6. If the effective interest rate is i per period and you invest b dollars at time0, then the value at time n is b(1 + i)”. For example, if at time 0 youinvest $100 at an effective interest rate of 6% per year, then the value attime 1 is $100(1+0.06) = $106. dollars. At time 2, the value will be$100 (1 + 0.06)² = $112.36. Similarly, the present value (value at time 0)of $100 received 2 years from now would be $100(1+0.06)-²≈ $88.99.Sometimes it's convenient to define d = 1/(1 + i) so that we would have$100(1+i)n = $100d".If the interest rate per year is i compounded semiannually, then the effectiveannual interest rate is(¹ + 2)²³ - 1₁1.For example, if the interest rate is 6% per year compounded semiannually,the effective annual interest rate is1 +.0622- 1 ≈ 6.09%.If the interest rate per year is i compounded quarterly, then the effectiveannual interest rate is4(¹ + 4) * - 11.(a) If the interest rate is 6% per year compounded quarterly, what is theeffective annual interest rate?(b) Once upon a time, a bank was advertising an interest rate for savingsaccount of i% per year. A competitor advertised a rate of i% per yearcompounded semiannually. The first bank advertised that they paid i%per year compounded quarterly. The other bank changed theirs to beingcompounded monthly. The first bank changed theirs to compoundeddaily. Finally, a bank said their interest rate was i% per year com-pounded continuously. What is the effective annual rate of 6% per yearcompounded semiannually? monthly? and continuously? (c) Suppose the effective annual interest rate per period is i and that you willreceive payments an amount a at times 1, 2, ..., n. Thus, the presentvalue is2b = ad + ad²+...+ ad" = ak=1where d = 1/(1+i). Give a closed form expression for the presentvalue b.(d) You wish to borrow $500,000 to purchase real estate. The bank isoffering a 30 year loan at an interest rate of 6%. What would themonthly payments be? (Hint: In the U.S., it's understood that this is6% per year compounded monthly. Treat the period as one month andthat there are 360 payments. The first payment is due 1 month fromreceiving the loan. The present value of this stream of payments shouldbe the loan amount. If you or someone you know has a mortgage,you can check to see whether the monthly payment is correct. Thepayment we computed is for principal and interest. Sometimes anadditional payment is required, which is escrowed for property taxesand/or insurance.)(e) Supppose you are considering a 15 year loan instead of the 30 year loan.Usually, a 15 year loan would have a lower interest rate, but let's assumethat it's the same 6%. You might be guessing that the monthly paymentwould be roughly double the monthly payment amount for the 30 yearloan. Please determine the monthly principal and interest payment forthe same loan except for 15 years. Was the monthly payment for 15years close to double the monthly payment for 30 years?

Question

6. If the effective interest rate is i per period and you invest b dollars at time0, then the value at time n is b(1 + i)”. For example, if at time 0 youinvest $100 at an effective interest rate of 6% per year, then the value attime 1 is $100(1+0.06) = $106. dollars. At time 2, the value will be$100 (1 + 0.06)² = $112.36. Similarly, the present value (value at time 0)of $100 received 2 years from now would be $100(1+0.06)-²≈ $88.99.Sometimes it's convenient to define d = 1/(1 + i) so that we would have$100(1+i)n = $100d&#34;.If the interest rate per year is i compounded semiannually, then the effectiveannual interest rate is(¹ + 2)²³ - 1₁1.For example, if the interest rate is 6% per year compounded semiannually,the effective annual interest rate is1 +.0622- 1 ≈ 6.09%.If the interest rate per year is i compounded quarterly, then the effectiveannual interest rate is4(¹ + 4) * - 11.(a) If the interest rate is 6% per year compounded quarterly, what is theeffective annual interest rate?(b) Once upon a time, a bank was advertising an interest rate for savingsaccount of i% per year. A competitor advertised a rate of i% per yearcompounded semiannually. The first bank advertised that they paid i%per year compounded quarterly. The other bank changed theirs to beingcompounded monthly. The first bank changed theirs to compoundeddaily. Finally, a bank said their interest rate was i% per year com-pounded continuously. What is the effective annual rate of 6% per yearcompounded semiannually? monthly? and continuously? (c) Suppose the effective annual interest rate per period is i and that you willreceive payments an amount a at times 1, 2, ..., n. Thus, the presentvalue is2b = ad + ad²+...+ ad&#34; = ak=1where d = 1/(1+i). Give a closed form expression for the presentvalue b.(d) You wish to borrow $500,000 to purchase real estate. The bank isoffering a 30 year loan at an interest rate of 6%. What would themonthly payments be? (Hint: In the U.S., it's understood that this is6% per year compounded monthly. Treat the period as one month andthat there are 360 payments. The first payment is due 1 month fromreceiving the loan. The present value of this stream of payments shouldbe the loan amount. If you or someone you know has a mortgage,you can check to see whether the monthly payment is correct. Thepayment we computed is for principal and interest. Sometimes anadditional payment is required, which is escrowed for property taxesand/or insurance.)(e) Supppose you are considering a 15 year loan instead of the 30 year loan.Usually, a 15 year loan would have a lower interest rate, but let's assumethat it's the same 6%. You might be guessing that the monthly paymentwould be roughly double the monthly payment amount for the 30 yearloan. Please determine the monthly principal and interest payment forthe same loan except for 15 years. Was the monthly payment for 15years close to double the monthly payment for 30 years?

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Interest rate = 6% compounded quarterly

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(a) If the interest rate is 6% per year compounded quarterly, what is the effective annual interest rate?