume its cost rises by 6% at the end of each year for three years. What is its expected cost at the end of year three, rounded to the nearest rupee?
₹7,14,610 includes all three increases and the specified rounding. End of year 1: ₹6,00,000 × 1.06 = ₹6,36,000. End of year 2: ₹6,36,000 × 1.06 = ₹6,74,160. End of year 3: ₹6,74,160 × 1.06 = ₹7,14,609.60, which rounds to ₹7,14,610. Remember: Project the cost to the date on which the money will be needed before calculating how to fund it.
Hint Common mistake: Using today’s cost as the future funding target or adding percentage increases without compounding. The 6% here is hypothetical, not current inflation data.
X-A-010From the practice bank · 7
An investor deposits ₹10,000 at the end of each of years one, two and three. The account earns 10% per year, compounded annually. What is its value immediately after the third deposit?
₹33,100 uses two, one and zero growth periods, matching the deposit dates. The deposit at the end of year 1 grows for two years: ₹10,000 × 1.10^2 = ₹12,100. The deposit at the end of year 2 grows for one year: ₹10,000 × 1.10 = ₹11,000. The deposit at the end of year 3 is valued immediately, so it contributes ₹10,000 with no growth. Add: ₹12,100 + ₹11,000 + ₹10,000 = ₹33,100. Remember: Draw the payment dates before applying an annuity formula. The final end-of-period deposit has no time to grow at that instant.
Hint Common mistake: Giving all deposits the same number of growth periods merely because they belong to the same plan.
X-A-011From the practice bank · 8
An investor deposits ₹10,000 immediately and makes two further ₹10,000 deposits exactly one and two years from today. At 10% per year, compounded annually, what is the account value exactly three years from today?
₹36,410 correctly uses growth periods of three, two and one years. The deposit today, at time 0, grows for three years: ₹10,000 × 1.10^3 = ₹13,310. The deposit at time 1 grows for two years to ₹12,100. The deposit at time 2 grows for one year to ₹11,000. At time 3 the total is ₹13,310 + ₹12,100 + ₹11,000 = ₹36,410. Alternatively, multiply the equivalent ordinary-annuity value ₹33,100 by 1.10. Remember: Timing matters even when the total contribution and annual return are unchanged.
Hint Common mistake: Substituting the correct annuity formula for the wrong payment timing.
X-A-012From the practice bank · 9
A hypothetical investment pays ₹15,000 at the end of every year forever, starting one year from today. Payments do not grow. At a constant annual discount rate of 6%, what is its present value?
₹2,50,000 is the correctly timed level-perpetuity value. The annual payment C is ₹15,000, the annual discount rate r is 0.06 and the first receipt is one year away. PV = C ÷ r = ₹15,000 ÷ 0.06 = ₹2,50,000. The discounted infinite payment series has a finite value at this positive rate. An intuition under the model is that 6% of ₹2,50,000 is ₹15,000 per year, allowing the capital amount to remain intact while generating the stated payment. Remember: Check whether the first payment is today or one period ahead before using a perpetuity formula.
Hint Common mistake: Adding an immediate payment when none is promised, or treating an infinite undiscounted total as the present value.
X-A-013From the practice bank · 10
A client will receive ₹10,000 at the end of each of the next three years. At a discount rate of 10% per year, what is the present value of these receipts, rounded to the nearest rupee?
₹24,869 is the sum of the correctly discounted receipts after final rounding. Year-1 receipt: ₹10,000 ÷ 1.10 = ₹9,090.91 approximately. Year-2 receipt: ₹10,000 ÷ 1.10^2 = ₹8,264.46 approximately. Year-3 receipt: ₹10,000 ÷ 1.10^3 = ₹7,513.15 approximately. Using unrounded values, the total is ₹24,868.52, or ₹24,869 to the nearest rupee. Later receipts are worth less today at the same positive rate. Remember: For multiple cash flows, align dates first, calculate second and round at the end.
Hint Common mistake: Discounting the entire total for one year or compounding when the question asks for present value.
X-A-014From the practice bank · 11
A client needs ₹1,57,625 immediately after making the third of three equal annual deposits. Deposits are made at the ends of years one, two and three. Assuming 5% annual compounding and no initial investment, how large must each deposit be?
₹50,000 per deposit exactly reaches the target under the stated timing. Let each year-end deposit be P. The first earns two years of return, the second one year and the third none. The accumulated value is P × (1.05^2 + 1.05 + 1) = P × 3.1525. Set 3.1525P = ₹1,57,625. Then P = ₹50,000. Check: ₹55,125 + ₹52,500 + ₹50,000 = ₹1,57,625. Remember: The required payment equals the goal amount divided by the relevant accumulation factor, not simply by the number of deposits.
Hint Common mistake: Using beginning-of-period timing for deposits that occur at year end.
X-A-015From the practice bank · 12
At 10% per year compounded annually, how many complete years are required for ₹2,00,000 to become ₹2,66,200, assuming no additional cash flows?
Three years produce exactly ₹2,66,200. After one year: ₹2,00,000 × 1.10 = ₹2,20,000. After two: ₹2,42,000. After three: ₹2,66,200. The target is reached exactly after three complete years. For less convenient numbers, use n = log(FV ÷ PV) ÷ log(1 + r). Here log(1.331) ÷ log(1.10) = 3. The same logarithm base must be used in both places. Remember: Use repeated accumulation for a simple check and logarithms when the required horizon is not obvious.
Hint Common mistake: Dividing the total percentage gain by the annual rate, which ignores compounding.
X-A-016From the practice bank · 13
CASE: A course costs ₹6,00,000 today and is payable in three years. Its cost rises 5% a year, compounded. What will it cost in three years?
₹6,00,000 × 1.05³ = ₹6,00,000 × 1.157625 = ₹6,94,575. Remember: Expense inflation and investment return belong to different cash-flow projections.
Hint Common mistake: Using the portfolio’s growth rate to inflate the spending goal because both rates appear in the same case.
X-A-024From the practice bank · 14
CASE: A course costing ₹6,00,000 today is payable in three years, and its cost rises 5% a year, compounded. A household earmarks ₹5,00,000 today in mutual funds assumed to earn 8% a year, compounded. What shortfall or excess is projected when payment is due?
Fund value = ₹5,00,000 × 1.08³ = ₹6,29,856. Goal cost = ₹6,00,000 × 1.05³ = ₹6,94,575. Shortfall = ₹64,719. Remember: Compare projected assets and projected liabilities on one date. The relevant result is a rupee gap, not simply the difference between two rates.
Hint Common mistake: Subtracting 5% from 8% and assuming the 3-percentage-point spread means the goal is fully funded.
X-A-025CAGR expresses the change between a starting value and an ending value as a single constant annual rate.
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