Equation of Value

Borrow $80,000 today, repay with three unequal payments at years 2, 5, and 8. The equation of value pins down the final payment, an error propagates everywhere.

Set the value of all inflows equal to all outflows at a focal date:

$$\sum_k C_k^{(\text{in})} (1+i)^{t^* - t_k} = \sum_j C_j^{(\text{out})} (1+i)^{t^* - t_j}$$

Equivalently, treat inflows as positive and outflows as negative:

$$\sum_k C_k \cdot v^{t_k} = 0$$

HIGH-FREQUENCY: Setting up the equation of value correctly (every cash flow, its sign, its timing) is tested on virtually every multi-payment FM problem.

The equation is linear in $X$. Set the focal date at the time of $X$ to avoid dividing by a discount factor:

$$X = \text{accumulated flows opposite } X - \text{accumulated other flows in } X\text{'s direction}$$

This works from either side of the deal. For a borrower solving for a final repayment, $X$ is the accumulated loan minus the accumulated earlier repayments.

Common mistakes

• Wrong sign on a cash flow. Reversing inflow/outflow signs leads to a negative rate or nonsensical payment. If the equation gives nonsense, check signs first.
• Off-by-one timing. "End of year 5" = time 5. "Beginning of year 5" = time 4. For $30,000 at 6%, using $v^4 = 0.79209$ instead of $v^5 = 0.74726$ changes PV by $1,345. Trap: $23,763 instead of $22,418.
• Forgetting a cash flow. Omitting the $25,000 at time 2 gives $X = (80000 - 22418)/0.62741 = 91777$. Trap: $91,777.

Bottom line

• Equation of value: PV of inflows = PV of outflows at a chosen focal date.
• Compound interest: any focal date gives the same answer; under simple interest the focal date matters, so use the date the problem specifies.
• Unknown payment: set the focal date at the payment time so $X$ carries no discount factor.
• Unknown rate (IRR): focal date at time 0, write NPV as a polynomial in $v$, test two rates, interpolate; denominator is $NPV_L - NPV_H$.

Exam shortcut

Draw the timeline, then write the equation at time 0. Unknown payment (linear, isolate. Unknown rate) test two, interpolate. Budget 90 seconds for a straightforward equation, up to 3 minutes for IRR. Focal date rule: payment unknown at time $t$? Focal date at $t$. Rate unknown? Focal date at 0. IRR: "Low rate plus fraction of the gap."

The full lesson (about 2,534 words, 17 min read) adds 4 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1d

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