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Mathematics and finance questions

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1. $1225 is borrowed for one year at a discount rate of D, giving the use of an extra $1000. Find D and the annual interest rate that this is equivalent to.

2. A savings account earns compound interest at an annual effective interest rate of i. Given that i[3,5.5] = 15%, find d[1,4]

3. Money grows according to the simple interest accumulation function a(t)=1+0.04ta ( t ) = 1 + 0.04 t. How much money would you need to invest at time 2 in order to have $2,300 at time 9?

4. What is the present value of $10,000 due in ten years assuming that money grows according to compound interest and that the annual effective rated of interest is 1% for the first three years, 2% for the next two years, and 4% for the remaining five years?

5. Show that if the growth of money is from compound interest with an annual effective rate of i >0, then the sum of the current value of a payment of K made n periods ago and K to be made n periods from now is greater than 2K.

6. Suppose we know that for money in an account with accumulation function a(t), the sum of the current value of a payment of K made n periods ago and K to be made n periods from now is greater than 2K. What must be true about a(t). [Note that this is more general than the previous problem].

7. You have two options for repaying a loan:

A. $5,000 now and $4,500 in one year.

B. $10,000 in six months.

Find the annual effective interest rate(s) i at which both options have the same present value.

8. A payment of X on year from now and a payment 3X three years from now repays a debt of $10,000 at 5% annual effective compound interest. What is the value of X?

9. You invest P into an account earning simple discount at a rate of 6% or simple interest earning 9%. How long must you invest your money so that the simple discount account is preferable?

10. An account is governed by compound interest. The interest for two years on $2,500 is $250.  Find the amount of discount for two years on $1,000.

Edited by Ben & Jerry's
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Let's keep it simple and practical with the Rule of 72: Your money roughly doubles when the earnings rate x # of years = 72.  E.g., 8% earnings (or growth, such as equities) x 9 years = 72.  Be more conservative with a 6% earnings rate and it takes 12 years (6 x 12 = 72).  Lesson: Stick your retirement money in an equity index fund(s) and forget about it for a couple or few decades.

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