Fixed-income securities
Bond pricing formula
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P = C  { [ 1 – 1 / (1 + i)N ] / i } + FV / (1 + i)N .
FV is the face (par) value of the bond.
C is coupon payment.
i is the period discount rate.
If coupons are paid out annually, i = YTM. If coupons are paid
out semiannually, i = YTM/2.
N is the number of periods remaining.
The first term, C  { [ 1 – 1 / (1 + i)N ] / i }, is the present value of
coupon payments, i.e., an annuity.
The second term, FV / (1 + i)N, is the present value of the par.
Yield (yield to maturity, YTM)
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The (quoted, stated) discount rate over a
year.
YTM, like other discount rates, has 2
components: (1) risk-free component, and (2)
risk premium.
Determined by the market.
Time-varying.
Bond pricing example, I
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Suppose that you purchase on May 8 this year a Tbond matures on August 15 in 2 years. The coupon
rate is 9%. Coupon payments are made every
February 15 and August 15. That is, there are still 5
coupon payments to be collected: August this year, 2
payments next year, and 2 payments the year after
next year. The par is $1,000. The YTM is 10%.
What is the fair price of the bond?
Bond pricing example, II
08/15/Y0
02/15/Y1
08/15/Y1
02/15/Y2
08/15/Y2
Bond Quotation
Accrued Interest
20.52197802
Actual Payment
998.8745947
Cash flow
45
45
45
45
1045
Discount rate
0.05
PV
42.85714
40.81633
38.87269
37.02161
818.7848
978.3526
coupon payment * (days since last payment /
days in this period)
% Par
99.88745947
Bond pricing example, III
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Calculator: 45 PMT; 1000 FV; 5 N; 5 I/Y;
CPT PV. The answer is: PV = -978.3526.
Bond quotations ignore accrued interest.
Bond buyer will pay quoted price
($978.3526) and accrued interest
($20.5220), a total of $998.8746, to the
seller.
YTM example
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Northern Inc. issued 12-year bonds 2 years
ago at a coupon rate of 8.4%. The
semiannual-payment bonds have just make
its coupon payments. If these bonds
currently sell for 110% of par value, what is
the YTM?
Calculator: 42 PMT; 1000 FV; 20 N; -1100
PV; CPT I/Y. The answer is: I/Y = 3.4966.
YTM = 2 × 3.4966 = 6.9932 (%).
A few observations
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Bond price is a function of (1) YTM, (2) coupon (rate), and (3)
maturity.
The YTMs of various bonds move more or less in harmony
because the general interest rate environment (e.g., Fed
policies) exerts a market-wide force on every bonds (that is, the
risk-free component).
As YTMs move (in harmony), bond prices move by different
amounts.
The reason for this is that every bond has its unique coupon
(rate) and maturity specification.
It is therefore useful to study price-yield curves for different
coupon rates or different maturities.
Price-yield curves and coupon rates
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Negative slopes: price and YTM have an
inverse relation.
When people say “the bond market went
down,” they mean prices are down, but
interest rates (yields, YTMs) are up.
When coupon rate = YTM, the bond has a
price of 100%.
Price-yield curves and maturity
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Everything else being equal, bonds with
longer maturities have steeper price-yield
curves.
That is, the prices of long bonds are more
sensitive to interest rate changes, i.e., higher
interest rate risk.
Assignment
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Use Excel to duplicate both Figure 3.3 and
3.4.
Due in a week.
Relative performance vs. a benchmark
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Suppose that you are a bond manager and your
(your company’s) goal is to have good relative
performance with respect to a 20-year bond index.
After studying the interest rate environment, you
believe that interest rates will fall in the near future
(and your belief is not widely shared by investors
yet). Should you have a bond portfolio that has an
average maturity longer or shorter than 20 years?
What if you believe interest rates will rise in the near
future?