Consider the amortization formula for finding the amount. An one has to pay periodically (often monthly),over...

 

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Consider the amortization formula for finding the amount. An one has to pay periodically (often monthly),over n periods, to payoff a debt that starts with principle amount P and interest rate per period i. An P- = Assume that P> 0 and 0 < i < 1. For example consider a $1.5 million mortgage (about the average home price in Santa Cruz) at 7% annual interest, which corresponds to i= 7 monthly interest. A 30 year, or 360 month mortgage, would have a monthly payment of about A360 = $9979.54. i(1+i)n (1+i)n-1' (As another fun exercise for your own amusement, you can figure out how much you'd have to make a year to pay this mortgage and still have money for income taxes, property taxes, insurance, and maybe some food.) You will prove by induction that the formula for An is correct. To do this, we have to introduce the concept of discounting cash flows, because the bank needs the sum of discounted cash flows from the stream of payments to equal the principle. 72 A dollar today is worth more than one in the next period, since a dollar today would be worth 1 + i tomorrow if it accumulates interest. Equivalently, we can think of one dollar 1 period from now being today. A dollar collected two periods from now is worth (1+2 dollars today. This is how we discount future cash flows based on the interest rate. For our amortization formula to be right, the bank needs the sum of all the discounted cash flows to equal the priniciple P. Thus we need worth j=1 An 1 (1 + i)¹ P. Prove by induction that n and An satisfies equation (1) for all n E N. [5 points] Hint: Show i Au-Au-i [1-1-1] An An-1 (1+i)n Then, use this to relate E-1 An to Ej=1 An-1(1+i) n-1 (1) Consider the amortization formula for finding the amount. An one has to pay periodically (often monthly),over n periods, to payoff a debt that starts with principle amount P and interest rate per period i. An P- = Assume that P> 0 and 0 < i < 1. For example consider a $1.5 million mortgage (about the average home price in Santa Cruz) at 7% annual interest, which corresponds to i= 7 monthly interest. A 30 year, or 360 month mortgage, would have a monthly payment of about A360 = $9979.54. i(1+i)n (1+i)n-1' (As another fun exercise for your own amusement, you can figure out how much you'd have to make a year to pay this mortgage and still have money for income taxes, property taxes, insurance, and maybe some food.) You will prove by induction that the formula for An is correct. To do this, we have to introduce the concept of discounting cash flows, because the bank needs the sum of discounted cash flows from the stream of payments to equal the principle. 72 A dollar today is worth more than one in the next period, since a dollar today would be worth 1 + i tomorrow if it accumulates interest. Equivalently, we can think of one dollar 1 period from now being today. A dollar collected two periods from now is worth (1+2 dollars today. This is how we discount future cash flows based on the interest rate. For our amortization formula to be right, the bank needs the sum of all the discounted cash flows to equal the priniciple P. Thus we need worth j=1 An 1 (1 + i)¹ P. Prove by induction that n and An satisfies equation (1) for all n E N. [5 points] Hint: Show i Au-Au-i [1-1-1] An An-1 (1+i)n Then, use this to relate E-1 An to Ej=1 An-1(1+i) n-1 (1)