Essential Financial Concepts & Calculations

Financial Concepts and Calculation Practice

Expectation Theory and Two-Year Rates

Problem: If the one-year rate on an instrument is 9.0% and the expected rate for a one-year instrument one year from today is 9.89%, what is the expected two-year rate today, if the expectation theory holds?

• One-year rate: 9.00%
• One-year rate, one year forward: 9.89%

Formula: (1 + R1) * (1 + ER2) = (1 + R2)^2

Calculation:

• (1 + 0.09) * (1 + 0.0989) = (1 + R2)^2
• (1.09) * (1.0989) = (1 + R2)^2
• 1.197801 = (1 + R2)^2
• sqrt(1.197801) = 1 + R2
• 1.094449 = 1 + R2
• R2 = 0.094449

Expected Two-Year Rate: 9.44% (approximately 9.4%)

Understanding the Rule of 72

The Rule of 72 is a quick method to estimate the number of years it takes for an investment to double in value, given a fixed annual rate of return. Simply divide 72 by the annual interest rate (as a whole number, not a percentage).

Formula: Years to Double = 72 / Interest Rate (as a whole number)

Example: If an investment earns 8% annually, it will take approximately 72 / 8 = 9 years to double.

Weakness: The Rule of 72 becomes less accurate the higher the interest rate.

Preferred Stock Valuation

Problem: If you wish to purchase Apple preferred stock paying an annual dividend of $15.00 and want a return of 14%, how much should you pay for the stock?

Formula (for preferred stock with constant dividend): P0 = D / r (where P0 = current price, D = annual dividend, r = required rate of return)

Note: The provided formula Po=Do*(i+g) / (i-g) is for common stock with constant growth (Gordon Growth Model). For preferred stock, growth (g) is typically 0.

Calculation:

• Dividend (D): $15.00
• Required Return (r): 14% (0.14)
• P0 = $15.00 / 0.14

Stock Price: $107.14

Bond Current Yield Calculation

Problem: What is the current yield on a bond paying an 8% coupon with a quoted rate of 94.87?

Formula: Current Yield (CY) = Annual Interest Payment / Bond Price

Calculation:

• Assume Face Value (Par Value) = $1,000
• Annual Interest Payment: $1,000 * 8% = $80
• Bond Price: 94.87% of $1,000 = $948.70
• CY = $80 / $948.70

Current Yield: 8.43%

Investment Rate of Return

Problem: If you own a financial instrument that you paid $1,250 to purchase and it is worth $2,489 six years later, what was the rate of return?

This is a compound annual growth rate (CAGR) problem, or finding the interest rate (RATE) in a financial function.

Inputs for RATE function (e.g., in Excel):

• Nper (Number of periods): 6 years
• Pmt (Payment per period): 0 (no interim payments)
• Pv (Present Value): -$1,250 (initial investment, outflow)
• Fv (Future Value): $2,489 (final value, inflow)

Calculation (using financial calculator or software): RATE(6, 0, -1250, 2489)

Rate of Return: 12.16% (approximately 12%)

Debt-to-Equity Ratio Analysis

Problem: If a company's debt-to-equity ratio is 1.5 and the company has $100,000 in debt, how much equity does the company have?

Formula: Debt-to-Equity Ratio = Total Debt / Total Equity

Calculation:

• Debt-to-Equity Ratio: 1.5
• Total Debt: $100,000
• 1.5 = $100,000 / Total Equity
• Total Equity = $100,000 / 1.5

Company Equity: $66,666.67

Tax Equivalent Yield for Bonds

Problem: If you can purchase a municipal bond which returns 3.85% and you are in the 40% tax bracket, what rate of return would you require on a corporate bond?

Formula: Tax Equivalent Yield = Municipal Bond Yield / (1 - Tax Rate)

Calculation:

• Municipal Bond Yield: 3.85% (0.0385)
• Tax Bracket: 40% (0.40)
• Tax Equivalent Yield = 0.0385 / (1 - 0.40)
• Tax Equivalent Yield = 0.0385 / 0.60

Required Corporate Bond Return: 6.42% (approximately 6.43%)

Effective Annual Rate (EAR)

Problem: If a credit card offers you a card with 20% APR and monthly payments, what is the EAR?

Formula: EAR = (1 + (APR / m))^m - 1 (where APR = Annual Percentage Rate, m = number of compounding periods per year)

Calculation:

• APR: 20% (0.20)
• Compounding periods (m): 12 (monthly)
• EAR = (1 + (0.20 / 12))^12 - 1

Effective Annual Rate: 21.94%

Net Working Capital Calculation

Problem: In reviewing a balance sheet you find the following numbers. What is the net working capital?

Balance Sheet Items:

• Cash and marketable securities: $120,000 (Asset)
• Accounts Receivable: $365,000 (Asset)
• Inventory: $589,000 (Asset)
• Accruals: $90,500 (Liability)
• Accounts Payable: $479,000 (Liability)
• Notes Payable: $120,000 (Liability)

Formula: Net Working Capital = Current Assets - Current Liabilities

Calculation:

• Current Assets (CA): $120,000 + $365,000 + $589,000 = $1,074,000
• Current Liabilities (CL): $90,500 + $479,000 + $120,000 = $689,500
• Net Working Capital = $1,074,000 - $689,500

Net Working Capital: $384,500

Annuity Due Valuation

Problem: What is the value of an annuity due if the present value of a 10-year ordinary annuity that is returning 12% is $7,700?

Concept: An annuity due's payments occur at the beginning of each period, making it worth more than an ordinary annuity (payments at the end of the period) because each payment earns interest for one additional period.

Formula: PV of Annuity Due = PV of Ordinary Annuity * (1 + Interest Rate)

Calculation:

• PV of Ordinary Annuity: $7,700
• Interest Rate: 12% (0.12)
• PV of Annuity Due = $7,700 * (1 + 0.12)
• PV of Annuity Due = $7,700 * 1.12

Value of Annuity Due: $8,624

Additional Financial Concepts

Defining an Annuity

An annuity is a series of equal payments made at regular intervals over a specified period. For an ordinary annuity, payments occur at the end of each period. For an annuity due, payments occur at the beginning of each period.

Future Annuity Problem (Timeline Request)

Problem: Find the present value (PV) of an annuity paid monthly for the next 7 years. The interest rate is 8%, the yearly payment is $100, and the future value (FV) is $5,000.

Note: This problem statement has a potential inconsistency: "yearly pmt. Is 100" but "paid monthly". I will interpret "yearly pmt. Is 100" as the annual payment amount, which is then distributed monthly. So, monthly payment = $100/12. The interest rate is 8% *yearly*, so monthly rate is 8%/12. Number of periods is 7 years * 12 months/year = 84 months.

Inputs for PV function (e.g., in Excel):

• Rate (monthly): 8% / 12
• Nper (total months): 7 * 12 = 84
• Pmt (monthly payment): $100 / 12
• Fv (Future Value): $5,000
• Type: 0 (end of period, ordinary annuity)

A. Draw a timeline for this info:

A financial timeline visually represents cash flows over time. For this problem:

Time (Months): 0   1   2   ...   83   84
Cash Flow:     PV  -$8.33 -$8.33 ... -$8.33 -$8.33 + $5,000 (FV)

Where PV is the unknown present value, and -$8.33 is the monthly payment ($100/12).