Need assistance with 4 - 6
You are an investor, interested in real estate. Part of your interest stems from the fact that you understand that real estate appreciates at about 10% per year. This project will investigate this and related phenomena. What does 10% appreciation per year mean? Presumably, it means that the value of a property after year n is 10% more than it was at the end of the year 11 1. This may be expressed mathematically by the difference equation v(n) = 1.10-v(n 1). $100,000. Find the value of the house at the end of each of the 1. Suppose that the initial value v(0) of a house next six years if the rate of appreciation is: (a) 10% (b) 4% (C) -2% 2. The term difference equation suggests that you might want to consider the difference vn) v (n 1). Write an appropriate difference equation in each of the three cases. 3. Difference equations are closely related to "lifferential equations-that is, equations involving derivatives. Write an appropriate differential equation for v(l) for each of the three cases. Solve this differential equation. (Is there any difference between a difference equation and the associated differential equation? Is there any difference belween the solutions to a difference equation and to the associated differential equation?) Let's return to the 10% appreciation case. Normally, when you buy a house, you don't pay for the entire house in cash-you make a down payment, say 20%. (We will assume that you rent the house and the rent payments cover mortgage payments and any other expenses.) As an investor, you are interested in your rate of return-that is, how much your investment is worth at the end of year n compared to how much it was worth at the end of the year n - 1. 4. Calculate your rate of return for each of the first six years assuming: (a) 50% down payment (b) 20% down payment (c) 10% down payment 5. Write a difference equation that describes your rate of return in each of the three cases. What happens to your rate of return? 6. Suppose you want a constant rate of retum of 20%. What will have to happen to the value of the house? You are an investor, interested in real estate. Part of your interest stems from the fact that you understand that real estate appreciates at about 10% per year. This project will investigate this and related phenomena. What does 10% appreciation per year mean? Presumably, it means that the value of a property after year n is 10% more than it was at the end of the year 11 1. This may be expressed mathematically by the difference equation v(n) = 1.10-v(n 1). $100,000. Find the value of the house at the end of each of the 1. Suppose that the initial value v(0) of a house next six years if the rate of appreciation is: (a) 10% (b) 4% (C) -2% 2. The term difference equation suggests that you might want to consider the difference vn) v (n 1). Write an appropriate difference equation in each of the three cases. 3. Difference equations are closely related to "lifferential equations-that is, equations involving derivatives. Write an appropriate differential equation for v(l) for each of the three cases. Solve this differential equation. (Is there any difference between a difference equation and the associated differential equation? Is there any difference belween the solutions to a difference equation and to the associated differential equation?) Let's return to the 10% appreciation case. Normally, when you buy a house, you don't pay for the entire house in cash-you make a down payment, say 20%. (We will assume that you rent the house and the rent payments cover mortgage payments and any other expenses.) As an investor, you are interested in your rate of return-that is, how much your investment is worth at the end of year n compared to how much it was worth at the end of the year n - 1. 4. Calculate your rate of return for each of the first six years assuming: (a) 50% down payment (b) 20% down payment (c) 10% down payment 5. Write a difference equation that describes your rate of return in each of the three cases. What happens to your rate of return? 6. Suppose you want a constant rate of retum of 20%. What will have to happen to the value of the house
