Mathematics of Finance: formula sheet

Key formulas

Simple interest

SI = P × R × T ÷ 100

R is the rate per year in %. T must be in years. 9 months = 0.75 years.

Amount under simple interest

A = P + SI = P(1 + RT/100)

Use this when the question asks for the total repayable.

Amount under compound interest (yearly)

A = P(1 + R/100)^n

n is the number of years. Interest is added once a year.

Compound interest

CI = A − P

Always subtract the principal if the question asks for interest.

Compounding m times a year

A = P(1 + R/(100m))^(mn)

Half-yearly: m = 2. Quarterly: m = 4. Rate per period = R ÷ m. Periods = m × n.

CI − SI for 2 years

CI − SI = P(R/100)²

Valid for yearly compounding over exactly 2 years.

CI − SI for 3 years

CI − SI = P(R/100)² × (3 + R/100)

Valid for yearly compounding over exactly 3 years.

Effective annual rate

Effective rate = (1 + R/(100m))^m − 1

Gives the equivalent yearly rate when compounding is more than once a year.

Effective rate of interest

E = (1 + i/m)^m − 1

i = nominal annual rate as a decimal, m = number of compounding periods in a year. Multiply E by 100 for a percentage.

Rate per period

Rate per period = i ÷ m

Half-yearly: m = 2. Quarterly: m = 4. Monthly: m = 12. Yearly: m = 1.

Amount after one year

A = P × (1 + i/m)^m = P × (1 + E)

Interest for one year = P × E.

Continuous compounding

E = e^i − 1

Use only when the question says compounded continuously.

Approximation for elimination

E ≈ i + (m − 1) × i² ÷ (2m)

Good for small i. It slightly underestimates E, so use it only to discard options.

Future value (annual compounding)

FV = P × (1 + i)^n

P = amount today, i = rate per year as a decimal, n = number of years.

Future value (m times a year)

FV = P × (1 + r ÷ m)^(m × n)

r = nominal annual rate, m = compounding periods per year. Use the rate per period (r ÷ m) and the number of periods (m × n).

Present value of a single amount

PV = FV ÷ (1 + i)^n

Same as FV × (1 + i)^(−n). Discounting is the reverse of compounding.

Net present value

NPV = C₁ ÷ (1 + i) + C₂ ÷ (1 + i)² + … + Cₙ ÷ (1 + i)ⁿ − C₀

C₁ to Cₙ are cash inflows at the end of years 1 to n. C₀ is the initial outlay at time 0.

NPV decision rule

NPV > 0: accept | NPV < 0: reject | NPV = 0: indifferent

Accepting means the project earns more than the discount rate. When choosing between projects, prefer the higher positive NPV.

FV of ordinary annuity

FV = C × [(1 + i)ⁿ − 1] ÷ i

Payments at the end of each period. FV is measured at the end of period n, right when the last payment is made.

PV of ordinary annuity

PV = C × [1 − (1 + i)⁻ⁿ] ÷ i

Payments at the end of each period. PV is measured one period before the first payment.

FV of annuity due

FV = C × (1 + i) × [(1 + i)ⁿ − 1] ÷ i

Payments at the start of each period. FV is measured at the end of period n.

PV of annuity due

PV = C × (1 + i) × [1 − (1 + i)⁻ⁿ] ÷ i

Payments at the start of each period. The first payment is made today, so it is not discounted.

Link between due and ordinary

Value of annuity due = Value of ordinary annuity × (1 + i)

Holds for both PV and FV, for the same C, i and n.

Finding the instalment

C = FV × i ÷ [(1 + i)ⁿ − 1] or C = PV × i ÷ [1 − (1 + i)⁻ⁿ]

Rearranged ordinary formulas. For an annuity due, also divide the answer by (1 + i).

Present value of perpetuity (immediate)

PV = C ÷ i

C is the payment per period. The first payment comes at the end of the first period. i is the rate per period as a decimal.

Present value of perpetuity due

PV = C + C ÷ i = C × (1 + i) ÷ i

Use when the first payment is made immediately, at time 0.

Rate per period

i = annual rate ÷ number of compounding periods per year

For quarterly payments with quarterly compounding, divide the annual rate by 4.

Future value of an annuity

S = A × [(1+i)^n − 1] ÷ i

A is the instalment at the end of each period. n is the number of periods.

Sinking fund instalment

A = S × i ÷ [(1+i)^n − 1]

Gives the equal end-of-period deposit needed to reach the target sum S.

Amount to be accumulated for replacement

S = cost of new asset − scrap value of old asset

Use when the question asks for a fund to replace an asset and gives scrap value.

Payback period (equal annual inflows)

Payback = Initial investment ÷ Annual cash inflow

Use only when every year's inflow is the same.

Payback period (uneven inflows)

Payback = Years fully recovered + (Unrecovered amount ÷ Cash inflow of the next year)

Build a cumulative cash inflow column. This assumes inflows come evenly through the year.

Net present value

NPV = Σ [CFt ÷ (1 + r)^t] − Initial outlay

IRR is the value of r for which this NPV is zero.

IRR by interpolation

IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L)

L is the lower trial rate (positive NPV) and H is the higher trial rate (negative NPV). The answer is an approximation.

EMI

EMI = P × r × (1 + r)^n ÷ [(1 + r)^n − 1]

P is the loan amount. r is the rate per instalment period (monthly rate = annual rate ÷ 12 for monthly EMI). n is the number of instalments.

Loan as present value of EMIs

P = EMI × [1 − (1 + r)^−n] ÷ r

Same relation as the EMI formula, rearranged. Use it to find the loan amount when EMI is given.

Amortization schedule rows

Interest = Opening balance × r; Principal = EMI − Interest; Closing balance = Opening balance − Principal

Closing balance of one period is the opening balance of the next.

Straight-line depreciation per year

D = (Cost − Scrap value) ÷ n

n is the useful life in years. If there is no scrap value, D = Cost ÷ n.

Book value under straight line

Book value after t years = Cost − t × D

Valid for t up to n. At t = n the book value equals the scrap value.

Book value under reducing balance

Book value after n years = Cost × (1 − r)^n

r is the depreciation rate as a decimal, charged on the opening book value each year.

Rate from scrap value (reducing balance)

r = 1 − (Scrap ÷ Cost)^(1/n)

Use when the scrap value after n years is given and the rate is asked.

Compound growth

Pₙ = P₀ × (1 + g)^n

Use for population, sales or investment growing at a constant rate g per period. For a decline, use (1 − r)^n.

CAGR

CAGR = (Ending value ÷ Starting value)^(1/n) − 1

n is the number of years between the two values, not the number of data points.

Quick revision

• Simple interest: SI = P × r × t, and amount A = P(1 + rt).
• Compound interest: A = P(1 + r/n)^(nt), and CI = A − P.
• Effective rate: E = (1 + r/n)ⁿ − 1, where r is the nominal yearly rate and n is compounding periods per year.
• Future value of a single sum = PV × (1 + i)ⁿ, and present value = FV ÷ (1 + i)ⁿ.
• NPV = present value of cash inflows − initial investment. Accept a project if NPV > 0.
• Future value of an ordinary annuity = C × [(1 + i)ⁿ − 1] ÷ i, with payments at period end.
• Annuity due = ordinary annuity × (1 + i), because each payment comes one period earlier.
• Perpetuity present value = C ÷ i for an ordinary perpetuity. For a perpetuity due (first payment now), PV = C + C ÷ i, which is the same as (C ÷ i) × (1 + i).
• Sinking fund payment = target amount × i ÷ [(1 + i)ⁿ − 1].
• IRR is the rate at which NPV = 0. Payback period is the time taken to recover the initial investment.
• Reducing balance depreciation: book value = cost × (1 − rate)ⁿ. Straight line: (cost − scrap) ÷ life per year.
• CAGR = (ending value ÷ beginning value)^(1/n) − 1.

Common mistakes

• Using the full yearly rate for half-yearly or quarterly compounding. Fix: Always change both together. Half-yearly: R ÷ 2 and n × 2. Quarterly: R ÷ 4 and n × 4.
• Giving the amount when the question asks for compound interest. Fix: Underline 'interest' or 'amount' in the question. For CI, subtract P as the last step.
• Using the full nominal rate in each period, for example (1 + 0.12)^4. Fix: Always write i ÷ m first. Quarterly 12% means 3% per quarter.
• Forgetting to subtract 1 and reporting (1 + i/m)^m as the rate. Fix: The bracket value is the growth factor. The rate is the factor minus 1, so 1.1255 gives 12.55%.
• Discounting the initial outlay Fix: The outlay at time 0 has a discount factor of 1. Use it as it is.
• Multiplying instead of dividing when finding PV Fix: Ask whether the money is moving to a later date or an earlier date. Earlier means divide by (1 + i)^n.
• Using the ordinary formula for an annuity due, or the reverse. Fix: Underline the timing phrase first. Due means multiply the ordinary value by (1 + i).
• Dividing the instalment by (1 + i) when the question is ordinary, or forgetting to divide when finding C for an annuity due. Fix: Remember that due values are higher. To get C from a given due value, divide by (1 + i).
• Using the annual rate for a perpetuity of quarterly or monthly payments. Fix: Convert the rate to the period rate first. For quarterly payments at 12% compounded quarterly, use i = 3%.
• Using C ÷ i for a perpetuity due. Fix: For a due, add one payment: PV = C + C ÷ i.

Exam tips

• Questions on this topic are usually direct. Practise the multipliers for 5%, 10% and 20% so that powers take seconds.
• Read whether the compounding is yearly, half-yearly or quarterly before anything else. Examiners often set a trap option that uses the wrong period.
• Check what is asked: amount, interest, principal or rate. Wrong options are often the amount when interest is asked.
• Use CI − SI formulas to find an unknown principal quickly, but only for 2 or 3 years of yearly compounding.
• If a question needs a long power at an awkward rate, leave it for the end. A wrong guess costs 0.25 marks.
• Questions are usually one-step: give i and m, ask for E. Practise the common cases 1.02⁴, 1.03⁴, 1.05² and 1.01¹² until squaring feels instant.
• Watch the wording. The question may ask for the effective rate, the amount, or the interest. Answer exactly what is asked.
• Options often include the half-yearly, quarterly and monthly results of the same nominal rate. Confirm m before picking.