This note provides a detailed explanation of key finance concepts, including risk management, capital budgeting, bond valuation, inventory management, and cost of capital. It is designed for BiM students to understand financial decision-making and its implications.

BIM / Sixth Semester / FIN 201: Business Finance

Candidates are required to give their answers in their own words as far as practicable.

Group "A"

Indicate whether the following statements are 'True' or 'False'. Support your answer with

[10 × 1 = 101

Finance is only concerned with raising of funds.

If the equity multiplier is 2, the debt ratio must be 0.5.

If we identify perfectly negative correlated assets, we can completely eliminate the risk.

Simple annual interest rate and effective annual interest rate become equal if interest compounds annually.

There is inverse relationship between market interest rate and value of bond.

If project are mutually exclusive, we can choose all the projects which have positive NPV,

Generally, cost of retained earnings is higher than cost of external equity.

NPV is a better capital budgeting technique than IRR.

If a firm places 24 times order in a year with Rs 300 costs per order placed, total cost associated to the ordering of inventory is Rs 3,600.
10. If ordering frequency is less than lead time of a firm, there is existence
 

Finance is only concerned with raising of funds.

False – Finance involves not only raising funds but also managing and allocating them efficiently through investment and financial decision-making.

If the equity multiplier is 2, the debt ratio must be 0.5.

True – The equity multiplier = Total Assets / Equity. If it is 2, then equity funds 50% of total assets, and the debt ratio is 0.5 (or 50%).

If we identify perfectly negatively correlated assets, we can completely eliminate the risk.

True – A correlation of -1 allows risk elimination through diversification. Gains in one asset offset losses in another.

Simple annual interest rate and effective annual interest rate become equal if interest compounds annually.

True – Effective Annual Rate (EAR) considers compounding. When compounded annually, EAR = simple annual interest rate.

There is an inverse relationship between market interest rate and value of bond.

True – When interest rates rise, bond prices fall, and vice versa. This happens because bonds with fixed payments lose value when newer bonds offer higher yields.

If projects are mutually exclusive, we can choose all the projects which have positive NPV.

False – Mutually exclusive projects require choosing the best alternative. Selecting all positive NPV projects is incorrect.

Generally, cost of retained earnings is higher than cost of external equity.

False – Retained earnings are internally generated, avoiding flotation costs. External equity is more expensive due to issuance costs.
 

NPV is a better capital budgeting technique than IRR.


True – NPV provides a clear value addition to shareholders, while IRR may be misleading in cases of multiple cash flow reversals.
 

If a firm places 24 times order in a year with Rs 300 costs per order placed, total cost associated with ordering inventory is Rs 3,600.

True – Ordering cost formula: Number of Orders × Cost per Order = 24 × 300 = Rs 3,600.
 

If ordering frequency is less than lead time of a firm, there is existence of goods in transit.
 

False – If ordering frequency is lower than lead time, there could be stockouts rather than goods in transit.

                                                                                                               Group "B"                                                                                  16 × 5 = 30]

Short Answer Questions:

Define business finance. Discuss the role of financial manager in an organizations.

Define business finance. Discuss the role of financial manager in an organizations.

Definition of Business Finance

Business finance refers to the management of funds and financial resources within an organization to ensure smooth operations, profitability, and growth. It involves planning, acquiring, managing, and controlling financial resources efficiently to meet the company's objectives. Business finance covers a wide range of activities, including investment decisions, capital budgeting, risk management, financial planning, and capital structure management.

Role of a Financial Manager in an Organization

A financial manager plays a critical role in ensuring that a company maintains financial stability and achieves its strategic goals. The key responsibilities of a financial manager include:

1. Financial Planning and Forecasting

Develops financial strategies to achieve business goals.

Prepares budgets and forecasts to ensure effective fund allocation.

Analyzes market trends to predict future financial conditions.

2. Investment Decisions

Determines the best investment opportunities to maximize returns.

Evaluates projects using capital budgeting techniques like Net Present Value (NPV), Internal Rate of Return (IRR), and Payback Period.

Ensures that investments align with the company's risk tolerance and financial objectives.

3. Capital Structure Management

Decides the right mix of debt and equity financing for the business.

Balances leverage to minimize financial risk while maximizing profitability.

Determines the cost of capital and its impact on financial decision-making.

4. Risk Management

Identifies and mitigates financial risks such as market risk, credit risk, and liquidity risk.

Uses hedging techniques and insurance policies to protect the organization from unexpected losses.

5. Fundraising and Capital Allocation

Raises capital through equity financing, debt financing, or retained earnings.

Manages working capital by optimizing accounts receivable, accounts payable, and inventory.

Ensures efficient cash flow management to meet short-term and long-term obligations.

6. Profit Maximization and Cost Control

Enhances profitability by minimizing unnecessary expenses and improving operational efficiency.

Implements cost control measures to maintain financial health.

Analyzes financial statements to identify areas for improvement.

7. Dividend Policy Decision

Decides the dividend payout ratio and whether to reinvest profits or distribute them to shareholders.

Ensures a balance between rewarding investors and retaining capital for future growth.

8. Compliance and Financial Reporting

Ensures adherence to financial regulations, tax laws, and corporate governance.

Prepares financial reports and statements as per accounting standards.

Works with auditors and regulatory authorities to maintain transparency and compliance.

Conclusion

The financial manager acts as a strategic partner in an organization, ensuring that financial resources are used effectively to drive business success. By managing risks, optimizing investment decisions, and ensuring financial stability, the financial manager contributes to the long-term growth and sustainability of the company.


Solutions to the Given Questions

12. Scenario Analysis Calculations

We are given the probability and returns for Stock A and Stock B in different economic conditions. We will compute:

(a) Expected Rate of Return for Stock A and Stock B

The formula for Expected Return (E(R)E(R)):

E(R)=∑Pi×RiE(R) = \sum P_i \times R_i

where:

PiP_i = Probability of scenario

RiR_i = Return in that scenario

E(RA)=(0.4×10%)+(0.3×15%)+(0.3×20%)E(R_A) = (0.4 \times 10\%) + (0.3 \times 15\%) + (0.3 \times 20\%) E(RB)=(0.4×−10%)+(0.3×20%)+(0.3×30%)E(R_B) = (0.4 \times -10\%) + (0.3 \times 20\%) + (0.3 \times 30\%)

(b) Standard Deviation for Stock A and Stock B

The variance formula:

σ2=∑Pi×(Ri−E(R))2\sigma^2 = \sum P_i \times (R_i - E(R))^2

The Standard Deviation (σ\sigma):

σ=σ2\sigma = \sqrt{\sigma^2}

(c) Coefficient of Variation (CV)

CV=σE(R)CV = \frac{\sigma}{E(R)}

This measures the risk per unit of return.

13. Stock Valuation Using Dividend Discount Model

The Gordon Growth Model:

P0=D1r−gP_0 = \frac{D_1}{r - g}

where:

P0P_0 = Present stock price

D1D_1 = Dividend next year

rr = Required rate of return (15%)

gg = Growth rate (20% for 3 years, then 10%)

Dividend Yield Calculation

Dividend Yield=D1P0×100\text{Dividend Yield} = \frac{D_1}{P_0} \times 100

14. Bond Valuation

Given:

Coupon Rate = 6.5%

Time to Maturity = 10 years

Yield to Maturity (YTM) = 7%

(a) Interest Payments & Current Yield

Annual Coupon Payment:

C=Coupon Rate×Face Value=6.5%×1000=65C = \text{Coupon Rate} \times \text{Face Value} = 6.5\% \times 1000 = 65

Current Yield:

Current Yield=CCurrent Price\text{Current Yield} = \frac{C}{\text{Current Price}}

(b) Bond Price Calculation

P=∑C(1+r)t+F(1+r)TP = \sum \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^T}

where:

CC = Coupon payment

FF = Face value (Rs 1000)

rr = YTM (7%)

TT = Time to maturity (10 years)

(c) Impact of YTM Reduction

If YTM drops to 6%, the bond price increases because bond values move inversely with interest rates.

(d) Relationship between YTM and Bond Price

When YTM increases, bond price decreases.

When YTM decreases, bond price increases.

I will now calculate and display these values.

I have calculated the expected return, standard deviation, coefficient of variation, stock price, dividend yield, bond price at different YTMs, and current yield. You can review the results in the table provided. Let me know if you need further explanations or additional calculations!

📘 Business Finance Notes for Bachelor Students (BIM) 📘

Topic: Financial Analysis & Balance Sheet Completion

🔹 Understanding Financial Ratios & Balance Sheet Components

Before diving into calculations, let’s break down the key financial ratios and their significance in analyzing a business’s financial health.

📊 Given Financial Ratios & What They Mean

These ratios help us understand how efficiently a business is operating and provide insights into its liquidity, profitability, and debt structure.

1️⃣ Long-term Debt to Net Worth (0.5 to 1)

Shows how much long-term debt a company has compared to its equity (net worth).

Formula: Long-term Debt=0.5×Net Worth\text{Long-term Debt} = 0.5 \times \text{Net Worth}

2️⃣ Total Assets Turnover (2.5 times)

Measures how efficiently a company uses its assets to generate revenue.

Formula: Sales=Total Assets×Total Asset Turnover\text{Sales} = \text{Total Assets} \times \text{Total Asset Turnover}

3️⃣ Average Collection Period (18 days)

Tells how long customers take to pay after making a credit purchase.

Formula: Accounts Receivable=(Sales360)×Average Collection Period\text{Accounts Receivable} = \left(\frac{\text{Sales}}{360}\right) \times \text{Average Collection Period}

4️⃣ Inventory Turnover (9 times)

Indicates how fast inventory is sold and replaced.

Formula: Inventory=Cost of Goods Sold (COGS)Inventory Turnover\text{Inventory} = \frac{\text{Cost of Goods Sold (COGS)}}{\text{Inventory Turnover}}

5️⃣ Gross Profit Margin (10%)

Tells us how much profit a company makes after deducting the cost of goods sold.

Formula: COGS=Sales×(1−Gross Profit Margin)\text{COGS} = \text{Sales} \times (1 - \text{Gross Profit Margin})

6️⃣ Acid Test Ratio (1:1)

A liquidity ratio that checks if a company can pay short-term liabilities using only cash & accounts receivable (without selling inventory).

Formula: Cash=(Acid Test Ratio×Current Liabilities)−Accounts Receivable\text{Cash} = (\text{Acid Test Ratio} \times \text{Current Liabilities}) - \text{Accounts Receivable}

📑 Given Information (Partial Balance Sheet)

We are provided with some elements of the balance sheet but need to calculate the missing values using the given ratios.

Given:

Notes Payable (Short-term Liabilities) = Rs 100,000

Common Stock = Rs 100,000

Retained Earnings = Rs 100,000

Long-term Debt = ?

Inventory = ?

Accounts Receivable = ?

Cash = ?

Plant & Equipment = ?

Total Assets = ?

Total Liabilities & Equity = ?

📌 Step-by-Step Calculation Process

🔸 Step 1: Calculate Net Worth (Equity)

Net Worth is the sum of Common Stock + Retained Earnings

Net Worth=100,000+100,000=200,000\text{Net Worth} = 100,000 + 100,000 = 200,000

🔸 Step 2: Calculate Long-term Debt

Using Long-term Debt to Net Worth Ratio (0.5 to 1):

Long-term Debt=0.5×200,000=100,000\text{Long-term Debt} = 0.5 \times 200,000 = 100,000

🔸 Step 3: Calculate Total Assets

Total Assets = Net Worth + Long-term Debt + Notes Payable

Total Assets=200,000+100,000+100,000=400,000\text{Total Assets} = 200,000 + 100,000 + 100,000 = 400,000

🔸 Step 4: Calculate Sales

Using Total Asset Turnover Ratio (2.5 times):

Sales=400,000×2.5=1,000,000\text{Sales} = 400,000 \times 2.5 = 1,000,000

🔸 Step 5: Calculate Cost of Goods Sold (COGS)

Using Gross Profit Margin (10%):

COGS=1,000,000×(1−0.10)=900,000\text{COGS} = 1,000,000 \times (1 - 0.10) = 900,000

🔸 Step 6: Calculate Inventory

Using Inventory Turnover (9 times):

Inventory=900,0009=100,000\text{Inventory} = \frac{900,000}{9} = 100,000

🔸 Step 7: Calculate Accounts Receivable

Using Average Collection Period (18 days):

Accounts Receivable=(1,000,000360)×18=50,000\text{Accounts Receivable} = \left(\frac{1,000,000}{360}\right) \times 18 = 50,000

🔸 Step 8: Calculate Cash

Using Acid Test Ratio (1:1):

Cash=(1×100,000)−50,000=50,000\text{Cash} = (1 \times 100,000) - 50,000 = 50,000

🔸 Step 9: Calculate Plant & Equipment

Using Total Assets - (Cash + Accounts Receivable + Inventory)

\text{Plant & Equipment} = 400,000 - (50,000 + 50,000 + 100,000) = 200,000

🔸 Step 10: Verify Total Liabilities & Equity

\text{Total Liabilities & Equity} = 400,000

(✅ Matches Total Assets)

📋 Final Balance Sheet (Completed)

📌 Key Takeaways

✅ Financial Ratios Help Analyze Business Health – They show how efficiently a company is operating.
✅ Balance Sheet Balances! – Total Assets = Total Liabilities + Equity always holds true.
✅ Understanding Cash Flow & Liquidity is Crucial – A business should have enough cash and accounts receivable to meet short-term liabilities.
✅ Inventory & Receivables Play a Key Role – These affect liquidity and overall financial stability.

15. Using the following information, complete the balance sheet.

Assume a 360 day year and all sales on credit

16. The following information for inventory purchased and storage cost has been provided fo he Marpha Apple Juice Company. Assume 50 weeks in a year

Annual demand are 180,000 units

Purchase price is Rs 2

Carrying cost is 50% of purchase price

Cost per order placed is Rs 400

Safety stock is Rs 10,000 units

Lead time is 1 week

What is the economic order quantity (EOQ)?

What is the optimal number of order to be placed?

At what inventory level should a reorder be made (ROL)?

 

📘 Business Finance Notes for Bachelor Students (BIM) 📘

Topic: Balance Sheet Completion & Inventory Management (EOQ, ROL, Orders)

📝 15. Completing the Balance Sheet (Step-by-Step Guide)

🔹 Given Financial Information & Key Ratios

Before we proceed, let's understand the key financial ratios and values provided:

1️⃣ Long-term Debt to Net Worth Ratio (0.5 to 1)

Long-term debt is 50% of net worth.

Formula: Long-term Debt=0.5×Net Worth\text{Long-term Debt} = 0.5 \times \text{Net Worth}

2️⃣ Total Assets Turnover (2.5 times)

Formula: Sales=Total Assets×2.5\text{Sales} = \text{Total Assets} \times 2.5

3️⃣ Average Collection Period (18 days)

Formula: Accounts Receivable=(Sales360)×18\text{Accounts Receivable} = \left(\frac{\text{Sales}}{360}\right) \times 18

4️⃣ Inventory Turnover (9 times)

Formula: Inventory=COGSInventory Turnover\text{Inventory} = \frac{\text{COGS}}{\text{Inventory Turnover}}

5️⃣ Gross Profit Margin (10%)

Formula: COGS=Sales×(1−Gross Profit Margin)\text{COGS} = \text{Sales} \times (1 - \text{Gross Profit Margin})

6️⃣ Acid Test Ratio (1:1)

Formula: Cash=(Acid Test Ratio×Current Liabilities)−Accounts Receivable\text{Cash} = (\text{Acid Test Ratio} \times \text{Current Liabilities}) - \text{Accounts Receivable}

📌 Step-by-Step Balance Sheet Completion

🔸 Step 1: Calculate Net Worth (Equity)

Net Worth is the sum of Common Stock + Retained Earnings

Net Worth=100,000+100,000=200,000\text{Net Worth} = 100,000 + 100,000 = 200,000

🔸 Step 2: Calculate Long-term Debt

Using Long-term Debt to Net Worth Ratio (0.5 to 1):

Long-term Debt=0.5×200,000=100,000\text{Long-term Debt} = 0.5 \times 200,000 = 100,000

🔸 Step 3: Calculate Total Assets

Total Assets = Net Worth + Long-term Debt + Notes Payable

Total Assets=200,000+100,000+100,000=400,000\text{Total Assets} = 200,000 + 100,000 + 100,000 = 400,000

🔸 Step 4: Calculate Sales

Using Total Asset Turnover Ratio (2.5 times):

Sales=400,000×2.5=1,000,000\text{Sales} = 400,000 \times 2.5 = 1,000,000

🔸 Step 5: Calculate Cost of Goods Sold (COGS)

Using Gross Profit Margin (10%):

COGS=1,000,000×(1−0.10)=900,000\text{COGS} = 1,000,000 \times (1 - 0.10) = 900,000

🔸 Step 6: Calculate Inventory

Using Inventory Turnover (9 times):

Inventory=900,0009=100,000\text{Inventory} = \frac{900,000}{9} = 100,000

🔸 Step 7: Calculate Accounts Receivable

Using Average Collection Period (18 days):

Accounts Receivable=(1,000,000360)×18=50,000\text{Accounts Receivable} = \left(\frac{1,000,000}{360}\right) \times 18 = 50,000

🔸 Step 8: Calculate Cash

Using Acid Test Ratio (1:1):

Cash=(1×100,000)−50,000=50,000\text{Cash} = (1 \times 100,000) - 50,000 = 50,000

🔸 Step 9: Calculate Plant & Equipment

Using Total Assets - (Cash + Accounts Receivable + Inventory)

\text{Plant & Equipment} = 400,000 - (50,000 + 50,000 + 100,000) = 200,000

🔸 Step 10: Verify Total Liabilities & Equity

\text{Total Liabilities & Equity} = 400,000

(✅ Matches Total Assets)

📋 Final Balance Sheet (Completed)

📌 16. Inventory Management for Marpha Apple Juice Company

Let’s calculate EOQ, Number of Orders, and Reorder Level (ROL).

📑 Given Information

Annual Demand = 180,000 units

Purchase Price = Rs 2 per unit

Carrying Cost = 50% of purchase price

Ordering Cost per Order = Rs 400

Safety Stock = 10,000 units

Lead Time = 1 week

Weeks per Year = 50

📌 Step-by-Step Inventory Management Calculations

🔸 Step 1: Calculate Economic Order Quantity (EOQ)

EOQ=2×Annual Demand×Ordering CostCarrying Cost per unitEOQ = \sqrt{\frac{2 \times \text{Annual Demand} \times \text{Ordering Cost}}{\text{Carrying Cost per unit}}}

Using the given values:

EOQ=2×180,000×4001=12,000 unitsEOQ = \sqrt{\frac{2 \times 180,000 \times 400}{1}} = 12,000 \text{ units}

🔸 Step 2: Calculate Optimal Number of Orders Per Year

Orders Per Year=Annual DemandEOQ\text{Orders Per Year} = \frac{\text{Annual Demand}}{EOQ} =180,00012,000=15 orders per year= \frac{180,000}{12,000} = 15 \text{ orders per year}

🔸 Step 3: Calculate Reorder Level (ROL)

ROL=(Weekly Demand×Lead Time)+Safety Stock\text{ROL} = (\text{Weekly Demand} \times \text{Lead Time}) + \text{Safety Stock} =(180,00050×1)+10,000= \left(\frac{180,000}{50} \times 1\right) + 10,000 =3,600+10,000=13,600 units= 3,600 + 10,000 = 13,600 \text{ units}

📋 Final Inventory Management Results

🎯 Key Takeaways

✅ Balance Sheet Always Balances! – Ensuring Total Assets = Total Liabilities & Equity is a fundamental check.
✅ EOQ Helps Minimize Costs – It ensures the best quantity to order reducing carrying & ordering costs.
✅ Reorder Level Prevents Stockouts – Ensuring the company never runs out of stock while keeping inventory optimized.
✅ Understanding Finance Helps in Business Success! 📈

🚀 Final Thought

By following a step-by-step approach, financial concepts become easier to understand. These calculations help businesses optimize their resources and improve decision-making! 💰📊

Group "C"

Comprehensive answer questions:

Read the following information and answer the questions given below:

2 × 10 = 20

17. Tara transportation Pvt. Ltd is Considering to run tourist bus from chitwan to

Kathmandu. A tourist bus costs Rs 2,000,000 and it will provide service for 5 years to come. Annual net cash inflows for five years will be as follows:

a. What is the payback period of the project? Should Tara Transportation run tourist bus from Chitwan to Kathmandu if its maximum cost recovery period is 4 years?

9. What is the NPV of the project? If required rate of return of the project is 10 percent. should Tara Transportation run the tourist bus service?

c. What is the IRR of the project? Should Tara Transportation run the tourist bus?

📘 Business Finance Notes for Bachelor Students (BIM) 📘

Topic: Capital Budgeting – Payback Period, NPV & IRR for Tara Transportation Pvt. Ltd.

🚌 Project Overview

Tara Transportation Pvt. Ltd. is considering running a tourist bus from Chitwan to Kathmandu. The company wants to evaluate whether this investment is financially viable based on three capital budgeting techniques:

✅ Payback Period (PP) – Measures how quickly the initial investment is recovered.
✅ Net Present Value (NPV) – Measures the profitability of the project by considering the time value of money.
✅ Internal Rate of Return (IRR) – Measures the rate at which the project breaks even in terms of NPV.

📊 Given Data

Initial Investment (Year 0): Rs 2,000,000 (negative as it’s an outflow).

Required Rate of Return (Discount Rate for NPV & IRR): 10%.

Maximum Payback Period Accepted: 4 years.

📌 Step 1: Payback Period Calculation

The Payback Period tells us how long it takes to recover the initial investment.
Formula:

Payback Period=Years before full recovery+(Remaining Investment to RecoverCash Flow in Next Year)\text{Payback Period} = \text{Years before full recovery} + \left(\frac{\text{Remaining Investment to Recover}}{\text{Cash Flow in Next Year}}\right)

🔸 Step-by-step Recovery Calculation:

The investment is fully recovered between Year 2 and Year 3.

Remaining amount to recover in Year 3=500,000\text{Remaining amount to recover in Year 3} = 500,000 Fractional Year=500,000650,000=0.77 years\text{Fractional Year} = \frac{500,000}{650,000} = 0.77 \text{ years} Payback Period=2+0.77=2.77 years≈2.8 years\text{Payback Period} = 2 + 0.77 = 2.77 \text{ years} \approx 2.8 \text{ years}

✅ Decision: ACCEPT (Since the payback period of 2.8 years is within the 4-year limit).

📌 Step 2: Net Present Value (NPV) Calculation

Net Present Value (NPV) measures the total profitability of the project by discounting future cash flows.

Formula:

NPV=∑Ct(1+r)t−C0NPV = \sum \frac{C_t}{(1 + r)^t} - C_0

Where:

CtC_t = Cash flow in year tt

rr = Discount rate (10%)

C0C_0 = Initial Investment (Rs 2,000,000)

tt = Year (1 to 5)

🔸 Step-by-step Discounted Cash Flow Calculation

Total Present Value of Cash Inflows:

727,272+578,512+488,160+409,800+496,800=2,700,544727,272 + 578,512 + 488,160 + 409,800 + 496,800 = 2,700,544

NPV Calculation:

NPV=2,700,544−2,000,000=∗∗Rs700,684.88∗∗NPV = 2,700,544 - 2,000,000 = **Rs 700,684.88**

✅ Decision: ACCEPT (Since NPV > 0, the project is profitable).

📌 Step 3: Internal Rate of Return (IRR) Calculation

The IRR is the discount rate that makes NPV = 0.

Using trial and error or financial calculator, we find:

IRR=23.21% (or 0.2321)IRR = 23.21\% \text{ (or 0.2321)}

Since IRR (23.21%) > Required Rate of Return (10%), the project is financially viable.

✅ Decision: ACCEPT (Since IRR is greater than the required return).

📋 Final Project Evaluation Summary

🎯 Key Takeaways

✅ Payback Period Measures Risk – Since the initial investment is recovered within 4 years, the project is acceptable.
✅ NPV Measures Profitability – A positive NPV (Rs 700,684.88) indicates that the project will increase the company's value.
✅ IRR Measures Return – Since IRR (23.21%) is greater than the required rate of return (10%), the project is a good investment.

📢 Final Decision: Tara Transportation should run the tourist bus service from Chitwan to Kathmandu! 🚍💰

18. Assume that it is now January 1, 2021. On January 1, 2022, you will deposit Rs 1,000 in to a saving account that pays 8%.

If the bank compounded interest annually, how much will you have in your account on January 1, 2025?

What would your January 01, 2025, balance be if the bank used quarterly compounding rather than annual compounding ?

Suppose you deposit the Rs 1,000 in 4 payments of Rs 250 each on January 1 of 2022, 2023, 2024 and 2025. How much would you have in your account on January 1, 2025,based on 8%annual compounding?

2022, 2023, 2024, and 2025. Assuming as 8% interest outs ho anue would

each of your payments have to be for you to obtain the same ending balance as you calculated in part (a)?

📘 Business Finance Notes for Bachelor Students (BIM) 📘

Topic: Future Value of Savings with Different Compounding Methods

📌 Given Scenario

Current Date: January 1, 2021

First Deposit: Rs 1,000 on January 1, 2022

Interest Rate: 8% per year

Time Period: 3 years (From January 1, 2022, to January 1, 2025)

Compounding Methods:
✅ Annual Compounding
✅ Quarterly Compounding
✅ Four Equal Deposits of Rs 250 each year

We will calculate how much money will be in the account under different conditions.

📌 Step 1: Future Value with Annual Compounding

🔹 Understanding Annual Compounding

The formula for Future Value (FV) with annual compounding is:

FV=P×(1+r)tFV = P \times (1 + r)^t

Where:

FVFV = Future Value

PP = Principal amount (Rs 1,000)

rr = Annual interest rate (8% or 0.08)

tt = Number of years (3 years: 2022 → 2025)

🔸 Calculation

FV=1,000×(1.08)3FV = 1,000 \times (1.08)^3 FV=1,000×1.2597FV = 1,000 \times 1.2597 FV=∗∗Rs1,259.71∗∗FV = **Rs 1,259.71**

✅ Final Amount in Account (Annual Compounding): Rs 1,259.71

📌 Step 2: Future Value with Quarterly Compounding

🔹 Understanding Quarterly Compounding

Interest is compounded every 3 months (4 times a year).

The formula for Future Value (FV) with quarterly compounding is:

FV=P×(1+rn)n×tFV = P \times \left(1 + \frac{r}{n}\right)^{n \times t}

Where:

FVFV = Future Value

PP = Principal amount (Rs 1,000)

rr = Annual interest rate (8% or 0.08)

nn = Number of times interest is compounded per year (4 times per year)

tt = Number of years (3 years)

🔸 Calculation

FV=1,000×(1+0.084)4×3FV = 1,000 \times \left(1 + \frac{0.08}{4}\right)^{4 \times 3} FV=1,000×(1.02)12FV = 1,000 \times (1.02)^{12} FV=1,000×1.2682FV = 1,000 \times 1.2682 FV=∗∗Rs1,268.24∗∗FV = **Rs 1,268.24**

✅ Final Amount in Account (Quarterly Compounding): Rs 1,268.24

📌 Step 3: Future Value with 4 Equal Deposits (Rs 250 per year)

🔹 Understanding Deposits Over Time

Instead of one deposit of Rs 1,000, now Rs 250 is deposited at the beginning of each year for 4 years (2022, 2023, 2024, 2025).

Each deposit earns interest for a different number of years.

The formula for Future Value of Annuity (FVA) with annual compounding is:

FVA=P×∑t=0n(1+r)n−tFVA = P \times \sum_{t=0}^{n} (1 + r)^{n-t}

Where:

PP = Annual deposit (Rs 250)

rr = Interest rate (8% or 0.08)

nn = Number of years (3 years)

🔸 Calculation

Each Rs 250 deposit grows at different rates:

Total Future Value:

314.93+292.50+270.00+250.00=∗∗Rs1,126.53∗∗314.93 + 292.50 + 270.00 + 250.00 = **Rs 1,126.53**

✅ Final Amount in Account (4 Equal Deposits): Rs 1,126.53

📌 Step 4: Finding Required Annual Payment for Same Ending Balance as (a)

🔹 Understanding Required Payment

We need to find an annual payment (Rs X) that results in the same final balance as (a) → Rs 1,259.71.

The formula to find Required Annual Payment (PP) is:

P=FV∑t=0n(1+r)n−tP = \frac{FV}{\sum_{t=0}^{n} (1 + r)^{n-t}}

🔸 Calculation

Using the future value sum formula, the denominator is:

(1.08)3+(1.08)2+(1.08)1+(1.08)0=3.255(1.08)^3 + (1.08)^2 + (1.08)^1 + (1.08)^0 = 3.255

So the required annual payment:

P=1,259.713.255P = \frac{1,259.71}{3.255} P=∗∗Rs279.56∗∗P = **Rs 279.56**

✅ Each Payment Must Be Rs 279.56 to Match Rs 1,259.71 in 2025

📋 Final Summary of Calculations

🎯 Key Takeaways

✅ More Compounding → More Growth – Quarterly compounding results in a slightly higher final balance than annual compounding.
✅ Regular Deposits vs. One-Time Deposit – Four equal deposits grow slower because some amounts earn interest for a shorter time.
✅ Finding Equal Payments is Essential – We can adjust annual deposits to match a specific future value.

📢 Conclusion:

If you have a lump sum of Rs 1,000, invest it at once for maximum growth!

If you can only invest yearly, you need Rs 279.56 per year to match a one-time Rs 1,000 deposit in growth.