Section II.C. Fixed Income Flashcards
Municipal Bond Yields
• To choose between taxable and tax-exempt bonds, compare after-tax returns on each bond.
• Let t equal the investor’s marginal tax bracket
• Let r equal the before-tax return on the taxable bond and r m denote the municipal bond rate.
• If r (1 - t ) > r m then the taxable bond gives a higher return; otherwise, the municipal bond is preferred.
Calculate Pre-Tax Equivalent
Question: A muni-bond offers a 2.4% annual yield. Your client’s federal tax bracket is 35%. What is the pre-tax
equivalent yield on this bond all else held equal and not considering state income tax?
Answer: 3.69%
= after-tax yield / (1 – tax rate)
= 2.4% / (1 – 35%)
= 2.4% / (.65)
= 3.69%
Characteristics of fixed income investments including basic features (priority of claims with capital structure), coupon structures, payment methods, and options based on several parameters:
• quality
• maturity, duration, and convexity
• issue size
• fixed or floating rate coupons
• call features
• YTM, YTW, YTC
What is Duration?
• a risk measurement for fixed income securities expressed in years
• duration is calculated using factors such as coupon, maturity, present value, and call features
• duration measures the sensitivity of a bond’s price to changes in interest rates
Named for creator Frank Macaulay, duration measures the present value of future interest payments and the length of time needed to reach par value.
Think of Macaulay Duration as the average time to receive the present value of a bond’s cash flows (aka “weighted average term to maturity”).
Understanding the concept behind Duration can help you answer questions without having to actually perform a calculation. That saves valuable test time.
What are Important Duration Observations?
• Duration is normally expressed in years.
• All else being equal, the greater the present value of the cash flow that is received as part of the coupon payment (i.e., versus principal received at maturity), the lesser the duration.
• All else being equal (credit risk, maturity), the higher the coupon rate, the lesser the
duration.
• All else being equal, a longer maturity bond (relative to a shorter maturity one) will have a longer duration.
• A zero-coupon bond’s duration will normally equal its maturity.
• The greater the duration, the greater the percentage price change from a change in
interest rates. Frank Macaulay developed Duration to measure how sensitive a bond is to interest rate changes.
• Duration assumes yields remain constant and reinvestment risk does not exist, which is actually only true during the life of the bond for zero-coupon bonds.
• Assume semi-annual payments unless told otherwise.
• The Cash Flow function of a calculator can be used to calculate the numerator.
What is Macaulay Duration of a Portfolio?
The Macaulay Duration of a portfolio is the average duration of each of the assets, weighted by its allocation to the
portfolio.
Consider a portfolio made up of bond A (from prior example) & B ($3,000
bond maturing in 3 years, 4% coupon, with similar bonds yielding 8%).
Test yourself: Confirm B’s value & duration as $2,685.47 & 2.85 years.
The portfolio’s duration is then 2.61 years.
What is Modified Duration?
Modified Duration measures price sensitivity when there is a change in interest rates or a change in yield to maturity.
Macaulay Duration (D) vs. Modified Duration (D*)?
Macaulay Duration (D) is the weighted average time to receive a bond’s cash flows and is a proxy for a bond’s sensitivity to interest rates.
Modified Duration (D*) is a variation that better approximates a bond’s change in price (i.e., sensitivity) due to interest rate changes.
Mod Dur D* = Mac Dur D / [1 + (Y/k)]
in which Y = current yields
k = frequency of compounding (e.g., 2 for semi-annual bonds)
Note: for zero-coupon bonds, use 2 or the same frequency used by the bonds for which the yield Y is based.
Calculation of Modified Duration
From a prior example, Bond A (5% coupon paying maturing in 2 years & paying semi-annually when similar bonds yield 6%) has a Mac Duration D = 1.93 years.
Mod Dur D*=D / [1 + (Y/k)]
= 1.93 yrs / [1 + (0.06/2)] = 1.93 yrs/1.03 = 1.87 yrs
Interest Rate Changes and Modified Duration
For a given change in interest rates (∆y), the estimated change (∆P) in a
bond’s price (P) is calculated as:
∆P/P ≈ -D* ∆y OR
∆P/P ≈ -D {∆y/[1+(y/k)]} (this equation just substitutes D for D*)
Note the negative sign, illustrating the inverse relationship between changes in
interest rates and changes in bond prices.
Calculation of Price Changes
Assume interest rates rise from 6.0% to 6.1%. Calculate the percentage change in the price of earlier example Bond A with Mod Dur = 1.87 years:
Percentage Change in Price = ∆P/P ≈ -D* ∆y
≈ -1.87 * 0.1%
≈ -0.187%
Confirm this by calculating the new price of the bond. This estimate is more accurate for smaller interest rate changes because of convexity.
Q: A 9% coupon, 16-year bond pays semi-annually and has a yield to maturity of 11%. Its duration is 8.18 years. If the market yield changes by 32 basis points, how much change will there be in the bond’s price?
Solution: The question is ambiguous in which case I would assume the duration refers to Macaulay duration, though this can be confirmed by calculation. In that case,
∆P/P ≈ -D* ∆y = -D {∆y/[1+(y/k)]
≈ -8.18 {0.0032/[1+(0.11/2)]}
≈ -2.48%
If given the price, you could calculate the actual dollar change in value.
Modified Duration and Interest Rate Risk
Example of two bonds with same duration, thus similar interest rate risk:
Bond 1: Coupon Bond
Pays 3% annual coupon, matures in exactly 4 years, similar bonds yield 4%.
You should be able to calculate:
•Price = $963.70 (TVM calc)
•Duration = 3.825 years
Bond 2: Zero Coupon Bond
$1,000 face value, no coupon, has exactly same duration as Bond 1, similar bonds yield 4%.
You should be able to calculate:
•Duration = 3.825 years (given)
•Maturity = 3.825 years (b/c it’s a zero)
•Price = $860.69 (TVM Calc) or
•Price = 1,000/(1.043.825)=$860.69
If yields decrease 0.01%, the value of the bond will rise.
Estimate the Percentage Price Change based on Duration:
% Change = ΔP/P ≈ (-3.825 * -0.0001) / (1.04) = 0.00037 = 0.037%
Confirm percentage price change of each directly via calculation:
•Price = $964.06 (TVM calc)
•% Change = 0.36/963.70 = 0.037%
•Price = $861.01 (TVM Calc) or
•Price = 1,000/(1.03993.825)=$860.01
•% Change = 0.32/860.69 = 0.037%
Q: Duration is important in bond portfolio management because
I) it can be used in immunization strategies.
II) it provides a gauge of the effective average maturity of the portfolio.
III) it is related to the interest rate sensitivity of the portfolio.
IV) it is a good predictor of interest rate changes.
a. I and II
b. I and III
c. III and IV
d. I, II, and III
Answer: d.
Solution: Duration can be used to calculate the approximate effect of
interest rate changes on prices but is not used to forecast interest rates.
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 higher when the coupon rate is lower
– Rule 3
• Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity
– Rule 4
• Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
– Rules 5
• The duration of a level perpetuity is equal to:
(1 + y) / y
What are interest rates sensitive to?
Interest Rate Sensitivity
1. Bond prices and yields are inversely related
2. An increase in a bond’s yield to maturity
results in a smaller price change than a
decrease of equal magnitude
3. Long-term bonds tend to be more price
sensitive than short-term bonds
Interest Rate Sensitivity
4. As maturity increases, price sensitivity
increases at a decreasing rate
5. Interest rate risk is inversely related to the bond’s coupon rate
6. Price sensitivity is inversely related to the yield to maturity at which the bond is selling
Yield to Maturity (YTM)
• YTM is the rate of return of a bond if it is held to maturity
• expressed as an annual rate
• assumes all interest payments are reinvested at the bond’s current yield
• YTM computes bond yields using maturity, coupon, current market price and par value
• the calculation allows for easier comparison of bonds with various coupons and maturities
