Time Value of Money (TVM): Compounding, Discounting, Annuities and Present Value Basics Questions

Question 1

Define time value of money (TVM).

Accepted Answer

Time value of money means a rupee today is worth more than a rupee in future due to its **earning capacity**, inflation and risk.

Question 2

What is present value (PV)?

Accepted Answer

Present value is the value **today** of a future amount discounted at a given rate.

Question 3

What is future value (FV)?

Accepted Answer

Future value is the value in **future** of an amount invested today after compounding for a given period.

Question 4

Define discounting.

Accepted Answer

Discounting is the process of finding the **present value** of future cash flows using a discount rate.

Question 5

What is an annuity?

Accepted Answer

An annuity is a series of **equal cash flows** occurring at regular intervals for a fixed number of periods.

Question 6

Write the formula for discount factor (DF).

Accepted Answer

Discount factor (DF) = **1/(1+r)^n**.

Question 7

Differentiate between compounding and discounting (any three points) using a table.

Accepted Answer

Compounding vs discounting:

| Basis | **Compounding** | **Discounting** |
|---|---|---|
| Direction | present → future | future → present |
| Result | gives **FV** | gives **PV** |
| Formula | FV = PV(1+r)^n | PV = FV/(1+r)^n |

Thus, compounding grows money; discounting brings it back to today.

Question 8

Compute PV of ₹10,000 receivable after 3 years at 10% p.a. (show working).

Accepted Answer

Formula: **PV = FV/(1+r)^n**

PV = 10,000/(1.10)^3 = 10,000/1.331 ≈ **₹7,513** (approx.)

| Item | Value |
|---|---:|
| FV | 10,000 |
| r | 10% |
| n | 3 |
| **PV** | **7,513 (approx.)** |

Question 9

Compute FV of ₹5,000 invested for 2 years at 12% p.a. compounded annually.

Accepted Answer

Formula: **FV = PV(1+r)^n**

FV = 5,000(1.12)^2 = 5,000 × 1.2544 = **₹6,272**

| PV | r | n | **FV** |
|---:|---:|---:|---:|
| 5,000 | 12% | 2 | **6,272** |

Question 10

Compute EAR for 12% nominal rate compounded monthly (show formula).

Accepted Answer

EAR =

**(1 + r/m)^m − 1**

= (1 + 0.12/12)^{12} − 1 = (1.01)^{12} − 1 ≈ 1.1268 − 1 = **12.68% (approx.)**

| Nominal r | m | **EAR (approx.)** |
|---:|---:|---:|
| 12% | 12 | **12.68%** |

Question 11

Explain PV of annuity (ordinary) and write the formula. Also state how annuity due differs.

Accepted Answer

PV of ordinary annuity is the present value of equal payments at **end of each period**.

Formula:
**PV = A × [1 − (1+r)^{−n}] / r**

Annuity due payments occur at **beginning** of each period, so:
PV(annuity due) = PV(ordinary annuity) × **(1+r)**.

| Type | Timing |
|---|---|
| Ordinary annuity | end of period |
| **Annuity due** | beginning of period |

Question 12

Compute PV of ₹10,000 due after 2 years at 8% p.a. and show the discount factor approach (basic).

Accepted Answer

Discount factor (DF) = 1/(1+r)^n.

Here r = 8% and n = 2.
DF = 1/(1.08)^2 = 1/1.1664 ≈ **0.8573**.

PV = FV × DF = 10,000 × 0.8573 ≈ **₹8,573**.

| FV | DF (approx.) | **PV (approx.)** |
|---:|---:|---:|
| 10,000 | 0.8573 | **8,573** |

Question 13

Explain why TVM is important in financial decision-making (exam-style) with a table.

Accepted Answer

### Meaning
TVM converts cash flows at different times into a common basis (usually **present value**), making decisions comparable.

### Importance
- helps evaluate projects using **NPV/IRR** (capital budgeting)
- supports valuation of bonds/shares (PV of future cash flows)
- used in loan EMI, sinking fund and retirement planning (annuities)
- considers **risk and timing** of cash flows

| Decision area | TVM use |
|---|---|
| Capital budgeting | **NPV** based on PV of cash flows |
| Financing | loan repayment/EMI (annuity) |
| Valuation | PV of dividends/interest |

### Conclusion
Therefore, TVM is the foundation for rational financial decisions because it incorporates **time, risk and cash flows**.

Question 14

Solve: Find PV of an ordinary annuity of ₹5,000 for 4 years at 8% p.a. (show working).

Accepted Answer

### Formula
PV(ordinary annuity) = **A × [1 − (1+r)^{−n}] / r**

Given: A=5,000; r=0.08; n=4

PV factor = [1 − (1.08)^{−4}]/0.08
(1.08)^4 ≈ 1.3605 ⇒ (1.08)^{−4} ≈ 1/1.3605 ≈ 0.7350
So PV factor ≈ (1 − 0.7350)/0.08 = 0.2650/0.08 = 3.3125

PV ≈ 5,000 × 3.3125 = **₹16,563 (approx.)**

| Item | Value |
|---|---:|
| A | 5,000 |
| r | 8% |
| n | 4 |
| PV factor (approx.) | 3.3125 |
| **PV (approx.)** | **16,563** |

### Conclusion
Thus, the PV of the annuity is about **₹16.6k** at 8% discount rate.

Question 15

Explain how to avoid unit mismatch errors in TVM problems (exam-style) with a short flow.

Accepted Answer

### Core rule
Always match the **rate per period** with the **number of periods**.

### Steps (flow)
Given data → identify compounding/discounting frequency → convert rate to **periodic rate** (r/m) → convert time to **number of periods** (n×m) → apply correct formula → write answer with units

| Given | Convert to |
|---|---|
| 12% p.a. monthly compounding | 1% per month |
| 3 years monthly | 36 months |

### Conclusion
Following this flow prevents the most common TVM mistakes and ensures correct PV/FV results.

Basis	Compounding	Discounting
Direction	present → future	future → present
Result	gives FV	gives PV
Formula	FV = PV(1+r)^n	PV = FV/(1+r)^n

Time Value of Money (TVM): Compounding, Discounting, Annuities and Present Value Basics

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