## Billy’s Pianos receives a loan of $180,000 today. The stated annual interest rate is 8.4%, compounded monthly. Payments are monthly, starting one month from today. The loan is amortized over 30 years. ## Billy’s Pianos receives a loan of $180,000 today. The stated annual interest rate is 8.4%, compounded monthly. Payments are monthly, starting one month from today. The loan is amortized over 30 years.
## If Billy pays an equal amount of principal each month, how much will the first payment be? - Monthly rate = .0084 / 12 = 0.7%
- Amount of principal paid each month = $180,000 / 360 = $500
- Amount of interest accrued in first month = $180,000 * .007 = $1,260
- First payment = 500 + 1,260 = $1,760
## If Billy makes equal month payments each month, how much will the first payment be? ## If Billy makes equal month payments each month, how much will the first payment be? - 180,000 = C / .007 * [1 – 1 / (1.007)360]
- 180,000 = 131.262 * C
- C = $1,371.31
## If Billy pays an equal amount of principal each month, how much will the last payment be? ## If Billy pays an equal amount of principal each month, how much will the last payment be? - Principal owed in 359 months = 180,000 / 360 = 500
- Interest owed = 500 * .007 = 3.50
- Last payment = 500 + 3.50 = $503.50
## If Billy makes equal month payments each month, how much will the last payment be? ## If Billy makes equal month payments each month, how much will the last payment be? - Note: equal payments means first = last (so same answer as #2)
- 180,000 = C / .007 * [1 – 1 / (1.007)360]
- 180,000 = 131.262 * C
- C = $1,371.31
## Carly Rae pays $50,000 to open her dating service. She receives $2,700 per year in cash flow, starting in two years. Annual discount rate is 5%. What is the profitability index? ## Carly Rae pays $50,000 to open her dating service. She receives $2,700 per year in cash flow, starting in two years. Annual discount rate is 5%. What is the profitability index? - PV of benefits = 2700 / .05 * 1 / 1.05 = 51,429
- PV of costs = 50,000
- PI = 51,429 / 50,000 = 1.029
## If the effective annual discount rate is 15%, then what is the effective discount rate for 8 months? ## If the effective annual discount rate is 15%, then what is the effective discount rate for 8 months? - (1.15)8/12 – 1 = 9.76534%
## Wolfgang will receive royalty payments of $500 every year, starting 5 years from today and ending 25 years from today. What is the present value of these payments if the effective annual discount rate is 15%? ## Wolfgang will receive royalty payments of $500 every year, starting 5 years from today and ending 25 years from today. What is the present value of these payments if the effective annual discount rate is 15%? - Annuity formula for 21 payments, discounted by 4 years due to 1st payment in year 5
- 500/.15 * [1 – 1 / 1.1521] * 1 / 1.154 = $1,804.59
## If the inflation rate this year is 5% and the nominal interest rate is 15%, then what is the real interest rate? ## If the inflation rate this year is 5% and the nominal interest rate is 15%, then what is the real interest rate? - (1 + real)(1 + inflation) = (1 + nominal)
- (1 + real)(1.05) = 1.15
- 1 + real = 1.15 / 1.05 = 1.0952381
- Real = 9.52381%
## Reba’s Rabbits invests $50,000 today, and will earn $10,000 each year starting one year from today. The effective annual discount rate is 9%. ## Reba’s Rabbits invests $50,000 today, and will earn $10,000 each year starting one year from today. The effective annual discount rate is 9%. ## If Reba uses discounted cash flows, how many years is the payback period for this investment? - 50000 = 10000/.09 (1 – 1/1.09T)
- 61111 = (10000/.09)/(1.09T)
- 1.09T = (10000/.09)/61111 = 1.81818
- T = ln(1.81818)/ln(1.09) = 6.93726 ≈ 7
## If Reba uses undiscounted cash flows, how many years is the payback period for this investment? ## If Reba uses undiscounted cash flows, how many years is the payback period for this investment?
## Pyotr’s Beauty Products is considering buying a new device. This machine would cost $8,000 today, and require maintenance costs of $600 every three years, starting in 2 years and ending in 11 years. The machine lasts 12 years, and the effective annual discount rate is 14%. ## Pyotr’s Beauty Products is considering buying a new device. This machine would cost $8,000 today, and require maintenance costs of $600 every three years, starting in 2 years and ending in 11 years. The machine lasts 12 years, and the effective annual discount rate is 14%.
## What is the present value of all costs of the machine over its life? ## What is the present value of all costs of the machine over its life? - Purchase cost today and maintenance costs in years 2, 5, 8, and 11
- 8000 + 600/(1.142) + 600/(1.145) + 600/(1.148) + 600/(1.1411) = $9,125.61
## Pyotr pays $X per year for five years, starting today. These payments will have the same present value as the answer you got from part (a). Find X. ## Pyotr pays $X per year for five years, starting today. These payments will have the same present value as the answer you got from part (a). Find X. - X + X/1.14 + X/(1.142) + X/(1.143) + X/(1.144) = 9125.61
- 3.91371 * X = 9125.61
- X = $2,331.70
## A bond has a face value of $750. It pays a coupon of 10% today, one year from today, and two years from today. Two years from today, the bond matures. If the current selling price of the bond is $800, what is the yield to maturity (expressed as an effective annual discount rate)? ## A bond has a face value of $750. It pays a coupon of 10% today, one year from today, and two years from today. Two years from today, the bond matures. If the current selling price of the bond is $800, what is the yield to maturity (expressed as an effective annual discount rate)?
## 800 = 75 + 75/(1+r) + 825/(1+r)2 ## 800 = 75 + 75/(1+r) + 825/(1+r)2 ## 725(1+r)2 – 75(1+r) – 825 = 0 ## 725r2 + 1375r – 175 = 0 ## 29r2 + 55r – 7 = 0 ## Ignore negative root. r = 0.119716 so r = 11.97%. Or…
## 800 = 75 + 75/(1+r) + 825/(1+r)2 ## 800 = 75 + 75/(1+r) + 825/(1+r)2 ## 725(1+r)2 – 75(1+r) – 825 = 0 ## Let x = 1+r ## 29x2 – 3x – 33 = 0 ## Ignore negative root. x = 1.1197 so r = 11.97%
## Michael is taking out a loan of $1,000,000 today and he will pay $22,000 per month for the next 10 years (120 payments, starting one month from today). The stated annual interest rate is 24%, compounded monthly. 13 years from today, Michael will make one additional payment to pay off the loan. How much will this payment be? ## Michael is taking out a loan of $1,000,000 today and he will pay $22,000 per month for the next 10 years (120 payments, starting one month from today). The stated annual interest rate is 24%, compounded monthly. 13 years from today, Michael will make one additional payment to pay off the loan. How much will this payment be?
## PV of monthly payments: ## PV of monthly payments: - 22000/.02 * [1 – 1/(1.02120)] = 997,818.55
## PV of payment made in 13 years: - 1,000,000 – 997,818.55 = 2,181.45
## FV of payment made in 13 years: - 2,181.45 (1.02)12*13 = $47,904.10
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