Analysing Timings of Cash Flow

• The Discount Factor: Represented as $\frac{1}{(1+r)^n}$, this factor determines how much a future dollar is worth today. As the interest rate ($r$) or the number of periods ($n$) increases, the discount factor decreases, leading to a lower present value.

• Additivity Principle: The total present value of a series of cash flows is the sum of the present values of each individual cash flow. This allows for the analysis of complex, uneven payment streams by breaking them into manageable components.

• Compounding Frequency: The timing analysis must account for how often interest is applied (annually, semi-annually, or monthly). The Effective Annual Rate (EAR) should be used to compare flows with different compounding frequencies.

1. Discounting Single Sums

• To find the value today of a single future payment, use the formula: $PV = \frac{FV}{(1+r)^n}$
• This is the most basic building block for all timing analysis.

2. Annuity Calculations

• Ordinary Annuity: Payments occur at the end of each period. The PV is calculated using: $PV = PMT \times \left[ \frac{1 - (1+r)^{-n}}{r} \right]$
• Annuity Due: Payments occur at the beginning of each period. Because each payment happens one period earlier, the PV is higher than an ordinary annuity: $PV_{due} = PV_{ordinary} \times (1+r)$

• The Timeline Method: Always draw a physical timeline for complex problems. Visualizing whether a payment falls at $t=0$ or $t=1$ prevents the most common formula selection errors.

• Check the 'n' Value: Ensure that $n$ represents the total number of payments, not just the number of years. If payments are monthly over 5 years, $n$ must be 60.

• Rate Consistency: The interest rate $r$ must match the period of the cash flow. If cash flows are monthly, you must use a monthly interest rate ($r_{annual} / 12$), not the annual rate.

• Sanity Check: Remember that discounting always reduces the value. If your calculated Present Value is higher than the sum of the nominal cash flows, you have likely compounded instead of discounted.

• Year 0 Confusion: Students often forget that 'immediate' or 'today' means $t=0$. In an ordinary annuity, the first cash flow is at the end of the first year ($t=1$), which means it is discounted for one full period.

• Mid-Period Flows: Assuming all cash flows occur at year-end when they actually occur mid-year can lead to significant valuation errors. Mid-year flows should be discounted using $n = 0.5, 1.5, 2.5$, etc.

• Perpetuity Timing: For a perpetuity starting at $t=1$, the formula $PV = \frac{C}{r}$ gives the value at $t=0$. If the perpetuity starts at $t=5$, the formula gives the value at $t=4$, which then must be discounted back to $t=0$.