FIN 300 Lecture Notes - Lecture 5: Effective Interest Rate, Real Interest Rate, Annuity
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Assume that you are nearing graduation and that you have appliedfor a job with a local bank. As part of the bankâs evaluationprocess, you have been asked to take an examination that coversseveral financial analysis techniques. The first section of thetest addresses time value of money analysis. See how you would doby answering the following questions:
a. Drawcash flow time lines for (1) a $100 lump-sum cash flow at the endof Year 2, (2) an ordinary annuity of $100 per year for threeyears, (3) an uneven cash flow stream of $50, $100, $75, and $50 atthe end of Years 0 through 3.
b. (1)What is the future value of an initial $100 after three years if itis invested in an account paying 10%annual interest?
(2) What is the present value of $100 to bereceived in three years if the appropriate interest rate is10%?
c. Wesometimes need to find how long it will take a sum of money (oranything else) to grow to some specified amount. For example, if acompanyâs sales are growing at a rate of 20%per year, approximatelyhow long will it take sales to triple?
d. Whatis the difference between an ordinary annuity and an annuity due?What type of annuity is shown in the following cash flow time line?How would you change it to the other type of annuity?
0 1 2 3
100 100 100
e. (1)What is the future value of a 3-year ordinary annuity of $100 ifthe appropriate interest rate is 10%?
(2) What is the present value of the annuity?
(3) What would the future and present values be ifthe annuity were an annuity due?
f. What is the present value of the following uneven cash flow stream?The appropriate interest rate is 10%, compounded annually.
0 1 2 3 4
100 300 300 150
g. Whatannual interest rate will cause $100 to grow to $125.97 in 3years?
h. (1)Will the future value be larger or smaller if we compound aninitial amount more often than annuallyâfor example, every 6months, or semiannuallyâholding the stated interest rateconstant? Why?
(2) Define the stated, or quoted, or simple, rate,(rSIMPLE), annual percentage rate (APR), the periodicrate (rPER), and the effective annual rate(rEAR).
(3) What is the effective annual rate for a simplerate of 10%, compounded semiannually? Compounded quarterly?Compounded daily?
(4) What is the future value of $100 after threeyears under 10%semiannual compounding? Quarterly compounding?
i. Will the effective annual rate ever be equal to the simple (quoted)rate? Explain.
j. (1) What is the value at the end of Year 3 of thefollowing cash flow stream if the quoted interest rate is 10%,compounded semiannually?
0 1 2 3
100 100 100
(2) What is the PV of the same stream?
(3) Is the stream an annuity?
(4) An important rule is that you should nevershow a simple rate on a time line or use it in calculations unlesswhat condition holds? (Hint: Think of annual compounding,when rSIMPLE = rEAR = rPER.) Whatwould be wrong with your answer to parts (1) and (2) if you usedthe simple rate 10%rather than the periodic raterSIMPLE/2 = 10%/2 = 5%?
k. (1)Construct an amortization schedule for a $1,000 loan that has a10%annual interest rate that is repaid in three equalinstallments.
(2) What is the annual interest expense for theborrower, and the annual interest income for the lender, duringYear 2?
l. Suppose on January 1 you deposit $100 in an account that pays asimple, or quoted, interest rate of 11.33463%, with interest added(compounded) daily. How much will you have in your account onOctober 1, or after 9 months?
m. Now suppose youleave your money in the bank for 21 months. Thus, on January 1 youdeposit $100 in an account that pays a 11.33463%compounded daily.How much will be in your account on October 1 of the followingyear?
n. Suppose someone offered to sell you a note that calls for a $1,000payment 15 months from today. The person offers to sell the notefor $850. You have $850 in a bank time deposit (savings instrument)that pays a 6.76649%simple rate with daily compounding, which is a7%effective annual interest rate; and you plan to leave this moneyin the bank unless you buy the note. The note is not riskyâthat is,you are sure it will be paid on schedule. Should you buy the note?Check the decision in three ways: (1) by comparing your futurevalue if you buy the note versus leaving your money in the bank,(2) by comparing the PV of the note with your current bankinvestment, and (3) by comparing the rEAR on the notewith that of the bank investment.
o. Suppose the note discussed in part n, above, costs $850, but callsfor five quarterly payments of $190 each, with the first paymentdue in 3 months rather than $1,000 at the end of 15 months. Wouldit be a good investment?
Assume that you have $1,000 to invest, so insert 1000 as your Present Value in the following table. Assume that you want to invest your money for 5 years (insert 5 for Number of Periods). Assume an annual interest rate of 3.00% (insert 3 for Interest Rate per Period). The table will determine the Future Value of your investment. If you input the numbers correctly, your Future Value is computed to be $1.159, which is what your investment will be worth in 5 years. Now revise the input to reflect your actual savings and the prevailing interest rate so that you can see how your savings will grow in 5 years. Even if you have no savings now, you can at least see how the interest rate affects the future value of savings by revising your input in the Interest Rate per Period and then observing the change in the Future Value. Future Value of a Present Amount Present Value $1,500 Number of Periods 5 Interest Rate per Period 3.0% FV = PV*(1+R)^N Future Value $1,739 2. Assume that you have $1,000 to invest at the end of each of the next 5 years, so insert 1000 as your Payment per Period in the following table. Assume that you want to invest your money for 5 years (insert 5 for Number of Periods). Assume an annual interest rate of 3.00% (insert 3 for Interest Rate per Period). The following table will determine the Future Value of your investment. If you input the numbers correctly, your Future Value is computed to be $5,309, which is what your investments will be worth in 5 years. Now revise the input to reflect your actual expected savings per year over the next 5 years, and existing interest rate quotations so that you can estimate how your savings will grow in 5 years. You can now revise the table to fit your own desired level of saving. Future Value of an Annuity Payment per Period $1,500 Number of Periods 5 Interest Rate per Period 3.0% FV = FV(R, N, PMT, (PV), beginning=1, end=0) Future Value $7,964 3. Assume that you want to deposit savings that will be worth $10,000 in 5 years, so insert 10000 as the Future Amount and 5 as the Number of Periods in the following table. Assume an annual interest rate of 3.00% (insert 3 for Interest Rate per Period). The following table will determine the Present Value, which represents the amount of savings you need today that would accumulate to be worth $10,000 in 5 years. If you input the numbers correctly, the Present Value is estimated in the table to be $8,606. Now revise the input to reflect your own desired savings amount in 5 years so that you can estimate how much you need now to achieve your savings goal in 5 years. Present Value of a Future Amount Future Amount $20,000 Number of Periods 5 Interest Rate per Period 3.0% PV = FV / (1+R)^N Present Value $17,252 4. Assume that you want to deposit savings at the end of each of the next 5 years so that you will have $10,000 in 5 years. So insert 10000 as the Future Amount and 5 for Number of Periods. Assume an annual interest rate of 3.00% (insert 3 for Interest Rate per Period). The following table will determine the Annual Payment, which represents the annual payments that will accumulate to your future desired investment. If you input the numbers correctly, your Annual Payment is computed to be $1,884. Now revise the input to reflect your own desired savings amount in 5 years so that you can estimate how much you need to save per year to achieve your savings goal in 5 years. Compute Payment Needed to Achieve Future Amount Future Amount $20,000.00 Number of Periods 5.00 Interest Rate per Period 3.00% PMT = FV / [FV(R, N, -1)] Annual Payment $3,767
Decisions 1. Using the above formulas and understanding of the impact of interest rates and time on your savings, report on how much you must save per year and the return you must earn to meet your savings goal for graduation, and your savings goal in your first three years of post-graduation life.
I need a report on how much to save per year and the return to earn to meet savings goal for graduation, and savings goal in the first three years of post graduation. Can you please use the numbers above that are already calculated in the formula. I have had an answer on this below. I don't understand why the periods don't stay the same for 5 years. The annuity is 7964 I took that divided b y 60 = 132.7 per month and multiplied it by 12 for a year and got 1592.4. Is that the savings for the answer to saving for a year. IF not I need help figuring out the calculation for the return to meet after gradutaion and the next three years post graduation.
| Goal 1 | Savings Goal for graduation, FV | $ 20,000 | |||||
| Time till graduation (Number of periods) | 5 | ||||||
| Present value of savings | $ - | ||||||
| Expected interest rates | 3% | ||||||
| Savings needed per year, PMT | $3,767.09 | =PMT(3%,5,0,20000,) | |||||
| Goal 2 | Savings Goal for 1st year of post graduation, FV | $ 15,000 | |||||
| Time till post graduation year 1 (Number of periods) | 6 | ||||||
| Present value of savings | $ - | ||||||
| Expected interest rates | 3% | ||||||
| Savings needed per year, PMT | $2,318.96 | =PMT(3%,6,0,15000,) | |||||
| Goal 3 | Savings Goal for 2nd year of post graduation, FV | $ 15,300 | |||||
| Time till post graduation year 1 (Number of periods) | 7 | ||||||
| Present value of savings | $ - | ||||||
| Expected interest rates | 3% | ||||||
| Savings needed per year, PMT | $1,996.75 | =PMT(3%,7,0,15300,) | |||||
| Goal 4 | Savings Goal for 3rd year of post graduation, FV | $ 15,606 | |||||
| Time till post graduation year 1 (Number of periods) | 8 | ||||||
| Present value of savings | $ - | ||||||
| Expected interest rates | 3% | ||||||
| Savings needed per year, PMT | $1,754.99 | =PMT(3%,8,0,15606,) | |||||