Equation of Value Principle: How to Set Up and Solve

Understand Equation of Value Principle

Key rules to remember

Equation of value (discrete, constant i)

Σ Aⱼ × (1 + i)^(t − tⱼ) over income = Σ Bₖ × (1 + i)^(t − sₖ) over outgo

Aⱼ is income at time tⱼ, Bₖ is outgo at time sₖ, t is the valuation date. A factor (1 + i)^(t − time) accumulates if the cashflow is before t and discounts if it is after t.

Discount factor

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

Using time 0 as the valuation date gives PV of income = PV of outgo with factors v^t.

Net cashflow form

Σ cₜ × v^t = 0

cₜ is the net cashflow at time t (income positive, outgo negative). This is the same equation, written as net present value equal to zero.

Continuous cashflows

∫ ρ(s) × exp(−∫₀^s δ(r) dr) ds over income = same expression over outgo

ρ(s) is the rate of payment at time s and δ(r) is the force of interest. With constant δ, the discount factor is e^(−δs).

Equation of value for life contingencies

EPV of income = EPV of outgo

This is the equivalence principle. Each cashflow is multiplied by its probability of being paid and by its discount factor.

Independence of valuation date

Multiplying both sides by (1 + i)^k leaves the solution unchanged

Holds for a constant rate or a consistent discount function. Do not mix different bases on the two sides.

How to solve Equation of Value Principle questions

1. 1Draw a timeline. Mark every cashflow with its amount and time, and label each as income or outgo from one person's point of view.
2. 2Write down the interest basis: effective annual rate, nominal rate with its frequency, or force of interest. Convert it so you have an effective rate for your time unit.
3. 3Name the unknown (X, n or i). Put it on the timeline at its own time.
4. 4Choose the valuation date. Pick the date that avoids awkward factors, usually the date of the unknown payment or time 0.
5. 5Write the equation: value of income at that date = value of outgo at that date. Use (1 + i)^(t − time) for each cashflow.
6. 6Solve for the unknown. For a payment, rearrange. For a time, use logs. For a rate, use algebra in v if the equation is a quadratic, or trial and error with interpolation otherwise.
7. 7Check by substituting your answer, or by redoing the equation at a different date. Give the answer in the units asked, with sensible rounding.

Quickest way: Valuation at the unknown's date

Common mistakes in Equation of Value Principle

• Mixing up income and outgo, or leaving out a payment.

  Students rush and do not draw a timeline, or they view the cashflows from two different people's viewpoints.

  Fix: Draw the timeline first. Label every flow as in or out from one person's view. Count the flows on the timeline against the question.

• Discounting a cashflow that should be accumulated, or the reverse.

  The valuation date is not time 0, and students apply v^t out of habit.

  Fix: Use (1 + i)^(t − time). If the cashflow is after the valuation date, the exponent is negative and you discount. If before, you accumulate.

• Using a nominal rate directly as the effective rate.

  The question states i⁽ᵐ⁾ and students put it straight into (1 + i).

  Fix: Convert first: (1 + i) = (1 + i⁽ᵐ⁾ ÷ m)^m. Then work in the effective rate for the time unit you are using.

• Believing the answer changes if you pick a different valuation date.

  The two equations look different, so students think they are different problems.

  Fix: With a constant rate or one consistent discount function, the answer is identical at any date. Use this to check your answer, and pick the date that gives the easier algebra.

• Counting time wrongly, for example a payment at the start of year 3 placed at time 3.

  Confusion between 'start of year n' and 'end of year n'.

  Fix: Start of year n is time n − 1. End of year n is time n. Mark exact times on the timeline.

• Rounding discount factors too early.

  Students round v^n to 3 or 4 decimals, so the final answer is off.

  Fix: Keep full calculator accuracy until the last step, and round only the final answer.

Worked examples