Practice Exam 1

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Feb 20, 2024

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Exam 1 Practice Exam 1. What is the relationship between future value and present value? FV = PV ( 1 + r ) t 2. For a given time period and future value – the lower the interest rate, ___________ the present value. For a given time period and future value – the lower the interest rate, higher is the present value. 3. Your best friend from FIN324 invested $3,000 five years ago and earns 2 percent annual interest. By leaving her interest earnings in her account, she increases the amount of interest she earns each year. The way she is handling her interest income is referred to as: a. simplifying. b. compounding. c. aggregating. d. accumulating. e. discounting. Solution: b. compounding 4. If the monthly interest rate is 1%. What is the annual interest rate? r monthly = 1% r annual = ( 1 + r monthly ) 12 1 r annual = ( 1.01 ) 12 1 = 12.68% 5. Calculate the present value of the following cash flows discounted at 10 percent per year. a. $100,000 received 10 years from today. PV = FV ( 1 + r ) t = 100,000 1.1 10 = $ 38,554.33 b. $100,000 received 2 years from today. PV = ¿ 100,000 1.1 2 = $ 82,644.63 c. $100,000 received 20 years from today. PV = ¿ 100,000 1.1 20 = $ 14,864.36
6. If you put up $36,000 today in exchange for a 7.00 percent, 18-year annuity, what will the annual cash flow be? PV = C r ( 1 1 ( 1 + r ) t ) 36,000 = C .07 ( 1 1 1.07 18 ) C = $ 3,578.85 7. You deposit $2,000 at the end of each year into an account paying 10.6 percent interest. a. How much money will you have in the account in 24 years? FV = C r [ ( 1 + r ) t 1 ] = 2000 0.106 [ ( 1.106 ) 24 1 ] = $ 192,893.66 b. How much will you have if you make deposits for 48 years? FV = C r [ ( 1 + r ) t 1 ] = 2000 0.106 [ ( 1.106 ) 48 1 ] = $ 2,357,809.41 8. Shao Professional Basketball Agents issued a 17-year bond 2 years ago at a coupon rate of 10%. The bond makes semi-annual payments. The bond currently sells for 102% of par value. If the YTM changes from 9.75% to 12%, where will the bond be traded? a) The bond will be traded above par. b) The bond will be traded at par. c) The bond will be traded below par. d) The change in rate will not impact the price at which the bond is traded. e) The bond could be traded below, at, or above par. 9. Live Forever Life Insurance Company is selling a perpetuity contract that pays $1,550 monthly. The contract currently sells for $74,000. a. What is the monthly interest rate? We need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation:   PV =  C  / r $74,000 = $1,550/ r   We can now solve for the interest rate as follows:   r  = $1,550/$74,000  r  = .0209, or 2.09% per month b. What is the APR? The interest rate is 2.09% per month. To find the APR, we multiply this rate by the
number of months in a year, so:   APR = 12(2.09%)  APR = 25.14% c. What is the EAR? Using the equation to find an EAR:   EAR = [1 + (APR/ m )] m  − 1 EAR = [1 + 0.0209] 12  − 1  EAR = .2824, or 28.24% 10. Given a market interest rate of 12 percent per year, what is the value at date t = 4 (i.e., the end of year 4) of a perpetual stream of $50 annual payments that begin at date t = 10 (i.e., at the end of year 10 and continue forever)? We have a perpetuity r = 12% C = $50 The payments begin at t=10 At the end of t=9, the value of the perpetuity is: V 9 = C r = 50 0.12 = $ 416.67 The value at t=4 of a cash flow of $416.67 at t=9 is: V 4 = $ 416.67 ( 1.12 ) 5 = $ 236.43 11. The appropriate discount rate for the following cash flows is 7 percent compounded quarterly. Year Cash Flow 1 $ 900 2 900 3 0 4 1,100 What is the present value of the cash flows? The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR to make the interest rate comparable with the timing of the cash flows. Using the equation for the EAR, we get:   EAR = [1 + (APR/ m )] m  − 1 EAR = [1 + (.07/4)] 4  − 1 
EAR = .0719, or 7.19%   And now we use the EAR to find the PV of each cash flow as a lump sum and add them together:   PV = $900/1.0719 + $900/1.0719 2  + $1,100/1.0719 4 PV = $2,456.41 12. Your job pays you only once a year for all the work you did over the previous 12 months. Today, December 31, you just received your salary of $46,000 and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 2 percent of your annual salary in an account that will earn 9 percent per year. Your salary will increase at 6 percent per year throughout your career. How much money will you have on the date of your retirement 41 years from today? Since your salary grows at 6 percent per year, your salary next year will be:   Next year's salary = $46,000 (1 + .06) Next year's salary = $48,760   This means your deposit next year will be:   Next year's deposit = $48,760(.02) Next year's deposit = $975   Since your salary grows at 6 percent, you deposit will also grow at 6 percent. We can use the present value of a growing perpetuity equation to find the value of your deposits today. Doing so, we find: PV = C r g { 1 [ 1 + g 1 + r ] t } PV = $ 975 .09 .06 { 1 [ 1 + .06 1 + .09 ] 41 } PV = $22,154.61   Now, we can find the future value of this lump sum in 41 years. We find:   FV = PV(1 +  r ) t FV = $22,154.61(1 + .09) 41 FV = $758,491.28
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