3-2 The Time Value of Money u The Interest Rate u Simple Interest u Compound Interest u Amortizing a Loan u Compounding More Than Once per Year u The Interest Rate u Simple Interest u Compound Interest u Amortizing a Loan u Compounding More Than Once per Year
3-3 $10,000 today Obviously, $10,000 today. TIME VALUE TO MONEY You already recognize that there is TIME VALUE TO MONEY!! The Interest Rate $10,000 today $10,000 in 5 years Which would you prefer -- $10,000 today or $10,000 in 5 years?
3-4 Main Idea about Time Value of Money uMuMoney that the firm or an individual has in its possession today is more valuable than future payments because the money it now can be invested and earn positive returns. A bird in hand worth more than two in the bush
3-5 Why Money has Time Value? The existence of interest rates in the economy results in money with its time value. Sacrificing present ownership requires possibility of having more future ownership. Inflation in economy is one of the major cause. Risk or uncertainty of favorable outcome.
3-6 TIME INTEREST TIME allows you the opportunity to postpone consumption and earn INTEREST. Why TIME? TIME Why is TIME such an important element in your decision?
3-7 Major Importance of Time Valueof Money It is required for accounting accuracy for certain transactions such as loan amortization, lease payments, and bond interest. In order to design systems that optimize the firms cash flows. For better planning about cash collections and disbursements in a way that will enable the firm to get the greatest value from its money.
3-8 Major Importance of Time Value of Money Funding for new programs, products, and projects can be justified financially using time –value-of- money techniques. Investments in new equipment, in inventory, and in production quantities are affected by time-value- of-money techniques.
3-9 Types of Interest u Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). u Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent).
3-10 Simple Interest Formula Formula FormulaSI = P 0 (i)(n) SI:Simple Interest P 0 :Deposit today (t=0) i:Interest Rate per Period n:Number of Time Periods
3-11 $140 u SI = P 0 (i)(n) = $1,000(.07)(2) = $140 Simple Interest Example u Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?
3-12 FV $1,140 FV = P 0 + SI = $1,000 + $140 = $1,140 u Future Value u Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (FV) Future Value FV u What is the Future Value (FV) of the deposit?
3-13 The Present Value is simply the $1,000 you originally deposited. That is the value today! u Present Value u Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV) Present Value PV u What is the Present Value (PV) of the previous problem?
3-14 FV 1 P 0 $1,000 $1,070 FV 1 = P 0 (1+i) 1 = $1,000 (1.07) = $1,070 Compound Interest You 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. Future Value Single Deposit (Formula)
3-15 FV 1 P 0 $1,000 $1,070 FV 1 = P 0 (1+i) 1 = $1,000 (1.07) = $1,070 FV 2 P 0 $1,000 P 0 $1,000 $1,144.90 FV 2 = FV 1 (1+i) 1 = P 0 (1+i)(1+i) = $1,000(1.07)(1.07) = P 0 (1+i) 2 = $1,000(1.07) 2 = $1,144.90 $4.90 You earned an EXTRA $4.90 in Year 2 with compound over simple interest. Future Value Single Deposit (Formula) Future Value Single Deposit (Formula)
3-16 FV 1 FV 1 = P 0 (1+i) 1 FV 2 FV 2 = P 0 (1+i) 2 Future Value General Future Value Formula: FV n FV n = P 0 (1+i) n FV n FVIFSee Table I or FV n = P 0 (FVIF i,n ) -- See Table I General Future Value Formula etc.
3-17 FVIF FVIF i,n is found on Table I at the end of the book. Valuation Using Table I
3-18 FV 2 FVIF $1,145 FV 2 = $1,000 (FVIF 7%,2 ) = $1,000 (1.145) = $1,145 [Due to Rounding] Using Future Value Tables
3-19 $10,000 5 years Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. Story Problem Example 5 0 1 2 3 4 5 $10,000 FV 5 10%
3-20 FV 5 FVIF $16,110 u Calculation based on Table I: FV 5 = $10,000 (FVIF 10%, 5 ) = $10,000 (1.611) = $16,110 [Due to Rounding] Story Problem Solution FV n FV 5 $16,105.10 u Calculation based on general formula: FV n = P 0 (1+i) n FV 5 = $10,000 (1+ 0.10) 5 = $16,105.10
3-21 The result indicates that a $1,000 investment that earns 12% annually will double to $2,000 in 6.12 years. Note: 72/12% = approx. 6 years Solving the time (N) Problem NI/YPVPMTFV Inputs Compute 12 -1,000 0 +2,000 6.12 years
3-22 $1,000 2 years. Assume that you need $1,000 in 2 years. Lets examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 2 0 1 2 $1,000 7% PV 1 PV 0 Present Value Single Deposit (Graphic)
3-24 PV 0 FV 1 PV 0 = FV 1 / (1+i) 1 PV 0 FV 2 PV 0 = FV 2 / (1+i) 2 Present Value General Present Value Formula: PV 0 FV n PV 0 = FV n / (1+i) n PV 0 FV n PVIFSee Table II or PV 0 = FV n (PVIF i,n ) -- See Table II General Present Value Formula etc.
3-25 PVIF PVIF i,n is found on Table II at the end of the book. Valuation Using Table II
3-26 PV 2 $1,000 $1,000 $873 PV 2 = $1,000 (PVIF 7%,2 ) = $1,000 (.873) = $873 [Due to Rounding] Using Present Value Tables
3-27 $10,0005 years Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%. Story Problem Example 5 0 1 2 3 4 5 $10,000 PV 0 10%
3-28 PV 0 FV n PV 0 $10,000 $6,209.21 u Calculation based on general formula: PV 0 = FV n / (1+i) n PV 0 = $10,000 / (1+ 0.10) 5 = $6,209.21 PV 0 $10,000PVIF $10,000 $6,210.00 u Calculation based on Table I: PV 0 = $10,000 (PVIF 10%, 5 ) = $10,000 (.621) = $6,210.00 [Due to Rounding] Story Problem Solution
3-29 Self-Test Problem Practice Q 1: Suppose you will receive $2000 after 10 years and now the interest rate is 8%, calculate the present value of this amount. Practice Q 2: You have $1500 to invest today at 9% interest compounded semi-annually. Find how much you will have accumulated in the account at the end of 6 years.
3-30 Self Test Problems: Solving for interest rate (i) & period (n) Practice Q.3 Suppose you can buy a security at a price of Tk78.35 that will pay you Tk100 after five years. What will be the rate of return, if you purchase the security? Practice Q.4 Suppose you know that a security will provide a 10% return per year, its price is Tk68.30 and you will receive Tk.100 at maturity. How many years does the security take to mature?
3-31 Types of Annuities u Ordinary Annuity u Ordinary Annuity: Payments or receipts occur at the end of each period. u Annuity Due u Annuity Due: Payments or receipts occur at the beginning of each period. u An Annuity u An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
3-32 Examples of Annuities u Student Loan Payments u Car Loan Payments u Insurance Premiums u Mortgage Payments u Retirement Savings
3-33 Parts of an Annuity 0 1 2 3 $100 $100 $100 (Ordinary Annuity) End End of Period 1 End End of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart End End of Period 3
3-34 Parts of an Annuity 0 1 2 3 $100 $100 $100 (Annuity Due) Beginning Beginning of Period 1 Beginning Beginning of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart Beginning Beginning of Period 3
3-35 Hint on Annuity Valuation end beginning The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.
3-36 FVA n FVA 3 $3,215 FVA n = R (FVIFA i%,n ) FVA 3 = $1,000 (FVIFA 7%,3 ) = $1,000 (3.215) = $3,215 Valuation Using Table III
3-37 FVAD n FVAD n = R (FVIFA i%,n )(1+i) FVAD 3 $3,440 FVAD 3 = $1,000 (FVIFA 7%,3 )(1.07) = $1,000 (3.215)(1.07) = $3,440 Valuation Using Table III
3-38 PVA 3 PVA 3 = $1,000/(1.07) 1 + $1,000/(1.07) 2 + $1,000/(1.07) 3 $2,624.32 = $934.58 + $873.44 + $816.30 = $2,624.32 Example of an Ordinary Annuity -- PVA $1,000 $1,000 $1,000 3 0 1 2 3 4 $2,624.32 = PVA 3 7% $934.58 $873.44 $816.30 Cash flows occur at the end of the period
3-39 Hint on Annuity Valuation beginning end The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the future value of an annuity due can be viewed as occurring at the end of the first cash flow period.
3-40 Example for Annuity (cont.) Example-5: (P. 166) Cute Baby Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular ordinary annuity. Find the present value if the annuity consists of cash flows of $700 at the end of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%.
3-41 Perpetuity # A stream of equal payments expected to continue forever. Formula: Payment PMT PV (Perpetuity) = = Interest Rate i The present value of this special type of annuity will be required when we value perpetual bonds and preferred stock.
3-42 1. Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) Steps to Solve Time Value of Money Problems
3-43 General Formula: PV 0 FV n = PV 0 (1 + [i/m]) mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FV n,m : FV at the end of Year n PV 0 PV 0 : PV of the Cash Flow today Frequency of Compounding
3-44 $1,000 Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%. 1,000 1,254.40 Annual FV 2 = 1,000(1+ [.12/1]) (1)(2) = 1,254.40 1,000 1,262.48 Semi FV 2 = 1,000(1+ [.12/2]) (2)(2) = 1,262.48 Impact of Frequency
3-46 Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. (1 + [ i / m ] ) m - 1 Effective Annual Interest Rate
3-47 EAR Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)? EAR 6.14%! EAR= ( 1 + 6% / 4 ) 4 - 1 = 1.0614 - 1 =.0614 or 6.14%! BWs Effective Annual Interest Rate
3-48 Example-7 u Mr. Mahin has Tk.10000 that he can deposit any of three savings accounts for 3-year period. Bank A compounds interest on an annual basis, bank B twice each year, and bank C each quarter. All 3 banks have a stated annual interest rate of 4%. a. What amount would Mr. Mahin have at the end of the third year in each bank? b. What effective annual rate (EAR) would he earn in each of the banks? c. On the basis of your findings in a and b, which bank should Mr. Mahin deal with? Why?
3-49 Example-8 u A municipal savings bond can be converted to $100 at maturity 6 years from purchase. If this state bonds are to be competitive with B.D Government savings bonds, which pay 8% annual interest (compounded annually), at what price must the state sell its bonds? Assume no cash payments on savings bond prior to redemption.
3-50 Solve for interest rate u You can buy a security at a price Tk. 7835, that will pay you Tk. 10,000 after three years. Calculate the interest rate, you will earn on your investment.
3-51 Solve for time (n) u You know that a particular deposit will provide a return of 12% per year. It will cost Tk. 6830 and that you will receive Tk. 10,000 at maturity. How long it will take?
3-52 Class Work u If a firms earnings increase from Tk. 260 per share to Tk. 340 over a 6-year period, what is the rate of growth?
3-53 Class Work u A first talker representative from Prime Bank just comes and offers to you an attractive rate of return on a particular deposit at 9.5% per year. You need to pay Tk. 50,000 now and will receive Tk. 75,000 at the time of maturity. Calculate the maturity time to receive the stated amount.