Present Worth Analysis: PW, NPV, MARR, Cash Flow Diagrams

Key Concepts and Definitions

Present Worth (PW) and Net Present Value (NPV)

Present Worth (PW) is the current value of a series of future cash flows, adjusted for the time value of money. It is calculated by discounting each cash flow back to the present using a discount rate. The formula for PW is:

$$PW=\sum _{t=0}^n\frac{C{F}_t}{(1+i{)}^t}$$

Where:

• $C{F}_t$ = Cash flow at time $t$
• $i$ = Discount rate (or interest rate)
• $n$ = Number of periods

Net Present Value (NPV) is the difference between the present worth of cash inflows and outflows. It is used to determine whether an investment is financially viable.

$$NPV=P{W}_{\text{inflows}}-P{W}_{\text{outflows}}$$

If $NPV\ge 0$, the investment is acceptable.

Minimum Attractive Rate of Return (MARR)

MARR is the minimum rate of return that a project must achieve to be considered acceptable. It is often set by the company or investor and is used as the discount rate in PW and NPV calculations.

Cash Flow Diagrams

Cash flow diagrams visually represent the timing and magnitude of cash flows associated with a project. They help in understanding the sequence of cash inflows and outflows over time.

Worked Example: Investment in a New Machine

Suppose a company is considering investing in a new machine that costs Rs. 500,000 and is expected to generate cash inflows of Rs. 150,000 per year for 5 years. The MARR is 10%. Calculate the NPV.

Step 1: Draw the Cash Flow Diagram

Year 0: -500,000 (Initial Investment)
Year 1: 150,000
Year 2: 150,000
Year 3: 150,000
Year 4: 150,000
Year 5: 150,000

Step 2: Calculate Present Worth of Cash Inflows

Using the PW formula for each year:

$$PW=\frac{150,000}{(1+0.10{)}^1}+\frac{150,000}{(1+0.10{)}^2}+\frac{150,000}{(1+0.10{)}^3}+\frac{150,000}{(1+0.10{)}^4}+\frac{150,000}{(1+0.10{)}^5}$$

$$PW=136,364+123,967+112,700+102,455+93,140=568,626$$

Step 3: Calculate NPV

$$NPV=P{W}_{\text{inflows}}-P{W}_{\text{outflows}}$$ $$NPV=568,626-500,000=68,626$$

Since $NPV>0$, the investment is acceptable.

Comparison of PW and NPV

Advantages and Disadvantages of Present Worth Analysis

Advantages

• Time Value of Money: Considers the time value of money, making it more accurate than methods that do not account for it.
• Comprehensive: Evaluates all cash flows over the life of the project.
• Comparability: Allows comparison of projects with different lifespans by using equivalent annual worth or other techniques.

Disadvantages

• Sensitivity to Discount Rate: The choice of discount rate can significantly impact the results.
• Complexity: Requires accurate estimation of future cash flows and discount rates.
• Ignores Risk: Does not explicitly account for risk or uncertainty in cash flows.

In the Real World

1. eSewa and Khalti: Digital Payment Systems

eSewa and Khalti use present worth analysis to evaluate the financial viability of expanding their services. For example, when deciding whether to invest in new servers or upgrade their payment infrastructure, they calculate the NPV of expected transaction fees over time. If the NPV is positive, the investment is justified.

2. Ncell and NTC: Telecom Infrastructure Investment

Telecom companies like Ncell and NTC use PW analysis to determine whether to invest in new cell towers or upgrade existing ones. They consider the cost of infrastructure and the present worth of future revenue from increased subscribers. For instance, if a new tower costs Rs. 20 million and generates Rs. 5 million annually for 10 years at a 12% discount rate, the NPV helps decide if the investment is worthwhile.

3. Daraz: E-Commerce Logistics

Daraz, an e-commerce platform, uses PW analysis to evaluate logistics investments. For example, if Daraz considers purchasing new delivery vehicles costing Rs. 10 million each, generating savings of Rs. 3 million annually in fuel and maintenance costs over 5 years, the NPV at a 10% discount rate would be calculated to decide if the investment is profitable.

Visualizing Cash Flows and NPV

Cash Flow Diagram for Daraz's Investment

flowchart LR
    A["Year 0"] -->|"-10,000,000"| B["Year 1"]
    B -->|"3,000,000"| C["Year 2"]
    C -->|"3,000,000"| D["Year 3"]
    D -->|"3,000,000"| E["Year 4"]
    E -->|"3,000,000"| F["Year 5"]

NPV Calculation for Daraz

Using a 10% discount rate:

$$PW=\frac{3,000,000}{(1+0.10{)}^1}+\frac{3,000,000}{(1+0.10{)}^2}+\frac{3,000,000}{(1+0.10{)}^3}+\frac{3,000,000}{(1+0.10{)}^4}+\frac{3,000,000}{(1+0.10{)}^5}$$

$$PW=2,727,273+2,479,339+2,253,944+2,049,040+1,863,198=11,373,794$$

$$NPV=11,373,794-10,000,000=1,373,794$$

Since $NPV>0$, Daraz should invest in the new delivery vehicles.

Present Worth Index (PWI)

The Present Worth Index (PWI) is another tool used in present worth analysis. It is the ratio of the present worth of cash inflows to the present worth of cash outflows.

$$PWI=\frac{P{W}_{\text{inflows}}}{P{W}_{\text{outflows}}}$$

A PWI greater than 1 indicates that the project is acceptable.

Worked Example: PWI for a Solar Panel Installation

Suppose a company is considering installing solar panels that cost Rs. 800,000 and are expected to generate savings of Rs. 200,000 annually for 6 years. The MARR is 8%. Calculate the PWI.

Step 1: Calculate PW of Savings

$$PW=\frac{200,000}{(1+0.08{)}^1}+\frac{200,000}{(1+0.08{)}^2}+\frac{200,000}{(1+0.08{)}^3}+\frac{200,000}{(1+0.08{)}^4}+\frac{200,000}{(1+0.08{)}^5}+\frac{200,000}{(1+0.08{)}^6}$$

$$PW=185,185+171,461+159,454+148,886+139,512+131,255=935,853$$

Step 2: Calculate PWI

$$PWI=\frac{935,853}{800,000}=1.17$$

Since $PWI>1$, the investment is acceptable.

Exam Tip

• Understand the Concepts: Ensure you understand the difference between PW and NPV, and how they are used to evaluate projects.
• Practice Calculations: Be comfortable with discounting cash flows and calculating NPV. Practice with different scenarios.
• Diagrams: Always draw cash flow diagrams to visualize the problem. This helps in understanding the timing and magnitude of cash flows.
• MARR: Remember that MARR is the benchmark rate used for discounting. It is crucial to use the correct MARR provided in the question.
• Comparison: Know how to compare multiple projects using NPV and PWI. Projects with higher NPV or PWI are generally preferred.
• Real-World Application: Relate theoretical concepts to real-world examples, such as investments in technology, infrastructure, or services by companies like eSewa, Ncell, or Daraz.