Time value of money: compound interest and discounting: formula sheet

Key formulas

Simple interest accumulation

A(t) = C(1 + it)

i is the annual rate, t is time in years. Interest earned = C × i × t.

Compound interest accumulation

A(t) = C(1 + i)^t

i is the effective annual rate. Works for fractional t when interest is compounded at rate i per year.

Accumulation factor

A(t) ÷ C = (1 + i)^t

Value at time t of one rupee invested at time 0.

Present value (compound)

C = A(t) × (1 + i)^(-t) = A(t) ÷ (1 + i)^t

Discounting is the reverse of accumulation. Define v = 1 ÷ (1 + i), so C = A(t) × v^t.

Present value (simple)

C = A(t) ÷ (1 + it)

Use only when the question states simple interest.

Interest earned in the nth year (compound)

A(n) − A(n − 1) = C × i × (1 + i)^(n − 1)

For simple interest the interest in every year is Ci.

Rate from two values (compound)

i = (A(t) ÷ C)^(1/t) − 1

Use to find an unknown effective annual rate.

Discount factor

v = 1 ÷ (1 + i)

i is the effective rate of interest per time unit. Then 1 + i = 1/v.

Present value of a single payment

PV = C × vⁿ = C ÷ (1 + i)ⁿ

C is paid at time n, with constant effective rate i per period.

Accumulation (the reverse)

A = P × (1 + i)ⁿ

Present value and accumulated value are inverse operations over the same period.

Relation to effective rate of discount

d = i ÷ (1 + i) = 1 − v

d is the effective rate of discount per period.

Equation of value

Σ (payments in) × v^t = Σ (payments out) × v^t

All terms are valued at the same date, using the same rate.

Discount factor

v = 1 ÷ (1 + i)

Present value of 1 due in one year.

Rate of discount from i

d = i ÷ (1 + i)

Valid for effective annual rates with compound interest.

Rate of discount from v

d = 1 − v

Equivalent to 1 − d = v.

Interest from discount

i = d ÷ (1 − d)

Needs d < 1.

Link between i and d

i − d = i × d

Useful check on your answers.

Discount and interest factors

(1 − d)^−1 = 1 + i

So v = 1 − d and 1 + i = 1 ÷ (1 − d).

Compound discount over n years

PV = S × (1 − d)^n = S × v^n

S is the amount due at time n.

Discount as d × v^−1

d = i × v

Interest at the end, discounted back one year.

Nominal to effective (interest)

1 + i = (1 + i(m)/m)^m

i(m) is the nominal annual rate convertible m times a year. Each period earns i(m)/m.

Nominal to effective (discount)

1 - d = (1 - d(m)/m)^m

Also 1 - d = v = (1 + i)^(-1).

Link between nominal interest and discount

(1 + i(m)/m)^m = (1 - d(p)/p)^(-p)

Both sides equal 1 + i. Use this to convert between i(m) and d(p) with different m and p.

Constant force of interest

δ = ln(1 + i), so 1 + i = e^δ and v = e^(-δ)

ln is the natural logarithm.

Force of interest from the accumulation function

δ(t) = A'(t) ÷ A(t) = d/dt ln A(t)

A(t) is the accumulated value at time t of 1 invested at time 0.

Accumulation with varying force

A(t1, t2) = exp(∫ δ(s) ds from t1 to t2)

This is the value at t2 of 1 invested at t1.

Discount factor with varying force

v(t1, t2) = exp(-∫ δ(s) ds from t1 to t2)

This is the present value at t1 of 1 paid at t2.

Limits as m → ∞

lim i(m) = lim d(m) = δ

Also d < d(m) < δ < i(m) < i for m > 1, when i > 0.

Relation of i and d

d = i ÷ (1 + i) = iv, and i - d = id

Useful for quick conversions.

Quick revision

• Simple interest: A(t) = P(1 + it). Compound interest: A(t) = P(1 + i)^t.
• Discount factor: v = 1 ÷ (1 + i), and PV = payment × v^t.
• Effective rate of discount: d = i ÷ (1 + i) = iv = 1 − v.
• Link: i − d = id, so 1 − d = v.
• Nominal rate: (1 + i^(m) ÷ m)^m = 1 + i, where i^(m) is paid m times a year.
• Nominal discount: (1 − d^(m) ÷ m)^(−m) = 1 + i.
• Force of interest: δ = ln(1 + i), so e^δ = 1 + i and v = e^(−δ).
• Generally δ(t) = (dA/dt) ÷ A(t), and A(t) = exp(∫ δ(s) ds) from 0 to t.
• For a constant force of interest, accumulation over t years is e^(δt).
• As m → ∞, both i^(m) and d^(m) tend to δ, and for m > 1, d < d^(m) < δ < i^(m) < i.
• Compare rates only after converting them to the same effective basis.
• State the time unit, rate basis and valuation date in written answers.

Common mistakes

• Using the simple interest formula when the question is compound. Fix: Look for the word 'compound' or 'effective'. In CM1 assume compound unless the question says simple.
• Writing (1 + i)t instead of (1 + i)^t. Fix: Remember that the exponent signals compounding. Check by testing t = 2: the answer should be C(1 + i)², not C(1 + 2i).
• Multiplying by (1+i)ⁿ instead of dividing when discounting. Fix: Ask: am I moving to an earlier date? Then divide by (1+i)ⁿ, or multiply by vⁿ.
• Using a nominal rate directly as if it were effective. Fix: Convert first: i = (1 + i⁽⁴⁾/4)⁴ − 1, or use the rate per quarter with time in quarters.
• Using d = i ÷ (1 − i) Fix: Remember d = i ÷ (1 + i) and i = d ÷ (1 − d). The sign matches the rate being divided into.
• Treating d as a rate paid at year end Fix: d is paid at the start and measured against the final amount. Think of paying 1 − d now to receive 1 later.
• Treating i(m) as the effective annual rate and using (1 + i(m))^n. Fix: Always divide by m inside the bracket and raise to the power m × n. Per-period rate is i(m)/m.
• Using 1 - d(m) instead of 1 - d(m)/m when converting discount rates. Fix: Write (1 - d(m)/m)^m = 1 - d. Check that d(m) > d for m > 1; if you get d(m) < d, you have made an error.

Exam tips

• Always state your assumption on the interest basis and the time unit in one line. Examiners award marks for clear method even if arithmetic slips.
• In MCQs, test the answer options with t = 1 and t = 2 to catch simple versus compound traps quickly.
• Show the formula in standard notation, such as A(t) = C(1 + i)^t, before substituting numbers.
• For written questions, compare simple and compound by reasoning about (1 + i)^t versus 1 + it, not only by numbers.
• In Paper B, use R or Excel to compute accumulation factors, and write down the formula used in the cell or code.
• Always write the timeline and the valuation date before computing. Written questions give marks for the set-up even if the arithmetic slips.
• In multiple-choice questions, estimate first. If i > 0, a present value must be smaller than the future payment, which removes some options quickly.
• State the formula in standard notation, such as PV = C vⁿ, and show the substituted values in written answers.