Formula Sheet: Rate of Return

Rate of Return Fundamentals

Rate of Return (ROR) Definition

The Rate of Return (ROR), also known as Internal Rate of Return (IRR), is the interest rate at which the present worth (or net present value) of a cash flow series equals zero.

\[PW = \sum_{t=0}^{n} \frac{F_t}{(1 + i^*)^t} = 0\]

• PW = Present Worth
• F_t = Net cash flow at time t (receipts - disbursements)
• i* = Rate of Return (the unknown interest rate to be determined)
• t = Time period
• n = Total number of time periods

Note: ROR is the interest rate that makes the present worth of benefits equal to the present worth of costs.

Alternative Formulations

Annual Worth Formulation:

\[AW = 0\]

The rate of return is the interest rate that makes the annual worth equal to zero.

Future Worth Formulation:

\[FW = \sum_{t=0}^{n} F_t(1 + i^*)^{n-t} = 0\]

• FW = Future Worth

Basic ROR Equation Forms

Simple Two-Cash-Flow ROR

For a single investment P at time 0 and a single return F at time n:

\[P = \frac{F}{(1 + i^*)^n}\]

Solving for i*:

\[i^* = \left(\frac{F}{P}\right)^{1/n} - 1\]

• P = Initial investment (present value)
• F = Future return
• n = Number of periods
• i* = Rate of return

Uniform Series ROR

For an initial investment P and uniform annual receipts A for n periods:

\[P = A \cdot \frac{(1 + i^*)^n - 1}{i^*(1 + i^*)^n}\]

Or equivalently:

\[P = A(P/A, i^*, n)\]

• A = Uniform annual receipt or payment
• (P/A, i*, n) = Uniform series present worth factor

General Cash Flow ROR

For a general series of cash flows:

\[\sum_{t=0}^{n} \frac{F_t}{(1 + i^*)^t} = 0\]

Note: This equation typically requires iterative or numerical solution methods (trial and error, interpolation, or calculator/software).

Solution Methods for ROR

Trial and Error Method

1. Select a trial interest rate
2. Calculate PW, AW, or FW at that rate
3. If result is positive, increase the trial rate; if negative, decrease the trial rate
4. Continue until PW ≈ 0

Linear Interpolation

When PW values at two interest rates bracket zero:

\[i^* = i_L + \frac{PW_L}{PW_L - PW_H}(i_H - i_L)\]

• i_L = Lower trial interest rate (where PW is positive)
• i_H = Higher trial interest rate (where PW is negative)
• PW_L = Present worth at i_L (positive value)
• PW_H = Present worth at i_H (negative value)

Note: This method assumes a linear relationship between interest rate and PW, which is an approximation.

Decision Rules and Criteria

Single Project Evaluation

Decision Rule:

• If i* ≥ MARR, the project is acceptable
• If i* <>, the project is not acceptable

Where:

• MARR = Minimum Attractive Rate of Return (required rate of return)

Multiple Alternatives Comparison

Incremental Analysis Required: When comparing mutually exclusive alternatives using ROR, incremental analysis must be performed.

General Principle: Do not select among alternatives based solely on which has the highest ROR. Instead, use incremental ROR analysis.

Incremental Rate of Return Analysis

Incremental Cash Flow

The incremental cash flow between two alternatives (Higher cost - Lower cost):

\[\Delta F_t = F_{H,t} - F_{L,t}\]

• ΔF_t = Incremental cash flow at time t
• F_H,t = Cash flow for higher-cost alternative at time t
• F_L,t = Cash flow for lower-cost alternative at time t

Incremental ROR Equation

\[\sum_{t=0}^{n} \frac{\Delta F_t}{(1 + \Delta i^*)^t} = 0\]

• Δi* = Incremental rate of return

Alternative form:

\[PW_H(i) - PW_L(i) = 0\]

Where i is solved to find Δi*

Incremental Analysis Decision Rules

For Two Alternatives (B = Higher cost, A = Lower cost):

1. Calculate the incremental ROR: Δi* (B - A)
2. If Δi* ≥ MARR, select the higher-cost alternative B
3. If Δi* <>, select the lower-cost alternative A

Note: Both alternatives must individually have ROR ≥ MARR to be considered.

Multiple Alternatives Procedure

1. Rank alternatives by increasing initial cost
2. Eliminate any alternative with ROR <>
3. Compare remaining alternatives incrementally, starting with the lowest cost
4. Compare the defender (current best) with the next challenger
5. If Δi* ≥ MARR, the challenger becomes the new defender
6. If Δi* < marr,="" retain="" the="" current="">
7. Continue until all alternatives are evaluated

Special Cases and Considerations

Multiple Rates of Return

Descartes' Rule of Signs: The maximum number of positive real roots (rates of return) equals the number of sign changes in the cash flow series.

Conditions for Multiple RORs:

• Non-conventional cash flow patterns (more than one sign change)
• Cash flows that are not a simple investment followed by returns

Example: A cash flow series: -100, +300, -200 has two sign changes and may have two positive ROR values.

Recommendation: When multiple RORs exist, use Present Worth or Annual Worth method instead for decision-making.

No Rate of Return Solution

Some cash flow series have no real, positive ROR solution.

Example: All positive cash flows or all negative cash flows will not yield a meaningful ROR.

External Rate of Return (ERR)

Used to resolve multiple ROR problems by assuming an external reinvestment rate.

Procedure:

1. Move all negative cash flows to time 0 using MARR as the discount rate
2. Move all positive cash flows to time n using MARR as the compound rate
3. Solve for the interest rate that equates these two amounts

\[P_{negatives} = \frac{F_{positives}}{(1 + i_E)^n}\]

• i_E = External rate of return
• P_negatives = Present worth of all negative cash flows
• F_positives = Future worth of all positive cash flows

ROR for Bonds

Bond Yield to Maturity

The ROR for a bond is called the yield to maturity (YTM).

\[P_{bond} = \sum_{t=1}^{n} \frac{I}{(1 + i^*)^t} + \frac{V}{(1 + i^*)^n}\]

• P_bond = Current bond price (purchase price)
• I = Periodic interest payment (coupon payment)
• V = Face value (par value) of the bond
• n = Number of periods to maturity
• i* = Yield to maturity (ROR on the bond)

Bond Interest Payment

\[I = V \times r\]

• r = Bond coupon rate (stated interest rate)
• V = Face value of the bond

Relationship Between ROR and Other Methods

ROR vs. Present Worth

At i = i*:

\[PW(i^*) = 0\]

• If i <>, then PW > 0 (for conventional investment)
• If i > i*, then PW <> (for conventional investment)

ROR vs. Annual Worth

At i = i*:

\[AW(i^*) = 0\]

ROR vs. Future Worth

At i = i*:

\[FW(i^*) = 0\]

Break-Even Analysis Using ROR

Break-Even ROR

The break-even ROR is the interest rate at which two alternatives are economically equivalent:

\[PW_A(i_{BE}) = PW_B(i_{BE})\]

Or equivalently:

\[PW_A(i_{BE}) - PW_B(i_{BE}) = 0\]

• i_BE = Break-even interest rate

Note: The break-even ROR is identical to the incremental ROR (Δi*).

Conventional vs. Non-Conventional Cash Flows

Conventional Cash Flow

Definition: An investment with one or more initial cash outflows followed only by cash inflows.

Sign Pattern: Negative cash flows followed by positive cash flows (one sign change only).

Example: -1000, +300, +300, +300, +300

Characteristic: Unique, positive ROR solution exists.

Non-Conventional Cash Flow

Definition: Cash flow series with more than one sign change.

Example: -1000, +500, +500, -200

Characteristic: May have multiple positive ROR values or no ROR solution.

Important Assumptions and Limitations

ROR Method Assumptions

• All net receipts (positive cash flows) are reinvested at the calculated ROR
• All net disbursements (negative cash flows) are financed at the calculated ROR
• The project can be terminated at any time with the calculated return

Limitations of ROR Method

• Cannot be used for ranking mutually exclusive alternatives without incremental analysis
• May yield multiple or no solutions for non-conventional cash flows
• Assumes reinvestment at the ROR, which may not be realistic
• Requires iterative solution for most cash flow patterns

When to Use ROR Method

• Evaluating single projects against MARR
• Comparing mutually exclusive alternatives (with incremental analysis)
• When decision-makers prefer a rate-based criterion

When NOT to Use ROR Method

• Non-conventional cash flows with multiple ROR solutions
• When quick decisions are needed (PW/AW methods are faster)
• Comparing projects of significantly different scales without incremental analysis

ROR for Leasing vs. Purchasing

Incremental Analysis for Lease/Purchase

To find the ROR on the extra capital required to purchase instead of lease:

\[\sum_{t=0}^{n} \frac{(CF_{purchase,t} - CF_{lease,t})}{(1 + \Delta i^*)^t} = 0\]

Decision Rule:

• If Δi* ≥ MARR, purchasing is preferred
• If Δi* <>, leasing is preferred

Tax Considerations in ROR

After-Tax ROR

When tax effects are considered, use after-tax cash flows (ATCF):

\[\sum_{t=0}^{n} \frac{ATCF_t}{(1 + i^*_{AT})^t} = 0\]

• ATCF_t = After-tax cash flow at time t
• i*_AT = After-tax rate of return

Note: After-tax ROR is generally lower than before-tax ROR for profitable projects.