Financial Mathematics

Simple interest is calculated only on the original principal amount. Interest does not earn interest.

Simple interest = Principal × Rate × Time

Future value (simple) = P × (1 + r × n)

where P = principal, r = annual interest rate (as a decimal), n = number of years.

Compound interest is calculated on the principal plus accumulated interest from prior periods. Interest earns interest.

Future value (compound) = P × (1 + r)ⁿ

Example: £10,000 invested at 8% for 3 years:

• Simple: £10,000 × (1 + 0.08 × 3) = £10,000 × 1.24 = £12,400. Interest = £2,400.
• Compound: £10,000 × (1.08)³ = £10,000 × 1.2597 = £12,597. Interest = £2,597.

Compound interest gives a higher return because interest earned in Year 1 earns additional interest in Years 2 and 3.

Non-annual compounding: If interest is compounded more frequently than annually (e.g., quarterly, monthly):

FV = P × (1 + r/m)^(m×n)

where m = number of compounding periods per year.

Example: £10,000 at 8% compounded quarterly for 3 years:

FV = £10,000 × (1 + 0.08/4)^(4×3) = £10,000 × (1.02)^12 = £10,000 × 1.2682 = £12,682

When interest is compounded more than once per year, the stated (nominal) annual rate understates the true annual return. The effective annual rate (EAR) converts the nominal rate to its annual equivalent, allowing comparison between rates with different compounding frequencies.

EAR = (1 + r/m)^m − 1

where r = nominal annual rate, m = compounding periods per year.

Example: A bank offers 6% nominal compounded monthly. What is the EAR?

EAR = (1 + 0.06/12)^12 − 1 = (1.005)^12 − 1 = 1.06168 − 1 = 6.168%

The true annual return is 6.168%, not 6%.

Uses of EAR:

• Comparing savings accounts or loans with different compounding frequencies (monthly vs quarterly vs annual)
• Determining the true cost of borrowing
• APR (annual percentage rate) for consumer lending is essentially the EAR including fees

The time value of money is the principle that money received today is worth more than the same amount received in the future, because money received today can be invested to earn a return. Three reasons:

• Opportunity cost: Money today can be invested to grow
• Inflation: Purchasing power erodes over time — £100 today buys more than £100 in five years
• Risk: Future cash flows are uncertain — there is a risk they may not materialise

Discounting is the reverse of compounding. It calculates the present value (PV) of a future cash flow — what that future amount is worth today.

PV = FV ÷ (1 + r)ⁿ = FV × 1/(1 + r)ⁿ

The term 1/(1 + r)ⁿ is the discount factor (DF). Discount factors for common rates and periods are provided in present value tables in exams.

Example: What is the present value of £15,000 receivable in 4 years at a discount rate of 10%?

PV = £15,000 × 1/(1.10)⁴ = £15,000 × 1/1.4641 = £15,000 × 0.6830 = £10,245

£15,000 received in 4 years is worth only £10,245 today at a 10% discount rate.

The discount rate reflects the required return — the minimum return an investor demands to compensate for the time value of money, inflation, and risk. For a company, it is typically the cost of capital (the weighted average cost of its debt and equity finance).

Annuity: A series of equal cash flows at regular intervals for a fixed number of periods. Examples: lease payments, loan repayments, pension payments.

Present value of an annuity:

PV = Annual cash flow × Annuity factor

Annuity factor (AF) = [1 − 1/(1 + r)ⁿ] ÷ r

The annuity factor is also called the cumulative discount factor or present value of an annuity factor. It is the sum of the individual discount factors for each year. Annuity factor tables are provided in exams.

Example: A project generates £20,000 per year for 5 years. The discount rate is 8%.

AF at 8% for 5 years = [1 − 1/(1.08)⁵] ÷ 0.08 = [1 − 0.6806] ÷ 0.08 = 0.3194 ÷ 0.08 = 3.9927

PV = £20,000 × 3.9927 = £79,854

Perpetuity: An annuity that continues indefinitely (forever). Examples: preference share dividends, endowment fund income.

PV of a perpetuity = Annual cash flow ÷ r

Growing perpetuity: Cash flows grow at a constant rate g per year forever.

PV = Cash flow in Year 1 ÷ (r − g)

where r > g (the discount rate must exceed the growth rate).

Example: An investment pays £5,000 per year in perpetuity. The discount rate is 10%.

PV = £5,000 ÷ 0.10 = £50,000

NPV is the sum of the present values of all cash flows (both inflows and outflows) associated with a project, discounted at the company's cost of capital (or required rate of return).

NPV = Σ [Cash flow in year t ÷ (1 + r)^t] − Initial investment

Or: NPV = PV of future cash inflows − PV of cash outflows (including initial investment)

Decision rule:

• NPV > 0 (positive): Accept the project — it generates a return in excess of the cost of capital, adding value to the company
• NPV = 0: Indifferent — the project generates exactly the required return
• NPV < 0 (negative): Reject the project — it destroys value, generating a return below the cost of capital

Advantages of NPV:

• Considers the time value of money (unlike payback and ARR)
• Uses all cash flows over the project's life (unlike payback which ignores cash flows after the payback period)
• Gives an absolute measure of value added (in £) — directly linked to shareholder wealth
• Theoretically the most correct method — maximising NPV maximises shareholder wealth

Disadvantages:

• Requires estimation of future cash flows (uncertain) and the appropriate discount rate
• More complex to calculate and explain than payback or ARR
• Does not give a percentage return (unlike IRR) — managers may find it harder to compare projects of different sizes
• Assumes cash flows can be reinvested at the cost of capital (which may not be realistic)

The IRR is the discount rate at which the NPV of a project equals zero. It is the rate of return the project generates — the breakeven cost of capital.

Decision rule:

• IRR > Cost of capital: Accept — the project earns more than the required return
• IRR < Cost of capital: Reject — the project earns less than required

Calculation by interpolation:

IRR is found by trial and error. Calculate NPV at two different discount rates (one giving a positive NPV, one giving a negative NPV), then interpolate:

IRR ≈ r₁ + [NPV₁ ÷ (NPV₁ − NPV₂)] × (r₂ − r₁)

where r₁ = lower rate (with positive NPV₁), r₂ = higher rate (with negative NPV₂).

Advantages of IRR:

• Considers the time value of money
• Uses all cash flows
• Expressed as a percentage return — intuitive and easy to compare with the cost of capital
• Does not require the cost of capital to be known in advance (unlike NPV)

Disadvantages:

• May give multiple IRRs if cash flows change direction more than once (non-conventional cash flows)
• Assumes reinvestment at the IRR rate (which may be unrealistic if IRR is very high or very low)
• Does not reflect the scale of the project — a small project with a high IRR may add less value than a large project with a lower (but still acceptable) IRR
• When comparing mutually exclusive projects, IRR and NPV may give conflicting rankings — NPV is preferred in this case

The payback period is the length of time it takes for the cumulative net cash inflows from a project to equal the initial investment — i.e., how long until the investment "pays for itself."

Simple payback uses undiscounted cash flows.

Discounted payback uses discounted (present value) cash flows — more accurate because it accounts for the time value of money.

Decision rule: Accept the project if the payback period is less than or equal to the company's target payback period. Choose the project with the shortest payback if comparing alternatives.

Calculation:

• If annual cash flows are equal: Payback = Initial investment ÷ Annual cash flow
• If annual cash flows are unequal: Calculate cumulative cash flows year by year until the investment is recovered. Interpolate within the year for precision.

Advantages:

• Simple to calculate and easy to understand
• Focuses on liquidity — when will the cash come back?
• Favours low-risk projects (shorter payback = less time exposed to uncertainty)
• Useful as a quick screening tool, especially in rapidly changing environments

Disadvantages:

• Ignores cash flows after the payback period — a project with a 3-year payback generating £1 million in Year 4 is treated the same as one generating nothing after payback
• Simple payback ignores the time value of money (discounted payback addresses this but is less commonly used)
• Does not measure profitability — only speed of recovery
• The target payback period is arbitrary — there is no theoretical basis for choosing a specific cutoff
• Ignores the total return on the investment

The ARR (also called Return on Investment — ROI) measures the average annual accounting profit generated by a project as a percentage of the average investment.

ARR = Average annual accounting profit ÷ Average investment × 100

Average annual accounting profit = Total profit over the project life ÷ Number of years. Note: this is profit (after depreciation), not cash flow.

Average investment = (Initial investment + Residual value) ÷ 2. If the residual value is nil: Average investment = Initial investment ÷ 2.

Decision rule: Accept if ARR exceeds the company's target return. Choose the project with the highest ARR if comparing alternatives.

Advantages:

• Simple to calculate and understand
• Uses profit (familiar to managers and consistent with financial reporting)
• Considers all years of the project (unlike payback)
• Can be compared to ROCE or other return measures

Disadvantages:

• Ignores the time value of money — treats all years' profits as equally valuable
• Uses accounting profit, not cash flows (profit is affected by depreciation methods and accounting policies)
• The average can be misleading if profits vary significantly from year to year
• The target ARR is arbitrary (like payback's cutoff)
• Different definitions exist (some use initial investment rather than average investment — watch the exam question)

Overall ranking (theoretical preference): NPV is the preferred method because it directly measures value creation. IRR is a useful complement. Payback and ARR are secondary tools — useful for quick screening but should not be the sole basis for decisions.