PORTFOLIO MANAGEMENT

Lecturer: Pham Ha Phuong, MSc

Department: Faculty of Banking and Finance
Email: phamhaphuong@ftu.edu.vn
Course materials: bit.ly/3O3IwkF
 CHAPTER 10:
Bond management
 1. The term structure of interest rate

• bond stripping and bond reconstitution

• pure yield curve vs on-the-run yield curve

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1. The term structure of interest rate

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The yield curve and future interest rate

➢The Yield Curve under Certainty

Consider two 2-year bond strategies:

• The first strategy entails buying the 2-year zero offering a 2-year yield to
maturity of y2 = 6%, and holding it until maturity. The zero has face value
$1,000, so it is purchased today for $1,000/1.062 = $890 and matures in
two years to $1,000. The total 2-year growth factor for the investment is
therefore $1,000/$890 = 1.062 = 1.1236.
• Invest the same $890 in a 1-year zero coupon bond with a yield to
maturity of 5%. When that bond matures, reinvest the proceeds in another
1-year bond.

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➢The Yield Curve under Certainty

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➢The Yield Curve under Certainty
Short rate

Spot rate

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➢ Forward rates

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 ➢ Forward rates

Recognizing that future interest rates are uncertain, we call the interest rate that we infer in
this matter the forward interest rate rather than the future short rate because it need not be
the interest rate that actually will prevail at the future date.

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 ➢ Liquidity Preference Theory

Advocates of the liquidity preference theory of the term structure believe that
short-term investors dominate the market so that the forward rate will generally
exceed the expected short rate.
The excess of f2 over E(r2), the liquidity premium, is predicted to be positive.
(Fig. 15.2)

(?) What about “long-term” investors?

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➢ Liquidity Preference Theory

Panel A: ………. expected short rate. Liquidity premium of 1%.

Result: a ………… yield curve 11
➢ Liquidity Preference Theory

Panel B: ………. expected short rate. ……….. liquidity premiums.

Result: a ………… yield curve despite falling expected interest rates. 12
➢ Liquidity Preference Theory

Panel C: ………. expected short rates. ………..liquidity premiums.

Result: a hump-shaped yield curve 13
➢ Liquidity Preference Theory

Panel B: ………. expected short rates. ……….. liquidity premiums.

Result: a sharply rising yield curve 14
 2. Managing Bond Portfolios
➢ Interest Rate Sensitivity
1. Bond prices and yields are inversely related: As yields increase, bond prices fall; as yields
fall, bond prices rise.
2. An increase in a bond’s yield to maturity results in a smaller price change than a decrease
in yield of equal magnitude
3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of
short-term bonds.
4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as
maturity increases. In other words, interest rate risk is less than proportional to bond
maturity.
5. Interest rate risk is inversely related to the bond’s coupon rate. Prices of low-coupon
bonds are more sensitive to changes in interest rates than prices of high-coupon bonds.
6. The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to
maturity at which the bond currently is selling.
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➢ Interest Rate Sensitivity

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 ➢ Duration

Macaulay’s duration (D): weighted average of the times to each coupon or

principal payment = effective maturity

Where:

Example:

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 ➢ Duration
• Summary statistic of the effective maturity of the portfolio
• Essential tool in immunizing portfolios from interest rate risk
• A measure of the interest rate sensitivity of a portfolio:

∆𝑃 ∆(1+𝑦) ∆𝑃
=-Dx = - D* x ∆𝑦
𝑃 1+𝑦 𝑃

Where: Modified Duration = D* = D/(1+y)

Note: ∆(1 + 𝑦) = ∆𝑦

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➢ What determines Duration?

Rule 1: The duration of a zero-coupon bond equals its time to maturity

Rule 2: Holding maturity constant, a bond’s duration is lower when the coupon rate is
higher
Rule 3: Holding the coupon rate constant, a bond’s duration generally increases with its
time to maturity. Duration always increases with maturity for bonds selling at par or
premium
Rule 4: Holding other factors constant, the duration of a coupon bond is higher when the
bond’s yield to maturity is lower
Rule 5: The duration of a level perpetuity is:
1 +𝑦
Duration of perpetuity =
𝑦
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➢ Convexity

Convexity allows us to improve the duration approximation for bond

price changes. Accounting for convexity, Equation 16.3 can be
modified as follows:

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➢ Convexity

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➢ Why do investors like Convexity?

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➢ Duration and Convexity of Callable bond

“Heads
I lose,
Tails
I don’t
win”

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➢ Duration and Convexity of Mortgage-Backed Securities

Callable amortizing loans

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➢ Mortgage-Backed Securities?

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 ➢ Mortgage-Backed Securities?

Mortgage: a loan secured by the collateral of some specified real estate

property that obliges the borrower to make a predetermined series of payments
to the lender (often initially a bank or mortgage company).
The mortgage gives the lender the right to foreclose on the loan if the borrower
defaults; that is, a foreclosure allows the lender to take possession of the
mortgaged property and then sell it in order to recover funds toward satisfying
the debt obligation.

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➢ Mortgage-Backed Securities?

Prepayment risk:
• Contraction risk: interest rate fall => more prepayment than expected =>
shorter maturity for the MBS
• Extension risk: interest rates rise => less prepayment than expected =>
longer maturity for the MBS

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➢ Duration and Convexity of Mortgage-Backed Securities

The following table is an example of a very simple CMO structure.

The underlying mortgage pool is divided into three tranches, each
with a different effective maturity and therefore interest rate risk
exposure. Suppose the original pool has $10 million of 15-
yearmaturity mortgages, each with an interest rate of 10.5%, and is
subdivided into three tranches as follows:

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➢ Duration and Convexity of Mortgage-Backed Securities

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➢ Duration and Convexity of Mortgage-Backed Securities

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2.1. Passive Bond Management

➢ Immunization

Immunization techniques: refer to strategies used by such investors to

shield their overall financial status from interest rate risk.
Concern: interest rate risk
=> The net worth of the firm or the ability to meet future obligations
fluctuates with interest rates
Example: banks, pension fund

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2.1. Passive Bond Management

➢ Immunization

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2.2. Passive Bond Management

➢ Immunization

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 2.1. Passive Bond Management

➢ Immunization

• Price risk
• Reinvestment risk
• At the 5-year horizon date,
equal to D, the two effect
just cancel

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2.1. Passive Bond Management

➢ Immunization

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➢ Immunization

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2.1. Passive Bond Management

➢ Immunization

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2.1. Passive Bond Management
➢ Bond-Index Funds

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 2.2. Active Bond Management

➢ Sources of Potential Profit

• The substitution swap
• The intermarket spread swap
• The rate anticipation swap
• The pure yield pickup swap
• Tax swap
➢ Horizon Analysis

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