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1. TVM Topics Be able to compute the future value of multiple cash flows
Be able to compute the present value of multiple cash flows
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand how interest rates are quoted
Understand how loans are amortized or paid off
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2. Table 6.2
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3. Multiple Cash Flows –Future Value Example 6.1
Find the value at year 3 of each cash flow and add them together
Today (year 0): FV = 7000(1.08)3 = 8,817.98
Year 1: FV = 4,000(1.08)2 = 4,665.60
Year 2: FV = 4,000(1.08) = 4,320
Year 3: value = 4,000
Total value in 3 years = 8, , , ,000 = 21,803.58
Value at year 4 = 21,803.58(1.08) = 23,547.87
The students can read the example in the book. It is also provided here.
You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years?
Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem, finding the future value at the end for each cash flow and then adding. Point out that you can find the value of a set of cash flows at any point in time, all you have to do is get the value of each cash flow at that point in time and then add them together.
I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive, and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.
Calculator:
Today (year 0 CF): 3 N; 8 I/Y; -7,000 PV; CPT FV = 8,817.98
Year 1 CF: 2 N; 8 I/Y; PV; CPT FV = 4,665.60
Year 2 CF: 1 N; 8 I/Y; PV; CPT FV = 4,320
Year 3 CF: value = 4,000
Total value in 3 years = = 21,803.58
Value at year 4: 1 N; 8 I/Y; PV; CPT FV = 23,547.87
I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.
6F-3

4. Multiple Cash Flows – FV Example 2
Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?
FV = 500(1.09) (1.09) = 1,248.05
Calculator:
Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV =
Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV =
Total FV = = 1,248.05
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5. Multiple Cash Flows – Example 2 Continued
How much will you have in 5 years if you make no further deposits?
First way:
FV = 500(1.09) (1.09)4 = 1,616.26
Second way – use value at year 2:
FV = 1,248.05(1.09)3 = 1,616.26
Calculator:
First way:
Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV =
Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV =
Total FV = = 1,616.26
Second way – use value at year 2:
3 N; -1, PV; 9 I/Y; CPT FV = 1,616.26
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6. Multiple Cash Flows – FV Example 3
Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?
FV = 100(1.08) (1.08)2 = =
Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV =
Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV =
Total FV = =
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7. Multiple Cash Flows – Present Value Example 6.3
Find the PV of each cash flows and add them
Year 1 CF: 200 / (1.12)1 =
Year 2 CF: 400 / (1.12)2 =
Year 3 CF: 600 / (1.12)3 =
Year 4 CF: 800 / (1.12)4 =
Total PV = = 1,432.93
The students can read the example in the book.
You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?
Point out that the question could also be phrased as “How much is this investment worth?”
Calculator:
Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV =
Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV =
Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV =
Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV =
Total PV = = 1,432.93
Remember the sign convention. The negative numbers imply that we would have to pay 1, today to receive the cash flows in the future.
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8. Example 6.3 Timeline 1 2 3 4
200
400
600
800
178.57
318.88
427.07
508.41
1,432.93
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9. Multiple Cash Flows – PV Another Example
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?
PV = 1000 / (1.1)1 =
PV = 2000 / (1.1)2 = 1,652.89
PV = 3000 / (1.1)3 = 2,253.94
PV = , , = 4,815.92
Calculator:
N = 1; I/Y = 10; FV = 1000; CPT PV =
N = 2; I/Y = 10; FV = 2000; CPT PV = -1,652.89
N = 3; I/Y = 10; FV = 3000; CPT PV = -2,253.94
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10. Multiple Uneven Cash Flows – Using the BAII
Another way to use the financial calculator for uneven cash flows is to use the cash flow keys
Press CF and enter the cash flows beginning with year 0.
You have to press the “Enter” key for each cash flow
Use the down arrow key to move to the next cash flow
The “F” is the number of times a given cash flow occurs in consecutive periods
Use the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then compute
Clear the cash flow keys by pressing CF and then CLR Work
The next example will be worked using the cash flow keys.
Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells.
Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero.
Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.
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11. Decisions, Decisions
Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?
Use the CF keys to compute the value of the investment
CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1
NPV; I = 15; CPT NPV = 91.49
**No – the broker is charging more than you would be willing to pay.
You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get – You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.
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12. Saving For Retirement
You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%?
Use cash flow keys:
CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5; NPV; I = 12; CPT NPV = 1,084.71
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13. Saving For Retirement Timeline…
… K 25K 25K 25K 25K
Notice that the year 0 cash flow = 0 (CF0 = 0)
The cash flows in years 1 – 39 are 0 (C01 = 0; F01 = 39)
The cash flows in years 40 – 44 are 25,000 (C02 = 25,000; F02 = 5)
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14. Annuities and Perpetuities Defined
Annuity – finite series of equal payments that occur at regular intervals
If the first payment occurs at the end of the period, it is called an ordinary annuity
If the first payment occurs at the beginning of the period, it is called an annuity due
Perpetuity – infinite series of equal payments
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15. Annuities and Perpetuities – Basic Formulas
Perpetuity: PV = C / r
Annuities:
Lecture Tip: The annuity factor approach is a short-cut approach in the process of calculating the present value of multiple cash flows and that it is only applicable to a finite series of level cash flows. Financial calculators have reduced the need for annuity factors, but it may still be useful from a conceptual standpoint to show that the PVIFA is just the sum of the PVIFs across the same time period.
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16. Annuities and the Calculator
You can use the PMT key on the calculator for the equal payment
The sign convention still holds
Ordinary annuity versus annuity due
You can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II Plus
If you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity due
Most problems are ordinary annuities
Other calculators also have a key that allows you to switch between Beg/End.
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17. Annuity – Example 6.5
You borrow money TODAY so you need to compute the present value.
48 N; 1 I/Y; -632 PMT; CPT PV = 23, ($24,000)
Formula:
The students can read the example in the book.
After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.
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18. Annuity – Sweepstakes Example
Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
Calculator:
30 N; 5 I/Y; 333, PMT; CPT PV = 5,124,150.29
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19. Buying a House
You are ready to buy a house, and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000, and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?
It might be good to note that the outstanding balance on the loan at any point in time is simply the present value of the remaining payments.
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20. Buying a House - Continued
Bank loan
Monthly income = 36,000 / 12 = 3,000
Maximum payment = .28(3,000) = 840
PV = 840[1 – 1/ ] / .005 = 140,105
Total Price
Closing costs = .04(140,105) = 5,604
Down payment = 20,000 – 5604 = 14,396
Total Price = 140, ,396 = 154,501
You might point out that you would probably not offer 154,501. The more likely scenario would be 154,500, or less if you assumed negotiations would occur.
Calculator:
30*12 = 360 N
.5 I/Y
840 PMT
CPT PV = 140,105
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21. Finding the Payment
Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4 year loan, what is your monthly payment?
20,000 = C[1 – 1 / ] /
C =
Note if you do not round the monthly rate and actually use 8/12, then the payment will be
Calculator:
4(12) = 48 N; 20,000 PV; I/Y; CPT PMT =
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22. Finding the Number of Payments – Example 6.6
Start with the equation, and remember your logs.
1,000 = 20(1 – 1/1.015t) / .015
.75 = 1 – 1 / 1.015t
1 / 1.015t = .25
1 / .25 = 1.015t
t = ln(1/.25) / ln(1.015) = months = 7.76 years
And this is only if you don’t charge anything more on the card!
You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000.
This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students don’t understand how it works, and it is rarely discussed. This is something that students can take away from the class, even if they aren’t finance majors.
Calculator:
1.5 I/Y
1,000 PV
-20 PMT
CPT N = MONTHS = 7.75 years
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23. Finding the Number of Payments – Another Example
Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $ How long before you pay off the loan?
2,000 = (1 – 1/1.05t) / .05
= 1 – 1/1.05t
1/1.05t =
= 1.05t
t = ln( ) / ln(1.05) = 3 years
Sign convention matters!!!
5 I/Y
2,000 PV
PMT
CPT N = 3 years
6F-23

24. Finding the Rate
Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate?
Sign convention matters!!!
60 N
10,000 PV
PMT
CPT I/Y = .75%
The next slide talks about how to do this without a financial calculator.
6F-24

25. Annuity – Finding the Rate Without a Financial Calculator
Trial and Error Process
Choose an interest rate and compute the PV of the payments based on this rate
Compare the computed PV with the actual loan amount
If the computed PV > loan amount, then the interest rate is too low
If the computed PV < loan amount, then the interest rate is too high
Adjust the rate and repeat the process until the computed PV and the loan amount are equal
6F-25

26. Future Values for Annuities
Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?
FV = 2,000( – 1)/.075 = 454,513.04
Remember the sign convention!!!
40 N
7.5 I/Y
-2,000 PMT
CPT FV = 454,513.04
Lecture Tip: It should be emphasized that annuity factor tables (and the annuity factors in the formulas) assumes that the first payment occurs one period from the present, with the final payment at the end of the annuity’s life. If the first payment occurs at the beginning of the period, then FV’s have one additional period for compounding and PV’s have one less period to be discounted. Consequently, you can multiply both the future value and the present value by (1 + r) to account for the change in timing.
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27. Annuity Due
You are saving for a new house, and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?
FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12
Note that the procedure for changing the calculator to an annuity due is similar on other calculators.
Calculator
2nd BGN 2nd Set (you should see BGN in the display)
3 N
-10,000 PMT
8 I/Y
CPT FV = 35,061.12
2nd BGN 2nd Set (be sure to change it back to an ordinary annuity)
What if it were an ordinary annuity? FV = 32,464 (so you receive an additional by starting to save today.)
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28. Annuity Due Timeline
32,464
If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
35,016.12
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29. Perpetuity – Example 6.7 Perpetuity formula: PV = C / r
Current required return:
40 = 1 / r
r = .025 or 2.5% per quarter
Dividend for new preferred:
100 = C / .025
C = 2.50 per quarter
This is a good preview to the valuation issues discussed in future chapters. The price of an investment is just the present value of expected future cash flows.
Example statement:
Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?
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30. Growing Annuity A growing stream of cash flows with a fixed maturity

31. Growing Annuity: Example
A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?
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32. Growing Perpetuity A growing stream of cash flows that lasts forever
Lecture Tip: To prepare students for the chapter on stock valuation, it may be helpful to include a discussion of equity as a growing perpetuity.
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33. Growing Perpetuity: Example
The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of this promised dividend stream?
It is important to note to students that in this example the year 1 cash flow was given. If the current dividend were $1.30, then we would need to multiply it by one plus the growth rate to estimate the year 1 cash flow.
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34. Effective Annual Rate (EAR)
This is the actual rate paid (or received) after accounting for compounding that occurs during the year
If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.
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35. Annual Percentage Rate
This is the annual rate that is quoted by law
By definition APR = period rate times the number of periods per year
Consequently, to get the period rate we rearrange the APR equation:
Period rate = APR / number of periods per year
You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate
6F-35

36. Computing APRs What is the APR if the monthly rate is .5%?
.5(12) = 6%
What is the APR if the semiannual rate is .5%?
.5(2) = 1%
What is the monthly rate if the APR is 12% with monthly compounding?
12 / 12 = 1%
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37. Things to Remember
You 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.
If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly
6F-37

38. Computing EARs - Example
Suppose you can earn 1% per month on $1 invested today.
What is the APR? 1(12) = 12%
How much are you effectively earning?
FV = 1(1.01)12 =
Rate = ( – 1) / 1 = = 12.68%
Suppose if you put it in another account, you earn 3% per quarter.
What is the APR? 3(4) = 12%
FV = 1(1.03)4 =
Rate = ( – 1) / 1 = = 12.55%
Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.
6F-38

39. EAR - Formula Remember that the APR is the quoted rate, and
m is the number of compounding periods per year
Using the calculator:
The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.
2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.
6F-39

40. Decisions, Decisions II
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?
First account:
EAR = ( /365)365 – 1 = 5.39%
Second account:
EAR = ( /2)2 – 1 = 5.37%
Which account should you choose and why?
Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.
Calculator:
2nd I conv 5.25 NOM Enter up arrow 365 C/Y Enter up arrow CPT EFF = 5.39%
5.3 NOM Enter up arrow 2 C/Y Enter up arrow CPT EFF = 5.37%
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41. Decisions, Decisions II Continued
Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?
First Account:
Daily rate = / 365 =
FV = 100( )365 =
Second Account:
Semiannual rate = / 2 = .0265
FV = 100(1.0265)2 =
You have more money in the first account.
It is important to point out that the daily rate is NOT .014, it is
First Account:
365 N; 5.25 / 365 = I/Y; 100 PV; CPT FV =
Second Account:
2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV =
Lecture Tip: Here is a way to drive the point of this section home. Ask how many students have taken out a car loan. Now ask one of them what annual interest rate s/he is paying on the loan. Students will typically quote the loan in terms of the APR. Point out that, since payments are made monthly, the effective rate is actually more than the rate s/he just quoted, and demonstrate the calculation of the EAR.
6F-41

42. Computing APRs from EARs
If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:
6F-42

43. APR - Example
Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?
On the calculator: 2nd I conv down arrow 12 EFF Enter down arrow 12 C/Y Enter down arrow CPT NOM
6F-43

44. Computing Payments with APRs
Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding. What is your monthly payment?
Monthly rate = .169 / 12 =
Number of months = 2(12) = 24
3,500 = C[1 – (1 / )24] /
C =
2(12) = 24 N; 16.9 / 12 = I/Y; 3,500 PV; CPT PMT =
See the Lecture Tip in the IM for a discussion of situations that present a mismatch between the payment period and the compounding period.
6F-44

45. Future Values with Monthly Compounding
Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?
Monthly rate = .09 / 12 = .0075
Number of months = 35(12) = 420
FV = 50[ – 1] / = 147,089.22
35(12) = 420 N
9 / 12 = .75 I/Y
50 PMT
CPT FV = 147,089.22
6F-45

46. Present Value with Daily Compounding
You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?
Daily rate = .055 / 365 =
Number of days = 3(365) = 1,095
FV = 15,000 / ( )1095 = 12,718.56
3(365) = 1095 N
5.5 / 365 = I/Y
15,000 FV
CPT PV = -12,718.56
6F-46

47. Continuous Compounding
Sometimes investments or loans are figured based on continuous compounding
EAR = eq – 1
The e is a special function on the calculator normally denoted by ex
Example: What is the effective annual rate of 7% compounded continuously?
EAR = e.07 – 1 = or 7.25%
6F-47

48. Pure Discount Loans – Example 6.12
Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.
If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?
PV = 10,000 / 1.07 = 9,345.79
Remind students that the value of an investment is the present value of expected future cash flows.
1 N; 10,000 FV; 7 I/Y; CPT PV = -9,345.79
6F-48

49. Interest-Only Loan - Example
Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.
What would the stream of cash flows be?
Years 1 – 4: Interest payments of .07(10,000) = 700
Year 5: Interest + principal = 10,700
This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.
6F-49

50. Chapter 7 Important bond features and bond types
Bond values and why they fluctuate
Bond ratings and what they mean
Impact of inflation on interest rates
Term structure of interest rates
Determinants of bond yields
Remember, as with any asset, the value of a bond is simply the present value of its future cash flows.
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6.50

51. Bond Definitions Bond Par value (face value) Coupon rate
Coupon payment
Maturity date
Yield or Yield to maturity
Although the coupon is typically paid in cash, examples exist of firms paying investors with product (e.g., Isle of Arran Distillers discussed in the Real World Tip in the IM).
Yield to maturity, required return, and market rate are used interchangeably.
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6.51

52. Table 7.1
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53. Present Value of Cash Flows as Rates Change
Bond Value = PV of coupons + PV of par
Bond Value = PV of annuity + PV of lump sum
As interest rates increase, present values decrease
So, as interest rates increase, bond prices decrease and vice versa
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54. Valuing a Discount Bond with Annual Coupons
Consider a bond with a coupon rate of 10% and annual coupons. The par value is $1,000, and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?
Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 – 1/(1.11)5] / ,000 / (1.11)5
B = =
Using the calculator:
N = 5; I/Y = 11; PMT = 100; FV = 1,000
CPT PV =
Remember the sign convention on the calculator. The easy way to remember it with bonds is we pay the PV (-) so that we can receive the PMT (+) and the FV(+).
Slide 7.8 discusses why this bond sells at less than par
Lecture Tip: You should stress the issue that the coupon rate and the face value are fixed by the bond indenture when the bond is issued (except for floating-rate bonds). Therefore, the expected cash flows don’t change during the life of the bond. However, the bond price will change as interest rates change and as the bond approaches maturity.
Lecture Tip: You may wish to further explore the loss in value of $115 in the example in the book. You should remind the class that when the 8% bond was issued, bonds of similar risk and maturity were yielding 8%. The coupon rate was set so that the bond would sell at par value; therefore, the coupons were set at $80 per year. One year later, the ten-year bond has nine years remaining to maturity. However, bonds of similar risk and nine years to maturity are being issued to yield 10%, so they have coupons of $100 per year. The bond we are looking at only pays $80 per year. Consequently, the old bond will sell for less than $1,000. The mathematical reason for that is discussed in the text. However, many students can intuitively grasp that you wouldn’t be willing to pay as much for a bond that only pays $80 per year for 9 years as you would for a bond that pays $100 per year for 9 years.
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6.54

55. Valuing a Premium Bond with Annual Coupons
Suppose you are reviewing a bond that has a 10% annual coupon and a face value of $1000. There are 20 years to maturity, and the yield to maturity is 8%. What is the price of this bond?
Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 – 1/(1.08)20] / / (1.08)20
B = =
Using the calculator:
N = 20; I/Y = 8; PMT = 100; FV = 1000
CPT PV = -1,196.36
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56. Bond Prices: Relationship Between Coupon and Yield
If YTM = coupon rate, then par value = bond price
If YTM > coupon rate, then par value > bond price
Why? The discount provides yield above coupon rate
Price below par value, called a discount bond
If YTM < coupon rate, then par value < bond price
Why? Higher coupon rate causes value above par
Price above par value, called a premium bond
There are the purely mechanical reasons for these results.
We know that present values decrease as rates increase. Therefore, if we increase our yield above the coupon, the present value (price) must decrease below par. On the other hand, if we decrease our yield below the coupon, the present value (price) must increase above par.
There are also more intuitive ways to explain this relationship. Explain that the yield to maturity is the interest rate on newly issued debt of the same risk and that debt would be issued so that the coupon = yield. Then, suppose that the coupon rate is 8% and the yield is 9%. Ask the students which bond they would be willing to pay more for. Most will say that they would pay more for the new bond. Since it is priced to sell at $1,000, the 8% bond must sell for less than $1,000. The same logic works if the new bond has a yield and coupon less than 8%.
Another way to look at it is that return = “dividend yield” + capital gains yield. The “dividend yield” in this case is just the coupon rate. The capital gains yield has to make up the difference to reach the yield to maturity. Therefore, if the coupon rate is 8% and the YTM is 9%, the capital gains yield must equal approximately 1%. The only way to have a capital gains yield of 1% is if the bond is selling for less than par value. (If price = par, there is no capital gain.) Technically, it is the current yield, not the coupon rate + capital gains yield, but from an intuitive standpoint, this helps some students remember the relationship and current yields and coupon rates are normally reasonably close.
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6.56

57. The Bond Pricing Equation
This formalizes the calculations we have been doing.
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6.57

58. Example 7.1 Find present values based on the payment period
How many coupon payments are there?
What is the semiannual coupon payment?
What is the semiannual yield?
B = 70[1 – 1/(1.08)14] / ,000 / (1.08)14 =
Or PMT = 70; N = 14; I/Y = 8; FV = 1,000; CPT PV =
The students can read the example in the book. The basic information is as follows:
Coupon rate = 14%, semiannual coupons
YTM = 16%
Maturity = 7 years
Par value = $1,000
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6.58

59. Interest Rate Risk Price Risk Reinvestment Rate Risk
Change in price due to changes in interest rates
Long-term bonds have more price risk than short-term bonds
Low coupon rate bonds have more price risk than high coupon rate bonds
Reinvestment Rate Risk
Uncertainty concerning rates at which cash flows can be reinvested
Short-term bonds have more reinvestment rate risk than long-term bonds
High coupon rate bonds have more reinvestment rate risk than low coupon rate bonds
Real-World Tip: Upon learning the concept of interest rate risk, students sometimes conclude that bonds with low interest-rate risk (i.e. high coupon bonds) are necessarily “safer” than otherwise identical bonds with lower coupons. In reality, the contrary may be true: increasing interest rate volatility over the last two decades has greatly increased the importance of interest rate risk in bond valuation. The days when bonds represented a “widows and orphans” investment are long gone.
You may wish to point out that one potentially undesirable feature of high-coupon bonds is the required reinvestment of coupons at the computed yield-to-maturity if one is to actually earn that yield. Those who purchased bonds in the early 1980s (when even high-grade corporate bonds had coupons over 11%) found, to their dismay, that interest payments could not be reinvested at similar rates a few years later without taking greater risk. A good example of the trade-off between interest rate risk and reinvestment risk is the purchase of a zero-coupon bond – one eliminates reinvestment risk but maximizes interest-rate risk.
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6.59

60. Figure 7.2
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6.60

61. Computing Yield to Maturity
Yield to Maturity (YTM) is the rate implied by the current bond price
Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding r with an annuity
If you have a financial calculator, enter N, PV, PMT, and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign)
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62. YTM with Annual Coupons
Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1,000. The current price is $
Will the yield be more or less than 10%?
N = 15; PV = ; FV = 1,000; PMT = 100
CPT I/Y = 11%
The students should be able to recognize that the YTM is more than the coupon since the price is less than par.
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6.62

63. YTM with Semiannual Coupons
Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1,000, 20 years to maturity and is selling for $1,
Is the YTM more or less than 10%?
What is the semiannual coupon payment?
How many periods are there?
N = 40; PV = -1,197.93; PMT = 50; FV = 1,000; CPT I/Y = 4% (Is this the YTM?)
YTM = 4%*2 = 8%
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64. Current Yield vs. Yield to Maturity
Current Yield = annual coupon / price
Yield to maturity = current yield + capital gains yield
Example: 10% coupon bond, with semiannual coupons, face value of 1,000, 20 years to maturity, $1, price
Current yield = 100 / 1, = = 8.35%
Price in one year, assuming no change in YTM = 1,193.68
Capital gain yield = (1, – 1,197.93) / 1, = = -.35%
YTM = = 8%, which is the same YTM computed earlier
This is the same information as the YTM calculation on slide The YTM computed on that slide was 8%
Lecture Tip: You may wish to discuss the components of required returns for bonds in a fashion analogous to the stock return discussion in the next chapter. As with common stocks, the required return on a bond can be decomposed into current income and capital gains components. The yield-to-maturity (YTM) equals the current yield plus the capital gains yield. A further example is provided in the IM.
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6.64

65. Bond Pricing Theorems
Bonds of similar risk (and maturity) will be priced to yield about the same return, regardless of the coupon rate
If you know the price of one bond, you can estimate its YTM and use that to find the price of the second bond
This is a useful concept that can be transferred to valuing assets other than bonds
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66. Differences Between Debt and Equity
Not an ownership interest
Creditors do not have voting rights
Interest is considered a cost of doing business and is tax deductible
Creditors have legal recourse if interest or principal payments are missed
Excess debt can lead to financial distress and bankruptcy
Equity
Ownership interest
Common stockholders vote for the board of directors and other issues
Dividends are not considered a cost of doing business and are not tax deductible
Dividends are not a liability of the firm, and stockholders have no legal recourse if dividends are not paid
An all equity firm can not go bankrupt merely due to debt since it has no debt
The portion of long-term debt that comes due within one year is treated as a current liability.
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6.66

67. The Bond Indenture
Contract between the company and the bondholders that includes
The basic terms of the bonds
The total amount of bonds issued
A description of property used as security, if applicable
Sinking fund provisions
Call provisions
Details of protective covenants
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68. Bond Classifications Registered vs. Bearer Forms Security Seniority
Collateral – secured by financial securities
Mortgage – secured by real property, normally land or buildings
Debentures – unsecured
Notes – unsecured debt with original maturity less than 10 years
Seniority
This is standard terminology in the US – but it may not transfer to other countries. For example, debentures are secured debt in the United Kingdom.
Lecture Tip: Domestically issued bearer bonds will become obsolete in the near future. Since bearer bonds are not registered with the corporation, it is easier for bondholders to receive interest payments without reporting them on their income tax returns. In an attempt to eliminate this potential for tax evasion, all bonds issued in the US after July 1983 must be in registered form. It is still legal to offer bearer bonds in some other nations, however. Some foreign bonds are popular among international investors particularly due to their bearer status.
Lecture Tip: Although the majority of corporate bonds have a $1,000 face value, there are an increasing number of “baby bonds” outstanding, i.e., bonds with face values less than $1,000. The use of the term “baby bond” goes back at least as far as 1970, when it was used in connection with AT&T’s announcement of the intent to issue bonds with low face values. It was also used in describing Merrill Lynch’s 1983 program to issue bonds with $25 face values. More recently, the term has come to mean bonds issued in lieu of interest payments by firms unable to make the payments in cash. Baby bonds issued under these circumstances are also called “PIK” (payment-in-kind) bonds, or “bunny” bonds, because they tend to proliferate in LBO circumstances.
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6.68

69. Bond Characteristics and Required Returns
The coupon rate depends on the risk characteristics of the bond when issued
Which bonds will have the higher coupon, all else equal?
Secured debt versus a debenture
Subordinated debenture versus senior debt
A bond with a sinking fund versus one without
A callable bond versus a non-callable bond
Debenture: secured debt is less risky because the income from the security is used to pay it off first
Subordinated debenture: will be paid after the senior debt
Bond without sinking fund: company has to come up with substantial cash at maturity to retire debt, and this is riskier than systematic retirement of debt through time
Callable – bondholders bear the risk of the bond being called early, usually when rates are lower. They don’t receive all of the expected coupons and they have to reinvest at lower rates.
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6.69

70. Bond Ratings – Investment Quality
High Grade
Moody’s Aaa and S&P AAA – capacity to pay is extremely strong
Moody’s Aa and S&P AA – capacity to pay is very strong
Medium Grade
Moody’s A and S&P A – capacity to pay is strong, but more susceptible to changes in circumstances
Moody’s Baa and S&P BBB – capacity to pay is adequate, adverse conditions will have more impact on the firm’s ability to pay
Lecture Tip: The question sometimes arises as to why a potential issuer would be willing to pay rating agencies tens of thousands of dollars in order to receive a rating, especially given the possibility that the resulting rating could be less favorable than expected. This is a good place to remind students about the pervasive nature of agency costs and point out a real-world example of their effects on firm value. You may also wish to use this issue to discuss some of the consequences of information asymmetries in financial markets.
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6.70

71. Bond Ratings - Speculative
Low Grade
Moody’s Ba and B
S&P BB and B
Considered possible that the capacity to pay will degenerate.
Very Low Grade
Moody’s C (and below) and S&P C (and below)
income bonds with no interest being paid, or
in default with principal and interest in arrears
It is a good exercise to ask students which bonds will have the highest yield to maturity (lowest price) all else equal.
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6.71

72. Government Bonds Treasury Securities Municipal Securities
Federal government debt
T-bills – pure discount bonds with original maturity of one year or less
T-notes – coupon debt with original maturity between one and ten years
T-bonds – coupon debt with original maturity greater than ten years
Municipal Securities
Debt of state and local governments
Varying degrees of default risk, rated similar to corporate debt
Interest received is tax-exempt at the federal level
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73. Example 7.4
A taxable bond has a yield of 8%, and a municipal bond has a yield of 6%
If you are in a 40% tax bracket, which bond do you prefer?
8%(1 - .4) = 4.8%
The after-tax return on the corporate bond is 4.8%, compared to a 6% return on the municipal
At what tax rate would you be indifferent between the two bonds?
8%(1 – T) = 6%
T = 25%
You should be willing to accept a lower stated yield on municipals because you do not have to pay taxes on the interest received. You will want to make sure the students understand why you are willing to accept a lower rate of interest. It may be helpful to take the example and illustrate the indifference point using dollars instead of just percentages. The discount you are willing to accept depends on your tax bracket.
Consider a taxable bond with a yield of 8% and a tax-exempt municipal bond with a yield of 6%
Suppose you own one $1,000 bond in each and both bonds are selling at par. You receive $80 per year from the corporate and $60 per year from the municipal. How much do you have after taxes if you are in the 40% tax bracket? Corporate: 80 – 80(.4) = 48; Municipal = 60
Why should the federal government exempt munis from taxation? It provides an incentive for local governments to raise capital on their own.
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6.73

74. Zero Coupon Bonds
Make no periodic interest payments (coupon rate = 0%)
The entire yield-to-maturity comes from the difference between the purchase price and the par value
Cannot sell for more than par value
Sometimes called zeroes, deep discount bonds, or original issue discount bonds (OIDs)
Treasury Bills and principal-only Treasury strips are good examples of zeroes
Most students are familiar with Series EE savings bonds. Point out that these are actually zero coupon bonds. The investor pays one-half of the face value and must hold the bond for a given number of years before the face value is realized. As with any other zero-coupon bond, reinvestment risk is eliminated, but an additional benefit of EE bonds is that, unlike corporate zeroes, the investor need not pay taxes on the accrued interest until the bond is redeemed. Further, it should be noted that interest on these bonds is exempt from state income taxes. And, savings bonds yields are indexed to Treasury rates.
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6.74

75. Floating-Rate Bonds Coupon rate floats depending on some index value
Examples – adjustable rate mortgages and inflation-linked Treasuries
There is less price risk with floating rate bonds
The coupon floats, so it is less likely to differ substantially from the yield-to-maturity
Coupons may have a “collar” – the rate cannot go above a specified “ceiling” or below a specified “floor”
Lecture Tip: Imagine this scenario: General Motors receives cash from a lender in return for the promise to make periodic interest payments that “float” with the general level of market rates. Sounds like a floating-rate bond, doesn’t it? Well, it is, except that if you replace “General Motors” with “Joe Smith,” you have just described an adjustable-rate mortgage.
Whereas there is less price risk, there is greater reinvestment (or refinancing) risk.
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6.75

76. Other Bond Types Disaster bonds Income bonds Convertible bonds
Put bonds
There are many other types of provisions that can be added to a bond and many bonds have several provisions – it is important to recognize how these provisions affect required returns
It is a useful exercise to ask the students if these bonds will tend to have higher or lower required returns compared to bonds without these specific provisions.
Disaster bonds – issued by property and casualty companies. Pay interest and principal as usual unless claims reach a certain threshold for a single disaster. At that point, bondholders may lose all remaining payments Higher required return
Income bonds – coupon payments depend on level of corporate income. If earnings are not enough to cover the interest payment, it is not owed. Higher required return
Convertible bonds – bonds can be converted into shares of common stock at the bondholders discretion Lower required return
Put bond – bondholder can force the company to buy the bond back prior to maturity Lower required return
A Lecture Tip in the IM discusses other unique bonds such as death puts and Bowie Bonds.
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6.76

77. Bond Markets
Primarily over-the-counter transactions with dealers connected electronically
Extremely large number of bond issues, but generally low daily volume in single issues
Makes getting up-to-date prices difficult, particularly on small company or municipal issues
Treasury securities are an exception
The reported volume of bonds traded is not indicative of total activity due to off exchange transactions.
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6.77

78. Treasury Quotations Highlighted quote in Figure 7.4
8 Nov
What is the coupon rate on the bond?
When does the bond mature?
What is the bid price? What does this mean?
What is the ask price? What does this mean?
How much did the price change from the previous day?
What is the yield based on the ask price?
The difference between the bid and ask prices is called the bid-ask spread and it is how the dealer makes money.
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6.78

79. Clean vs. Dirty Prices Clean price: quoted price
Dirty price: price actually paid = quoted price plus accrued interest
Example: Consider a T-bond with a 4% semiannual yield and a clean price of $1,282.50:
Number of days since last coupon = 61
Number of days in the coupon period = 184
Accrued interest = (61/184)(.04*1000) = $13.26
Dirty price = $1, $13.26 = $1,295.76
So, you would actually pay $ 1, for the bond
Assuming that the November maturity is November 15, then the coupon dates would be November 15 and May 15. Therefore, July 15 would be = 61 days since the last coupon
The number of days in the coupon period would be = 184
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6.79

80. Inflation and Interest Rates
Real rate of interest – change in purchasing power
Nominal rate of interest – quoted rate of interest, change in actual number of dollars
The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation
Be sure to ask the students to define inflation to make sure they understand what it is.
For computations, students need to insure that they are matching real rates with real dollars or nominal rates with nominal dollars.
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6.80

81. The Fisher Effect
The Fisher Effect defines the relationship between real rates, nominal rates, and inflation
(1 + R) = (1 + r)(1 + h), where
R = nominal rate
r = real rate
h = expected inflation rate
Approximation
R = r + h
The approximation works pretty well with “normal” real rates of interest and expected inflation. If the expected inflation rate is high, then there can be a substantial difference.
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6.81

82. Example 7.5
If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate?
R = (1.1)(1.08) – 1 = .188 = 18.8%
Approximation: R = 10% + 8% = 18%
Because the real return and expected inflation are relatively high, there is significant difference between the actual Fisher Effect and the approximation.
Lecture Tip: In late 1997 and early 1998 there was a great deal of talk about the effects of deflation among financial pundits, due in large part to the combined effects of continuing decreases in energy prices, as well as the upheaval in Asian economies and the subsequent devaluation of several currencies. How might this affect observed yields? According to the Fisher Effect, we should observe lower nominal rates and higher real rates and that is roughly what happened. The opposite situation, however, occurred in and around 2008.
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6.82

83. Term Structure of Interest Rates
Term structure is the relationship between time to maturity and yields, all else equal
It is important to recognize that we pull out the effect of default risk, different coupons, etc.
Yield curve – graphical representation of the term structure
Normal – upward-sloping; long-term yields are higher than short-term yields
Inverted – downward-sloping; long-term yields are lower than short-term yields
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84. Figure 7.6 – Upward-Sloping Yield Curve
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85. Figure 7.6 – Downward-Sloping Yield Curve
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86. Figure 7.7 Insert new Figure 7.7 here
www: Click on the web surfer to go to Bloomberg to get the current Treasury yield curve
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6.86

87. Factors Affecting Bond Yields
Default risk premium – remember bond ratings
Taxability premium – remember municipal versus taxable
Liquidity premium – bonds that have more frequent trading will generally have lower required returns
Anything else that affects the risk of the cash flows to the bondholders will affect the required returns
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88. Next Time Turn in Ch 6 Minicase Turn in Ch 7 Minicase
Ch 7: Interest Rates & Bond Valuation
Ch 8: Stock Valuation