Chapter 3: the Time Value of Money
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Chapter 3: The Time Value of Money
Study objectives The interest rate Simple interest Compound interest Compounding more than once a year Amortizing a loan 1.Interest Interest: money paid (earned) for the use of money How does interest come into being? Aversion of future uncertainty Opportunity to invest and make a profit Why shall we know interest in studying FM? In FM, a lot of decisions can be made only according to computations using interest (interest can cause cash flow, and this affect the value of a firm) 2.Interest rate Interest rate is The rate between interest paid (earned) and principal Interest rate is a very important index, both in macro-economy and FM, because both state and company want to make decision by it. 3.Simple Interest Simple interest is interest that is paid on only the original amount borrowed (lent) SI=P(i)(n) The assumption under simple interest: interest earned cannot be invested immediately (1)Future value 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 Why does it appear in FM? Under simple interest, future value is computed as: FV=PV+PV(i)(n) Example Miss Lin wants to buy a small house in three years time, so she leads a thrifty life and saves all her spare money in deposit. As her salary is 1000 a month, she tries to save 7000 yuan each year and the interest rate for this kind of deposit is 2% a year. How much will she get after three years? FV= 7000×(1+2%×3)+ 7000×(1+2%×2)+ 7000×(1+2%×1)= 7000×(3+12%) =21840 yuan Present Value The present value is the current value of a future amount of money, or a series of payments, evaluated at a given interest Under simple interest, the present value is computed as: PV=FV/(1+(i)(n)) example Miss Lin has a friend who works in an insurance company. One day Miss Lin asked her friend if she wants to buy a kind of life insurance, how much she should pay and how much she will get after forty years. Her friend told her that if she pays a premium of 50000 yuan now, she will get 100000 yuan after 40 years. Now the interest rate for one- year deposit is 2%, shall Miss Lin buy the insurance? 100000/(1+2%×40)=55555 Miss Lin should buy the insurance Does the insurance company suffer losses in sell such insurances? Investment opportunity Why shall we know the computation of future value and present value? We make many decisions according to future value or present value in FM For example, if you want to make an investment decision, how can you determine whether the investment project is worth carrying out? By comparing the present value of its future cash inflow with its current cash outflow 4.Compound Interest Compound interest is interest paid (earned) on any previous interest paid (earned), as well as on the principal borrowed We will use compound interest to compute future value and present value in later study, because it is more realistic
FV=PV(1+i)n
PV=FV/(1+i)n
Example Under compound interest, should Miss Lin buy the insurance?
PV=100000/(1+2%)40=100000/2.208=45289yuan
Miss Lin should not buy the insurance The Value of a Firm Assume the firm is a machine to make money n CF V i i i1 (1 r)
5.Some Issues of Compound Interest Discount Rate: interest rate used to convert future values to present values We can use this rate to assess the feasibility of a project Use the learned knowledge, we can solve: unknown interest rate and unknown number of compounding Example How do you determine the future value (present value) of an investment over a time span that contains a fractional period? (Page 45) Unknown interest: (Page 45), Growth rate Unknown number of compounding periods: (page 46) Growth rate It is frequently useful to estimate growth rates from financial data. Suppose Emery Brothers had sales of $15 million in 2002 and estimate that by 2006 its sales will be $35 million. What compound annual rate of growth is Emery predicting? (PV?%,4)×35000=15000 PV?%,4=0.429, ?%=23.6% 6.Annuities An annuity is a series of equal payments or receipts occurring over a specified number of periods. Example of annuity such as the Nobel Prize In FM, annuities often appear as pension funds and cash inflow of a project (1)Ordinary annuity In an ordinary annuity, payments or receipts occur at the end of each period
FVAn=R[(1+i)n-1]/i, FVIFAi,n= [(1+i)n-1]/i
PVAn=R[(1+i)n-1]/[i (1+i)n], PVIFAi,n= [(1+i)n-1]/ [i (1+i)n]
Example Linda Jackson recently took out an $80000 loan at 12% when she purchased a new house. The loan will be pay back in 30 equal annual installments. However, the lender charged fees and closing costs that amounted to $5552. Consequently, Linda received only $74448. What is the yearly percentage cost of the loan? The size of the loan is based on the $80000, so: PV0=R(PVIFA12%,30), R=$80000/8.055=$9931.72 Since Linda received only $74448 net, her yearly cost is PV0=R(PVIFA?%,30) PVIFA?%,30=$74448/$9931.72=7.496, ?=13% Perpetuity A perpetuity is an ordinary annuity whose payments or receipts continue forever FVA∞= ∞ PVA ∞=R/i Case: the value of a fixed dividend company Annuity Due In contrast to an ordinary annuity where payments or receipts occur at the end of each period, an annuity due calls for a series of equal payments occurring at the beginning of each period. FVADn=R(FVIFAi,n)(1+i) PVADn=R(PVIFAi,n-1+1) Case of annuity due in FM such as: prepaid rent Example Mr. Brown, a shrewd businessman of tractor, is thinking about the advice given by one of his customers. The man want to rent Mr. Brown’s 10 tractors for 10 years. During these 10 years, he will pay Mr. Brown $2000 at the beginning of each year. Mr. Brown sells his tractor at a price of $1500, and he knows that after 10 years of use his tractor was almost worthless. If Mr. Brown wants a investment return at 5% per year, should he accept the advice? PV of 10 years rent revenue: $2000× (PVIFA5%,10-1+1)=2000×8.108=$16216>$15000 Mr. Brown should accept this advice 7.Mixed flows Mixed flows are cash flows that are not a single amount, neither a series of equal payments or receipts, but a series of unequal payments or receipts The mixed flows can only be computed year by year Example In January, Mr. Jackson wants to buy a new machine to replace the old one to improve the quality of his products. Before purchasing the machine, Mr. Jackson makes estimation for the investment. The purchasing price for the machine is $2000, and at the end of the third year, the machine needs a thorough repair, the expense for this repair will be $1000. What is more, the machine needs good maintenance, the expense for this kind of machine is $500 a year. The machine has a 6-year useful life. Assume that all the expenditures on this machine happen at the end of each year and the capital cost of Mr. Jackson’s company is 5% per year. What is the present value of expenditure on this machine? PVm=2000+1000×PV5%,3+PVIFA5%,5 × 500 =2000+2164.5+864=5028.5 8.Compounding more than once a year Up to now, we have assumed that interest is paid annually. But in fact, there are situations in which interest is paid semiannually, quarterly or monthly. So the real interest rate is different from nominal rate Nominal interest rate is a rate of interest quoted from a year that has not been adjusted for the frequency of compounding If interest is compounded more than once a year, the effective interest rate will be higher than the nominal rate Example Ellesmere Corporation issues 1 million $1 par value bonds. The stated interest rate is 6% per year and the interest is paid twice a year. What is the real interest rate of the bond?
FN=PV(1+i/m)mn, ir=(1+i/m)m-1
PV=FN/(1+i/m)mn,
The real interest rate of Ellesmere Corporation is ir=(1+i/m)m-1=6.09%
9.Amortization a loan An important use of present value concept is in determining the payments required for an installment-type loan. This is called amortizing a loan Example Suppose you borrow $22000 at 12% compounded annual interest to be repaid over the next six years. Equal installment payments are required at the end of each year. In addition, these payments must be sufficient in amount to repay the $22000 together with providing the lender with a 12% return To determine the annual payment R, we set up the problem as follows: $22000=R(PVIFA12%,6)=R(4.111) R=$5351 End of year Installment Annual interest Principal Principal payment payment amounting 0 $22000 1 $5351 $2640 $2711 19289 2 $5351 2315 3036 16253 3 $5351 1951 3400 12853 4 $5351 1542 3809 9044 5 $5351 1085 4266 4788 6 $5351 573 4788 total $32106 $10106 $22000