Investment Analysis Final (Ch15, Ch16 so far...)

Stripped treasuries are

zero-coupon bonds created by selling each coupon or principal payment from a whole treasury bond as a separate cash flow

Pure yield curve

The relationship between yield to maturity and time to maturity for zero-coupon bonds

On-the-run yield curve

Relationship between yield to maturity and time to maturity for newly issued bonds selling at or near par velue

Under certainty, the yield on the zero 2-year coupon bonds is greater than that on the 1-year zero. Why?

This upward sloping curve is evidence that short term rates are going to be higher next year than they are now.

The yield to maturity on zero-coupon bonds is called the

Spot rate

The short rate refers to ____.

The interest rate for that interval available available at different points in time (a one period interest rate)

Expectations hypothesis

Theory that forward interest rates are unbiased estimates of expected future interest rates.

Short term investers will be unwilling to hold long-term bonds unless the forward rate expected short interest rate_____, whereas long-term investors will be unwilling to hold short term bonds unless,

(F2 > E(r2)); E(r2)>f2

Noth short term and long term investors require a

premium to hold bonds with maturities different from their investment horizons

Advocates of the ________ of the term structure believe that short term iinvestors dominate the market so that the forward rate will generally exceed the expected short rate. The excess of f2 over E(r2), the risk premium, is predicted to be positive.

liquidity preference theory

Market segmentation theory holds that

long- and short-maturity bonds trade in essentially distinct and unconnected markets, each of which finds its own equilibrium independently.

Under uncertainty, 1 plus the yield to maturity on a zero-coupon bond is simply:

The geometric average of 1 plus the future short rates that will prevail over the life of the bond

What happens if the yield curve is rising?

Forward rates must be greater than current yields

Key Rule (IMPORTANT)
If yield curve slopes upward →

fn + 1 > yn

Why does the yield curve rise?

Because new forward rates added are higher than the previous average

Intuition (Test Score Analogy)

To raise an average → the new value must be above the current average

Forward Rate Decomposition (VERY IMPORTANT)

fn = E(rn) + liquidity premium

Two Reasons Forward Rates Are High

1. Expected future interest rates are high

2. Liquidity premium is positive

BIG EXAM TRAP ⚠
A rising yield curve does NOT necessarily mean rates will increase

---

Why is that a trap?

Because upward slope could be due to liquidity premium, not expectations

Liquidity Premium Meaning

Extra return investors require for holding long-term bonds

Can liquidity premium be negative?

Yes (if investors prefer long-term bonds)

Empirical Fact

Long-term yields are usually higher than short-term yields

Exception to Normal Pattern

When short-term rates exceed long-term → often precedes economic downturns

Suppose you want to make a loan at some future date:

The interest rate on such a “forward loan” would be the forward rate of interest for the period of the loan.

REMEMBER THAT BOND PRICES AND YIELDS ARE

INVERSELY RELATED

The "break-even" interest rate for year n that equates the return on an n-period zero-coupon bond to that of an n−1 period zero-coupon bond rolled over into a one-year bond in year n is defined as:

the forward rate.

The __________blank yield curve is created from stripped treasuries.

pure

Treasury STRIPS are:

created by selling each coupon or principal payment from a whole Treasury bond as a separate cash flow.

_________ are created from coupon paying treasuries, where the coupon and principal are separated.

Stripped treasuries

When computing yield to maturity, the implicit reinvestment assumption is that the interest payments are reinvested at the:

yield to maturity at the time of the investment.

The on the run yield curve is:

a plot of yield as a function of maturity for recently-issued coupon bonds trading at or near par.

Which of the following are possible explanations for the term structure of interest rates?

The expectations theory and the liquidity preference theory

Duration Rule 1

• The duration of a zero-coupon bond equals its time to

maturity.

Duration Rule 2

Holding maturity constant, a bond’s duration is lower when

the coupon rate is higher. (Higher weights on early

payments)

Duration rule 3

Holding the coupon rate constant, a bond’s duration

generally increases with its time to maturity(distant

payments have greater PVs, accounting for greater share)

Duration Rule 4

Holding other factors constant, the duration of a coupon

bond is higher when the bond’s yield to maturity is lower

(lower yield reduces the PV of the longer dated maturities

by a lesser amount)

Duration Rule 5

The duration of a level perpetuity is equal to:
1 + y / y

What is the forward interest rate?

The rate of interest for a future period, inferred from the term structure, that equates the return of holding a long-term bond with rolling over short-term bonds.

The liquidity premium compensates _____-____ investors for the uncertainty about the price at which they will be able to sell their ____-_____ bonds at the end of the year.

short-term; long-term

What would contribute to the negative slope of a yield curve?

Decreasing expected short rates

Negative liquidity premium

Consider a bond that makes a series of annual coupon payments and matures in four years. How should the term structure be used to value the bond?

Use zero-coupon bonds with maturities matching each payment date.

Explanation:
Each cash flow should be discounted using its own spot rate (from the term structure) because interest rates vary by maturity.

The idea that the forward rate differs from the expected short rate because of supoky abd demand issues, with the forward rate usually higher, is termed, _____ _____ theory.

Liquidity preference

For an upward sloping yield curve, a(n) ____ average forward rate must be added to the other previously observed rates in order to increase the yield to maturity.

above-

If you want to obtain a forward-loan that begins in year 3 and ends in year 5, you would issue a _____-year zero-coupon bond and buy a ____-year zero.

5; 3

If a bond is issued with a 6% coupon when competitive yields are 6% then it will sell at par value. If the market rises to 7%, however, who would be willing to pay par value for a bond only offering 6%?

The bond price must fall

Immunization needs rebalancing but _____ doesn’t.

dedication

Duration formula

%ΔP ≈ − Dmod × Δy

Zero-coupon bond duration

Duration = maturity

Modified duration meaning

Measures price sensitivity to interest rate changes

Duration approximation error

Ignores convexity → linear estimate only

Convexity effect

Actual price change is curved, not linear

When rates decrease (convexity effect)

Actual price increases MORE than duration predicts

When rates increase (convexity effect)

Actual price decreases LESS than duration predicts

Duration vs convexity

Duration = linear
Convexity = curvature

Portfolio immunization condition

Investment horizon = portfolio duration

Portfolio duration formula

Weighted average of individual durations

Immunization meaning

Protects against interest rate risk

What risks are balanced in immunization

Price risk and reinvestment risk

Price risk

Bond price changes when interest rates change

Reinvestment risk

Coupons reinvested at new interest rates

Key immunization idea

Price risk ↓ + reinvestment risk ↑ = cancel out

Why “equal risk” is wrong wording

They offset, not necessarily identical

Duration over time

Decreases as bond approaches maturity

Why portfolios must be rebalanced

Duration changes over time

Immunization limitation

Does NOT protect against credit risk

Dedication strategy

Matches cash flows exactly to liabilities

Why dedication is not widely used

Too restrictive on bond selection

Immunization vs dedication

Immunization = flexible
Dedication = strict matching

Bond index behavior

Bonds removed as they near maturity

Bond market characteristic

Many bonds are thinly traded

Active management condition

Requires superior information

Abnormal return

Return above expected (alpha)

Substitution swap

Swap one bond for similar but better-priced bond

Horizon analysis

Forecast total return over holding period

Horizon analysis includes

Future price + coupon income

Convexity is related to

Second-order price sensitivity

Quick rule for duration

Longer maturity → higher duration (generally)

Quick rule for coupons

Lower coupon → higher duration

Quick rule for yields

Lower yield → higher duration

Remember that bond prices and yields are ____ related; as yields increase, price falls - as yields fall, price rises

inversely

An increase in a bond’s yield to maturity results in____________.

a smaller price change than a decrease in yield of equal magnitude.

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

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

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

Zero-coupon vs coupon bond sensitivity

Zero-coupon bonds are MORE sensitive to interest rate changes than coupon bonds with the same maturity because all cash flows occur at maturity, giving them a longer effective duration.

In the previous chapter, we pointed out that it can be useful to view a coupon bond as a “portfolio” of coupon payments. The effective maturity of the bond is therefore some sort of average of the maturities of all the cash flows. The zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is, therefore, _______.

well defined.

Higher-coupon-rate bonds have a higher fraction of value tied to coupons rather than final payment of par value, and so the “portfolio of payments” is more heavily weighted toward the earlier, short-maturity payments, which gives it _____.

lower “effective maturity.”

Similar logic explains our sixth rule, that price sensitivity falls with yield to

maturity. A higher yield reduces the present value of all of the bond’s payments, but

more so for more-distant payments. Therefore, at a higher yield, a higher proportion

of the bond’s value is due to its earlier payments, so effective maturity and interest

rate sensitivity are

lower.

______________ equals the weighted average of the times to each coupon or principal payment, with weights related to the “importance” of that payment to the value of the bond. 

Macaulay’s duration

Duration is a key concept in fixed-income portfolio management for at least three reasons

First, as we have noted, it is a simple summary statistic of the average maturity of the portfolio. Second, it turns out to be an essential tool in immunizing portfolios from interest rate risk. We explore this application in Section 16.3. Third, duration is a measure of the interest rate sensitivity of a portfolio, which we explore here.

For example, if we wish to speculate on interest rates, duration tells us

how strong a bet we are making. 

Conversely, if we wish to remain “neutral” on rates, and simply match the interest rate sensitivity of a chosen bond-market index, duration allows us to ____________.

measure that sensitivity and mimic it in our own portfolio. 

At lower yields, more distant payments have relatively greater present values and account for a greater share of total value. Thus, in the weighted-average calculation, the distant payments receive greater weights, which results in ______.

a higher duration

The formula for the duration of a perpetuity makes it obvious that maturity and duration can differ substantially. The maturity of the perpetuity is infinite, _______________.

whereas its duration at a 10% yield is only 11 years.

The percentage price change is directly proportional to

the change in the bond’s yield.

The duration rule is a good approximation for small changes in bond yield, but it is ______ for larger changes.

less accurate

The duration rule becomes progressively less accurate!

The duration rule becomes progressively less accurate!

The duration approximation (the straight line) always _______ the value of the bond; it ______ the increase in bond price when the yield falls, and it _______ the decline in price when the yield rises

understates; underestimates; overestimates

We measure convexity as the __________.

rate of change of the slope of the price-yield curve, expressed as a fraction of the bond price.

The convexity of noncallable bonds such as that in Figure 16.3 is ______: The slope increases (i.e., becomes less negative) at higher yields

positive