IE 343 Midterm Exam 1 Feb 17, 2012 Version A Closed book, closed notes. Write your printed name in the spaces provided above on every page. Show all of your work in the spaces provided. Interest rate tables are provided for you to use in questions that require numerical answers. For problems requiring expressions as answers, carry your solution to the point where the equation for each problem is specified. For example, 1,000 (P/A, 4%, 5) + 2,500 (P/F, 4%, 5) 4,000. If the question asks you to decompose the cash flow into Basic Components, you can only decompose the cash flow into the Seven Basic Components: 1) Single Cash Flow, 2) Annuity, 3) Deferred Annuity, 4) Uniform (Arithmetic) Gradient Series, 5) Deferred Uniform (Arithmetic) Gradient Series, 6) Geometric Gradient Series, 7) Deferred Geometric Gradient Series. Exam 1 has 3 Parts totally 5 Problems with 5 Points: Part I 3 Old Problems, totally 80 Points. The old problems are selected from Homework, Quizzes, Lecture Notes Examples and Textbook Examples with numbers changed. Part II 1 New Problem, 20 Points. Part III 1 Bonus Problem, 5 Points. You are suggested to do Part I first, and then part II. Part III is optional. Version A 1
Part (I) 3 Old Problems, totally 80 Points: old problems selected from Homework, Quizzes, Lecture Notes Examples or Textbook Examples Question 1 (20 Points) A company produces a memory chip that is used in manufacturing cell phones. The fixed cost is $5000 per month, and the variable cost is $50 per chip. The selling price per unit is P = 600 5D, where D is monthly demand. Maximum output of the plant is 80 units per month. (a) Determine the demand quantity that maximizes profit per month? (8 Points) (a), so maximizes profits. (b) What is/are the breakeven point(s) of the firm per month? (8 Points) (b) Solving it quadratically, D = or 0 Since maximum output is 80, breakeven can only occur at D =. (c) What is the range of profitable demand per month? (4 Points) (c) Find the values of D which makes. The range is simply (, 80]. Version A 2
Question 2 (22 Points) Suppose you took out a bank loan of $,000 at an annual interest rate of 12% compounded quarterly. The loan is repayable over a period of years. Quarterly payments are made at the end of every quarter and the first payment is made at the end of the 1 st quarter. (a) What is the effective interest rate per quarter? (6 Points) (a) For quarterly compounding, the effective interest rate per quarter = nominal interest rate per quarter. So (b) Calculate your quarterly payment? Please refer to the interest rate tables for numerical answers. (8 Points) (b) (c) After making 30 such payments, you could pay a lump sum now (right after the 30 th payment) to close out the loan. How much do you need to pay? Please refer to the interest rate tables for numerical answers. (8 Points) (c) To find out the lump sum amount right after the 30 th payment, we just need to find out the present value of the last payments at the end of the 30 th quarter. Version A 3
Question 3 (38 Points) The following two cash flows Cash Flow (A) and Cash Flow (B) are economically equivalent. The effective interest rate is % per period. Please follow the questions to find out the unknown X. Cash Flow (A) A = $2,000 X/3 X/3 X/3 Cash Flow (B) X X X $5,000 (a) What is the present equivalent value of Cash Flow (A). Please refer to the interest rate tables for numerical answers. (6 Points) (a) Version A 4
(b) Decompose the Cash Flow (B) into several Basic Components. (9 Points) (b) Cash Flow (B) can be decomposed into 3 Basic Components as follows: Standard Annuity X/3 X/3 X/3 Deferred Annuity X X X Single Cash Flow $5,000 (c) Write an expression in terms of the unknown X: What is the present equivalent value of Cash Flow (B) based on your decomposition from part (b). Just write down the (c) expression like e.g.. You don t need to calculate the final numerical answer. (9 Points) Version A 5
(d) Based on your decompositions from part (c), use the interest rate tables to calculate the numerical answer of the present equivalent value in terms of the unknown X. e.g.. (8 Points) (d) From the table, (e) Since Cash Flow (A) and Cash Flow (B) are economically equivalent, solve for X. (6 Points) (e) Since Cash Flow (A) and Cash Flow (B) are economically equivalent, so. Version A 6
Part(II), 1 Problems, 20 Points: New Problems Question 4 (20 Points) Consider the following cash flow diagram. Assume the interest rate is % per period. $1,000 $3,000 $2,000 $4,000 $5,000 -$1,000 -$2,000 -$3,000 -$4,000 -$5,000 (a) Decompose the Cash Flow into several Basic Components. (12 Points) (a) The cash flow diagram can be decomposed into the following three Basic Components Single Cash Flow -$5,000 Version A 7
Standard Annuity A = -$4,000 Standard Uniform Gradient Series $5,000 $4,000 $3,000 $2,000 $1,000 $9,000 $8,000 $7,000 $6,000 (b) Write an expression: Based on your cash flow decompositions in part (a), what is the annual equivalent value over the -year period given an interest rate of % per period. Just write down the expression like e.g. A = 1,000 (A/P, %, ) + 2,500 (A/F, %, ) 4,000. You don t need to calculate the final numerical answer. (8 Points) (b). Version A 8
Part(III), 1 Bonus Problem, 5 Points Question 5 (5 Points) Consider the following cash flow diagram. Assume the interest rate is % per period. 0 80 60 40 20 0 1 2 3 4 5 20 6 40 7 60 8 80 9 0 Write an expression: what is the present equivalent value of the cash-flow diagram? Just write down the expression like e.g. P = 1,000 (P/A, %, ) + 2,500 (P/F, %, ) 4,000. You don t need to calculate the final numerical answer. (5 Points) The cash flow diagram can be decomposed into the following FOUR Basic Components 0 0 1 2 3 4 5 6 7 8 9 80 80 80 80 0 1 2 3 4 5 6 7 8 9 Version A 9
0 1 2 3 4 5 6 7 8 9 20 40 60 0 1 2 3 4 5 20 6 40 7 60 8 80 9 0 Version A
Formulas: Roots of a quadratic function:. Find F given P: Find P given F: Find F given A: N (1 i) 1 F A i Find P given A: Find P given G: P G 1 (1 i) N 1 N i i(1 i) N (1 i) N Find A given G: A G 1 i N (1 i) N 1 Find F given G: N 1 (1 i) 1 G NG F G N ( F / A, i%, N) i i i i Present Equivalent Value P of the Geometric Gradient Series: or equivalently Compute effective interest rate when interest compounds more frequently: Version A 11
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