Discrete Cashflow Streams and Net Present Value

Understand Discrete Cashflow Streams and Net Present Value

Key rules to remember

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).

How to solve Discrete Cashflow Streams and Net Present Value questions

1. 1Draw a timeline. Mark each payment with its time and sign (inflow positive, outflow negative).
2. 2Identify the valuation date. It is time 0 for PV, time n for accumulated value, or any date the question names.
3. 3Get the interest rate in the right form. Convert nominal rates or discount rates to an effective annual rate i, or use δ if the force of interest is given.
4. 4Write the value as a sum: each payment times (1 + i)^(valuation date − payment time). Check the sign of each power.
5. 5Calculate each term separately and keep full precision until the end.
6. 6Add the terms. For NPV, make sure outflows are subtracted.
7. 7Check by sense: PV should be less than the sum of payments if all are positive and i > 0. Accumulated value should be more than the sum for payments before n.
8. 8State the answer with units, such as ₹, and the date of valuation.

Quickest way: Factor-by-factor table, then Horner-style roll-forward

Common mistakes in Discrete Cashflow Streams and Net Present Value

• Using the wrong sign for the power, such as multiplying by v^t when you should accumulate.

  Students memorise the PV formula and apply it even when the valuation date is later than time 0.

  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.

  The question says '8% per annum convertible half-yearly' and the student uses 0.08.

  Fix: Convert first: i = (1 + i⁽²⁾ ÷ 2)² − 1. Then use this effective annual i.

• Adding outflows instead of subtracting them in NPV.

  Costs are written as positive numbers in the problem text.

  Fix: Put a negative sign on every outflow in the timeline before you calculate anything.

• Counting the wrong number of years between payments.

  Words such as 'end of the third year' and 'beginning of the third year' are confused.

  Fix: Convert every date to a time on the number line. End of year 3 is t = 3. Start of year 3 is t = 2.

• Rounding v or intermediate values too early.

  Students copy 0.9091 from a table or screen and reuse it.

  Fix: Store v in memory. Round only the final answer, usually to the nearest rupee or to the precision asked.

• Concluding a project is good because the sum of cashflows is positive.

  Time value is ignored in the final judgement.

  Fix: Judge by NPV at the stated rate. A project with a positive total can have a negative NPV.

Worked examples