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A bank accepts a 20,000 deposit from a customer on which it guarantees to pay an annual effective interest rate of 10% for two years. The customer needs to withdraw half of the accumulated value at the end of the first year. The customer will withdraw the remaining value at the end of the second year. The bank has the following investment options available, which may be purchased in any quantity: Bond H: A one-year zero-coupon bond yielding 10% annually Bond I: A two-year zero-coupon bond yielding 11% annually Bond J: A two-year bond that sells at par with 12% annual coupons Any portion of the 20,000 deposit that is not needed to be invested in bonds is retained by the bank as profit. Determine which of the following investment strategies produces the highest profit for the bank and is guaranteed to meet the customer’s withdrawal needs.

(A) 9,091 in Bond H, 8,264 in Bond I, 2,145 in Bond J

(B) 10,000 in Bond H, 10,000 in Bond I

(C) 10,000 in Bond H, 9,821 in Bond I

(D) 8,910 in Bond H, 731 in Bond I, 10,000 in Bond J

(E) 8,821 in Bond H, 10,804 in Bond J

I am trying to solve the question above. The official solution is as follows:

The correct answer is the lowest cost portfolio that provides for \$11,000 at the end of year one and provides for $12,100 at the end of year two. Let H, I, and J represent the face amount of each purchased bond. The time one payment can be exactly matched with H + 0.12J = 11,000. The time two payment can be matched with I + 1.12J = 12,100. The cost of the three bonds is H/1.1 + I/1.2321 + J. This function is to be minimized under the two constraints. Substituting for H and I gives (11,000 – 0.12J)/1.1 + (12,100 – 1.12J)/1.2321 + J = 19,820 – 0.0181J. This is minimized by purchasing the largest possible amount of J. This is 12,100/1.12 = 10,803.57. Then, H = 11,000 – 0.12(10,803.57) = 9703.57. The cost of Bond H is 9703.57/1.1 = 8,821.43.

I understand that the bank needs to pay \$11,000 at the end of year 1 because $\frac{20,000(1.1)}{2} = 11,000$ and that the bank needs to pay \$12,100 at the end of year 2 because $20,000(1.1)^2-11000(1.1) = 12,100$. Now, I also see that the official solution is using the technique of exact matching to get to the solution. Beyond that, the official solution does not make much sense to me. Could someone please explain how "The time one payment can be exactly matched with H + 0.12J = 11,000. The time two payment can be matched with I + 1.12J = 12,100. The cost of the three bonds is H/1.1 + I/1.2321 + J"?

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H, I and J are the face amounts. So, at time 1, you need to pay out 11,000. You can get Bond H and Bond J to get payments at year 1. So at year 1, you can receive face amount of H and a coupon payment of .12J. At year 2, you need 12,100. So you can get a face amount of I and J plus the coupon payment from J, which is I + J + .12J = I + 1.12J. For the last part, the cost of purchasing the bonds to receive H + .12J at time 1 and I + 1.12J at time 2, the cost will be H(1.1^-1) + I(1.11^-2) + J. I(1.11^-2) = I/(1.11)^2 = I/1.2321.

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