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1、The Interest RateSimple InterestCompound InterestCompounding More Than Once per YearAmortizing a LoanPresent ValueValue today of a future cash flow.Future ValueAmount to which an investment will grow after earning interestDiscount RateInterest rate used to compute present values of future cash flows
2、.Discount FactorPresent value of a $1 future payment.Obviously, .The reason is that there is !Which would you prefer - or ? allows you the opportunity to postpone consumption and earn .Why is such an important element in your decision? Interest paid (earned) on any previous interest earned, as well
3、as on the principal borrowed (lent). Interest on InterestInterest paid (earned) on only the original amount, or principal, borrowed (lent).SI = P0(i)(n)SI: Simple InterestP0: Deposit today (t=0)i: Interest Rate per Periodn:Number of Time PeriodsSI = P0(i)(n)= $1,000(.07)(2)= Assume that you deposit
4、$1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? = P0 + SI = $1,000 + $140 = is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.What is the () of the depos
5、it?The Present Value is simply the $1,000 you originally deposited. That is the value today!P0 = - SI is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.What is the () of the previous problem?Future Value (U.S. Dollars)Assume that you deposi
6、t at a compound interest rate of 7% for . 0 1 7% = (1+i)1 = (1.07) = Compound InterestYou earned $70 interest on your $1,000 deposit over the first year.This is the same amount of interest you would earn under simple interest. = P0(1+i)1 = P0(1+i)2General Formula: = P0 (1+i)n or = P0 (i,n) i,n is on
7、 Table I at the end of the bookPeriod6%7%8%11.0601.0701.08021.1241.1451.16631.1911.2251.26041.2621.3111.36051.3381.4031.469Julie Miller wants to know how large her deposit of today will become at a compound annual interest rate of 10% for . 0 1 2 3 4 10%Calculation based on Table I: = $10,000 (10%,
8、5)= $10,000 (1.611)= Due to RoundinguCalculation based on general formula: = P0 (1+i)n = $10,000 (1+ 0.10)5= We will use the Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?a) 72 / 12% = 6 Years or b) 72/ 6 years = 12%Assume that you need in Lets examine th
9、e process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 7%PV1General Formula:= (1+i)n=(i,n)General Formula:= / (1+i)n or = (i,n) - i,n is on Table II at the end of the bookPeriod 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .7
10、63 .735 5 .747 .713 .681 Julie Miller wants to know how large of a deposit to make so that the money will grow to in at a discount rate of 10%. 0 1 2 3 4 10%Calculation based on general formula: = / (1+i)n = / (1+ 0.10)5= Calculation based on Table 2: = (10%, 5)= (.621)= Due to RoundingIf you invest
11、 $1,000 today, you will receive $3,000 in exactly 8 years. What is the compound interest rate implicit in this situation?=(i,n) (i,8) = = 3,000/1,000 =3 i= 14.68%How long would it take for an investment of $1,000 to grow to $1,900 if we invested it at a compound annual interest rate of 10 percent? =
12、(i,n) (10%,n) = = 1,900/1,000 =1.9 n = 6.72 yearsWhat is the future value of $1million invested at 10 percent for 25 years?=(i,n)=(10%,25) =$1,000,000 * 10.835 = $10,835,000You need $30,000 in cash to buy a house 4 years from today. You expect to earn 5 percent on your savings. How much do you need
13、to deposit today if this is the only money you save for this purpose? = (i,n) = (5%,4) = $30,000 * 0.823 = $246,900Your firm has been told that it needs $74,300 today to fund a $120,000 expense 6 years from now. What rate of interest was used in the computation?=(i,n) (i,6) = : Payments or receipts
14、occur at the end of each period.: Payments or receipts occur at the beginning of each period. represents a series of equal payments (or receipts) occurring over a specified number of equal distant periods. Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Saving
15、s0 1 2 3 $100 $100 $100 ofPeriod 1 ofPeriod 2Today Cash Flows Each 1 Period Apart ofPeriod 30 1 2 3$100 $100 $100 ofPeriod 1 ofPeriod 2Today Cash Flows Each 1 Period Apart ofPeriod 3It is an ordinary annuity whose payments or receipts continue forever.PVA = R / IThe ABS Co. wants to offer preferred
16、stock for sale at a price of $60 a share. If the company wants their investors to earn at least a 7.5 percent rate of return, what is the minimum annual dividend they will need to pay per share? R = PVA * i = 60 * 7.5% = $4.5= (i,n)= (i,n)= (i,n) (1+i)= (1+i) (i,n)R is the period receiptWhat is the
17、future value of annual payments of $6,500 for eight years at 12 percent?= (i,n)= $6,500 *(12%,8) = $6,500 * 12.30 = $79,9501. Read problem thoroughly2. Create a time line3. Put cash flows and arrows on time line4. Determine if it is a PV or FV problem5. Determine if solution involves a single CF, an
18、nuity stream(s), or mixed flow6. Solve the problem7. Check with financial calculator (optional)Julie Miller will receive the set of cash flows below. What is the at a discount rate of . 0 1 2 3 4 1. Solve a “” by discounting each back to t=0.2. Solve a “” by firstbreaking problem into groups of annu
19、ity streams and any single cash flow groups. Then discount each back to t=0. 0 1 2 3 4 10% 0 1 2 3 4 10%$600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57$100 (PVIF10%,5) = $100 (0.621) = $62.10 0 1 2 3 4 equals 0 1 2 0 1 2 3 4 5Julie Miller will rec
20、eive the set of cash flows below. What is the at a discount rate of 7%.(PIECE AT A TIME AND GROUP AT A TIME) 0 1 2 3 4 44This is the annual rate that is quoted by law that has not been adjusted for frequency of compounding.If interest is compounded more that once a year, the effective interest rate
21、will be higher that the nominal rate.45This is the actual rate paid (or received) after adjusting for compounding that occurs during the yearIf you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.46Future Values w
22、ith Monthly CompoundingSuppose you deposit $50 a month into an account that has an interest rate of 12%, based on monthly compounding. How much will you have in the account in 1 years? = P0 (1+i/m)mn = / (1+i/m)mn = $50 (1+12%/12)12*1 = $50 * 1.127 = $56.3547Sometimes investments or loans are figure
23、d based on continuous compounding = PV0 (e)in 48You ALWAYS need to make sure that the interest rate and the time period match. If you are looking at annual periods, you need an annual rate. If you are looking at monthly periods, you need a monthly rate.491 mi 1 EARmRemember that the i is the quoted
24、ratem is the number of compounding periods per yearIf a savings plan offered a nominal interest rate of 8% compounded quarterly on a one-year investment, what s the effective annual interest rate?EAR=1+ (8%/4)4-1 = 8.243%Pure discount loans A borrower receives money today and repays a single lump su
25、m at some time in the future.Interest-only loans A borrower pays interest each period and repays the entire principle at some point in the future.Amortized loans A lender requires the borrower to repay parts of the loan amount over time.2022-4-23Essentials of Corporate Finance51The process of paying
26、 off a loan by making regular principle reductions is called amortizing the loan.Installment-type Loan It is repaid in equal periodic payments that include both interest and principle. These payments can be made monthly, quarterly, semiannually, or annually. Common examples: mortgage loans, auto loa
27、ns, consumer loans, and certain business loans.1. Calculate the payment per period.2. Determine the interest in Period t. (Loan Balance at t-1) x (i% / m)3. Computein Period t.(Payment - Interest from Step 2)4. Determine ending balance in Period t.(Balance - from Step 3)5. Start again at Step 2 and
28、repeat.Julie Miller is borrowing at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years.Step 1: Payment on Table IV = R (PVIFA i%,n) = R (PVIFA 12%,5) = R (3.605) = / 3.605 = Last Payment Slightly Higher Due to Rounding1.Understand what is meant by the t
29、ime value of money. 2.Understand the relationship between present and future value.3.Describe how the interest rate can be used to adjust the value of cash flows both forward and backward to a single point in time. 4.Calculate both the future and present value of: (a) an amount invested today; (b) a
30、 stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows. 5.Distinguish between an “ordinary annuity” and an “annuity due.” 6.Use interest factor tables and understand how they provide a shortcut to calculating present and future values. 7.Use interest factor tables to find an
31、unknown interest rate or growth rate when the number of time periods and future and present values are known. 8.Build an “amortization schedule” for an installment-style loan.57You have $10,000 to invest for five years.1. How much is the interest on interest when the investment provides 10% annual r
32、eturn?2. How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4% annual return?3. How long will it take your $10,000 to double in value if it earns 5% annually?4. What annual rate has been earned if $1,000 grows into $4,000 in 20 years?Page 67-Q12