1. Valuation (discounted cash flow, interest rates and bond, stock)

2. Annuity & Perpetuity
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 (P-121 Q18)
Perpetuity – infinite series of equal payments

3. Annuities & Perpetuities – Basic Formulas1
Annuities & Perpetuities – Basic Formulas
Perpetuity: PV = C / r
Annuities: 3

8. Annuity – Sweepstakes Example1
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 8

9. 1
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 = 9

10. 1
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 year 40. How much would you be willing to invest today if you desire an interest rate of 12%? 10

11. Saving For Retirement Time1
Saving For Retirement Time …
… K 25K 25K 25K 25K
Notice that the year 0 cash flow = 0 (CF0 = 0)
The cash flows years 1 – 39 are 0 (C01 = 0; F01 = 39)
The cash flows years 40 – 44 are 25,000 (C02 = 25,000; F02 = 5) 11

12. Annuity – Finding the Rate Without a Financial Calculator1
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 12

13. Future Values for Annuities1
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 year 40?
FV = 2,000( – 1)/.075 = 454,513.04 13

14. 1
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 year 3?
FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12 14

15. 1
Annuity Due Timeline
32,464
35,061.12 15

17. Growing Perpetuity
A perpetuity paying 100 a year with 8% interest rate;
Without growing, the present value of the future CFs is 100/0.08 = 1,250;
With 4% growth of payment, the present value of future CFs is
100 /( ) = 2,500

18. Growing Perpetuity Textbook example 4.18
Cash flows of perpetuity occur on date 1 instead of date 0. Present value must incorporate the cash flows occurring in date 0.
66.6 = /( )
Stock price = dividend received today + PVFCF

19. Growing Annuity C is the first payment; R is interest rate T is period
g is growth rate

20. Growing Annuity
Your salary is $80,000 with expected growth rate of 9 % for 40 years. The interest rate is 20%
Present value of your lifetime salary is
$80,000 x [ 1-(1.09/1.20)^40 / ( ) ]
= $ 711,731

21. Infrequent Annuity
A $450 annuity is paid once every 2 years for 20 years. The first payment is made at date 2 with 6% interest rate.
Interest rate over 2 years = (1.06)^2 -1
= 12.36%
The present value of the annuity is
450 x [ 1- 1/( )^10 / ] =

22. Annuity and annuity due
Compare the textbook example 4.19 and 4.22
PV of annuity due is greater than that of annuity by 1 year interest
PV of annuity due = PV of annuity x (1+r)
PV = PV x 1.08
530, ,905 x 1.08

23. Computing Payments with APRs1
Computing Payments with APRs
Suppose you want to buy a new computer system and the store is willing to sell it 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 = 23

24. Future Values with Monthly Compounding1
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 24

25. Continuous Compounding1
Continuous Compounding
Sometimes investments or loans are figured based on continuous compounding
EAR = er – 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% 25

26. Loan types & Loan amortization
Pure discounted loans
Interest-only loans
Amortized loans

27. 1
Pure Discount Loans
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 27

28. 1
Interest-Only Loan
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. 28

29. Amortized loans
Amortizing loan – the process of providing for a loan to be paid off by making regular principal reduction
Way of amortizing loan (example P-112)

30. What is a firm worth
Suppose you are appraising a small firm that expected future cash flows are $5000 in the first year, $2000 in 2-6 year and $10000 in the 7th year with 10% interest rate.
The firm’s value will be
5000/1.1 + (2000xPVIFA)/ /(1.1)^7
= $16569

31. Bond & Stock Valuation

34. Bond features and prices
Beck Corporation to borrow $1,000 for 30 years at 12 percent
Coupons = $120 (level coupon bond)
Par value/face value = $1,000 (par value bond)
Coupon rate = 120/1,000
Maturity = 30 years

35. Basic Terminology Bond, note, debenture Par Value
Coupon Interest Payment
Coupon Interest Rate
Maturity Date
Bond’s Market Rate of Interest, kd

42. Yield-to-Maturity
YTM is the rate of return earned on a bond held to maturity, also called “promised yield.”
It is the discount rate that equates the present value of the interest and principal payments to the price of the bond.

43. Yield to Maturity A Graphical Interpretation
Consider a U.S. Treasury bond that has a coupon rate of 10%, a face value of $1,000 and matures exactly 10 years from now.
Market price of $1,500, implies a yield of 3.91% (semi-annual compounding); for B=$1,000 we obviously find R=10%. 43

44. YTM Example
X company issues a bond with 10 years to maturity. This bond has an annual coupon of $80. Similar bonds have a yield to maturity of 8%. In 10 years, X will pay $1000 to the owner of the bond.
What is the bond price?
If the market interest rate rises to 10% next year, what is the bond price?
What if the interest rate falls to 6%?

45. YTM Example 1. PV of the par: 1000/(1.08)^10 = $463.19
PV of coupon (remember it is annuity)
80 * (1-(1/1.08^10))/0.08 = $536.81
Bond value = $ $ = $1000

46. YTM Example 2. PV of par: 1000/(1.1)^9 = $424.1 PV of coupon
Bond value = $424.1+$ = $884.82

47. Another way to get the price
At 10% interest rate, you can get $100 per year if you put your $1000 in bank.
If you buy this bond, you lose $20 per year for 9 years.
Your total loss during this period is
= 20 x (1-1/1.1^9)/0.1 = $115.18
You want the bond price to reflect this loss as a compensation.

48. YTM Example Coupon rate = 8%, interest rate falls to 6%
Present value of Par
= 1000/(1.06)^9 = $591.89
Present value of coupon payment
= 80 x (1-1/0.06^9)/ = $544.14
Bond price = = $

49. Another way to get the price
At 6% interest rate, you can get $60 per year if you put your $1000 in bank.
If you buy this bond, you get additional $20 per year for 9 years.
Additional gain form the coupon payment
= 20 x (1-1/0.06^9)/0.06 = $136.03
This is the premium you must pay.

50. Required Rate of Return
The discount rate (ki) is the opportunity cost of capital, i.e., the rate that could be earned on alternative investments of equal risk.
ki = k* + IP + LP + MRP + DRP

51. Value of Bond       ... $100 $100 $1 , 000 V  + . . . + + 1 + k1 2 10
10%
...
V = ?
100
100
,000
$100
$100 $1 ,
000 V  + . . . + + B   1   10   10 1 + k 1 + k 1 + k d d d
= $ $ $385.54
= $1,000.

52. General Observations About Bond Values
If coupon rate < kd, bond sells at a discount.
If coupon rate > kd, bond sells at a premium.
If coupon rate = kd, bond sells at its par value.
If kd rises, price falls; if kd falls, price rises.
At maturity, the value of a bond equals par.

53. Semiannual coupons
Usually US bonds make coupon payments twice a year. If a coupon rate is 14%, this means you will receive $70 every 6 months. Assume the yield to maturity is 16%.
Assume the bond matures in 7 years.
What is the bond price?
What is the effective annual rate?

54. Semiannual coupons Bond price = PV of par + PV of coupon
= 1000/1.08^ x (1-1/1.08^14)/0.08 = =
Effective annual rate = (1.08)^2 -1 = 16.64%

55. Changes in Bond Values Over Time
At maturity, the value of any bond must equal its par value.
The value of a premium bond would decrease to $1,000.
The value of a discount bond would increase to $1,000.
A par bond stays at $1,000 if kd remains constant.

56. Time Path of Bond Value Bond Value ($) Years remaining to Maturity
1,372
1,211
1,000
837
775
kd = 7%.
kd = 13%.
kd = 10%. 56

57. Bond Prices and Interest Rates
Bond prices are inversely related to IR
Longer term bonds are more sensitive to IR changes than short term bonds
The lower the IR, the more sensitive the price.
Graphical illustration of previous comments
1) relationship b/w price and interest rate: across all maturities
2) relationship b/w price and maturity: fix IR
3) Longer bonds are more sensitive to changes in interest rates (both absolute and relative) bigger price change for a change in IR, longer maturity is
4) Convexity (the lower the interest rate the, the greater the change in price) 57

58. Interest Rate Risk kd 1-year Change 10-year Change 5% $1,048 $1,386
Rising interest rates have an adverse effect on bond values.
The longer the maturity of a bond, the greater the exposure to interest rate risk. kd
1-year
Change
10-year
Change 5%
$1,048
$1,386
4.8%
38.6%
10%
1,000
1,000
4.4%
25.1%
15%
956
749

59. Interest Rate Risk Value kd 10-year 1-year 1,500 1,000 500 0% 5% 10%0% 5%
10%
15%

60. Interest Rate Risk
When looking at a bond, you must remember that keep all other things equal
The longer the time to maturity, the greater the interest rate risk (because the risk of rate change);
The lower the coupon rate, the greater the interest rate risk (because the par value becomes the key factor determining the present value)

61. Current Yield Annual interest payment/Current value of bond
Provides information about cash income on bond.
Does not provide accurate measure of total expected return on bond.

62. Yield Example
A bond with $80 coupon for 6 years and $1000 face value (par value) is sold for $ It’s yield to maturity is
= 80*[1-1/(1+r)^6]/r +1000/(1+r)^6
Finding r = 9% using trial and error.
It’s current yield is 80/955.14=8.38%
CY<YTM because it ignores the built-in gain. That is, the benefit of buying cheaper is not considered.

63. Review of Annuity
Danielle will receive a four-year annuity of $500 per year, beginning at date 6. If the interest rate is 10 percent, what is the present value of her annuity?

64. Example 5.2 A bond has a quoted price = $1080.42 Face value = $1000
Semiannual coupon = $30
Maturity of 5 years
1: What is the current yield?
2: What is the yield to maturity?
3: Which one is bigger and why?

65. Example 5.2
1. Current yield = annual payment / price = 60/ = 5.55% 2: V = sum (payment/(1+r)^n) + par/(1+r)^n = annuity + PV of par V = , payment = 30, n = 10, finding r r = 2.1% (semiannual rate) = 4.2 % annual rate 3: CY > r because CY ignores the built-in loss of the premium.

66. Example 5.3 2 identical bond (A,B) except their coupons
Years to maturity = 12
A has 10% coupon and sell for $935.08
B has 12% coupon, what is B’s price?
(1) (2) < (3) >935.08
What information is required to proceed this calculation?

67. Example 5.3 Back to bond A: here we know LHS
= annuity$100,T=12,i=? + PV$1000, T=12,i=?
i = 11%
For bond B: Now you have all information in RHS, you can obtain its price
VB = annuity$120,T=12,i=11% + PV$1000, T=12,i=11% =

68. Valuing Coupon Bonds The General Formula
What is the market price of a bond that has an annual coupon C, face value F and matures exactly T years from today if the required rate of return is R, with m-periodic compounding?
Coupon payment is: c = C/m
Effective periodic interest rate is: i = R/m
number of periods N = Tm
… N
c c c c … … c+F
What happens when compounding is monthly? Continuous? (1+I)^N e^RT
Just change the discount factor: 1/(1+I)^N 68

69. Review of Amortized Loans
Suppose we have a $100,000 commercial mortgage with a 12 percent APR and a 20-year amortization. What will the monthly payment be?

71. About bond You need to remember
Debt is not an ownership interest in the firm (cannot vote)
Interest payment is one cost of doing business and thus, it is tax deductible
Unpaid debt is a liability of the firm. A firm may go bankruptcy if he cannot meet the obligation.

72. Protective Covenant Negative covenant – a firm cannot do
Limit of dividend payout;
Cannot pledge assets to other lenders;
Cannot issue additional long-term debt;
Positive covenant – a firm MUST do
Provide audited financial statements to the lender;
 Maintain any security in good condition.

73. Types of Bonds Treasury Bonds Municipal Bonds Zero Coupon Bonds
Floating-Rate Bonds
Income Bonds
Convertible Bonds
Put Bonds (the holder has the right, but not the obligation, to demand early repayment of the principal)

74. Taxable vs Municipal Bonds
Taxable bonds = 8%
Munis of comparable risk = 6%
Your tax bracket = 40%
1: Which one is more attractive?
2: Break-even tax rate?
3: How do you interpret this rate?

75. Taxable vs Municipal Bonds
1: you invest in 8%, you will receive 8% x (1-0.40)=4.8%
2: break-even rate = 8% x (1-t) = 6%
t = 0.25 = 25%
3: an investor in a 25% will be indifferent to either bond.

76. Zero Coupon Bond
Zero with par $1000 and 5 years to maturity is sold for $508.35
We can verify the yield to maturity (using semiannual periods) = 14%
excel formula (=pv(0.07,10,0,1000));
Total interest paid over the life
= = $491.65

77. Zero Coupon Bond Tax law requires
1. Issuers ( no interest is actually paid) can deduct interest expense
2. Owner of the bond pays tax on interest accrued (no interest is actually received)

78. Zero Coupon Bond
Before 1982, annual interest deduction is calculated as $491.65/ 5yrs = $98.33
Under current tax law, we must use the change of the present value of the bond.
V5yrs = and V4yrs =
Difference = $73.66 (interest deduction for issuer and taxable income for owner)
New method is less attractive. Why?