Finance AI Solver
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Expert help when you need it- Q1:Your firm needs to borrow $100,000. The bank is willing to lend your firm this amount as a 10-year loan and will charge an interest rate of 5.5%. Each payment is to be made annually at the end of the year. i. Determine the flat annual payment you will pay to the bank each year. ii. Setup a loan table to verify that the payment you computed in (i) will pay off the loan in full. iii. Compute the duration of your liability. Use the bank loan rate as your discount rate. Report your answer at 3 decimal places.Answer**i. Flat Annual Payment** When you take a loan, you have to pay the bank in equal amounts every year. This payment includes both interest and part of the principal (the amount you borrowed). To find the flat annual payment, we use this formula: PMT = P *r/{1 - (1 + r)^(-n)} Where: P = 100,000(loan amount) r = 5.5% = 0.055 (interest rate) n = 10 (number of years) Plugging in the numbers: PMT = (100,000 * 0.055)/{1 - (1 + 0.055)^{-10}} = 13,266.78 So, you’ll pay $13,266.78 every year for 10 years. **ii. Loan Amortization Table** This table helps you see how much of each payment goes toward interest and how much goes toward paying down the loan principal. It also shows the remaining balance after each year’s payment. Here’s the amortization schedule I generated for your loan: **Loan Amortization Table** - Year 1: - Payment: $13,266.78 - Interest: $5,500.00 - Principal: $7,766.78 - Remaining Balance: $92,233.22 - Year 2: - Interest: $5,072.83 - Principal: $8,193.95 - Remaining Balance: $84,039.27 This continues every year until the loan balance is fully paid off in Year 10. Each year, the interest decreases because the balance is shrinking, and more of your payment goes toward the principal. **iii. Macaulay Duration of the Loan** Think of duration as a weighted average of when the bank receives its money. It's not just about when the final payment happens (in 10 years), but how the payments are spread out over time. We can calculate the Macaulay duration with this formula: D = Weighted sum of present values of payments/P Using the formula, we found the duration to be 5.060 years. This means, on average, the bank will get most of its money back around 5.06 years into the loan. **Summary** 1. **Flat Annual Payment**: You will pay **$13,266.78** each year. 2. **Amortization Table**: Shows the breakdown of interest and principal every year. 3. **Duration**: The average time for repayment is **5.060 years**.
- Q2:The Expanding Capital Corporation has a current capital structure of $15 million in secured bonds paying 6.5% annual interest, $10 million in preferred stock with a par value of $50 per share and an annual dividend of $3.80 per share, and common stock with a book value of $75 million. It is about to issue new debentures in the amount of $10 million paying 7.5% annual interest. Its CFO says its marginal tax rate is 30% and its cost of common equity capital is 12%. Calculate the company’s Weighted Average Costs of Capital for the following: Before the new bond issue After the new bond issueAnswerLet’s solve for the **Weighted Average Cost of Capital (WACC)** for the two scenarios: 1. **Before the new bond issue** 2. **After the new bond issue** --- ### **Step 1: General Formula for WACC** WACC = E/V . r_E + D/V . r_D . (1 - T) + P/V . r_P Where: E = Value of Common Equity D = Value of Debt P = Value of Preferred Stock V = Total Value (E + D + P) r_E = Cost of Equity = 12% r_D = Cost of Debt (adjusted for tax) r_P = Cost of Preferred Stock T = Corporate Tax Rate = 30% --- ## **Step 2: Calculate WACC Before the New Bond Issue** ### 1. Calculate the Cost of Preferred Stock r_P r_P = Annual Dividend/Par Value = 3.80/50 = 7.6% ### 2. Calculate the After-Tax Cost of Debt r_D: r_D = 6.5% *1 - 0.30 = 6.5% *0.70 = 4.55% ### 3. Calculate the Total Value V Before the New Bond Issue: V = D + P + E = 15 + 10 + 75 = 100 million ### 4. Weight of Each Component: - **Weight of Debt**: D/V = 15/100 = 0.15 - **Weight of Preferred Stock**: P/V = 10/100 = 0.10 - **Weight of Equity**: E/V = 75/100 = 0.75 ### 5. WACC Before the New Bond Issue: WACC = 0.75 . 12% + 0.15 .4.55% + 0.10 .7.6% WACC = 9% + 0.6825% + 0.76% = 10.4425% --- ## **Step 3: Calculate WACC After the New Bond Issue** ### 1. Calculate the New Total Value V After the New Bond Issue: V = D + P + E = (15 + 10) + 10 + 75 = 110 million ### 2. After-Tax Cost of New Debt r_D: r_D = 7.5% * (1 - 0.30) = 7.5% *0.70 = 5.25% ### 3. Weight of Each Component: - **Weight of Debt (now including new debt)**: D/V = 25/110 = 0.2273 - **Weight of Preferred Stock**: P/V =10/110 = 0.0909 - **Weight of Equity**: E/V = 75/110 = 0.6818 ### 4. WACC After the New Bond Issue: WACC = 0.6818 . 12% + 0.2273 . 5.25% + 0.0909 . 7.6% WACC = 8.1816% + 1.1933% + 0.6908% = 10.0657% - **WACC Before the New Bond Issue:** 10.4425% - **WACC After the New Bond Issue:** 10.0657% The WACC decreases slightly after the new bond issue because debt is typically cheaper than equity, especially with the tax benefit of interest payments.
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