Let's break this complex real estate finance problem into manageable steps, as requested. **Given:** - **Purchase Price (PP):** $5,000,000 - **Loan-to-value (LTV):** 75% - **Loan Amount:** 0.75 × $5,000,000 = $3,750,000 - **Equity Investment (Down Payment):** $5,000,000 - $3,750,000 = $1,250,000 - **Loan Terms:** 10% annual interest, amortized monthly for 25 years - **NOI Year 1:** $475,000, growing at 3% per year for 5 years - **Sale Price at Year 5:** $6,000,000 - **Improvements %:** 80% of cost = 0.8 × $5,000,000 = $4,000,000 (for depreciation) - **Depreciation period:** 39 years - **Tax Bracket:** 28% (income and cap gains) - **Holding Period:** 5 years - **No prepayment before 5 years (lockout)** We need: a. **BTIRR and ATIRR on equity investment** b. **BEIR and cost of equity participation financing** --- ## Part a: **BTIRR and ATIRR on Equity Investment** ### **Step 1: Calculate Annual Debt Service (Monthly Payments)** - **Loan Amount:** $3,750,000 - **Interest Rate per month:** 10% / 12 = 0.83333% - **Number of months:** 25 × 12 = 300 **Monthly Payment (PMT):** \[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \] Where: - \(P\) = loan amount = $3,750,000 - \(r\) = 0.10/12 = 0.00833333 - \(n\) = 300 \[ PMT = \frac{3,750,000 \times 0.00833333}{1 - (1 + 0.00833333)^{-300}} \] Calculate denominator: \[ (1 + 0.00833333)^{-300} = (1.00833333)^{-300} \approx 0.08055 \] \[ Denominator = 1 - 0.08055 = 0.91945 \] Calculate numerator: \[ 3,750,000 \times 0.00833333 = 31,250 \] So, \[ PMT = \frac{31,250}{0.91945} = \$33,978.98 \text{ per month} \] **Annual Debt Service:** \[ 33,978.98 \times 12 = \$407,747.76 \text{ per year} \] --- ### **Step 2: Project NOI for 5 Years** - Year 1: $475,000 - Year 2: $475,000 × 1.03 = $489,250 - Year 3: $489,250 × 1.03 = $503,927.50 - Year 4: $503,927.50 × 1.03 = $519,045.33 - Year 5: $519,045.33 × 1.03 = $534,616.69 --- ### **Step 3: Calculate Interest and Principal Paid Each Year** Loan balance reduces over time; we need the balance after 5 years (60 months). #### **Find Remaining Loan Balance After 5 Years** \[ B_{60} = P \times (1 + r)^{60} - PMT \times \frac{(1 + r)^{60} - 1}{r} \] Alternatively, we can use the standard formula for remaining balance after \(k\) payments: \[ B_k = P \times (1 + r)^k - PMT \times \frac{( (1 + r)^k - 1 )}{r} \] Where \(k = 60\) Compute \( (1 + 0.00833333)^{60} \): \[ (1.00833333)^{60} = e^{60 \times \ln(1.00833333)} \approx e^{60 \times 0.008298} \approx e^{0.49788} \approx 1.6453 \] So, \[ B_{60} = 3,750,000 \times 1.6453 - 33,978.98 \times \frac{1.6453 - 1}{0.00833333} \] \[ = 6,169,875 - 33,978.98 \times 77.436 \] \[ 33,978.98 \times 77.436 = 2,630,676.80 \] \[ B_{60} = 6,169,875 - 2,630,676.80 = \$3,539,198.20 \] --- ### **Step 4: Cash Flows to Equity (Before Tax)** \[ \text{BTCF} = \text{NOI} - \text{Debt Service} \] - Year 1: $475,000 - $407,747.76 = $67,252.24 - Year 2: $489,250 - $407,747.76 = $81,502.24 - Year 3: $503,927.50 - $407,747.76 = $96,179.74 - Year 4: $519,045.33 - $407,747.76 = $111,297.57 - Year 5: $534,616.69 - $407,747.76 = $126,868.93 **Year 5 also includes sales proceeds:** \[ \text{Net Sales Proceeds} = \text{Sale Price} - \text{Loan Balance} = 6,000,000 - 3,539,198.20 = 2,460,801.80 \] **Total Year 5 BTCF:** \[ = 126,868.93 + 2,460,801.80 = 2,587,670.73 \] --- ### **Step 5: BTIRR Calculation** Initial equity investment: \(-1,250,000\) Cash flows: - Year 1: $67,252.24 - Year 2: $81,502.24 - Year 3: $96,179.74 - Year 4: $111,297.57 - Year 5: $2,587,670.73 BTIRR is the IRR for these cash flows. \[ \begin{array}{c|c} \text{Year} & \text{BTCF} \\ \hline 0 & -1,250,000 \\ 1 & 67,252.24 \\ 2 & 81,502.24 \\ 3 & 96,179.74 \\ 4 & 111,297.57 \\ 5 & 2,587,670.73 \\ \end{array} \] You can use Excel's **=IRR()** or financial calculator, but let's estimate manually: - The total cash inflows are much more than the outlay, so expect a high IRR. **Using a financial calculator or Excel:** \[ IRR = \text{(use calculator or Excel)} \] But for estimation, let's use a quick trial: - Year 5: $2.58M on $1.25M outlay, even ignoring all previous years, that's a 20%+ annualized return. - Including the prior years' cash flows, the IRR will be higher. Do a quick trial at 30%: \[ NPV = -1,250,000 + \frac{67,252.24}{1.3} + \frac{81,502.24}{1.3^2} + \frac{96,179.74}{1.3^3} + \frac{111,297.57}{1.3^4} + \frac{2,587,670.73}{1.3^5} \] Calculate denominators: - 1.3^1 = 1.3 - 1.3^2 = 1.69 - 1.3^3 = 2.197 - 1.3^4 = 2.856 - 1.3^5 = 3.713 So: - 67,252.24 / 1.3 = 51,732.49 - 81,502.24 / 1.69 = 48,236.09 - 96,179.74 / 2.197 = 43,791.91 - 111,297.57 / 2.856 = 38,974.75 - 2,587,670.73 / 3.713 = 696,854.68 Sum: -1,250,000 + 51,732.49 + 48,236.09 + 43,791.91 + 38,974.75 + 696,854.68 = -370,409.08 So at 30%, NPV is still negative. Try 20%: - 1.2^1 = 1.2 - 1.2^2 = 1.44 - 1.2^3 = 1.728 - 1.2^4 = 2.074 - 1.2^5 = 2.488 - 67,252.24 / 1.2 = 56,043.53 - 81,502.24 / 1.44 = 56,597.39 - 96,179.74 / 1.728 = 55,667.38 - 111,297.57 / 2.074 = 53,677.79 - 2,587,670.73 / 2.488 = 1,040,131.56 Sum: -1,250,000 + 56,043.53 + 56,597.39 + 55,667.38 + 53,677.79 + 1,040,131.56 = 12,117.65 So, at 20%, NPV is slightly positive. Try 21%: - 1.21^1 = 1.21 - 1.21^2 = 1.4641 - 1.21^3 = 1.771561 - 1.21^4 = 2.14358 - 1.21^5 = 2.59373 - 67,252.24 / 1.21 = 55,612.60 - 81,502.24 / 1.4641 = 55,667.37 - 96,179.74 / 1.771561 = 54,317.65 - 111,297.57 / 2.14358 = 51,918.75 - 2,587,670.73 / 2.59373 = 997,650.78 Sum: -1,250,000 + 55,612.60 + 55,667.37 + 54,317.65 + 51,918.75 + 997,650.78 = -34,832.85 So, at 21%, NPV is negative; at 20%, it's positive. So IRR is a bit above 20%. Let's interpolate: \[ IRR \approx 20\% + \frac{12,117.65}{12,117.65 + 34,832.85} \times (21\%-20\%) = 20\% + \frac{12,117.65}{46,950.50} \times 1\% \] \[ = 20\% + 0.2583 \% = 20.26\% \] **Final BTIRR:** \[ \boxed{20.3\%} \] --- ### **Step 6: Calculate After-Tax Cash Flows (ATCF)** We need to account for: - Depreciation - Interest deduction - Taxes on operating income - Taxes on sale (capital gains, depreciation recapture) #### **6a. Find Depreciation Expense** - Depreciable base = $4,000,000 - Annual depreciation = $4,000,000 / 39 = $102,564.10 #### **6b. Find Interest Paid Each Year** First, find interest paid each year (the loan amortizes, so interest declines): ##### **Year 1 Interest:** - Start balance: $3,750,000 - First month's interest: $3,750,000 × 0.00833333 = $31,250 - Each month, principal paid increases, so average for the year: But for simplicity, let's calculate **total interest paid in Year 1**: Using an amortization table, the **interest paid in the first year** can be found: - Total payments in Year 1: $33,978.98 × 12 = $407,747.76 - The principal paid in the first year: Use the formula for principal paid in n payments: \[ \text{Principal paid in year 1} = \text{Total payments} - \text{Interest paid} \] But to get interest, we can sum up the interest portions: - Month 1: $3,750,000 × 0.00833333 = $31,250.00 - Month 2: ($3,750,000 - principal1) × 0.00833333 -