Fixed Income Technicals
A bond's spread relative to the benchmark curve has two components:
A bond's spread relative to the benchmark curve has two components: 1. A premium for credit risk. 2. A premium for lack of liquidity relative to the benchmark securities.
A positive duration gap (Macaulay duration greater than the investment horizon) exposes the investor to:
A positive duration gap (Macaulay duration greater than the investment horizon) exposes the investor to market price risk from increasing interest rates.
Compared to a non-callable and non-puttable bond, is the interest rate sensitivity of a callable and putable bond higher or lower?
Interest rate sensitivity of callable and puttable bonds is lower.
In response to a 0.1% increase in YTM and a Modified Duration of 3.567, the price of the bond should:
Approximate percentage change in bond price = -(ModDur) * YTM. Percentage change in bond price = -3.567*0.1 => Fall in price by approx 0.3567%
Given an expected change in a bond's price of -1.65% due to change in yield to maturity (YTM) by 20 basis points comes a 0.02% change in the bond's price from the convexity effect. What is the percentage contribution to the change in the bond's price owed to the duration effect?
Change in bond price = Duration effect + Convexity effect Duration effect = Convexity effect - Change in bond price = -1.65 - 0.02 = -1.67%
A bond selling for par currently has a 9% yield. If the bond price increases to 101 when yields fall 10 basis points and the price falls to 98 when yields rise by 10 basis points, then what is this bond's effective duration?
Deff = (P- - P+ )/ (2xΔCurvexPV) = (101 - 98)/(2x0.001x100) = 15
A 9% bond has a full price of $905 and a YTM of 10%. Estimate the percentage change in the full price of the bond for a 30 basis point increase in YTM assuming the bond's duration is 9.42 and its convexity is 68.33.
Duration effect = -9.42 * 0.003 = 0.02826 = -2.826% Convexity effect = 0.5 * 68.33 * 0.0032 = 0.000307 = 0.0307% Expected change in bond price = (-0.02826 + 0.000307) = -2.79530%
A bond investor has an investment horizon of 9 years. He recently calculated that the Macaulay duration of his portfolio is 11 years. What is the duration gap?
Duration gap = Macaulay duration - Investment horizon = 11 - 9
For a given change in yields, the difference between the actual change in a bond's price and the predicted change in price using the duration measure will be greater for:
Duration is a linear measure of the relationship between a bond's price and the yield. The true relationship is not linear as measured by convexity. When convexity is higher, duration will be less accurate in predicting a bond's price for a given change in interest rates. Note: Short-term bonds generally have low convexity.
What is Effective duration?
Effective duration is a measure of the duration for bonds with embedded options (e.g., callable bonds). Unlike the modified duration and Macaulay duration, effective duration considers fluctuations in the bond's price movements relative to the changes in the bond's yield to maturity (YTM). In other words, the measure takes into account possible fluctuations in the expected cash flows of a bond.
For a bond that was purchased at a premium, the YTM of the bond is:
For a bond purchased at a premium, the YTM is less than the coupon rate because both the amortization of the premium and the reduction in interest earned on reinvestment of its cash flows decrease the bond's return.
Investor 'A' has an investment horizon of 2 years and has invested in a 5% coupon paying bond with a YTM of 7%. Investor 'B' has an investment horizon of 5 years and has invested in a 6% coupon paying bond with a YTM 7%. Which investor is facing higher market price risk as compared to reinvestment risk?
For an investor with a short investment horizon, the interest rate risk increases and the reinvestment risk decreases. Investor A, who has a shorter investment horizon, will have a higher market price risk as compared to reinvestment risk.
Michael holds a bond whose YTM is 6%. However, the YTM of the bond increases to 6.5% before the first coupon date. His rate of return on the bond will:
If the YTM, which is also the reinvestment rate for the bond, increases (decreases) after the purchase, the return from coupon payments and reinvestment income will increase (decrease) as a result and increase (decrease) the investor's rate of return on the bond above (below) its YTM at purchase.
What is Macaulay Duration?
Macaulay duration is a weighted average of the times until the cash flows of a fixed-income instrument are received. It is a measure of a time required for an investor to be repaid the bond's price by the bond's total cash flows. The Macaulay duration is measured in units of time (e.g., years).
A $5 million par value bond has a modified duration of 7.42 and a full price of 101.32, expressed per $100 face value. Its money duration is closest to:
Money duration = Modified duration * Full value of the bond = Money duration per $100 of par value = 7.42 * 101.32 = $751.79 Note: If the question had not asked for the money duration per $100 of par value, the answer would then be: Money duration = 7.42 * $5,000,000 * 1.0132 = $37,589,720
An investor purchased a 6% three-year bond that has an investment horizon of 1 year. The bond has a YTM of 7%. Calculate the carrying value of the bond.
N=2, I/Y=7,PMT=60, FV=1000 CPT => PV = -981.92
The price value of a basis point (PVBP) for a 7-year, 10% semiannual bond with a par value of $1,000 and a yield of 6% is closest to:
PVBP = Initial price - Price if yield changes by 1 bps. Initial price: FV=1000; PMT=50; N=14; I/Y=3%; CPT -> PV =1225.92 Price after change: FV=1000; PMT=50; N=14; I/Y=3.005%; CPT -> PV =1225.28 PVBP = 1225.92 -1225.28 = 0.64 Note: The price value of a basis point (PVBP) is always the absolute value.
Calculate the expected percentage price gain (loss) from the following data: Reduction in yield-to-maturity: 20bps Annual modified duration: 23.657 Annual convexity: 678.98
Percentage change in full price = (-23.657)(-0.002) + 0.5x678.98x(-0.002)^2 = 4.86%
Assume that a city issues a $5 million bond to build a new arena. The bond pays 8% semiannual interest and will mature in 10 years. If current interest rates are 9%, then the present value of the bond and the estimated value of the bond seven years from today are closest to:
Present value using the financial calculator: FV=5,000,000; N=20; PMT=0.04*5,000,000=200,000; I/Y=4.5; CPT => PV = -4,674,802 Value in seven years using the financial calculator: FV=5,000,000; N=6; PMT=0.04*5,000,000=200,000; I/Y=4.5; CPT => PV = -4,871,053
A bond valued at $200,000 has a duration of 8 and a convexity of 20. Assuming that the bond's spread relative to the benchmark curve increases by 25 basis points due to a credit downgrade, then the approximate change in the bond's market value is closest to:
Price Change = (-Duration x Yield change) + (0.5 x Convexity x Yield change^2)) Price change = (-8 x 0.0025) + (0.5 x 20 x 0.0025^2) = -1.99% The bond's value will fall by approximately 1.990% * 200,000 = $3,988.
A corporate bond has the following characteristics: Price: 106.50 Coupon rate: 5% Duration: 7.5 years Convexity: 101 If the credit spreads narrow by 175 basis points, then the price of the bond is:
Return impact = (Duration x ΔSpread) + (1/2 x Convexity x (ΔSpread)^2) = -(7.5 x (-0.0175)) + (1/2 x 101 x (-0.0175)^2) =0.1313 + 0.0155 = 0.1468 = 14.68% Price = 106.5 x (1 + 0.1468) = 122.13
The duration-only based estimate of the decrease in price resulting from an increase in yield is:
The convexity adjustment to the price change is the same for both an increase and a decrease in yield. The duration-only based estimate of the increase in price resulting from a decrease in yield is too low for a bond with positive convexity and is improved by a positive adjustment for convexity. The duration-only based estimate of the decrease in price resulting from an increase in yield is larger than the actual decrease, so it's also improved by a positive adjustment for convexity.
The coupon reinvestment risk is directly correlated with what?
The coupon reinvestment risk increases with a higher coupon rate and a longer reinvestment time period.
A bond has a duration of 10.62 and a convexity of 91.46. For a 200 bps increase in yield, the bond's approximate percentage price change is closest to:
The estimated price change = -(Duration)(Change in yield) + (1/2)*(Convexity)(Change in yield)^2 = -10.62x0.02 + 0.5x91.46x0.02^2 = -19.41%
What is Modified duration?
The modified duration figure indicates the percentage change in the bond's value given an X% interest rate change. Unlike the Macaulay duration, modified duration is measured in percentages. The modified duration is often considered as an extension of the Macaulay duration.
While calculating duration, for which of the following bonds would an investor least likely use Effective Duration rather than Modified Duration?
The only difference between modified and effective duration is that Effective Duration is used for bonds with embedded options. Effective duration is a measure that considers the embedded options by considering the price sensitivity to an incremental increase and decrease in interest rates.
For a 10-year, 5% annual-pay bond currently trading at par, calculate the approximate modified duration based on a change in yield of 25 basis points.
The price of the bond at a yield of 5% + 0.25% is: N = 10; I/Y = 5.25; FV = 1,000; PMT = 50; CPT => PV = -981 The price of the bond at a yield of 5% - 0.25% is: N = 10; I/Y = 4.75%; FV = 1,000; PMT = 50; CPT => PV= -1020 The approximate modified duration = (1,020 - 981)/(2x1000x0.0025) = 7.8 Therefore, the approximate change in price for a 1% change in YTM is 7.8%.
The price value of a basis point (PVBP) is a measure of:
The price value of a basis point (PVBP) is a measure of the change in price given a 1 basis point change in the YTM
The coupon reinvestment risk dominates the market price risk when:
When the Macaulay duration is lower than the investment horizon, the coupon reinvestment risk dominates the market price risk. When the Macaulay duration is higher than the investment horizon, the market price risk dominates the coupon reinvestment risk.
