Bank: The bank's deposit pays 8 % per year with annual compounding. Bond: The price of the bond is $75. You will receive $100 five years later.

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1 ü 4.4 lternative Discounted Cash Flow Decision Rules ü Three Decision Rules (1) Net Present Value (2) Future Value (3) Internal Rate of Return, IRR ü (3) Internal Rate of Return, IRR Internal Rate of Return is a discount rate that makes PV of future cash inflows equal to PV of cash outflows. In other words, it is the interest rate at which the NPV is equal to zero. Suppose that you are considering to start a given project. You need initial investment for this project. You calculate IRR. If IRR of the project is greater than the opportunity cost of capital, then you should invest in a project. If the IRR is smaller than the opportunity cost of capital, you shouldn't invest. ü Example 1 : Choosing between bank deposit and bond Suppose that you have money in a bank account. You are considering something alternative; buying a bond. Bank: The bank's deposit pays 8 % per year with annual compounding. Bond: The price of the bond is $75. You will receive $100 five years later. Which is better? We are going to apply three decision rules; NPV, FV, IRR. To buy the bond, you give up bank deposit. So when we evaluate NPV of buying bond, the bank's interest rate becomes opportunity cost of capital. ü comparing FV FV value of money in your bank account is given by We compare this with FV of bond. N FV value of money in your bank account is greater than bond's FV. You had better keep your money at the bank account. ü comparing IRR of bond What is the definition of Internal Rate of Return? IRR is obtained by solving the following equation; x 5 Here, variable x is discount rate which makes PV equal to initial investment, i.e. $75. In other words, IRR is interest rate which makes NPV equal to zero. Clear x, solution, ans, IRR solution NSolve x 5 75, x x , x , x , x , x The answer is x = Compare IRR with opportunity cost of capital. IRR is smaller than interest rate of bank account. You had better not buy the bond. We have an equation of higher degree. The equation has a term of the 5th power. So the equation has 5 answers. The fifth answer is meaningful as interest rate. You can solve equation by NSolve[... ]. More specifically, NSolve[ expression = = expression, { unknown variable}] We can take out fifth answer in the following way: Ch04v2.nb 1

2 solution is name for the list of five answers of the equation. By solution[[ 5 ]], we specify the fifth value in the list. ans solution 5 x IRR x. ans Print "IRR ", IRR IRR Internal Rate of Return is equal to 5.922%. Compare IRR with opportunity cost of capital. IRR is smaller than interest rate of bank account. You had better not buy the bond. In order to take value of x from inside of {x Ø }, we need to go through inputting the following; IRR = x /. ans The meaning of the above expression is as follows; name of variable you like to assign = symbol of unknown /. name of variable which represents { } part. We need /. slush and period. ü Example 2 : Borrowing money to buy a car You need to borrow $5,000 to buy a car. Two types of loan are available to you; an example in p.2. type 1. bank offers you a four year loan with interest rate of % per year with annual compounding. You pay principal and interest all at once four years later. type 2. Mr. asks you to pay him back $9,000 in four years. We evaluate the second loan. What are NPV and IRR? The opportunity cost of capital is bank's interest rate. Inflow and outflow of Mr.'s loan are as follows. Cash inflow is $5,000 today. Cash outflow is $9,000 four years later. ü Net Present Value NPV of Mr.' s loan is negative as shown below. Clear NPV NPV N ü Internal Rate of Return IRR is value of x which satisfies the following equation; x 4. Clear x NSolve x 4 0, x x , x , x , x IRR is %. It is higher than the bank loan. Both decision rules conclude that you are better off borrowing from the bank Ch04v2.nb 2

3 ü 4.5 Multiple Cash Flows Suppose that an investment plan has multiple cash inflows as follows. Cash inflows are $1,000 a year from now and $2,000 two years from now. Cash outflow is $2,500 now. We describe multiple cash flows using a diagram called "time line." What value of NPV does this investment plan have? If you deposit money with a bank, interest rate is 10 % per year. So the opportunity cost of capital is 10%. NPV is obtained by calculating the following. Clear NPV NPV ; Print "NPV ", NPV NPV NPV is positive. You should undertake this investment. ü 4.6 nnuities ü PV and FV of nnuities Cash flows of the same amount is called "annuity". Paying 10,000 yen per month for gift certificate of the department store constitutes an annuity. If cash flow starts immediately, it is called an "immediate annuity." If the cash flow starts at the end of the current period, rather than immediately, it is call an "ordinary annuity." Let r be interest rate per month. Suppose you keep paying amount $ each month for a year. Starting today, you pay twelve times. pply monthly compounding. This is an immediate annuity. FV is given as a sum of geometric sequence; FV 1 r t, where 1 r t 1 r 1 r r Next suppose that you will receive amount $ per month times for a year. The first payment is a month from now. What is PV of this ordinary annuity? pply monthly compounding. PV of the annuity is expressed as a sum of geometric sequence. is the first term and 1 is the common ratio. 1 r 1+r PV of the first cash flow = 1 r PV of the second cash flow = ª PV of the last cash flow = 1 r 2 1 r Then PV of this annuity is given by. 1 r t Example Suppose that you will receive amount $100 per month times for a year, starting the next month. Interest rate per month is 1 percent. Then PV of this annuity is $1,5.51 as shown below. In terms of PV, having $1,5.51 today is equivalent to receiving $100 each month for a year Ch04v2.nb 3

4 Clear, r ; 100; r 0.01; Print " 1 r t ", 1 r t r t Mathematica Print[ expression you want to show] For example command if you input Print[ variable name ] then you will see value of variable name. Print[ "expression as you see ", expression to be calculated] You have comma here. You use comma to differentiate the end of one expression from the next one. ü 4.8 Loan mortization Many loans are repaid in equal periodic installments. Part of each payment is interest on the outstanding balance of the loan. nd the remaining part is repayment of principal. fter each payment, the outstanding balance is reduced by the amount of principal repaid. The process of paying off a loan s principal gradually over its term is called loan amortization. Example $100,000 home mortgage loan. nnual percentage rate 9%. To be paid in three equal annual installments. Let c be constant annual payment. This must satisfy the following equation c c c Clear c ; NSolve c c c , c c You must pay $39, per year. t the first payment, Out of this amount, $9,000 is interest; $100, The remaining 30,505.5 is for repayment of principal. The remaining outstanding balance is 100,000-30,505.5 = $69, Interest is calculated on this amount during the next year. 69, = 6, is the next interest payment. The remaining 39, , = 33,251 is for repayment of the principal. The similar calculation is repeated for the third payment ü Example: Buying a car by a one year loan of $1,000 at an PR of % per year to be repaid in equal monthly payments. What is the monthly payment? Clear c ; NSolve 1000 c c, c 1 0. t Mathematica and Mathematics Can we find constant monthly payment in another way, not using NSolve[ ]. Yes, we can. Try to find PV of 1 annuity of $1. Value of annuity of payment of $1 is given by. The constant payment c is equal 1 0. t to 1000 value of annuity Ch04v2.nb 4

5 Clear vannuity vannuity t ; Print "value of $1 annuity of payments of $1 $", vannuity 1000 vannuity value of $1 annuity of payments of $1 $ ü Homework No. 2, Due May 2nd Q1. Consider an example given in Change 80 years to 85 and change $10,000 to $18,000. Calculate NPV, IRR and the number of year long enough to be break even. Hint: You don t have to use financial calculator to solve this question. Q2. p.145, Problem 25 Q3 p.145, Problem 28 Q4. p.146, Problem 36. Hint: interest rate per month = PR. Consider initially there were 13 loans. Sammy paid back of them in a year. Q5. p.146, Problem Ch04v2.nb 5

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