Math 325
Ch 11 – Duration, Convexity and Immunization
Duration:
The goal of this section is to develop indices to measure the timing of future cash flows.
a) Term to maturity is the crudest index. It distinguishes a 10-yr bond from a 20-yr bond but would
not distinguish between two 10-yr bonds, one with 5% coupons and the other with 10% coupons.
b) Method of equated time is a better index. It is the weighted average of the various times of
payments, where the weights are the various amounts paid.
For example, a 10-yr bond with 5% coupons:
A 10-yr bond with 10% coupons:
In this case, a lower coupon rate implies ___________ term.
c) Macaulay Duration or simply, DURATION, is a much better index. It is the weighted average of
various times of payments, where the weights are the present values of the various payments amounts,
instead of just the payment amounts in case of method of equated time.
Notes:
1. If i = 0, then ________. So, method of equated time is a special case of duration that ignores interest.
2. The duration is a decreasing function of i.(As i increases,the payments at later times are discounted
more with the smaller i, giving less weight to later times).
3. If there is only one future cash flow, then duration is the point in time at which the cash flow is
made.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Example 1. Find the Macaulay duration of the following investments assuming the effective rate of
interest is 8%.
a) A 10-yr zero-coupon bond.
b) A 10-yr bond with 8% annual coupons.
c) A 10-yr mortgage repaid with level annual payments of principal and interest.
d) A preferred stock paying level annual dividends into perpetuity.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Interest Rate Sensitivity
Let P(i) = S Rt(1+ i) –t be the present value of a set of future cash flows. We are interested in the rate at
which P(i) changes as i changes.
We define volatility of this present value as:
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Math 325
Ch 11 – Duration, Convexity and Immunization
Modified and Macaulay duration are important analytical tools in financial analysis.
1. Reinvestment Risk
2. Modified duration provides a method to estimate the change in PV of a series of cash flows when the
yield rate changes.
Example 2. Estimate the price of the 10-yr bond with 8% coupons from Example 1, if the yield rate
rises from 8% to 9%.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Convexity
The convexity of the present value of a set of cash flows is defined to be
“Convexity” because the price curve is convex(concave up) in shape.
Example 3. The current price of an annual coupon bond is $100. The yield to maturity is an effective
rate of 7% and dP/di = – 650.
a) Calculate the Macaulay duration of the bond.
b) Using the given information, estimate the price of the bond when i = 8% instead of 7%.
c) Refine your price estimate by using both modified duration and convexity given that dv/di = –800.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Macaulay convexity: convexity based on the force of interest d.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Analysis of Portfolios
Multiple Securities
Consider a portfolio of m different securities. The modified duration of the entire portfolio is simply the
weighted average of modified durations of each security in the portfolio, where the weights are the
fraction of the entire portfolio applicable to each security.
Convexity of a portfolio:
Example 4. An investment fund wants to invest $100,000 in a mix of 5-yr zero coupon bonds yielding
6% and 10-yr zero coupon bonds yielding 7%, such that the modified duration of the portfolio will
equal 7. Find the amount which should be invested in each type of bond.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Duration and Passage of Time
As time apsses, the duration of a security with multiple future payments does decline, with a very
interesting exception:
The duration duration of a security immediately after a cash flow is actually larger than
immediately before that cash flow. (An effect of weighted avg. of times of CFs). This produces a graph
of duration vs time with jump discontinuities.
Interest Conversion Frequency
Many investors like to analyze what will happen if interest rates rise or fall by a certain amount, say
100 basis points. It is important to state clearly which interest you are talking about.
Changing a nominal rate on a bond with semiannual coupons is not equivalent to changing the nominal
rate on a monthly mortgage by the same number of basis points.
Consider the following example.
Example 5. You have an investment portfolio consisting of coupon bonds with a 7% yield rate
convertible semiannually and mortgages with a 6% yield rate convertible monthly. For anayltical
purposes, you are using annual effective rates on the entire portfolio and wish to examine the effects of
a 100 basis points increase and decrease in the effective rate of interest. Find the yield rates that should
be used for the (i)bonds, and (ii) mortgages.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Matching Assets and Liabilities
Assets generate a series of cash inflows, while liabilities generate a series of cash outflows.
Goal : to achieve an equilibrium or safe balance between these cash inflows and outflows, to account
for the risk of adverse effects created by changes in the level of interest rates.
As an example, suppose a bank issues a 1-yr CD with a guaranteed rate of interest. The bank must have
assets backing this contract holder's statement.
• If assets are invested “too long,” (a duration greater than 1 year): the bank is vulnerable to loss
if interest rates rise. If interest rates rise, contract holders are likely to withdraw their funds at
the end of one year, so the bank may have to sell some assets to pay these departing contract
holders. However, the assets that are sold have declined in value due to rise in interest rates.
• If assests are invested “too short,” the bank is vulnerable to losses if interest rates fall. Assets
invested very short term would mean interest earnings will decline and may not be sufficient to
pay the guaranteed interest rate on the CDs at the end of the year.
To address this issue, we will study 3 techniques:
1. Absolute matching (Dedication)
2. Redington Immunization
3. Full Immunization
Absolute Matching/Dedication
A technique used to protect a financial institution from movements in interest rates where the approach
is to structure an asset portfolio in such a way that the cash inflow that will be generated will exactly
match cash outflow from the liabilities in every period.
Example 6. Several years ago, ABC Corporation entered into a 10-year lease agreement under which
ABC committed to make 10 annual lease payments. The 1st payment was $840,000 and each payment
thereafter was to be adjusted for inflation with annual 4% increases(compounded). ABC has just made
its seventh lease payment and has enough extra funds on hand that it wishes to “pre-fund” its three
remaining lease payments. It decides to do so using an absolute matching strategy with zero coupon
bonds. The current yield curve for zero coupon bonds shows the following rates:
Term
Spot Rate
1-yr
7%
2-yr
8%
3-yr
8.75%
Find the amount ABC will need to invest to implement this strategy.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Example 7. Rework example 6 using coupon bonds with annual coupons, Assume coupon bonds all
have the same yield rate as the zero coupon bonds above and sell at par.
NOTE: We are construction a dedicated bond portfolio with coupon bonds.
Strategy: We must select coupon bonds with maturity dates matching each of the cash outflow dates
and display cash flows from each bond. Then we start at the end and match the last cash outflow with
the longest term bond and work backwards recursively to the 1st cash flow.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Immunization (or Redington Immunization)
Immunization is more flexible than absolute matching since it does not require exact matching of an
asset cash flow for each liability cash flow.
Major Assumptions of Redington Immunization
1. The yield curve is flat, i.e. spot rates si are equal for all i.
2. Interest rate changes are parallel shifts up or down in that yield curve.
Requirements for a portfolio to be Redington Immunized
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Math 325
Ch 11 – Duration, Convexity and Immunization
Example 7. A financial institution has to pay $1000 two years from now and $2000 four years from
now. The current market interest rate is 10% and the yield curve is assumed to be flat at any time. The
institution wishes to immunize the interest rate risk by purchasing zero coupon bonds which mature
after 1, 3, and 5 years. A financial risk consultant suggests the following strategy:
•
•
•
Purchase a 1-year zero coupon bond with a face value of $154.16.
Purchase a 3-year zero coupon bond with a face value of $2189.04.
Purchase a 5-year zero coupon bond with a face value of $660.18.
Show that this portfolio satisfies the three conditions of Redington immunization.
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Math 325
Ch 11 – Duration, Convexity and Immunization
Difficulties and Limitations of Immunization in Practice
1. What one interest rate should be used?
2. It only provides protection against small changes in i.
3. Yield curves are usually not flat, and shifts in rates are often not parallel.
4. Requires frequent re-balancing of portfolios.
5. Exact cash flows may not be known and may have to be calculated. (ex. Callable bonds)
The convexity condition seems to imply a profit can be made with interest rate movements in either
direction. This violates a principle of finance theory which states that risk-free arbitrage does not exist
for any significant period of time in efficient markets.
6. Assets may not exist in the right maturities to achieve immunization.
Full Immunization
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