Interest rates over different time periods: formula sheet

Key formulas

Simple interest accumulation

A(t) = 1 + it; amount = P(1 + it)

Interest is on the original capital only. t in years, i an annual rate.

Compound interest accumulation

A(t) = (1 + i)^t; amount = P(1 + i)^t

i is the effective annual rate. t can be fractional.

Discount factor (compound)

v = 1 ÷ (1 + i); v^t = (1 + i)^(−t)

Present value of ₹1 due in t years.

Discount factor (simple)

1 ÷ (1 + it)

Reciprocal of the simple accumulation factor.

Effective rate in year n under simple interest

i_n = i ÷ (1 + i(n − 1))

Falls as n rises. Under compound interest the effective rate stays at i.

Effective rate from accumulation function

i_n = [A(n) − A(n − 1)] ÷ A(n − 1)

Works for any accumulation function.

Present value and accumulation

PV = S × v^t; S = PV × (1 + i)^t

Move money forward by multiplying by the accumulation factor, back by the discount factor.

Nominal to effective annual

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

i(m) is a nominal annual rate convertible m times a year. i(m)/m is the effective rate per 1/m year.

Effective to nominal

i(m) = m × [(1 + i)^(1/m) − 1]

Use this to find the nominal rate that matches a given effective annual rate.

Effective rate per period

j = i(m)/m, and (1 + j)^m = 1 + i

j is the effective rate over each 1/m year.

Ordering of rates

i(m) < i for m > 1, and i(m) decreases as m increases

Holds for i > 0. As m grows, i(m) tends to the force of interest δ = ln(1 + i).

Equivalent rate for another period

(1 + i)^t = 1 + (effective rate over t years)

Gives an effective rate over any period of length t years, such as the effective quarterly rate (1 + i)^(1/4) − 1.

Definition of force of interest

δ(t) = A'(t) ÷ A(t) = d/dt [ln A(t)]

A(t) is the accumulation function. δ is an instantaneous rate, quoted per year.

Accumulation factor from t₁ to t₂

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

Works for any δ(s), constant or varying. The limits are times, not rates.

Discount factor from t₂ back to t₁

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

This is the reciprocal of the accumulation factor over the same interval.

Constant force of interest

A(t) = A(0) × e^(δt)

Holds only when δ does not change with time.

Link to effective annual rate

1 + i = e^δ, so δ = ln(1 + i)

Valid when δ is constant over the year. For varying δ, use the integral over that year.

Link to discount rate and discount factor

v = e^(−δ), d = 1 − e^(−δ)

v = 1 ÷ (1 + i) for a constant force.

Link to nominal rate

δ = lim of i^(m) as m → ∞

i^(m) is the nominal rate convertible m times a year. δ is its continuous limit.

Discount factor

v = 1 ÷ (1 + i)

One-year discount factor at effective annual rate i. PV of S due in n years = S × v^n.

Effective rate of discount

d = i ÷ (1 + i) = 1 − v

d is interest at the end of the year as a proportion of the amount at the end of the year.

Interest from discount

i = d ÷ (1 − d)

Use to convert d to i. Also 1 + i = 1 ÷ (1 − d).

Difference identity

i − d = i × d

Quick check that your i and d are consistent.

Nominal discount rate

(1 − d(m) ÷ m)^m = 1 − d = v

d(m) is payable m times a year in advance. Each 1/m year the factor is 1 − d(m)/m.

Link between nominal rates

(1 + i(m) ÷ m)^m = 1 + i = (1 − d(p) ÷ p)^(−p)

Use this to move between i(m) and d(p) for any m and p, through the effective annual rate.

Nominal link, same m

1 ÷ d(m) − 1 ÷ i(m) = 1 ÷ m

Valid when i(m) and d(m) have the same m and the same effective rate.

Force of interest

δ = ln(1 + i); v = e^(−δ)

Both d(m) and i(m) tend to δ as m → ∞.

PV with nominal discount rate

PV = S × (1 − d(m) ÷ m)^(mn)

S is paid at time n years. mn must be the number of discount periods.

Equation of value

Σ (inflows × v^t) = Σ (outflows × v^t)

Taken at one time point. Cashflows after the chosen point are discounted; those before it are accumulated.

Spot rate accumulation

Accumulation of ₹1 over t years = (1 + y_t)^t

y_t is the annual effective spot rate for term t, measured from time 0.

Discount factor from spot rate

v(t) = (1 + y_t)^(-t)

Present value of ₹1 due at time t. Also the price of a t-year zero-coupon bond paying ₹1.

No-arbitrage relationship

(1 + y_{t+r})^(t+r) = (1 + y_t)^t × (1 + f_{t,r})^r

f_{t,r} is the annual effective forward rate for the r years starting at time t.

Forward rate from spot rates

f_{t,r} = [ (1 + y_{t+r})^(t+r) ÷ (1 + y_t)^t ]^(1/r) − 1

Rearranged from the relationship above.

One-year forward rate

f_{t,1} = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t − 1

The case r = 1, the most common in exam questions.

Forward rate from discount factors

(1 + f_{t,r})^r = v(t) ÷ v(t+r)

Useful when zero-coupon prices are given.

Spot rate from one-year forwards

(1 + y_t)^t = (1 + f_{0,1})(1 + f_{1,1}) ... (1 + f_{t-1,1})

Here f_{0,1} = y_1. Spot rate is a geometric average of the one-year forwards.

Quick revision

• Simple interest: A(t) = P(1 + it). Compound interest: A(t) = P(1 + i)^t.
• v = 1/(1 + i) and d = i/(1 + i) = 1 − v.
• 1 + i = (1 + i(m)/m)^m, where i(m) is the nominal rate compounded m times a year.
• 1 − d = (1 − d(p)/p)^p for nominal discount rates.
• δ = ln(1 + i), so 1 + i = e^δ and v = e^(−δ).
• With varying force of interest, the accumulation from 0 to t is exp(∫ δ(s) ds) from 0 to t.
• Present value of a payment S due at time t is S × v^t, or S × exp(−∫ δ(s) ds) if δ varies.
• Always convert the rate to match the time unit of the cash flows before discounting.
• Spot rate y_t is the annual rate for an investment from time 0 to t; the discount factor is (1 + y_t)^(−t).
• Forward rate f from t to t + 1 satisfies (1 + y_(t+1))^(t+1) = (1 + y_t)^t × (1 + f).
• For a fixed effective rate i > 0, i(m) decreases as m increases, and i(m) < i for m > 1.
• Check answers for sense: δ < i(m) < i for m > 1 when i > 0.

Common mistakes

• Using the compound formula when the question says simple interest, or the reverse. Fix: Underline the word simple or compound before writing any formula. If neither appears, use compound.
• Treating the effective rate in each year of a simple interest investment as constant. Fix: Use i_n = i ÷ (1 + i(n − 1)). The rate falls because the same interest is earned on a growing balance.
• Treating i(m) as the effective annual rate Fix: Always divide by m to get the period rate, then compound m times to get the effective annual rate.
• Using i(m) × m instead of i(m) ÷ m Fix: The period rate is i(m)/m. Multiply by m only at the end when converting an effective period rate into a nominal rate.
• Using δ = i or treating δ as the effective rate. Fix: Remember δ = ln(1 + i). For positive rates, δ is smaller than i. Convert before you use either in an effective-rate formula.
• Integrating from 0 to t when the question asks for the interval t₁ to t₂. Fix: Always write the limits of the integral from the question. Circle the start and end times before you integrate.
• Using d(m) as if it were an effective rate over the year, for example PV = S × (1 − d(m))^n. Fix: Always divide by m and raise to the power mn: PV = S × (1 − d(m)/m)^(mn).
• Writing i = 1 − v or d = 1/v − 1. Fix: Remember d = 1 − v and i = 1/v − 1. Check with i − d = id.
• Subtracting rates, for example f = 2 × y_2 − y_1 for the second-year forward rate. Fix: Use accumulation factors: 1 + f_{1,1} = (1 + y_2)^2 ÷ (1 + y_1).
• Forgetting the power t on the spot rate, using (1 + y_t) instead of (1 + y_t)^t. Fix: Always write the exponent. Spot rates are annual, so a t-year accumulation needs the power t.

Exam tips

• Check the wording for simple or compound. A single missed word changes the whole answer.
• In written answers, state the formula in notation first, then the substituted numbers, then the result. Method marks depend on this.
• For comparison questions, say clearly which basis is larger and for which range of t.
• State your assumption when the question is silent, for example: I assume compound interest at an effective annual rate.
• Keep full precision on a calculator and round only the final answer, as small errors grow over long terms.
• Write the compounding frequency m next to the rate before you do anything else.
• State the formula in notation, such as 1 + i = (1 + i(m)/m)^m, then substitute. Marks are given for method.
• Use the sense check i(m) < i to catch calculator slips in MCQs.