# MATH 177 Practice Midterm Questions

Math 177 Midterm 1 – Page 2 of 5 04/30/2021 1. (25 points) Carloman and Charles are two brothers. Brothers’ finance advisor, Desiderius arranged two funds for these brothers, Fund YSDS for Carloman and Fund LLTE for Charles. At the beginning, both make deposits of 100, 000. Fund YSDS has a compound accumulation at an annual effective interest rate of 5%, whereas Fund LLTE has a simple accumulation with a yearly effective rate of 8%. t years into the future, the forces of interest on the two funds are equal. At this time t, find the difference between funds YSDS and LLTE, specifying Carloman/YSDS or Charles/LLTE will be greater (pun intended). Math 177 Midterm 1 – Page 3 of 5 04/30/2021 2. (25 points) Evariste Galois, aged exactly 20, purchases a perpetuity-due to act as a pension benefit that starts 40 years later, exactly at his 60th birthday.

According to this benefit he will be paid an annual total of 48, 000 with level payments at the start of every month. (The total annual payment of 48, 000 is to be equally distributed to each month.) To purchase this benefit, he will be making forty annual premium payments, first payment being at the time of policy purchase, at the 20th birthday. Every payment after the first will increase by 6%. Assuming a flat term structure and annual effective interest rate of 8% for all valuations, what should be the amount of initial payment? Math 177 Midterm 1 – Page 4 of 5 04/30/2021 3. (25 points) To repay her gambling debts, Tsunade borrows money from Jiraiya. This 25-year loan is being paid off via level amortization payments made at the end of each quarter. The nominal annual interest rate is 12%, convertible monthly. The amount of principal reduction in the 29th payment is 1, 860. Determine the amount of principal reduction in the 61st payment. Math 177 Midterm 1 – Page 5 of 5 04/30/2021 4. (25 points) A 50-year, 1, 000 par-valued bond with semiannual coupons, with a nominal coupon rate of 2X. Avicenna purchased it for 1, 630.31 to yield a nominal annual rate of X compounded semiannually.

Determine X. Math 177 Midterm 2 – Page 2 of 5 05/28/2021 1. (25 points) Petyr Baelish has an investment account in the Iron Bank of Braavos. At the start of the year, the account is worth 300, 000. After k months, the value has increased to 315, 000 and then Petyr withrew 15, 000. 2k months before the end of the year, the account is again worth 315, 000, and again, Petyr withdraws 15, 000. At the end of the year, the account is worth 315, 000 (again). The dollar-weighted yield rate turns out to be 16%. What is k? Math 177 Midterm 2 – Page 3 of 5 05/28/2021 2. (25 points) Consider the term structure with the following spot rates rt = 0.08 + 0.02t, for t ∈ {1, 2, 3, 4}. Assuming that the annual effective yield rate for a 4-year bond with annual coupons is 15%, what must be the underlying effective coupon rate? Math 177 Midterm 2 – Page 4 of 5 05/28/2021 3. (25 points) You are given the following information about an asset: (i) The present value of the cash flows at an annual effective interest rate of 5% is 100, 000. (ii) The estimate of the present value of the cash flows at an annual effective interest rate of 4.9% using the first order Macaulay approximation is 101, 000. Determine the modified duration of the asset at 5%. Math 177 Midterm 2 – Page 5 of 5 05/28/2021 4. (25 points) The price of a 1-year zero-coupon bond with maturity value of 1 is 0.943. The price of a 2-year zero-coupon bond with a maturity value of 1 is X. Realizing that the above prices describe the current term structure and assuming that at time 0, the constant notional level swap rate is 0.065826, find the value of X.

Math 177 Midterm 1 – Page 2 of 5 04/30/2021 1. (25 points) Carloman and Charles are two brothers. Brothers’ finance advisor, Desiderius arranged two funds for these brothers, Fund YSDS for Carloman and Fund LLTE for Charles. At the beginning, both make deposits of 100, 000. Fund YSDS has a compound accumulation at an annual effective interest rate of 5%, whereas Fund LLTE has a simple accumulation with a yearly effective rate of 8%. t years into the future, the forces of interest on the two funds are equal. At this time t, find the difference between funds YSDS and LLTE, specifying Carloman/YSDS or Charles/LLTE will be greater (pun intended). Math 177 Midterm 1 – Page 3 of 5 04/30/2021 2. (25 points) Evariste Galois, aged exactly 20, purchases a perpetuity-due to act as a pension benefit that starts 40 years later, exactly at his 60th birthday. According to this benefit he will be paid an annual total of 48, 000 with level payments at the start of every month. (The total annual payment of 48, 000 is to be equally distributed to each month.) To purchase this benefit, he will be making forty annual premium payments, first payment being at the time of policy purchase, at the 20th birthday.

Every payment after the first will increase by 6%. Assuming a flat term structure and annual effective interest rate of 8% for all valuations, what should be the amount of initial payment? Math 177 Midterm 1 – Page 4 of 5 04/30/2021 3. (25 points) To repay her gambling debts, Tsunade borrows money from Jiraiya. This 25-year loan is being paid off via level amortization payments made at the end of each quarter. The nominal annual interest rate is 12%, convertible monthly. The amount of principal reduction in the 29th payment is 1, 860. Determine the amount of principal reduction in the 61st payment. Math 177 Midterm 1 – Page 5 of 5 04/30/2021 4. (25 points) A 50-year, 1, 000 par-valued bond with semiannual coupons, with a nominal coupon rate of 2X. Avicenna purchased it for 1, 630.31 to yield a nominal annual rate of X compounded semiannually.

Determine X. Math 177 Midterm 2 – Page 2 of 5 05/28/2021 1. (25 points) Petyr Baelish has an investment account in the Iron Bank of Braavos. At the start of the year, the account is worth 300, 000. After k months, the value has increased to 315, 000 and then Petyr withrew 15, 000. 2k months before the end of the year, the account is again worth 315, 000, and again, Petyr withdraws 15, 000. At the end of the year, the account is worth 315, 000 (again). The dollar-weighted yield rate turns out to be 16%. What is k? Math 177 Midterm 2 – Page 3 of 5 05/28/2021 2. (25 points) Consider the term structure with the following spot rates rt = 0.08 + 0.02t, for t ∈ {1, 2, 3, 4}. Assuming that the annual effective yield rate for a 4-year bond with annual coupons is 15%, what must be the underlying effective coupon rate? Math 177 Midterm 2 – Page 4 of 5 05/28/2021 3. (25 points) You are given the following information about an asset: (i) The present value of the cash flows at an annual effective interest rate of 5% is 100, 000.

(ii) The estimate of the present value of the cash flows at an annual effective interest rate of 4.9% using the first order Macaulay approximation is 101, 000. Determine the modified duration of the asset at 5%. Math 177 Midterm 2 – Page 5 of 5 05/28/2021 4. (25 points) The price of a 1-year zero-coupon bond with maturity value of 1 is 0.943. The price of a 2-year zero-coupon bond with a maturity value of 1 is X. Realizing that the above prices describe the current term structure and assuming that at time 0, the constant notional level swap rate is 0.065826, find the value of X. (1) (10 pts) Jamie is given a savings account with 7% nominal discount rate, convertible quarterly. On the other hand, Cersei is given a savings account that compounds monthly. What is Cersei’s effective monthly interest rate, assuming that both accounts produce the same accumulation? (2) (10 pts) The master of coin, Petyr Baelish is offered an investment where interest is calculated according to the following force of interest: δt =  0.02 0≤t≤3 0.04 3

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