(Intrinsic Value) Deployment Specialists pays a current (annual) dividend of $1 and is expected
to grow at 20% for two years then at 4% thereafter. If the required return for Deployment
Specialists is 8.5%, what is the intrinsic value of its stock?
𝐷
𝐷
1
2
𝐼𝑛𝑠𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑉𝑎𝑙𝑢𝑒 = 𝑉0 = (1+𝑘)
+ (1+𝑘)
2 + ⋯+
($1×1.20)
($1×1.202 )
(𝐷𝐻 +𝑃𝐻 )
(1+𝑘)𝐻
($1×1.202 ×1.04)
𝑉0 = (1+0.085) + (1+0.085)2 + ((0.085−0.04)×(1+0.085)2)
𝑉0 = $30.60
(Required RoR Constant-Growth) Jand, Incorporated, currently pays a dividend of $1.22, which
is expected to grow indefinitely at 5%. If the current value of Jand’s shares based on the
constant-growth dividend discount model is $32.03, what is the required rate of return?
(𝐷 ×(1+𝑔)
𝐼𝑛𝑠𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑉𝑎𝑙𝑢𝑒 = 𝑉0 = 0(𝑘−𝑔)
$32.03 =
($1.22×1.05)
(𝑘−0.05)
𝑘 = 0.08994 𝑜𝑟 9.00%
(Required RoR DDM) A firm pays a current dividend of $1, which is expected to grow at a rate
of 4% indefinitely. If the current value of the firm’s shares is $35, what is the required return
applicable to the investment based on the constant-growth dividend discount model (DDM)?
𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑉𝑎𝑙𝑢𝑒 = 𝑉0 =
$35 =
($1×1.05)
(𝑘−0.05)
(𝐷0 ×(1+𝑔)
(𝑘−𝑔)
, 𝑘 = 0.08 𝑜𝑟 8%
(Market Capitalization Rate and Intrinsic Value) Miltmar Corporation will pay a year-end
dividend of $4, and dividends thereafter are expected to grow at the constant rate of 4% per
year. The risk-free rate is 4%, and the expected return on the market portfolio is 12%. The stock
has a beta of 0.75.
a. Calculate the market capitalization rate
𝑀𝑎𝑟𝑘𝑒𝑡 𝑐𝑎𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 = 𝑘 = 𝑟𝑓 + 𝛽 × [𝐸(𝑟𝑀 ) − 𝑟𝑓 ]
𝑘 = 0.04 + 0.75 × (0.12 − 0.04) = 0.10 𝑜𝑟 10%
b. What is the intrinsic value of the stock?
𝐷1
𝐼𝑛𝑠𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑣𝑎𝑙𝑢𝑒 = 𝑉0 = (𝑘−𝑔)
$4
𝑉0 = (0.10−0.04) = $66.67
(Dividend-Growth Price) The risk-free rate of return is 5%, the required rate of return on the
stock market is 10%, and High-Flyer stock has a beta coefficient of 1.5. If the dividend per share
expected during the coming year, 𝐷1 , is $2.50 and 𝑔 = 4%, at what price should a share sell?
𝑘 = 𝑟𝑓 + 𝛽 × [𝐸(𝑟𝑀 ) − 𝑟𝑓 ]
𝑘 = 0.05 + 1.5 × (0.10 − 0.05) = 0.125 𝑜𝑟 12.5%
𝐷1
$2.50
𝑃0 = (𝑘−𝑔)
, 𝑃0 = (0.125−0.04) = $29.41
(PVGO) Tri-coat Paints has a current market value of $41 per share with earnings of $3.64.
What is the present value of its growth opportunities (PVGO) if the required return is 9%
𝐸
𝑃𝑟𝑖𝑐𝑒 = 1 + 𝑃𝑉𝐺𝑂
𝑘
$3.64
$41 = 0.09 + 𝑃𝑉𝐺𝑂
𝑃𝑉𝐺𝑂 = $0.56
(Constant-Growth, No-Growth Price, and PVGO) Sisters Corporation expects to earn $6 per
share next year. The firm’s ROE is 15% and its plowback ratio is 60%. The firm’s market
capitalization rate is 10%.
a. Calculate the price with the constant dividend growth model
𝐺𝑖𝑣𝑒𝑛 𝐸𝑃𝑆 = $6, 𝑅𝑂𝐸 = 15%, 𝑝𝑙𝑜𝑤𝑏𝑎𝑐𝑘 𝑟𝑎𝑡𝑖𝑜 = 0.60, 𝑘 = 10%
𝐷
(𝐸𝑃𝑆×(1−𝑏))
𝑃0 = 1 =
(𝑘−𝑔)
(𝑘−𝑅𝑂𝐸×𝑏)
($6×(1−0.60))
𝑃0 = (0.10−0.15×0.60)
𝑃0 = $240
b. Calculate the price with no growth
𝐸
𝑃𝑟𝑖𝑐𝑒 = 𝑘1 + 𝑃𝑉𝐺𝑂
$6
𝑃𝑟𝑖𝑐𝑒 = 0.10 + 0
𝑃𝑟𝑖𝑐𝑒 = $60
c. What is the present value of its growth opportunities?
𝐸
𝑃𝑟𝑖𝑐𝑒 = 𝑘1 + 𝑃𝑉𝐺𝑂
$6
$240 = 0.10 + 𝑃𝑉𝐺𝑂
𝑃𝑉𝐺𝑂 = $180
(Constant No-Growth Value) A common stock pays an annual dividend per share of $2.10. The
risk-free rate is 7% and the risk premium for this stock is 4%. If the annual dividend is expected
to remain at $2.10, what is the value of the stock?
𝐶𝑜𝑠𝑡 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦 = 𝑟𝑓 + 𝐸(𝑅𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚)
𝐶𝑜𝑠𝑡 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦 = 7% + 4%
𝐶𝑜𝑠𝑡 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦 = 11%
𝑇ℎ𝑒 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠 𝑎𝑟𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑡𝑜 𝑏𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑒 𝑖𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑠 𝑛𝑜 𝑔𝑟𝑜𝑤𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒
𝐷
𝑃0 = 𝑘
𝑒
$2.10
𝑃0 = 0.11
𝑃0 = $19.09
(DDM, Price, and PE Ratio) Computer stocks currently provide an expected rate of return of
16%. MBI, a large computer company, will pay a year-end dividend of $2 per share.
a. If the stock is selling at $50 per share, what must be the market’s expectation of the
growth rate of MBI dividends?
𝐷1
𝐷𝐷𝑀; 𝑃0 = (𝑘−𝑔)
$2
$50 = (0.16−𝑔)
𝑔 = 0.12 𝑜𝑟 12%
b. If the dividend growth forecasts for MBI are revised downward to 5% per year, what will
happen to the price of MBI stock?
𝐷
𝑃0 = 1
𝑃0 =
(𝑘−𝑔)
$2
(0.16−0.05)
𝑃0 = $18.18
c. What (qualitatively) will happen to the company’s price-earnings ratio?
𝑃𝑟𝑖𝑐𝑒 𝑓𝑎𝑙𝑙𝑠 𝑤𝑖𝑡ℎ 𝑙𝑜𝑤 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑔𝑟𝑜𝑤𝑡ℎ. 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝑓𝑜𝑟 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑢𝑛𝑐ℎ𝑎𝑛𝑔𝑒𝑑. 𝑆𝑜 𝑃𝐸 𝑓𝑎𝑙𝑙𝑠
(DDM Price, ROE, and PVGO) The FI Corporation’s dividends per share expected to grow
indefinitely by 5% per year
a. If this year’s year-end dividend is $8 and the market capitalization rate is 10% per year,
what must the current stock price be according to the dividend discount model?
𝐷1
𝑃0 = (𝑘−𝑔)
$8
𝑃0 = (0.10−0.05)
𝑃0 = $160
b. If the expected earnings per share are $12, what is the implied value of ROE on future
investment opportunities?
8
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 12 = 0.67, 𝑝𝑙𝑜𝑤𝑏𝑎𝑐𝑘 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑏 = 0.33,
𝑔 = 𝑏 × 𝑅𝑂𝐸
0.05 = (0.33) × 𝑅𝑂𝐸
𝑅𝑂𝐸 = 0.15 𝑜𝑟 15%
c. How much is the market paying per share for growth opportunities (that is, for an ROE
on future investments that exceeds the market capitalization rate)?
𝑃𝑉𝐺𝑂 = (𝑃0 −
𝐸1
𝑘
)
$12
𝑃𝑉𝐺𝑂 = ($160 − 0.10)
𝑃𝑉𝐺𝑂 = $40
(Non-Constant Growth Price) Consider the following information: The firm’s dividends are
expected to grow at 𝑔 = 20% until 𝑡 = 3 𝑦𝑒𝑎𝑟𝑠. At the start of year four, growth slows to
𝑔𝑠 = 5%. The stock just paid a 𝐷0 = $1. Assume a market capitalization rate of 𝑘 = 12%.
The is the 𝑃0 of this stock?
𝑃0 =
𝑃0 =
𝐷0 ×(1+𝑔)
+
+
+
(1+𝑘)
$1×(1+0.20)
(1+0.12)
𝐷0 ×(1+𝑔)𝑡
(1+𝑘)𝑡
$1×(1+0.20)2
+⋯+
(1+0.12)2
𝐷0 ×(1+𝑔)𝑡 ×(1+𝑔𝑠 )
(1+𝑘)𝑡 ×(𝑘−𝑔𝑠 )
$1×(1+0.20)3
$1×(1+0.2)3 ×(1+0.5)
(1+0.12)3
+
(1+0.12)3 ×(.12−.05)
𝑃0 = $1.07 + $1.15 + $1.23 + $18.45
𝑃0 = $21.90
(Intrinsic Value, Price, and HPR) The risk-free rate of return is 8%, the expected rate of return
on the market portfolio is 15%, and the stock of Xyrong Corporation has a beta coefficient of
1.2. Xyrong pays out 40% of its earnings in dividends, and the latest earnings announced were
$10 per share. Dividends were just paid and are expected to be paid annually. You expect that
Xyrong will earn an ROE of 20% per year on all reinvested earnings forever.
a. What is the intrinsic value of a share of Xyrong stock?
𝑘 = 𝑟𝑓 + 𝛽 × [𝐸(𝑟𝑚 ) − 𝑟𝑓 ]
𝑘 = 0.08 + 1.2 × (0.15 − 0.08)
𝑘 = 0.164 𝑜𝑟 16.4%
𝑔 = 𝑏 × 𝑅𝑂𝐸
𝑔 = 0.6 × 0.20
𝑔 = 0.12 𝑜𝑟 12%
(𝐷 ×(1+𝑔)
𝑉0 = 0(𝑘−𝑔)
$4×(1+0.12)
𝑉0 = (0.164−0.12)
𝑉0 = $101.82
b. If the market price of a share is currently $100, and you expect the market price to be
equal to the intrinsic value one year from now, calculate the price of the share after one
year from now
𝑃1 = 𝑉1 = 𝑉0 × (1 + 𝑔)
𝑃1 = $101.82 × (1 + 0.12)
𝑃1 = $114.04
c. What is your expected one-year holding period return on Xyrong stock?
𝐸(𝑟) =
𝐸(𝑟) =
𝐷1 +(𝑃1 −𝑃0 )
𝑃0
($4.48+$114.04)−$100
$100
𝐸(𝑟) = 0.1852 𝑜𝑟 18.52%
(Price and Discounted Price) MF Corporation has an ROE of 16% and a plowback ratio of 50%.
a. If the coming year’s earnings are expected to be $2 per share, at what price will the
stock sell? The market capitalization rate is 12%
𝑅𝑂𝐸 = 16%, 𝑏 = 0.50, 𝐸𝑃𝑆 = $2, 𝑘 = 12%
𝐷1
(𝐸𝑃𝑆×(1−𝑏))
𝑃0 = (𝑘−𝑔)
= (𝑘−(𝑅𝑂𝐸×𝑏))
$2×(1−0.50)
𝑃0 = 0.12−(0.16×0.50)
$1
𝑃0 = (0.12−0.08)
𝑃0 = $25
b. What price do you expect MF shares to sell for in three years?
𝑃3 =
𝐸𝑃𝑆×(1−𝑏)×(1+𝑔)3
(𝑘−𝑔)
= 𝑃0 × (1 + 𝑔)3
𝑃3 = $25 × (1.08)3
𝑃3 = $31.49
(Market-To-Book Ratio) A firm has current assets that could be sold for their book value of $10
million. The book value of its fixed assets is $60 million, but they could be sold for $90 million
today. A firm has a total debt with a book value of $40 million, but interest rate declines have
caused the market value of the debt to increase to $50 million. What is this firm’s marketto=book ratio?
𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑖𝑟𝑚 = 𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡𝑠 = 𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑒𝑏𝑡𝑠
𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 = ($10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 + $90 𝑚𝑖𝑙𝑙𝑖𝑜𝑛) − $50 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 = $50 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑖𝑟𝑚 = 𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡𝑠 − 𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑒𝑏𝑡𝑠
𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 = ($10 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 + $60 𝑚𝑖𝑙𝑙𝑖𝑜𝑛) − $40 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 = $30 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
𝑀𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒
𝑀𝑎𝑟𝑘𝑒𝑡 𝑡𝑜 𝑏𝑜𝑜𝑘 𝑟𝑎𝑡𝑖𝑜 = 𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒
$50 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
𝑀𝑎𝑟𝑘𝑒𝑡 𝑡𝑜 𝑏𝑜𝑜𝑘 𝑟𝑎𝑡𝑖𝑜 = $30 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
𝑀𝑎𝑟𝑘𝑒𝑡 𝑡𝑜 𝑏𝑜𝑜𝑘 𝑟𝑎𝑡𝑖𝑜 = 1.67
(Growth Rate and PE Ratio) The market capitalization rate for Admiral Motors Company is 8%.
Its expected ROE is 10% and its expected EPS is $5. The firm’s plowback ratio is 60%
a. Calculate the growth rate
𝑔 = 𝑅𝑂𝐸 × 𝑏
𝑔 = 0.10 × 0.60
𝑔 = 0.06 𝑜𝑟 6%
b. What will be its P/E ratio?
𝑃𝐸 =
(1−𝑏)
(𝑘−𝑔)
(1−0.60)
𝑃𝐸 = (0.08−0.06)
𝑃𝐸 = 20
(FCFF) Eagle Products’ EBIT is $300, its tax rate is 21%, depreciation is $20, capital expenditures
are $60, and the planned increase in net working capital is $30. What is the free cash flow to
the firm?
𝐹𝐶𝐹𝐹 = 𝐸𝐵𝐼𝑇 × (1 − 𝑡𝑐 ) + 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 − 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑒𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒𝑠 − 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑁𝑊𝐶
𝐹𝐶𝐹𝐹 = $300 × (1 − 0.21) + $20 − $60 − $30
𝐹𝐶𝐹𝐹 = $167
(FCFF)
Suppose FCFF = $1m for years 1 to 4 and then is expected to grow at 3%. Assume WACC = 15%.
𝐹𝐶𝐹𝐹𝑡
𝑃𝑇
𝐹𝑖𝑟𝑚 𝑉𝑎𝑙𝑢𝑒 = ∑𝑇𝑡=1 (1+𝑊𝐴𝐶𝐶)
𝑡 + (1+𝑊𝐴𝐶𝐶)𝑇
𝐹𝑖𝑟𝑚 𝑉𝑎𝑙𝑢𝑒 =
$1,000,000
∑4𝑡=1
(1+0.15)1
+
(
$1,000,000×1.03
)
0.15−0.30
(1+0.15)4
𝐹𝑖𝑟𝑚 𝑉𝑎𝑙𝑢𝑒 = $7,762,527
If 500,000 shares are outstanding, what is the predicted price of this stock if the firm has
$5,000,000 of debt?
𝐹𝑖𝑟𝑚 𝑉𝑎𝑙𝑢𝑒−𝐷𝑒𝑏𝑡
𝑃0 = 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔
𝑃0 =
$7,762,527−$5,000,000
500,000
𝑃0 = $5.53
(FCFE)
Suppose 𝐹𝐶𝐹𝐸 = $900,000 for years 1 to 4 and then is expected to grow at a rate of 3%.
Assume 𝑘𝑒 = 18%
𝐹𝐶𝐹𝐸
𝑃
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦 = ∑𝑇𝑡=1 (1+𝑘 )𝑡 + (1+𝑘𝑇 )𝑡
𝑒
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦 =
𝑒
$900,000×1.03
( 0.18−0.03 )
$900,000
∑4𝑡=1
+
1
(1+0.18)
(1+0.18)4
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦 = $2,500,851
If there are 500,000 shares outstanding, what is the predicted price of this stock? Why can debt
be ignored?
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦
𝑃0 = 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔
𝑃0 =
$2,500,851
500,000
= $5.00
(Invoice Price) A coupon bond paying semiannual interest is reported as having an ask price of
117% of its $1,000 par value. If the latest interest payment was made one month ago and the
coupon rate is 6%, what is the invoice price of the bond? Assume that a month as 30 days.
𝑆𝑒𝑚𝑖 𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 = $1,000 × 6% × 0.5
𝑆𝑒𝑚𝑖 𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑖𝑝𝑜𝑛 = $30
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐷𝑎𝑦𝑠 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 =
×
2
𝐷𝑎𝑦𝑠 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑛𝑔 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
$60
30
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 2 × 182
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $4.945
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = 𝐴𝑠𝑘 𝑝𝑟𝑖𝑐𝑒 + 𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = 1,170 + $4.945
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = $1,174.95
(Invoice Price) A bond with a coupon rate of 7% makes semiannual coupon payments on
January 15 and July 15 of each year. The Wall Street Journal reports the ask price for the bond
on January 30 at 100.125. What is the invoice price of the bond. The coupon period has 182
days.
𝑅𝑒𝑝𝑜𝑟𝑡𝑒𝑑 𝑏𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = $1,001.25
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐷𝑎𝑦𝑠 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 =
× 𝐷𝑎𝑦𝑠 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑛𝑔 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
2
$70
15
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 2 × 182
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $2.8846
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = 𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝑅𝑒𝑝𝑜𝑟𝑡𝑒𝑑 𝑏𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = $1,001.25 + $2.8846
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = $1,004.13
(Invoice Price) Suppose that today’s date is April 15. A bond with a 10% coupon, paid
semiannually every January 15 and July 15 is quoted as selling at an ask price of 101.25. If you
buy the bond from a dealer today, what price will you pay for it?
𝐴𝑝𝑟𝑖𝑙 15 𝑖𝑠 𝑚𝑖𝑑𝑤𝑎𝑦 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑏𝑜𝑛𝑑
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 ℎ𝑖𝑔ℎ𝑒𝑟 𝑡ℎ𝑎𝑛 𝑡ℎ𝑒 𝑠𝑡𝑎𝑡𝑒𝑑 𝑎𝑠𝑘 𝑝𝑟𝑖𝑐𝑒 𝑏𝑦 0.50 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = 𝑄𝑢𝑜𝑡𝑒𝑑 𝑃𝑟𝑖𝑐𝑒 + (0.50 × 𝑆𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛)
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = $1,012.50 + (0.50 × $50)
𝐼𝑛𝑣𝑜𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 = $1,037.50
(Credit Default Swap) An investor believes that a bond may temporarily increase in credit risk.
Which of the following would be the most liquid method of exploiting this?
𝐼𝑛𝑣𝑒𝑠𝑡𝑜𝑟 𝑏𝑒𝑙𝑖𝑒𝑣𝑒𝑠 𝑏𝑜𝑛𝑑 𝑚𝑎𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑐𝑟𝑒𝑑𝑖𝑡 𝑟𝑖𝑠𝑘 𝑤ℎ𝑖𝑐ℎ 𝑟𝑎𝑖𝑠𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐𝑟𝑒𝑑𝑖𝑡 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑠𝑤𝑎𝑝𝑠
𝐷𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑤𝑖𝑑𝑒𝑛𝑖𝑛𝑔 𝑠𝑤𝑎𝑝 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡ℎ𝑒 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑜𝑓 𝑎 𝑐𝑟𝑒𝑑𝑖𝑡 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑠𝑞𝑢𝑎𝑑 𝑖𝑠 𝑏𝑒𝑠𝑡 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑
(Credit Default Swap) Which of the following most accurately describes the behavior of credit
default swaps?
𝐶𝑟𝑒𝑑𝑖𝑡 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑠𝑤𝑎𝑝𝑠 𝑎𝑟𝑒 𝑚𝑜𝑟𝑒 𝑣𝑎𝑙𝑢𝑎𝑏𝑙𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑜𝑓 𝑝𝑟𝑜𝑡𝑒𝑐𝑡𝑖𝑜𝑛
𝐶𝑟𝑒𝑑𝑖𝑡 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑠𝑤𝑎𝑝𝑠 𝑑𝑜 𝑛𝑜𝑡 𝑝𝑟𝑜𝑣𝑖𝑑𝑒 𝑝𝑟𝑜𝑡𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑟𝑖𝑠𝑘
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑤ℎ𝑒𝑛 𝑐𝑟𝑒𝑑𝑖𝑡 𝑟𝑖𝑠𝑘 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠, 𝑠𝑤𝑎𝑝 𝑝𝑟𝑒𝑚𝑖𝑢𝑚𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒
(YTM) A zero-coupon bond with face value $1,000 and maturity of five years sells for $746.22
a. What is its yield to maturity?
𝑃𝑉 = −746.22, 𝐹𝑉 = 1,000, 𝑛 = 5, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 6.03%
b. What will the yield to maturity be if the price falls to $730?
𝑃𝑉 = −730.00, 𝐹𝑉 = 1,000, 𝑛 = 5, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 6.50%
(YTM) Two bonds have identical times to maturity and coupon rates. One is callable at 105, the
other at 110. Which should have the higher yield to maturity?
𝑇ℎ𝑒 𝑏𝑜𝑛𝑑 𝑐𝑎𝑙𝑙𝑎𝑏𝑙𝑒 𝑎𝑡 105 𝑠ℎ𝑜𝑢𝑙𝑑 𝑠𝑒𝑙𝑙 𝑎𝑡 𝑎 𝑙𝑜𝑤𝑒𝑟 𝑝𝑟𝑖𝑐𝑒 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑐𝑎𝑙𝑙 𝑝𝑟𝑜𝑣𝑖𝑠𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑎𝑏𝑙𝑒 𝑡𝑜 𝑓𝑖𝑟𝑚
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑖𝑡𝑠 𝑦𝑖𝑒𝑙𝑑 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 ℎ𝑖𝑔ℎ𝑒𝑟
(YTM, Price and, Maturity) Fill in the table below for the following zero-coupon bonds, all of
which have par values of $1,000. Assume annual compounding
Price
$400.00
$500.00
$500.00
d
e
$400.00
Maturity (years)
20
20
10
10
10
f
Yield to maturity (%)
a
b
c
10.00
8.00
8.00
a. 𝑛 = 20, 𝐹𝑉 = 1,000, 𝑃𝑉 = −400, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 4.69%
b. 𝑛 = 20, 𝐹𝑉 = 1,000, 𝑃𝑉 = −500, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 3.53%
c. 𝑛 = 10, 𝐹𝑉 = 1,000, 𝑃𝑉 = −500, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 7.18%
d. 𝑛 = 10, 𝐹𝑉 = 1,000, 𝐼𝑌 = 10, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝑃𝑉 = $385.54
e. 𝑛 = 10, 𝐹𝑉 = 1,000, 𝐼𝑌 = 8, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝑃𝑉 = $463.19
f. 𝑃𝑉 = −400, 𝐼𝑌 = 8, 𝑃𝑀𝑇 = 0, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑛 = 11.91
(Accrued Interest) A bond at par value makes semiannual payments with a coupon rate of 6%.
If 45 days have passed since the last coupon payment, what is the accrued interest?
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 =
𝐷𝑎𝑦𝑠 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐷𝑎𝑦𝑠 𝑖𝑛 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑
45
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 =
× $30
182
𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $7.42
×
𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡
2
(Current Yield) A bond with an annual coupon rate of 4.8% sells for $970. What is the bond’s
current yield?
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 =
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒
($1,000×4.8%)
$970
= 4.95%
(EAY) Treasury bonds paying an 8% coupon rate with semiannual payments currently sell at par
value. What coupon rate would they have to pay in order to sell at par if they paid their
coupons annually?
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑡
𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 𝑦𝑖𝑒𝑙𝑑: 𝐸𝐴𝑌 = (1 +
) −1
2
0.08
𝐸𝐴𝑌 = (1 + 2 )2 − 1 = 8.16
𝐼𝑓 𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑏𝑜𝑛𝑑𝑠 𝑎𝑟𝑒 𝑡𝑜 𝑠𝑒𝑙𝑙 𝑎𝑡 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑡ℎ𝑒𝑦 𝑚𝑢𝑠𝑡 𝑜𝑓𝑓𝑒𝑟 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑟𝑎𝑡𝑒 𝑠𝑜 8.16%
(BEY and EAY) A 20-year maturity bond with par value $1,000 makes semiannual coupon
payments at a coupon rate of 8%
a. Find the bond equivalent and effective annual yield to maturity of the bond if the
bond price is $950?
𝑛 = 40, 𝐹𝑉 = 1,000, 𝑃𝑉 = −950, 𝑃𝑀𝑇 = 40, 𝐶𝑃𝑇 𝐼𝑌 = 4.26%
𝐵𝐸𝑌 = 𝐼𝑌 × 2
𝐵𝐸𝑌 = 4.26% × 2
𝐵𝐸𝑌 = 8.52%
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑡
𝐸𝐴𝑌 = (1 +
) −1
2
𝐸𝐴𝑌 = (1 +
0.0852 2
)
2
− 1 = 0.0870 𝑜𝑟 8.70%
b. Find the bond equivalent yield and effective annual yield to maturity of the bond if
the bond price is $1,000?
𝐵𝑜𝑛𝑑 𝑠𝑒𝑙𝑙𝑖𝑛𝑔 𝑎𝑡 𝑝𝑎𝑟, 𝑦𝑖𝑒𝑙𝑑 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 = 𝑠𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛
𝐵𝐸𝑌 = 𝐼𝑌 × 2
𝐵𝐸𝑌 = 4% × 2
𝐵𝐸𝑌 = 8%
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑡
𝐸𝐴𝑌 = (1 +
) −1
2
𝐸𝐴𝑌 = (1 +
0.08 2
)
2
− 1 = 0.0816 𝑜𝑟 8.16%
c. Find the bond equivalent and effective annual yield to maturity of the bond if the
bond price is $1,050?
𝑛 = 40, 𝐹𝑉 = $1,000, 𝑃𝑉 = −$1,050, 𝑃𝑀𝑇 = 40, 𝐶𝑃𝑇 𝐼𝑌 = 3.76%
𝐵𝐸𝑌 = 𝐼𝑌 × 2
𝐵𝐸𝑌 = 3.76% × 2 = 7.52%
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑡
𝐸𝐴𝑌 = (1 +
) −1
2
𝐸𝐴𝑌 = (1 +
0.0752 2
)
2
− 1 = 7.66%
(Capital Gain) A bond has a par value of $1,000, a time to maturity of 10 years, and a coupon
rate of 8%, with interest paid annually. If the current market price is $800, what will be the
approximate capital gain of this bond over the next year if its yield to maturity remains
unchanged?
𝑃𝑉 = −800, 𝐹𝑉 = 1,000, 𝑛 = 10, 𝑃𝑀𝑇 = 80, 𝐶𝑃𝑇 𝐼𝑌 = 11.46%
𝐹𝑉 = 1,000, 𝑛 = 9, 𝑃𝑀𝑇 = 80, 𝐼𝑌 = 11.46, 𝐶𝑃𝑇 𝑃𝑉 = 811.70
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 = 𝑃1 − 𝑃0
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 = $811.70 − $800.00
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 = $11.70
(Par Value) A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond selling
above or below par value?
𝐼𝑓 𝑡ℎ𝑒 𝑌𝑇𝑀 > 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑, 𝑏𝑜𝑛𝑑 𝑜𝑓𝑓𝑒𝑟𝑠 𝑝𝑟𝑖𝑐𝑒 𝑎𝑝𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑎𝑠 𝑖𝑡 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑏𝑜𝑛𝑑 𝑖𝑠 𝑠𝑒𝑙𝑙𝑖𝑛𝑔 𝑏𝑒𝑙𝑜𝑤 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒
(Bond Price and Total RoR) Consider a bond paying a coupon rate of 10% per year semiannually
when the market interest rate is only 4% per half-year. The bond has three years until maturity.
a. Find the bond’s price today and six months from now after the next coupon is paid
$1,000 × 10% × 0.5 = $50
𝑃0 = [𝑆𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 × 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 × 𝑛] + [$1,000 × 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 × 𝑛]
𝑃0 = [$50 × 4% × 6] + [$1,000 × 4% × 6]
𝑃0 = $1,052.42
𝑃1 = [$50 × 4% × 5] + [$1,000 × 4% × 5]
𝑃1 = $1,044.42
b. What is the total rate of return on the bond?
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 = 𝑅𝑜𝑅 =
𝑅𝑜𝑅 =
𝑆𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛+(𝑃1 −𝑃0 )
$50+($1,044.52−$1,052.42)
𝑃0
$1052.42
𝑅𝑜𝑅 = 0.0400 𝑜𝑟 4.00% 𝑝𝑒𝑟 𝑠𝑖𝑥 𝑚𝑜𝑛𝑡ℎ𝑠
(Stated and Expected YTM) A 10-year bond of a firm in severe financial distress has a coupon
rate of 14% and sells for $900. The firm is currently renegotiating the debt, and it appears that
the lenders will allow the firm to reduce coupon payments on the bond to one-half the
originally contracted amount. The firm can handle these lower payments. What are the stated
and expected yield the maturity of the bonds? The bond makes its coupon payments annually.
𝑛 = 10, 𝑃𝑉 = −900, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 140, 𝐶𝑃𝑇 𝐼𝑌 = 16.07%
𝑆𝑡𝑎𝑡𝑒𝑑 𝑌𝑇𝑀 = 𝐶𝑃𝑇 𝐼𝑌; 16.07%
𝑛 = 10, 𝑃𝑉 = −900, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 70, 𝐶𝑃𝑇 𝐼𝑌 = 8.53%
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑌𝑇𝑀 = 𝐶𝑃𝑇 𝐼𝑌
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑌𝑇𝑀 = 8.53%
(HPR) You buy an eight-year maturity bond that has a 6% current yield and a 6% coupon (paid
annually). In one year, promised yield to maturity has risen to 7%. What is your holding-period
return?
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑎𝑛 𝑎𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑜𝑓 6% 𝑖𝑚𝑝𝑙𝑦 𝑡ℎ𝑎𝑡 𝑏𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 𝑤𝑎𝑠 𝑎𝑡 𝑝𝑎𝑟 𝑎 𝑦𝑒𝑎𝑟 𝑎𝑔𝑜
𝐹𝑉 = −1,000, 𝑛 = 7, 𝐼𝑌 = 7, 𝐶𝑃𝑇 𝑃𝑉 = $946.11
𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑓𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒+𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒+𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝐹𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
−$1,000+$946.11+$60
$1,000
= 0.0061 𝑜𝑟 0.61%
(YTM and Realized Compound YTM) A two-year bond with par value $1,000 making annual
coupon payments of $100 is priced at $1,000
a. What is the yield to maturity of the bond
𝐵𝑜𝑛𝑑 𝑖𝑠 𝑠𝑒𝑙𝑙𝑖𝑛𝑔 𝑎𝑡 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒, 𝑖𝑡𝑠 𝑌𝑇𝑀 = 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒; 10%
b. What will be the realized compound yield to maturity if the one-year interest rate next
year turns out to be 8%, 10%, 12%
𝐶𝑜𝑢𝑝𝑜𝑛 𝑟𝑒𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑎𝑠 𝑟, 𝑡𝑜𝑡𝑎𝑙 𝑝𝑟𝑜𝑐𝑒𝑒𝑑𝑠 = [𝐶𝑜𝑢𝑝𝑜𝑛 × (1 + 𝑟) + (𝑃0 + 𝐶𝑜𝑢𝑝𝑜𝑛)]
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (8%) = [100 × (1 + 0.08) + (1,000 + 100)]
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (8%) = $1,208
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑌𝑇𝑀 = √
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑
𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒
−1
$1,208
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑌𝑇𝑀 (8%) = √$1,000 − 1 = 0.0991 𝑜𝑟 9.91%
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (10%) = [100 × (1 + 0.10) + (1,000 + 100)
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (10%) = $1,210
$1,210
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑌𝑇𝑀 (10%) = √$1,000 − 1 = 0.100 𝑜𝑟 10.00%
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (12%) = [100 × (1 + 0.12) + (1,000 + 100)
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑒𝑑 (12%) = $1,212
$1,212
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑌𝑇𝑀 (12%) = √$1,000 − 1
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑌𝑇𝑀 (12%) = 0.01009 𝑜𝑟 10.09%
(YTM and Interest Rates) Fincorp issues tow bonds with 20-year maturities. Bond bonds are
callable at $1,050. The first bond is issued at a deep discount with a coupon rate of 4% and a
price of $580 to yield 8.4%. The second bond is issued at par value with a coupon rate of 8.75%
a. What is the yield to maturity of the par bond?
𝑌𝑇𝑀 𝑜𝑓 𝑝𝑎𝑟 𝑏𝑜𝑛𝑑 = 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒; 8.75%
b. If you expect rates to fall substantially in the next two years, which bond would you
prefer to bond?
4% 𝑚𝑜𝑟𝑒 𝑎𝑡𝑡𝑟𝑎𝑐𝑡𝑖𝑣𝑒, 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑓𝑎𝑟 𝑏𝑒𝑙𝑜𝑤 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑚𝑎𝑟𝑘𝑒𝑡 𝑦𝑖𝑒𝑙𝑑𝑠, 𝑝𝑟𝑖𝑐𝑒 𝑓𝑎𝑟 𝑏𝑒𝑙𝑜𝑤 𝑐𝑎𝑙𝑙 𝑝𝑟𝑖𝑐𝑒
(Yield To Call) A 30-year maturity, 6% coupon bond paying coupons semiannually is callable in
five years at a call price of $1,100. The bond currently sells at a yield to maturity of 5% (2.5%
per half-year)
a. What is the yield to call
𝑛 = 60, 𝐼𝑌 = 2.5, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 30, 𝑃𝑉 = $1,154.5433
𝑛 = 10, 𝑃𝑉 = −11,54.54, 𝐹𝑉 = 1,100, 𝑃𝑀𝑇 = 30, 𝐶𝑃𝑇 𝐼𝑌 = 2.1703%
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 𝐼𝑌 × 2
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 2.1703% × 2 = 4.34%
b. What is the yield to call if the call price is only $1,050?
𝑛 = 10, 𝑃𝑉 = −1,154.54, 𝐹𝑉 = 1,100, 𝑃𝑀𝑇 = 30, 𝐶𝑃𝑇 𝐼𝑌 = 1.7625%
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 𝐼𝑌 × 2
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 1.7625% × 2 = 3.52%
c. What is the yield to call if the call price is $1,100 but the bond can be called in two years
instead of five years?
𝑛 = 4, 𝑃𝑉 = −1,154.54, 𝐹𝑉 = 1,100, 𝑃𝑀𝑇 = 30, 𝐶𝑃𝑇 𝐼𝑌 = 1.4426%
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 𝐼𝑌 × 2
𝑌𝑖𝑒𝑙𝑑 𝑡𝑜 𝑐𝑎𝑙𝑙 = 1.4426% × 2 = 2.89%
(Imputed Interest) A newly issued 20-year-maturity, zero-coupon bond is issued with a yield to
maturity of 8% and a face value $1,000. Find the imputed interest income in the first, second,
and last year of the bon’s life.
𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 = (1+𝑌𝑇𝑀)𝑇
$1,000
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 (0) = (1.08)20 = $214.55
$1,000
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 (1) = (1.08)19 = $231.71
$1,000
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 (2) = (1.08)18 = $250.25
$1,000
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 (19) = (1.08)1 = $925.93
$1,000
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 (2) = (1.08)0 = $1,000
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝑃𝑇 − 𝑃𝑇−1
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 (1) = $231.71 − $214.55 = $17.16
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 (2) = $250.25 − $231.71 = $18.54
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 (𝑙𝑎𝑠𝑡) = $1,000 − $925.92 = $74.07
(Bond Price and HPR) Assume you have a one-year investment horizon and are trying to choose
among three bonds. All have the same degree of default risk and mature in 10 years. The first is
a zero-coupon bond that pays $1,000 at maturity. The second has an 8% coupon rate and pays
the $80 coupon once per year. The third has a 10% coupon rate and pays the $1,00 coupon
once per year. Assume that all bonds are compounded annually.
a.
𝐼𝑌
𝐼𝑌
𝐼𝑌
If all three bonds are now priced to yield 8% to maturity, what are their prices?
= 8, 𝑛 = 10, 𝑃𝑀𝑇 = 0, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $463.19
= 8, 𝑛 = 10, 𝑃𝑀𝑇 = 80, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $1,000.00
= 8, 𝑛 = 10, 𝑃𝑀𝑇 = 100, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $1,134.20
b. If you expect their yield to maturity to be 8% at the beginning of next year, what will
their prices be then?
𝐼𝑌 = 8, 𝑛 = 9, 𝑃𝑀𝑇 = 0, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $500.25
𝐼𝑌 = 8, 𝑛 = 9, 𝑃𝑀𝑇 = 80, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $1,000.00
𝐼𝑌 = 8, 𝑛 = 9, 𝑃𝑀𝑇 = 1000, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $1,124.94
c. What is your rate of return on each bond during the one-year holding period?
(𝑃 −𝑃 +𝐶𝑜𝑢𝑝𝑜𝑛)
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 1 0
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝑃0
$500.25−$463.19+$0
$463.19
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 8%
$1,000−$1,000+$80
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
$1,000
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 8%
$1,124.94−$1,134.20+$100
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
$1,134.20
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 8%
(Forward Rate, Yield Curve, YTM, and Expected Total Return) The yield curve for default-free
zero-coupon bonds is currently as follows:
Maturity (years)
YTM
1
10%
2
11%
3
12%
a. What are the implied one-year forward rates?
Maturity (years)
YTM
1
10.0%
2
11.0%
3
12.0%
(1+𝑌𝑇𝑀)𝑡
𝑡−1
𝑡−1 )
2
(1.11)
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 = (1+𝑌𝑇𝑀
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 (2) =
(1.10)
(1.12)3
Forward rate
a
b
−1
− 1 = 12.01%
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 (3) = (1.11)2 − 1 = 14.03%
b. Assume that the pure expectations hypothesis of the term structure is correct. If the
market expectations are accurate, what will the pure yield curve (that is, the yields to
maturity on one and two-year zero-coupon bonds) be next year?
𝑆ℎ𝑖𝑓𝑡 𝑢𝑝𝑤𝑎𝑟𝑑 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑖𝑠 𝑦𝑒𝑎𝑟 ′ 𝑠𝑢𝑝𝑤𝑎𝑟𝑑 𝑠𝑙𝑜𝑝𝑖𝑛𝑔 𝑦𝑖𝑒𝑙𝑑 𝑐𝑢𝑟𝑣𝑒
c. What will be the yield to maturity on two-year zeros?
𝑃𝑎𝑟 𝑣𝑎𝑙𝑢𝑒
𝑃𝑟𝑖𝑐𝑒 = (1+𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 )×(1+𝑔𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 )
𝑡
1,000
𝑡−1
𝑃𝑟𝑖𝑐𝑒2 = (1.1403×1.1201) = $782.93
𝑛 = 2, 𝑃𝑉 = −782.92, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 13.01%
d. If you purchase a two-year zero-coupon bond now, what is the expected total rate of
return over the next year? Ignore taxes. Compute the three-year return as well.
𝑛 = 2, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐼𝑌 = 11.0%, 𝐶𝑃𝑇 𝑃𝑉 = $811.62
𝑛 = 3, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐼𝑌 = 12.0%, 𝐶𝑃𝑇 𝑃𝑉 = $711.78
1,000
𝑃𝑟𝑖𝑐𝑒1 =
= $892.78
𝐸(𝑟) =
(1.1201)
𝑃𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑅𝑎𝑡𝑒
−1
𝑃𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑
892.78
𝐸(𝑡𝑤𝑜 𝑦𝑒𝑎𝑟) = 811.62 − 1
𝐸(𝑡𝑤𝑜 𝑦𝑒𝑎𝑟) = 10.00%
782.93
𝐸(𝑡ℎ𝑟𝑒𝑒 𝑦𝑒𝑎𝑟) =
−1
711.78
𝐸(𝑡ℎ𝑟𝑒𝑒 𝑦𝑒𝑎𝑟) = 10.00%
(Forward Rate, Hypothesis, and Theory) The yield to maturity on one-year zero-coupon bonds
is 8%. The yield to maturity on two-year zero-coupon bonds is 9%.
a. What is the forward rate of interest for the second year?
𝑓2 𝑖𝑠 𝑟𝑎𝑡𝑒 𝑡ℎ𝑎𝑡 𝑚𝑎𝑘𝑒𝑠 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 1 𝑦𝑒𝑎𝑟 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 2 𝑦𝑒𝑎𝑟
(1 + 𝑌𝑇𝑀) × (1 + 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒) = (1 + 𝑌𝑇𝑀)𝑡
(1 + 8%) × (1 + 𝑓2 ) = (1 + 9%)2
𝑓2 = 0.1001 𝑜𝑟 10.01%
b. If you believe in the expectations hypothesis, what is your best guess as to the expected
value of the short-term interest rate next year?
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑠ℎ𝑜𝑟𝑟𝑡 𝑡𝑒𝑟𝑚 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒; 10.01%
c. If you believe in the liquidity preference theory, is your best guess as to next year’s
short-term interest rate higher or lower than in (b)?
𝐿𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 > 𝑠ℎ𝑜𝑟𝑡 − 𝑡𝑒𝑟𝑚 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡; 𝑠𝑜 𝑙𝑜𝑤𝑒𝑟
(Forward Rate and Expected Yields) Consider the following $1,000 par value zero-coupon
bonds:
Bond
Years until maturity
Yield to maturity
A
1
5%
B
2
6%
C
3
6.5%
D
4
7%
a. According to the expectations hypothesis, what is the market’s expectation of the oneyear interest rate three years from now?
(1+𝑌𝑇𝑀)𝑡
𝑓𝑡 = (1+𝑌𝑇𝑀
𝑓4 =
(1.07)4
𝑡−1
𝑡−1 )
(1.065)3
−1
− 1 = 8.51%
𝐸𝑥𝑝𝑒𝑐𝑡𝑎𝑡𝑖𝑜𝑛 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 𝐼𝑅 = 𝑓𝑡 ; 8.51%
b. What are the expected values of next year’s yields on bonds with maturities of 1 year, 2
years, 3 years?
𝑓2 =
𝑃𝑡 =
1.062
1.05
− 1 = 7.01%,
𝑃𝑎𝑟 𝑣𝑎𝑙𝑢𝑒
(1+𝑓𝑡+1 )×(1+𝑓𝑡)×(1+𝑓𝑛 )
1,000
𝑓3 =
,
1.0653
1.062
1,000
− 1 = 7.51%
𝑓4 =
𝑃1 = 1.0701 = $934.50
𝑃2 = 1.0701×1.0751 = $869.21
1,000
𝑃3 = 1.0701×1.0751×1.0851 = $801.04
𝑛 = 1, 𝑃𝑉 = −934.50, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 7.01%
𝑛 = 2, 𝑃𝑉 = −869.24, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 7.26%
𝑛 = 3, 𝑃𝑉 = −801.04, 𝐹𝑉 = 1,000, 𝑃𝑀𝑇 = 0, 𝐶𝑃𝑇 𝐼𝑌 = 7.68%
1.074
1.0653
− 1 = 8.51%
(HPR, OID, Realized Compound Yield) A newly issued bond pays its coupons once a year. Its coupon rate is 5%, its
maturity is 20 years, and its yield to maturity is 8%
a.
Find the holding-period return for a one-year investment period if the bond is selling at a yield to maturity
of 7% by the end of the year
𝑛 = 20, 𝑃𝑀𝑇 = 50, 𝐹𝑉 = 1,000, 𝐼𝑌 = 8, 𝐶𝑃𝑇 𝑃𝑉 = 705.46
𝑛 = 19, 𝑃𝑀𝑇 = 50, 𝐹𝑉 = 1,000, 𝐼𝑌 = 7, 𝐶𝑃𝑇 𝑃𝑉 = 793.29
𝐶𝑜𝑢𝑝𝑜𝑛+𝑃1 −𝑃0
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝑃
0
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 =
$50+$793.29−$705.46
$705.46
= 19.54%
b.
If you sell the bond after one year when its yield is 7%, what taxes will you owe if the tax rate on interest
income is 40% and the tax rate on capital gains income is 30%? The bond is subject to original-issue
discount (OID) tax treatment
𝑂𝐼𝐷 𝑡𝑎𝑥 𝑟𝑢𝑙𝑒, 𝑐𝑜𝑠𝑡 𝑏𝑎𝑠𝑖𝑠 𝑎𝑛𝑑 𝑖𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑢𝑛𝑑𝑒𝑟 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 𝑜𝑏𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝑎𝑡 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑦𝑖𝑒𝑙𝑑 𝑡𝑜 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑎𝑛𝑑 𝑟𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑏𝑦 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒
𝑛 = 19, 𝑃𝑀𝑇 = 50, 𝐹𝑉 = 1,000, 𝐼𝑌 = 8, 𝐶𝑃𝑇 𝑃𝑉 = 711.89
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑡𝑎𝑥 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝑃1 − 𝑃0
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑡𝑎𝑥 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $711.89 − $705.46 = $6.43
𝑇𝑎𝑥 𝑤𝑖𝑡ℎ 𝑐𝑜𝑢𝑝𝑜𝑛 = 𝑇𝑎𝑥 𝑟𝑎𝑡𝑒 × (𝐶𝑜𝑢𝑝𝑜𝑛 + 𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑡𝑎𝑥 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡)
𝑇𝑎𝑥 𝑤𝑖𝑡ℎ 𝑐𝑜𝑢𝑝𝑜𝑛 = 40% × ($50 + $6.43) = $22.57
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑝𝑟𝑖𝑐𝑒 𝑎𝑡 𝑌𝑇𝑀𝑁𝑒𝑤 − 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑌𝑖𝑒𝑙𝑑 𝑃𝑟𝑖𝑐𝑒 = 𝑃1 − 𝑃1 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑
𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 = $793.29 − $711.89 = $81.40
𝑇𝑎𝑥 𝑜𝑛 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛 = 30% × $81.40 = $24.42
𝑇𝑜𝑡𝑎𝑙 𝑡𝑎𝑥𝑒𝑠 = $22.57 + $24.42 = $46.99
c.
What is the after-tax holding-period return on the bond?
𝐴𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 𝐻𝑃𝑅 =
$50+($793.29−$705.46)−$46.99
$705.46
= 12,88%
d.
Find the realized compound yield before taxes for a two-year holding period assuming you sell the bond
after two years, the bond yield is 7% at the end of the second year, and the coupon can be reinvested for
one year at a 3% interest rate
𝑛 = 18, 𝑃𝑀𝑇 = 50, 𝐹𝑉 = 1,000, 𝐼𝑌 = 7, 𝐶𝑃𝑇 𝑃𝑉 = $798.82
𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒 𝑓𝑟𝑜𝑚 𝑡𝑤𝑜 𝑐𝑜𝑢𝑝𝑜𝑛𝑠 = (𝐶𝑜𝑢𝑝𝑜𝑛 × (1 + 𝑟𝑒𝑖𝑛𝑣𝑒𝑠𝑡 𝑟𝑎𝑡𝑒)) + 𝐶𝑜𝑢𝑝𝑜𝑛
𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒 𝑓𝑟𝑜𝑚 𝑡𝑤𝑜 𝑐𝑜𝑢𝑝𝑜𝑛𝑠 = ($50 × 1.03) + $50 = $101.50
𝑇𝑜𝑡𝑎𝑙 𝑎𝑓𝑡𝑒𝑟 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠 = 𝑃𝑉 + 𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑎𝑓𝑡𝑒𝑟 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠 = $798.82 + $101.50 = $900.32
𝑃0 × (1 + 𝑟)𝑡 = 𝑡𝑜𝑡𝑎𝑙 𝑎𝑓𝑡𝑒𝑟 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠
$705.46 × (1 + 𝑟)2 = $900.32
𝑟 = 12.97%
e.
Use the tax rates in part (b) to compute the after-tax two-year realized compound yield. Remember to
take account of OID tax rules
𝑁𝑒𝑡 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 = 𝐶𝑜𝑢𝑝𝑜𝑛 − 𝑇𝑎𝑥 𝑜𝑛 𝑐𝑜𝑢𝑝𝑜𝑛 − 𝑇𝑎𝑥 𝑜𝑛 𝑖𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑁𝑒𝑡 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 = $50 − 20 − (40% × $6.43) = $27.43
𝐼𝑓 𝑦𝑜𝑢 𝑟𝑒𝑖𝑛𝑣𝑒𝑠𝑡 𝑦𝑒𝑎𝑟 − 1 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑎𝑡 𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 𝑜𝑓 3% × (1 − 40%) = 1.8%
𝑌𝑒𝑎𝑟 2 = $27.43 × 1.018 = $27.92
𝑛 = 18, 𝐼𝑌 = 8, 𝑃𝑀𝑇 = 50, 𝐹𝑉 = 1,000, 𝐶𝑃𝑇 𝑃𝑉 = $718.84
𝐼𝑚𝑝𝑢𝑡𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $718.84 − $711.89 = $6.95
$829.97 = $798.82 − (40% × $6.95) + ($50 × (1 − 40%)) − (30% × ($798.82 − $718.84) + $27.92
𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑔𝑟𝑜𝑤𝑠 𝑓𝑟𝑜𝑚 $705.46 𝑡𝑜 $829.97 𝑎𝑓𝑡𝑒𝑟 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠
705.46 × (1 + 𝑟)2 = 829.97
𝑟 = 8.47%
Chapter Connect 13 Summary
-
-
-
-
One approach to firm valuation is to focus on the firm’s book value, either as it appears
on the balance sheet or adjusted to reflect the current replacement cost of assets or the
liquidation value. Another approach is to focus on the present value of expected future
dividends
The dividend discount model holds that the price of a share of stock should equal the
present value of all future dividends per share, discounted at an interest rate
commensurate with the risk of the stock
The constant-growth version of the DDM asserts that if dividends are expected to grow
at a constant rate forever, then the intrinsic value of the stock is given by the formula:
𝐷1
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑔𝑟𝑜𝑤𝑡ℎ 𝐷𝐷𝑀: 𝑉0 = 𝑘−𝑔
The constant-growth version of the DDM is simplistic in its assumption of a constant
value of g. There are more sophisticated multistage versions of the model for more
complex environments. When the constant-growth assumption is reasonably satisfied,
however, the formula can be inverted to infer the market capitalization for the stock:
𝐷
𝑘 = 𝑃1 + 𝑔
0
-
Stock market analysts devote considerable attention to a company’s price-earnings
ratio. The P/E ratio is a useful measure of the market’s assessment of the firm’s growth
opportunities. Firms with no growth opportunities should have a P/E ratio that is just
the reciprocal of the capitalization rate, k. As no growth opportunities become a
progressively more important component of the total value of the firm, the P/E ratio will
increase
- Many analysts form their estimates of a stock’s value by multiplying their forecast of
next year’s EPS by a predicted P/E multiple. Some analysts mix the P/E approach with
the dividend discount model. They use an earnings multiplier to forecast the terminal
value of shares at a future date an add the present value of that terminal value to the
present value of all interim dividend payments
- The free cash flow approach is the one used most in corporate finance. The analyst first
estimates the value of the firm as the present value of expected future free cash flows
to the entire firm and then subtracts the value of all claims other than equity.
Alternatively, the free cash flow to equity can be discounted at a rate appropriate to the
risk of the stock.
- The models presented in this chapter can be used to explain or to forecast the behavior
of the aggregate stock market. The key macroeconomic variables that determine the
level of stock prices in the aggregate are interest rates and corporate profit
𝐷1
𝐷2
𝐷𝐻 +𝑃𝐻
𝐼𝑛𝑠𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑣𝑎𝑙𝑢𝑒 = 𝑉0 = 1+𝑘
+ (1+𝑘)
2 + ⋯ + (1+𝑘)𝐻
𝐺𝑟𝑜𝑤𝑡ℎ 𝑜𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑖𝑒𝑠: 𝑃𝑟𝑖𝑐𝑒 =
𝑃0
𝐸1
𝑘
1
+ 𝑃𝑉𝐺𝑂
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡 𝑜𝑓 𝑃𝐸 𝑟𝑎𝑡𝑖𝑜: 𝐸 = 𝑘 (1 +
1
𝑃𝑉𝐺𝑂
𝐸
( 1)
𝑘
𝐹𝑟𝑒𝑒 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤: 𝐹𝐶𝐹𝐹 = 𝐸𝐵𝐼𝑇(1 − 𝑡𝑐 ) + 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 − 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐸𝑥𝑝 − 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑁𝑊𝐶
𝐹𝑟𝑒𝑒 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤: 𝐹𝐶𝐹𝐸 = 𝐹𝐶𝐹𝐹 − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑒𝑥𝑝𝑒𝑛𝑠𝑒 × (1 − 𝑡𝑐 ) + 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑛𝑒𝑡 𝑑𝑒𝑏𝑡
Chapter 13 Slides Summary
-
Equity Valuation
o Book value – net worth of common equity according to a firm’s balance sheet
o Alternatives to book value
▪ Liquidation value – net amt by selling assets of firm and paying debt
▪ Replacement cost – cost to replace firm’s assets
▪ Tobin’s q – ratio of firm’s market value to replacement cost
o 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝐻𝑃𝑅 = 𝐸(𝑟) =
-
𝐸(𝐷1 )+[𝐸(𝑃1 )−𝑃0
𝑃0
Intrinsic Value versus Market Price
o Intrinsic value – pv of firm’s expected future net cash flows disc by ror
o Market Capitalization Rate – est of appropriate discount rate for cash flows (k)
𝐷 +𝑃
o 𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑉𝑎𝑙𝑢𝑒 (𝑜𝑛𝑒 𝑝𝑒𝑟𝑖𝑜𝑑): 𝑉0 = 1 1
1+𝑘
𝐷
𝐷
𝐷 +𝑃
1
2
𝐻
𝐻
o 𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑉𝑎𝑙𝑢𝑒 (ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑): 𝑉0 = 1+𝑘
+ (1+𝑘)
2 + ⋯ + (1+𝑘) 𝐻
𝐷
𝐷
𝐷
1
2
3
o 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑀𝑜𝑑𝑒𝑙: 𝑉0 = 1+𝑘
+ (1+𝑘)
2 + (1+𝑘)3 …
-
Dividend Discount Models
o 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐺𝑟𝑜𝑤𝑡ℎ 𝐷𝐷𝑀: 𝑉0 =
𝐷1
𝑘−𝑔
o Constant Growth DDM has a higher value when more dividend, lower k, higher g
𝐷
𝑃 −𝑃
𝐷
o 𝐸(𝑟) = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛𝑠 𝑌𝑖𝑒𝑙𝑑 = 𝑃1 + 1𝑃 0 = 𝑃1 + 𝑔
0
-
0
0
o Dividend payout ratio – percentage of earnings paid as dividends
o Plowback/earnings retention ratio – proportion of firm’s earnings reinvested
o Present value of growth opportunities (PVGO)
𝐸
o 𝑃0 = 𝑁𝑜 𝑔𝑟𝑜𝑤𝑡ℎ 𝑉𝑎𝑙𝑢𝑒 𝑝𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 + 𝑃𝑉𝐺𝑂 = 𝑘1 + 𝑃𝑉𝐺𝑂
o Two-stage DDM – DDM in which dividend growth assumed to level off
o Multistage Growth Models – allow div to grow at different rates as firm matures
o As firm matures low reinvestment means a lower growth in dividends
o As firm matures high reinvestment means a higher growth in dividends
Price-Earnings Ratios
o Price-earnings multiple – ratio of stock’s price to earnings per share
𝑃
1
𝑃𝑉𝐺𝑂
o 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡 𝑜𝑓 𝑃𝐸 𝑟𝑎𝑡𝑖𝑜: 𝐸0 = 𝑘 [1 + 𝐸1 ]
1
(𝑘)
o 𝑃𝐸 𝑓𝑜𝑟 𝐹𝑖𝑟𝑚 𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝑎𝑡 𝐿𝑜𝑛𝑔 𝑅𝑢𝑛 𝑆𝑢𝑠𝑡𝑎𝑖𝑛𝑎𝑏𝑙𝑒 𝑃𝑎𝑐𝑒:
𝑃0
𝐸1
=
1−𝑏
𝑘−𝑔
=
1−𝑏
𝑘−𝑅𝑂𝐸×𝑏
𝑃
o PEG Ratio – ratio of P/E multiple to earnings growth rate:
𝑃
(𝐸0 )
1
𝑔
1−𝑏
o 𝑃𝐸 𝑅𝑎𝑡𝑖𝑜: 𝐸0 = 𝑘−𝑔
o
o
o
o
o
1
Riskier stocks have lower PE, higher required RoR, and a higher k
Pitfalls of P/E is acc flexibility to improve profit, large discretion in earnings
PE ratios decrease as inflation increases
Con Ed increases EPS over time, PepsiCo EPS remains flat over time
Con Ed PE ratio decreases over time, PepsiCo PE ratio increases over time
o
o
o
o
o
-
Cycl adjusted P/E ratio divide stock price by e(r), uses inflation-adjusted earnings
Combining PE Analysis and the DDM estimates stock price at horizon date
PE ratio highest before recession
CAPE lowest before recession, not inversely just gauged more accurately
Other comparative ratios
▪ Price-to-book – how aggressively market values firm
▪ Price-to-cash-flow – cash flows less affected by accounting then earnings
▪ Price-to-sales – for start-ups with no earnings
Free Cash Flow Valuation Approaches
o 𝐹𝑟𝑒𝑒 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤 (𝐹𝑖𝑟𝑚): 𝐹𝐶𝐹𝐹 = 𝐸𝐵𝐼𝑇 × (1 − 𝑡𝑐 ) + 𝐷𝑒𝑝 − 𝐶𝑎𝑝𝐸𝑥 − ∆𝑁𝑊𝐶
▪ 𝐸𝐵𝐼𝑇 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑎𝑛𝑑 𝑡𝑎𝑥𝑒𝑠
▪ 𝑡𝑐 = 𝐶𝑜𝑟𝑝𝑜𝑟𝑎𝑡𝑒 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒
o 𝐹𝐶𝐹𝐸 (𝐸𝑞𝑢𝑖𝑡𝑦): 𝐹𝐶𝐹𝐸 = 𝐹𝐶𝐹𝐹 − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝐸𝑥𝑝 × (1 − 𝑡𝑐 ) + ∆𝑁𝑒𝑡 𝐷𝑒𝑏𝑡
𝐹𝐶𝐹𝐹1
𝑃𝑇
o 𝑇𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐺𝑟𝑜𝑤𝑡ℎ: 𝐹𝑖𝑟𝑚 𝑉𝑎𝑙𝑢𝑒 = ∑𝑇𝑡=1 (1+𝑊𝐴𝐶𝐶)
𝑡 + (1+𝑊𝐴𝐶𝐶)𝑇
▪
𝐹𝐶𝐹𝐹
𝑇+1
Where 𝑃𝑇 = 𝑊𝐴𝐶𝐶−𝑔
𝐹𝐶𝐹𝐸𝑡
o 𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦: ∑𝑇𝑡=1 (1+𝑘
▪
-
Where 𝑃𝑇 =
𝐹𝐶𝐹𝐸𝑇+1
𝑘𝐸 −𝑔
𝐸
)𝑡
𝑃
+ (1+𝑘𝑇 )𝑇
𝐸
, 𝑘𝐸 = 𝐶𝑜𝑠𝑡 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
o Diff valuation models differ in practice and stem from simplifying assumptions
o Prob w DCF models: imprecise, investors like real assets over profit over growth
The Aggregate Stock Market
o Earnings mult at aggregate level, forecast profits for period, derive PE based on ir
o Earnings yield of S&P 500 and 10-Year Treasury Bond highly corr until recently
o S&P 500 Forecasts
▪ Treasury bond yield decreases as optimism increases
▪ Earnings yield decreases as optimism increases (T-bond yield plus 2.75%)
▪ Resulting P/E ratio increases as optimism increases (reciprocal of e(r))
▪ EPS forecast remains unchanged as optimism increases
▪ Forecast for S&P 500 increases as optimism increases
Chapter 10 Connect Summary
-
Debt securities are distinguished by their promise to pay a fixed or specified stream of income to
their holders. The coupon bond is a typical debt security
- Treasury notes and bonds have original maturities greater than one year. They are issued at or near
par value, with their prices quoted net of accrued interest
- Callable bonds should offer higher promised yield to maturity to compensate investors for the fact
that they will not realize full capital gains should the interest rate fall and the bonds be called away
from the at the stipulated call price. Bonds often are issued with a period of call protection. In
addition, discount bonds selling significantly below their call price offer implicit call protection
- Put bonds give the bondholder rather than the issuer the choice to terminate or extend the life of
the bond
- Floating-rate bonds pay a fixed premium over a reference short-term interest rate. Risk is limited
because the rate paid is tied to current market conditions
- The yield to maturity is the single discount rate that equates the present value of a security’s cash
flows to its price. Bond prices and yields are inversely related. For premium bonds, the coupon rate is
greater than the current yield, which is greater than the yield to maturity. These inequalities are
reversed for discount bonds
- The yield to maturity often is interpreted as an estimate of the average rate of return to an investor
who purchases a bond and holds it until maturity. However, when future rates are uncertain, actual
returns including reinvested coupons may diverge from yield to maturity. Related measures are yield
to call, realized compound yield, and expected (versus promised) yield to maturity
- Treasury bills are US government issued zero-coupon bonds with original maturities of up to one
year. Treasury STRIPS are longer-term default-free-zero-coupon bonds. The prices of zero-coupon
bonds rise exponentially over time, providing a rate of appreciation equal to the interest rate. The
IRS treats this price appreciation as imputed taxable interest income to the investor
- When bonds are subject to potential default, the stated yield to maturity is the maximum possible
yield to maturity that can be realized by the bondholder. In the event of default, however, that
promised yield will not be realized. To compensate bond investors for default risk, bonds must offer
default premiums, that is, promised yield more than those offered by default-free government
securities
- Bond safety often is measured using financial ratio analysis. Bond indentures offer Safeguards to
protect the claims of bondholders. Common indentures specify sinking fund requirements,
collateralization, dividend restrictions, and subordination of future debt
- Credit default swaps provide insurance against the default of a bond or loan. The swap buyer pays an
annual premium to the swap dealer but collects a payment equal to lost value if the loan later goes
into default
- The term structure of interest rates is the relationship between time to maturity and term to
maturity. The yield curve is a graphical depiction of the term structure. The forward rate is the breakeven interest rate that would equate the total return on a rollover strategy to that of a longer-term
zero-coupon bond
- The expectations hypothesis holds that forward interest rates are unbiased forecasts of future
interest rates. The liquidity preference theory, however, argues that long-term bonds will carry a risk
premium known as liquidity premium. A positive liquidity premium can cause the yield curve to slope
upward even if no increase in short rates is anticipated
1
1
1
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑎 𝐵𝑜𝑛𝑑: 𝑃𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑢𝑝𝑜𝑛 × 𝑟 [1 − (1+𝑟)𝑇 + 𝑃𝑎𝑟 𝑣𝑎𝑙𝑢𝑒 × (1+𝑟)𝑇
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑎 𝐵𝑜𝑛𝑑: 𝑃𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑢𝑝𝑜𝑛 × 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑇) + 𝑃𝑎𝑟 𝑣𝑎𝑙𝑢𝑒 × 𝑃𝑉 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑇)
(1+𝑦𝑛 )𝑛
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 1 + 𝑓𝑛 = (1+𝑦
𝑛−1 )
𝑛−1
𝐿𝑖𝑞𝑢𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚: 𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 − 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑠ℎ𝑜𝑟𝑡 𝑟𝑎𝑡𝑒
Chapter 10 Slides Summary
-
Bond Characteristics
o Bond – security that obligates issuer to make payments to holder over time
o Face Value = Par Value – payment to bondholder at maturity of bond
o Coupon Rate – bond’s annual interest payment per dollar of par value
o Zero-Coupon Bond – pays no coupons, sells at discount, pays par value at t
o Accrued interested and quoted bond prices – price no inclu int btw pmt dates
𝐴𝑛𝑛𝑢𝑎𝑙 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐷𝑎𝑦𝑠 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡
o 𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡(𝐴. 𝐼. ) =
× 𝐷𝑎𝑦𝑠 𝑏𝑡𝑤 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠
2
o
o
o
o
o
o
Callable bonds – corp bond provision, repurchased by issuer at specified p/n
Convertible bonds – let exchange bond for number of common stock shares
Puttable bonds – holder may choose to exchange for par value or more of n
Floating-rate bonds – coupon rates reset according to specified market date
Preferred stock – commonly pays fixed dividend
▪ Floating-rate preferred stock becoming more popular
▪ Dividends not normally tax deductible
▪ Corp that purchase other corp ref stock taxed only 30% of div received
Other domestic bond issuers
▪ Municipal bonds, FHLBB, GNMA (mae), FNMA (mae), FHLMC (mac), Farm
Foreign bonds – issued in diff country from sold country, denominated not usd
Eurobonds – denominated in foreign issue money different than that of market
Innovation in the bond market
▪ Maturity – usually 30, now can be 50 to 100 years
▪ Inverse floaters – coupon rate falls when interest rates rise
▪ Asset-backed bonds – income from assets used to service debt
▪ Pay-in-kind bonds – issuers can pay interest in cash or add bonds
▪ Catastrophe bonds – higher coupon rates to investors for risk
▪ Indexed bonds – pmt tied to general price index or price of commodity
▪ TIPS – par value increases with consumer price index
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡+𝑃𝑟𝑖𝑐𝑒 𝐴𝑝𝑟𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑃𝑟𝑖𝑐𝑒
o
𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 =
o
o
o
o
-
1+𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛
1+𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
Bond Pricing
o 𝐵𝑜𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 = 𝑝𝑣 𝑜𝑓 𝑐𝑜𝑢𝑝𝑜𝑛𝑠 + 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒
𝐶𝑜𝑢𝑝𝑜𝑛
𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒
o 𝐵𝑜𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 = ∑𝑇𝑡=1
𝑡 +
𝑇 ; T = maturity date; r = discount rate
(1+𝑟)
-
−1
(1+𝑟)
o
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝑐𝑜𝑢𝑝𝑜𝑛 × 𝑎𝑛𝑛𝑢𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑡) + 𝑝𝑎𝑟 × 𝑃𝑉 𝑓𝑎𝑐𝑡𝑜𝑟(𝑟, 𝑇)
o
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑢𝑝𝑜𝑛 ×
o
o
o
o
Prices fall as market interest rate rises
Interest rate fluctuations are primary source of bond market risk
Bonds with longer maturities more sensitive to fluctuations in interest rate
Bond pricing between coupon dates – invoice price = flat price + accrued interest
1−(1+𝑟)−𝑇
𝑟
1
+ 𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒 × (1+𝑟)𝑇
Bond Yields
o YTM – discount rate that makes pv of bond’s patments equal to price
o Current annual – annual coupon divided by bond’s price
o Premium bonds – bonds selling above par value
o Discount bonds – bonds selling below par value
o Yield to Call – calc like ytm, n until call replaces n until maturity, prem called
o Realized compound return – comp ror with all coupons reinvested until maturity
o Horizon analysis – bonds returns over horizon, based on forecasts of YTM +
o Reinvestment rate risk – uncertainty cumulative future value of reinv pmt
-
-
-
-
Bond Prices Over Time
o YTM versus Holding Period Return
▪ YTM measures ror if bond held to maturity
▪ HPR rate over period, depends on market price at end of the period
o Zero-Coupon bond – no coupon, all return is price appreciation
o STRIPS – creation of zero-coupon bonds from coupon-bearing notes and bonds
o After tax returns – built in price appreciation is interest, IRS calc
Default Risk and Bond Pricing
o Investment grade BBB and above by S&P or Baa and above by Moody’s
o Speculative BB or lower by S&P or Ba or lower by Moody’s
o Coverage ratios – company earnings to fixed costs
o Leverage ratios – debt to equity
o Liquidity ratios – current (assets to liabilities), quick (assets excl inv to liabilities)
o Profitability ratios – ror on assets or equity
o Cash flow-to-debt ratio – total cash flow to outstanding debt
o Indenture – contract between issuer and holder
o Sinking fund -indenture calling for issuer to repurchase bond before mature
o Subordination clause – restriction on additional borrowing
o Collateral – specific asset pledged against possible default
o Debenture – bond not backed by specific collateral
o YTM and Default – stated yield max yield, default premium incr promised yield
o Credit Default Swaps – insurance to buy protection against large losses
Yield Curve
o Graph of yield to maturity as a function of term to maturity
o Term structure of IR – relat btw ytm and terms across bonds
o Expectation hypothesis – ytm based on short-term interest rates
o Forward rate – inferred short-term roi for future period equal of rollover
o (1 + 𝑦𝑛 )𝑛 = (1 + 𝑦𝑛−1 )𝑛−1 × (1 + 𝑓𝑛 )
o Liquidity preference – risk prem for short extra demand, spread btw forward roi
o 𝑓𝑛 = 𝐸(𝑟𝑛 ) + 𝐿𝑖𝑞𝑢𝑑𝑖𝑡𝑦 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
Interest Rate Risk
o Bond prices and yield inversely related
o Increase in bond’s ytm smaller price chance than yield decrease of equal magnit
o Long-term bond prices more sensitive to ir changes than short-term
o As maturity increases, sensitive of bond p to changes in yield incre at lower rate
o IR risk inversely related to coupon rate, low-coupon more sensitive to IR
o Sensitivity of bond’ price-to-yield change inversely to current ytm
o Macaulay’s Duration (D)
▪ Measures effective bond maturity
▪ Weight average of the times until each payment weight prop to pv of pmt
𝐶𝐹𝑡
)
(1+𝑦)𝑡
𝐷 = ∑𝑇𝑡=1 𝑡 × 𝑤𝑡 where 𝑤𝑡 = 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒
▪
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑏𝑜𝑛𝑑 𝑝 𝑡𝑜 𝑦𝑡𝑚:
▪
o
(.
▪
∆𝑃
𝑃
∆(1+𝑦)
= 𝐷×[
𝐷
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 (𝐷 ∗) = 1+𝑦 ,
∆𝑃
𝑃
1+𝑦
]
= −𝐷 ∗× ∆𝑦
Determining Duration
▪ Zero-coupon bond’s duration is time to maturity
▪ Holding time and ytm same, duratio and ir sensitivity high when coup low
▪ Coup rate same, bond dur and ir sens inc w T, dur inc w T for bonds ± par
▪ All constant, durat and ir sens of coupon higher when bond ytm lower
1+𝑦
▪ 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝑦