Present value and accumulated value of cashflow streams: formula sheet

Key formulas

Compound interest accumulation

A(t) = (1 + i)^t, so accumulated value = C × (1 + i)^t

i is the effective annual rate and t is in years. The rate and the time unit must match.

Compound interest discount factor

v = 1 ÷ (1 + i); PV = C × v^t = C × (1 + i)^(-t)

PV is the value at time 0 of a payment C due at time t.

Simple interest accumulation

A(t) = 1 + it, so accumulated value = C × (1 + it)

Interest is earned only on the original amount C.

Simple interest present value

PV = C ÷ (1 + it)

This is the usual convention: the present value is the amount that accumulates to C under simple interest over time t.

Accumulating or discounting between two dates

Value at time s = C × (1 + i)^(s - t) for a payment C at time t

Under compound interest this works for any s, whether before or after t. A negative power discounts.

Interest earned over a period

Interest = C × [A(t) - 1]

Simple: C × it. Compound: C × [(1 + i)^t - 1].

Discount factor

v = 1 ÷ (1 + i), so v^t = (1 + i)^(−t)

i is the effective annual rate of interest. If the rate is nominal, convert it to an effective annual rate first.

Present value of a discrete stream

PV = Σ cₜ × v^t = Σ cₜ ÷ (1 + i)^t

Sum over all payment times t, measured from the valuation date at time 0. Outflows carry a negative sign.

Accumulated value at time n

AV(n) = Σ cₜ × (1 + i)^(n − t)

Valid for constant i. Payments made after time n would have a negative power, which means they are discounted back to n.

Link between PV and accumulated value

AV(n) = PV × (1 + i)^n

Use this as a check. It holds for a constant effective rate.

Net present value

NPV(i) = PV of inflows − PV of outflows = Σ cₜ v^t

Positive NPV means the cashflows are profitable at rate i.

Value at any time s

Value at s = Σ cₜ × (1 + i)^(s − t)

This is the general form. Setting s = 0 gives PV, and s = n gives accumulated value.

Varying rate version

PV = Σ cₜ × exp(−∫₀ᵗ δ(s) ds)

Use when the force of interest δ(s) changes with time. For constant δ this becomes cₜ × e^(−δt).

Total payment

Total paid from a to b = ∫ₐᵇ ρ(t) dt

No interest involved. For constant ρ this is ρ(b − a).

Discount factor from force of interest

v(t) = exp(−∫₀ᵗ δ(s) ds)

Gives the present value at time 0 of ₹1 paid at time t.

Accumulation factor between two times

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

Growth of ₹1 from time t₁ to time t₂, where t₁ < t₂.

Present value of a continuous stream

PV = ∫ₐᵇ ρ(t) exp(−∫₀ᵗ δ(s) ds) dt

Payments run from time a to time b, valued at time 0.

Accumulated value of a continuous stream

AV at time T = ∫ₐᵇ ρ(t) exp(∫ₜᵀ δ(s) ds) dt

Here T ≥ b. Each payment grows from t to T.

Constant ρ and constant δ

PV = ρ ∫₀ⁿ e^(−δt) dt = ρ (1 − e^(−δn)) ÷ δ

This equals ρ ā_n̅|. Here ā_n̅| = (1 − v^n) ÷ δ with δ = ln(1 + i).

Accumulated value, constant case

AV at time n = ρ (e^(δn) − 1) ÷ δ

This equals ρ s̄_n̅|.

Discount factor

v = 1 ÷ (1 + i), and v^t = (1 + i)^(−t)

Present value at time 0 of ₹1 paid at time t, for effective rate i per period.

Accumulation factor

(1 + i)^t

Value at time t of ₹1 invested at time 0.

Equation of value at time 0

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

Use any focal date. Rates and times must be in the same unit.

Equation of value at focal date T

Σ C_t × (1 + i)^(T − t) = 0, with inflows positive and outflows negative

Works for payments before and after T. Use (1 + i)^(T − t) for every payment.

Exact equated time

Σ C_t × v^t = (Σ C_t) × v^t̄, so t̄ = −ln[ Σ C_t v^t ÷ Σ C_t ] ÷ δ, where δ = ln(1 + i)

Exact for a single replacement payment equal to the total of the C_t.

Approximate equated time

t̄ ≈ Σ (t × C_t) ÷ Σ C_t

Weighted average of times. Only an approximation, and it is not exact for compound interest.

Net present value

NPV(i) = Σ C_t × v^t

Inflows positive and outflows negative. Equation of value is NPV(i) = 0 for the rate that makes it so.

Quick revision

• Accumulation factor over one year: 1 + i. Discount factor: v = 1 ÷ (1 + i).
• Discount rate d = i ÷ (1 + i) = iv, and 1 − d = v.
• Force of interest: δ = ln(1 + i), so 1 + i = e^δ.
• With a variable force, the accumulation factor from s to t is exp(∫ δ(r) dr) from s to t.
• Present value of a stream of payments: Σ C_t × v(t) over all payment times t.
• Net present value is the PV of inflows minus the PV of outflows at a stated rate.
• Continuous payment at rate ρ(t) has present value ∫ ρ(t) e^(−∫δ) dt, the inner integral running from 0 to t.
• Nominal rate i^(m) is convertible m times a year: 1 + i = (1 + i^(m) ÷ m)^m.
• The equation of value holds at any date, so choose the date that makes the working simplest.
• Yield (IRR) is the rate i at which the NPV of the cashflows equals zero.
• Always check the units of time and whether the rate is effective annual before you substitute.

Common mistakes

• Using simple interest formula when compound interest is intended, or the reverse. Fix: Look for the words 'simple' or 'compound' in the question. If neither appears, use compound and state that assumption.
• Writing 1 + i as 1.i, for example using 1.6 for i = 6%. Fix: Convert the rate to a decimal first: 6% = 0.06, so 1 + i = 1.06.
• Using the wrong sign for the power, such as multiplying by v^t when you should accumulate. Fix: Always write the exponent as (valuation date − payment time). If the answer is positive, accumulate; if negative, discount.
• Using a nominal rate directly as i. Fix: Convert first: i = (1 + i⁽²⁾ ÷ 2)² − 1. Then use this effective annual i.
• Using v^t = (1 + i)^(−t) with a time-varying force and ignoring the integral. Fix: Always compute ∫ δ(s) ds first. Only use e^(−δt) when δ is truly constant.
• Integrating δ from 0 to t when finding accumulation from t to T. Fix: Check the limits. For growth from t to T the inner integral runs from t to T.
• Discounting a payment that lies before the focal date, or accumulating one that lies after it Fix: Use (1 + i)^(T − t) for every payment. If t is before T the exponent is positive and the payment grows. If t is after T the exponent is negative and it is discounted.
• Using a nominal rate directly as the effective rate Fix: Convert first. The effective monthly rate is 0.06 ÷ 12 = 0.005. The effective annual rate is (1.005)^12 − 1.

Exam tips

• In multiple-choice questions, estimate first. Direction and rough size of the answer often eliminate two options at once.
• In written questions, show the formula, the substituted numbers and the factor before the final figure. Method marks are awarded even if the arithmetic slips.
• State your assumption when the type of interest is not given: write 'assuming compound interest'.
• When the rate changes over time, discount in stages from the later date back to the earlier date, one rate at a time.
• Watch for time in months. Convert to years (for example 18 months = 1.5 years) before using an annual rate.
• Always draw the timeline first. It earns method marks even if your arithmetic slips.
• In written answers, show the formula in the form PV = Σ cₜ v^t, then the numbers, then the result.
• Check the rate type in the question. Many marks are lost by using a nominal rate as effective.