Chapter 5: Introduction to Valuation (The Time Value of Money) (Problems pg 146-148)
Basic 4) Solve for the unknown interest rate in each of the following:
-to find r, we use: r=[(FV/PV)^(1/t)]-1 .1318 or 13.18% .0672 or 6.72% .0737 or 7.37% .1086 or 10.86%
Basic 5) Solve for the unknown number of years in each of the following:
-to find t, we use: t=ln(FV/PV)/ln(1+r) 15.59 years 9.40 years 26.41 years 24.52 years
CRCTQ 9) Suppose that TMCC offered the security for $24,099 the U.S. Treasury had offered an essentially identical security. Do you think it would have a higher or lower price Why?
-A higher price due to the fact that the Treasury is as reliable of a borrower as there is
Basic 1) First City Bank pays 8% simple interest on its savings account balances, whereas Second City Bank pays 8% interest compounded annually. If you made a deposit of $9,000 in each bank, how much more $ would you earn from your Second City Bank account @ the end of 7 years?
-First City Bank: $9,000*.08=$720 ==>$720*7=$5,040==> total balance of $14,040 -Second City Bank: use future value formula (FV=PV(1+r)^t) FV=$9,000(1.08)^7=$15,424.42 -diff. b/w two: 15,424.42-14,040=$1,384.42
CRCTQ 10) The TMCC security is bought & sold on the NYSE. If you looked @ the price today, do you think the price would exceed the $24,099 original price? Why? If you looked in the year 2019, do you think the price would be higher or lower than today's price? Why?
-Price would be higher b/c, as time passes, the price of the security will rise toware $100,000; rise=just of a reflection of the time value of $ -in 2019, price will probably be higher for same reason; can't be sure, b/c interest rates could be higher or b/c TMCC's financial position could fall
discount
-calculate the present value of some future amount
discounted cash flow (DCF) valuation
-calculating the present value of a future cash flow to determine its value today
CRCTQ 2) What is compounding? What is discounting?
-compounding: the growth of a dollar amount through time via reinvestment of interest earned; process of determining future value of an investment -discounting: process of determining the value today of an amount to be received in the future
CRCTQ 3) As you increase the length of time involved, what happens to future values? What happens to present values?
-future value RISES, present value SHRINKS ***just look @ equation
CRCTQ 4) What happens to a future value if you increase the rate, r? What happens to a present value?
-future value RISES, present value SHRINKS ***just look @ equation
compound interest
-interest earned on both the initial principal & the interest reinvested from prior periods
interest on interest
-interest earned on the reinvestment of previous interest payments
simple interest
-interest earned only on the original principal amount invested
CRCTQ 6) Why would TMCC be willing to accept such a small amount today ($24,099) in exchange for a promise to repay about four times that amount ($100,000) in the future?
-reflets the time value of $ -if the $ is used wisely, it will actually be worth far more than $100,000 in 30 years
future value (FV)
-the amount of $ an investment will grow to over some period of time @ some given interest rate
present value (PV)
-the current value of future cash flows discounted @ the appropriate discount rate
compounding
-the process of accumulating interest on an investment over time to earn more interest
discount rate
-the rate used to calculate the present value of future cash flows
Basic 13) In 1895, the first U.S. Open Golf Championship was held. The winner's prize $ was $150. In 2014, the winner's check was $1,620,000. What was the % increase per year in the winner's check over this period? If the winner's prize increases @ the same rate, what will it bein 2040?
-to find the % increase per year, we use: FV=PV(1+r)^t ==>r=[(FV/PV)^(1/t)]-1 r=[(1,620,000/150)^(1/119)]-1 =.0812 or 8.12% -to find the FV of the first prize in 2040, we use: FV=PV(1+r)^t FV=1,620,000(1.0812)^26 =$12,324,441.95
Basic 12) Your coin collection contains fifty 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2063, assuming they appreciate @ an annual rate of 4.3%?
-to find the FV of a lump sum, we use: FV=PV(1+r)^t FV=50(1.043)^111 =$5,352.15
Basic 2) For each of the following, compute the future value:
-to find the FV of a lump sum, we use: FV=PV(1+r)^t $7,575,83 $12,310.02 $397,547.04 $306,098.52
Basic 3) For each of the following, compute the present value:
-to find the PV of a lump sum, we use: PV=FV/[(1+r)^t] $5,039.79 $39,332.59 $1,730.78 $3.37
Basic 11) You have just received notification that you have won the $2 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you're around to collect) 80 years from now. What is the present value of your windfall if the appropriate discount rate is 7.5%?
-to find the PV of a lump sum, we use: PV=FV/[(1+r)^t] PV=2,000,000/(1.075)^80 =$6,142.61
Intermediate 17) Suppose you are still committed to owning a $225,000 Ferrari (see problem 9). If you believe your mutual fund can achieve a 12% annual rate of return & you want to buy the car in 9 years on the day you turn 30, how much must you invest today?
-to find the PV of a lump sum, we use: PV=FV/[(1+r)^t] PV=225,000/(1.12^9) =$81,13726
Basic 10) Imprudential, Inc., has an unfunded pension liability of $475 million that must be paid in 20 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 6.1%, what is the present value of this liability?
-to find the PV of a lump sum, we use: PV=FV/[(1+r)^t] PV=475,000,000/(1.061)^20 =$145,340,259.61
Basic 14) Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, Sotheby's sold the Edgar Degas bronze sculpture @ auction for a price of $10,311,500. Unfortunately for the previous owner, he had purchased it in 1999 @ a price of $12,377,500. What was his annual rate of return on this sculpture?
-trying to find "r" here -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(10,311,500/12,377,500)^(1/4)]-1 =-4.46% ***negative value b/c FV is less than PV
Basic 8) According to the Census Bureau, in January 2013, the average house price in the U.S. was $306,900. In January 2000, the average price was $200,300. What was the annual increase in selling price?
-trying to find "r" here -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(306,900/200,300)^(1/13)]-1 r=.0334 or 3.34%
Basic 6) Assume the total cost of a college education will be $320,000 when your child enters college in 18 years. You presently have $67,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education?
-trying to find "r" here -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(320,000/67,000)^(1/18)]-1 r=.0908 or 9.08%
Intermediate 15) The "Brasher doubloon," which was featured in the plot of the Raymond Chandler novel, was sold @ auction in 2014 for $4,582,500. The coin had a face value of $15 when it was first issued in 1787 & had been previously sold for $430,000 in 1979. At what annual rate did the coin appreciate from its minting to the 1979 sale? What annual rate did the 1979 buyer earn on his purchase? At what annual rate did the coin appreciate from its minting to the 2014 sale?
-trying to find "r" here -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(430,000/15)^(1/192)]-1 =5.49% _____________________________________________ -trying to find "r" here from FIRST sale to SECOND sale -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(4,582,500/430,000)^(1.35)]-1 =6.99% _____________________________________________ -trying to find "r" here from MINTING to SECOND sale -to find r, we use: FV=PV(1+r)^t -from here, we solve for r by rearranging to get: r=[(FV/PV)^(1/t)]-1 r=[(4,582,500/15)^(1/227)]-1 =5.72%
Basic 7) At 7.3% interest, how long do it take to double your $? To quadruple it?
-trying to find "t" here -to find t, we use: FV=PV(1+r)^t -from here, we solve for t by rearranging to get: t=ln(FV/PV)/ln(1+r) -to DOUBLE: FV=$2=$1(1.073)^t t=ln2/ln1.073=9.84 years -to QUADRUPLE: FV=$4=$1(1.073)^t t=ln4/ln1.073=19.68 years ***takes twice as long to quadruple $; BIG concept of time value of $
Basic 9) You're trying to save to buy a new $225,000 Ferrari. You have $45,000 today that can be invested @ your bank. The bank pays 4.8% annual interest on its accounts. How long will it be before you have enough to buy the car?
-trying to find "t" here -to find t, we use: FV=PV(1+r)^t -from here, we solve for t by rearranging to get: t=ln(FV/PV)/ln(1+r) t=ln(225,000/45,000)/ln(1.048)= 34.33 years
Intermediate 20) You expect to receive $15,000 @ graduation in two years. You plan on investing it @ 9% until you have $75,000. How long will you wait from now?
-trying to find "t" here -to find t, we use: FV=PV(1+r)^t -from here, we solve for t by rearranging to get: t=ln(FV/PV)/ln(1+r) t=ln(75,000/15,000)/ln(1.09) =18.68; ***HOWEVER: will NOT receive $ for another 2 years==>18.68+2 years =20.68 years
Intermediate 18) You have just made your first $5,000 contribution to your retirement account. Assuming you earn a return of 10% per year & make no additional contributions, what will your account be worth when you retire in 45 years? What if you wait 10 years before contributing? (Does this suggest an investment strategy?)
-trying to find the FV of a lump sum, we use: FV=PV(1+r)^t FV (if right away)=5,000(1.10)^45 =$364,452.42 FV (if 10 years later)=5,000(1.10)^35 =$140,512.18
Intermediate 19) You are scheduled to receive $20,000 in two years. When you receive it, you will invest it for six more @ 7.3% per year. How much will you have in eight year?
-trying to find the FV of a lump sum; ***remember: the $ will only be invested for 6 years, so the # of period is 6 FV=PV(1+r)^t FV=20,000(1.073)^6 =$30,523.08
CRCTQ 8) Would be willing to pay $24,099 today in exchange for $100,000 in 30 years? What would be the key considerations in answering yes or no? Would your answer depend on who is making the promise to repay?
1. Is the rate of return, r, attractive in this offer in comparison to other similarly risky investments? 2. How risky is the investment? *In general, it DOES matter who is promising to repay us
CRCTQ: 1) The basic present value equation has four parts. What are they?
1. present value (PV) 2. future value (FV) 3. discount rate (r) 4. life of investment (t)
future value equation
PV*(1+r)^(t) -PV=$ invested today a.k.a. "present value" -r=annual interest rate -time
present value equation
PV*1/[(1+r)^(t)] -PV=$ invested today a.k.a. "present value" -r=annual interest rate -time
Intermediate 16) Refer back to the Series EE savings bonds we discussed @ the very beginning of the chapter. a) Assuming you purchased a $50 face value bond, what is the exact rate of return you would earn if you held the bond for 20 years until it doubled in value? b) If you purchased a $50 face value bond in early 2014 @ the then current interest rate of .10% per year, how much would the bond be worth in 2024? c) In 2024, instead of cashing the bond in for its then current value, you decide to hold the bond until it doubles in face value in 2034. What rate of return will you earn over the last 10 years?
a) -trying to solve for r r=[(FV/PV)^(1/t)]-1 r=[(100/50)^(1/20)]-1 =3.53% b) -trying to solve for FV FV=PV(1+r)^t FV=50(1+.001)^10 =$50.50 c) -trying to solve for r r=[(FV/PV)^(1/t)]-1 r=[(100/50.50)^(1/10)]-1 =7.07%
