Reading 53 LOS's Flashcards
LOS 53a: Calculate a bond’s price given a market discount rate
To calculate a plain vanilla fixed-rate bond price, all we have to do is discount our future cash flows by the market discount rate or the required rate of return demanded by investors.
If the coupon rate is less than what investors require, the bond will sell at a discount.
If the coupon rate is more than what investors require, the bond will sell at a premium.
The relationship between the discount rate and the price of a fixed-income instrument is convex. This leads to bond pricing having a convexity effect, that is an increase in price from a decrease in the discount rate will be greater than a decrease in price from an equal increase in the discount rate.
It is important to note the frequency of coupons when discounting future cash flows. If a bond has a 10% coupon that pays semiannually, this means every six months the investor will receive 5%. If the market discount at the time is 9%, this means for everycash flow, we will want to discount at 4.5%.
Also of note, if we know the market price of the bond, the term of the bond, the coupon rate and frequency, we can compute yield to maturity.
LOS 53b: Identify the relationships among a bond’s price, coupon rate, maturity , and market discount rate (yield to maturity)
Relationships between the Bond Price and Bond Characteristics
- A bonds price is inversely related to the market discount rate
- Given the same coupon rate and maturity, the percentage price change is greater in terms of magnitude when the discount rate decreases than when it increases— convexity effect.
- For the same term to maturity, a lower coupon bond is more sensitive to changes in the market discount rate than a higher coupon bond (coupon effect)
- Generally speaking, for the same coupon rate, the longer term bond is more sensitive to changes in the market discount than the shorter term bond ( maturity effect)
Relationship between Price and Maturity
- A premium bond’s value decreases towards par as it nears maturity
- A discount bond’s value increases towards par as it nears maturity
- A par bond’s value remains unchanged as it nears maturity
LOS 53c: Define spot rates and calculate the price of a bond using spot rates
A spot rate ( or zero rate) is the yield on a zero-coupon bond for a given maturity. So the 5 year zero-coupon rate would be our spot rate that we would use to discount cash flows received in year 5.
The traditional method of discounting bonds at a constant discount rate is called yield-to-maturity discounting. The arbitrage-free valuation approach, uses the different spot rates for each year to discount cash flows. There is still the same concept of discounting future cash flows, its just that we will use a different discount rate for each cash flow.
LOS 53d: Describe and calculate the flat price, accrued interest, and the full price of a bond
When a coupon bond is sold between coupon payment dates, we must account for the interest or coupon that the seller has earned, but not yet received. The idea is that the seller has bore the risk and earned the interest throught part of the coupon period, but when they sell it to the buyer, the buyer will receive the full coupon. Since part of that full coupon was earned by the seller, the buyer must compensate the seller when buying.
When the price of the bond is computed for the settlement date, the computed value is known as the full price. In this full price is the accrued interest of the seller. To find the accrued interest, we simply multiply the coupon payment by the amount of days in the coupon period before the settlement date divided by total number of days in the coupon period. We can then subtract this accrued interest from the full price, to get of flat price (aka clean or quoted price).
The flat price is the price that dealers will quote, as it actually shows the true price trend of the bond (the flat price is the price that converges to par over time). The full price increases every coupon period day, and then suddenly drops when a coupon payment is made.
To find the full price, we multiply the Present value of the bond at the last coupon payment, by 1 + the coupon rate raised to the number of days passed in the coupon period divided by the total number of days in the coupon period
LOS 53e: Describe matrix pricing
Matrix pricing is a method used to estimate the market discount rate and price of bonds that are not actively traded. Essentially prices of comparable bonds ( in terms of terms to maturity, coupon rates, and credit qualtiy) are used to interpolate the price of the subject bond.
- Example. We want to know the value of a 6-year bond, and are given the values of 2 similar 5-year and 8-year bonds
- First figure out the yields on the 5 and 8 year bonds
- Then add the 5 year yield to (6-5 / 8-5) x (8 year yield - 5 year yield) to get the yield on the 6 year bond. From here we can figure out value.
- Note (6-5/8-5)– these numbers are simply the different maturities on the bonds
Matrix pricing can also be used when underwriting new bonds to estimate the required yield spread over the benchmark rate on the bonds to be issued
- Example. A company is issuing a 5-year bond. The only other debt they have has 4 years to maturity.
- We are given that there are no 4 year gov bonds.
- we are also given the yields on 3 and 5 year gov bonds
- We are also given that similar companies with similar credit ratings have a 50bps spread when comparing their 5 and 4 year bonds
- 1st figure out the yield on the Companies 4 year bond
- then estimate the 4 year gov bond, by taking the average of the 3 and year
- From this average find the spread between the 4 year gov bond and the companies 4 year bond
- Add onto this spread the 50bps spread from comparable companies 4 and 5 year bonds
- Then take this spread and add it onto the benchmark 5 year gov bond
LOS 53f: Calculate and interpret yield measures for fixed-rate bonds
Fixed-Rate Bonds- the effective annual rate or yield on a fixed-rate bond depends on the assumed number of periods in the year, which is know as the periodicity of the stated annual rate. If the stated annual yield is 4% but the pay is quarterly, we need to divide by 4 to get 1%, then add 1 and raise to the power of 4 to get our actual annual yield
To convert a stated annual rate (SAR) of M periods per year to a SAR for N periods per year, use the formula:
( 1 + SARM / M)M = ( 1+ SARN / N) N
Bonds are typically quoted using the street convention, where the yield represents the IRR on the bond’s cash flows assuming all payments are made on schedule dates regardless if they fall on weekends or holidays. The true yield uses when payments are actually made, meaning it can never be higher than the street convention, sa delays only lower yields.
For corporate bonds, a government equivalent yield is sometimes quoted that restates the YTM from a 30/360 day count to an actual/actual day count
Another common used yield is the current yield (aka income or interest yield) and it is calculated by dividing the annual cash coupon payment by the bond price. This is relatively a crude measure because :
- It neglects the frequency of coupon payments
- it neglects any accrued interest in the denominator
- it neglects an premium or discount when purchased
The simple yield accounts for the premium or discount. It is the coupon payments received plus or minus the straight line depreciation from any gain or loss from purchasing the bond at a premium or discount divided by the flat price of the bond
For callable bonds the yield by the investor will depend on if the bond is called or not. Investors compute the yield-to-call for each call date and then determine the yield-to-worst as the worst they can expect to receive.
A better way to evaluate callable bonds than yield to worst, is to use an option pricing model. With this model we will value the embedded call option and add it to the flat price of the bond to acheive an option-adjusted price. From this price we can calculate the option-adjusted yield
LOS 53f continued: Calculate and interpret yield measures on Floating Rate notes
As we know interest payments on FRNs are not fixed, but instead rely on a reference rate with some spread. An important thing to recognize regarding FRNs is that the effective coupon rate for a specified period is determined at the beginning of the period, but actually paid out at the end of the period. So if on July 1st the 90day LIBOR is 5%, the investor will realize this 5% being paid on September 30th, even though the 90day LIBOR may be different at that time.\
NOTE when working with LIBOR it is important to unannualize the coupon rate. If the effective rate for the 90 day LIBOR is 5%, then the coupon for the period is calculated as:
.05 x 90/360 x principal
The quoted margin is the spread offered to investors on top of the reference rate to compensate them for risk. The required margin is the margin that investors require to hold the note. If the company has changes in credit risk, liquidity, or tax status, this required margin can change.
- If the required margin is less than the quoted, the note will trade at a premium
- if the required margin is more than the quoted, the note will trade at a discount
A simplified FRN pricing model calculates the value of a FRN at a reset date by estimating future cash flows and discounting them to the present
- Each future coupon payment is assumed to be calculated based on the current level of the reference rate plus the quoted margin
- the discount rate applicable to each future payment is assumed to equal the current reference rate plus the required margin
LOS 53f continued: Calculate and interpret yield measures on money market instruments
Money market instruments are short term debt securities with maturities ranging from one day (repos) to one year (CDs). They differ from yields in the bond market in the following ways:
- Bonds YTM are annualized and compounded. Money market yields are simply annualized
- Bonds YTM can be calculated using standard time value of money analysis. Money market yields are often quoted in terms of nonstandard interest rates so users need to work with various pricing equations
- Bonds YTM are typically stated for a common periodicity for all terms-to-maturity. Money market instruments that have different times-to-maturity have different periodicities for the stated annual rate
LOS 53g: Define and compare the spot curve, yield curve on coupon bonds, par curve, and forward curve
Spot rate curve the ideal date set for analyzing the term structure of interest rates would be YTMs on zero-coupon governement bonds or spot rates. They are most ideal because they are not subject to reinvestment risk. Using spot rates therefore , provides a more accurate relationship between yields and terms to maturity relative to using yields to maturity on coupon bearing Treasuries
Yield Curve for Coupon Bonds shows the YTMs for coupon paying bonds of different maturities. Even though treasury securities are not available for every single maturity, using matrix pricing we can estimate the gaps
Par Curve- maturity structure can also be calculated using the par curve, which represents a series of YTMs such that each bond trades at par. The par curve is derived from the spot rate curve
Forward Curve this represents a series of forward rates, each having the same horizon. Typically the forward curve shows 1-year forward rates stated on semiannual bond basis
LOS 53h: Define forward rates and calculate spot rate from forward rates, forward rates from spot rates, and the price of a bond using forward rates
Forward rates can be described as the market’s current estimate of future spot rates. If we know 2 different spot prices for different maturities, we can figure out forward prices in between those spot prices.
- Example- We are given the 1-year spot rate and the 2 year spot rate. From this we can calculate the 1 year forward rate 1 year from now as:
- (1 + 1-year spot) ( 1 + 1 year forward) = ( 1 + 2-year spot)2
- In general we can use this methodoligy to figure out any spot or forward rate, using this basic formula:
- (1 + ySpot0)y (1 + xforwardy)x = ( 1 + x+y spot 0)x+y
NOTE with this formula we used xfy to stand for the x period forward rate y years from today ( or yso to stand for the spot rate y years from now). The CFA will note these as 2y5y to stand for the 5 year rate 2 years into the future or 3y2y for the 2 year rate 3 years into the future.
LOS 53i: Compare, calculate, and interpret yield spread measures
The benchmark yield is the base rate and is also referred to as the risk-free rate of return. This captures our macroeconomic factors such as inflation, monetary and fiscal policy. Changes here impact all bonds.
The spread (aka risk premium) refers to the difference between the YTM on a bond and the benchmark. This captures all microeconomic factors specific to the issuer such as credit rating, liquidity, and tax status.
