Using Maple:

We prepare the following formulas: the future value of a lumpsum A (if interest is compounded n times per year) is L := (A, r, n, t) -> A*(1 + r/n)^(n*t); the future value of a lumpsum A (if interest is compounded continuously) is Lc := (A, r, t) -> A*exp(r*t);

the future value of a stream of equal payments P is (with payments at the end of each period) S := (P, r, n, t) -> P*((1 + r/n)^(n*t) - 1)*n/r;

the Balance on a mortgage (or similar loan or annuity) is B := (A, r, n, t, P) -> A*(1 + r/n)^(n*t) - P*((1 + r/n)^(n*t) - 1)*n/r;

Problem 3: Suppose we wish to take out a 30-year mortgage of $200,000. The interest rate is 10%, compounded monthly. a. Find the monthly payment

b. Graph the balance on the loan at a time t, as time passes from now to time of amortization.

c. At what time is half the loan paid back?

Problem 4: Suppose we take out a 20-year mortgage of $200,000. The interest rate is 8%, compounded monthly. a. Find the monthly payment

b. After 5 years, the interest rate for a 15-year mortgage is now 6.5%. The cost of refinancing is $4000. If we choose to refinance, how long will it take to recoup the $4000 cost?

Problem 5: Suppose we can afford $1000 a month toward a mortgage. a. If the interest rate is r=0%, how big of a mortgage can we afford? (use common sense, the formula will not work for r=0) b. If the interest rate is r=5%, how big of a mortgage can we afford? c. If the interest rate is r=10%, how big of a mortgage can we afford? d. Draw a graph for the amount of mortgage we can afford as a function of the interest rate r, with r ranging from 0% to 20%.

Problem 6: Suppose you need a $10,000 loan from the car dealer to buy the car you wish. The dealer offers you a choice of three ways to pay back the loan a. $250 a month for 4 years b. $1000 every 6 months for 6 years.. c. No payments for the first year, then $600 quarterly for 5 years

Notice that the total payments amount to the same in each case: $12,000. But the deals are not equally good. Which is the best deal in terms of the implied interest rate? (Specify both the nominal and the effective rate for each arrangement) a. nominal: effective: b. nominal: effective: c. nominal: effective:

Problem 7: On Jan 1, 2021, a trust fund has an initial value of $1,000,000 and earns interest at an annual rate of 5%, compounded monthly. If a withdrawal of $10,000 is made monthly, beginning Feb 1, 2021, a) Estimate how long the trust fund will last (back-of-the-envelope estimate, no calculators). Explain:

b) on what day will the account fall below $500,000?

c) On what day will the last penny be withdrawn?

d) How much is the last withdrawal?