Not To Be Naïve about Duration
1. The duration D we have been discussing also known
as Macaulay duration.
2. First derivative of price-yield curve is
D
(1  i )
and is known as modified duration. Found in
%ΔPB formula.
3. Convexity is second derivative of price-yield curve.
Is a complicated expression (not studied here).
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Managing Interest Rate Risk (a)
Duration is the holding period for which reinvestment
risk exactly offsets price risk. Designed to give investor
the YTM that was in effect at time bond purchased.
A way duration is used: If have a $5 million liability 7.5
years from now, buy a bond (or a portfolio of bonds)
today that has a duration of 7.5 years.
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Example 18: Rebalancing Bond Portfolio
Consider the $20,000 portfolio ($4,000 in D = 5, $10,000
in D = 7, $6,000 in D = 9) of Example 17.
How much in D = 9 bonds should be sold, and how much
in D = 5 bonds should be purchased, to reduce Portfolio
D to 6.80?
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Eliminating Interest Rate Risk (b)
•
Zero-coupon approach (best way). Buy high quality
“zeros” with maturity equal to desired holding period.
Locks in YTM. No reinvestment risk because no
coupons payments, no price risk when held to maturity.
•
Duration matching (next best way). Selecting a portfolio
of bonds whose duration matches desired holding period.
Theoretically perfect, but only approximately perfect in
real world as per footnote 8 on p. 162.
•
Maturity matching (don’t use). That is, selecting bonds
with terms to maturity equal to desired holding period.
Don’t use. Doesn’t work for eliminating interest rate
risk.
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Three Bond Yields
1. Yield-to-maturity. Assumes
• Issuer makes all payments as promised
• Coupon payments are reinvested at the rate that
the bond yielded when purchased
• Investor holds bond to maturity
2. Realized yield. An ex post calculation of the bond’s
yield while holding it. For instance, holder sells a
bond before maturity.
3. Expected yield. An ex ante calculation of a bond’s
expected yield based upon anticipated cash flows.
All trial-and-error “number crunching” calculations
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Example 19: Yield-to-Maturity
Yield-to-maturity. The annual rate that causes all cash
flows to discount back to the bond’s market price. Solve
by trial-and-error.
What is the yield-to-maturity on a 12-year, 8% coupon
bond (semi-annual payments) whose price is $1,097.37?
40
40
1000
1097.37 
 

1
24
(1  r )
(1  r )
(1  r ) 24
Find r-value that fits. 25 terms (lot of work). Then
double to obtain yield-to-maturity answer.
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Example 19: (con’t)
40
1097.37 

1
(1  r )
40
1000


24
(1  r )
(1  r ) 24
Easier if we employ annuity formula. Only 2 terms.
1  (1  r ) 24 
24
 40 

1000(1

r
)

r


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Example 20: Realized Yield
Realized yield. Rate that causes all cash flows to
discount back to the purchase price. What did bond
project yield (annual rate) now that it is over?
Paid $995 for a new 6% coupon bond (semi-annual
payments). Sold after 3 years for $1,068 (minutes after
coupon payment). What was realized yield?
995.00 
30

(1  r )

30
1068

(1  r )6 (1  r )6
1  (1  r ) 6 
6
 30 

1068(1

r
)

r


Principle: If trial r makes RHS too low, decrease r
If trial r makes RHS too high, increase r
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Example 21: Expected Yield Calculation
Expected yield -- the discount rate that causes the sum of
the PV’s of all expected cash flows to equal purchase
price. Solve by trial-and-error.
Let’s do 6-month clock version of prob on p. 148 in book:
Purchased a new 8% 10-yr, semiannual coupon payment
bond at par. Plan to sell in 2 yrs when bond expected to
yield 6% (at which point PB = 1,126) . What is project’s
expected yield?
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Fixed Income Securities (a)
Fixed income security are liabilities to their issuers.
Assets
Liabilities
Capital
on issuer’s
books in here
Securities issued by governments and corporations
that are designed to pay contractually a specified
income over a specified time horizon.
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