Chapter 6 - Bonds Flashcards
How large is the US bond market?
Approximaltey the same as the US equity market actually. There is a great amount invested in both treasury and regular bonds
Why do we study bonds?
1) Bond prices can be used to determine the risk free interest rate. This allows us to create the yield curve. This yield curve is important predictor of future sentiment.
2) Firms typically issue bonds to raise funds for their own investments. Understanding the bonds is necessary to understand the firm’s cost of debt.
3) Easy understanding of how securities are priced in a compattiive market
Recall the definition of a bond
A bond is a security sold by governments and corporations in order to raise money that will fund projects. It is an obligation that force the issuer to return some promised future payment.
what is the final repayment date of a bond typically called?
Maturity date
What is the bond certificate?
bond certificate holds the details of amounts and dates of the payments.
What do we call the time remaining of a bond, until the maturity date?
The term of the bond
What payments do bonds typically pay?
2 types.
coupons and the principal.
The coupon is the promised interest payment.
The principal, also called FACE VALUE, is the amount we use to compute the coupons.
Usually, the face value is payed at maturity.
When we say “a $1000 bond”, what do we mean?
A bond with a face value of $1000.
elaborate on how the coupon is computed
We need the coupon rate and the face value. The coupon rate is set by the issuer of the bond. It is stated on the bond certificate.
Coupon rate is by convention APR.
We compute the coupon by the following formula:
coupon = (coupon rate x FaceValue) / (CPNs each year)
What is the simplest type of bond?
Zero coupon bonds
Name an exmapel of a zero coupon bond
US treasury bills.
Us treasury bills are zero coupon bonds with a time to maturity of up to one year.
Why are zero coupon bonds simple?
They dont have a coupon. Instead they have a face value, and will typically (most likely) trade at a price the is below the face value (discount, in other words).
Recall IRR
Internal rate of return is the discount rate that makes the present value of the cash flows equal to 0.
Requires at least one negative initial investment
What is YTM?
Yield to maturity.
It is a special name given to bonds. It is actually the same as IRR, but for bonds. It is also commonly just called “yield”.
Formally: The yield to maturity of a bond is the discount rate that sets the present value of the promised payments equal to the current market price of the bond.
Given a zero coupon bond with a time to maturity of several years, how do we find the yield to maturity?
The YTM is solved by transforming the IRR equation:
Price = FV / (IRR ^n)
IRR = (FV / (Price)) ^{1/n}
This is basically derived from the IRR formula AND the law of one price. we use the law of one price to say that the bonds market price should be exactly equal to the present value of its cash flows.
Therefore, if the bonds YTM is unknown, we use the current market price as well as the bond certificate to find the YTM.
If YTM is known, but the price is not, we can derive the price easily by using the established assumption of law of one price.
What perticular benefit can we use the zero coupon default-free bonds with time to maturity at date n for?
They are suitable at determining the risk free interest rate for various periods.
Since the zero coupon bond dont pay coupons, they are easily used to compute YTM. Then we use the law of one price to establish the fact that YTM of the zero coupon bond MUST be the same as the risk free rate for the same period.
This is something we can do because of the same level of risk, and same cash flows. This allows us to use the law.
What is the yield curve?
The yield curve is a function that maps a time to maturity to an interest rate. Risk free interest rates speciically.
Because of the relationship between risk free interest rate, yield curve, zero coupon US treasury bonds, we often refer to the yield curve as the ZERO COUPON YIELD CURVE.
What is spot interest rate?
Spot interest rate is the yield for some specific time to maturity on a zero coupon risk-free bond.
We can find it by using the YTM formula established earlier.
Elaborate on coupon bonds
Same as the zero coupon bonds, they too pay the face value at maturity. However, they also pay coupons during the timebetween.
Two types of US Treasury coupons currently active?
1) US treasury notes
2) US treasury bonds
The difference is the time frame. They differ in terms of hte original time to maturity.
The notes originally have a time to maturity of one to ten years.
The bonds have 10 years to 30 years typically, but the only formality is that they have longer than 10 year maturity.
Why is coupon bonds slightly tricky to work with?
They may have cash flows at intervals that makes it difficult, at least more difficult than the zero coupon bonds.
If we want to find the yield, or the market price, we must use the cash flows at all the various times and discount them correctly.
How do we go about when trying to find the YTM of a coupon bond?
We make use of the fact that bonds typically are annuities. Because of this, we can use the annuity formula, and smack the face value with the same discount rate at the end as well.
This requires the coupons to be the same always. However, they typically are because they are computed using the face value. The face value will not change, which means that the annuity formula is perfectly valid.
Again, the YTM is equal to the discount rate that would equate the current market price with the promised cash flows of the bond.
P = CPN x 1/y (1 - 1/ (1+y)^n) + FV/(1+y)^n
The formula for finding the market price of the coupon bond has one signifcant drawback….?
Because the equaiton is transcendental, there is no way to solve for the YTM without without using a trial-and-error apporahc, or solving numerically. Because of this, I suppose it would be more common with tasks that is given the YTM and the case is to compute the price.
do coupon bonds trade at a discount?
Coupon bonds can trade at a discount, at par, or at a premium. This will depend on a varity of factors. There are some very basic rules that govern the dynamic nature of bond prices and determine whether they will trade at par, premium or at a discount
In general, if the coupon rate is lower than the risk free rate we use to discount, then the bond will trade at a discount.
If the coupon rate is very large, and larger than the discount rate, then the bond will trade at a premium.
What happens when a coupon bond is trading at a discount?
When a bond is trading at a discount, the investor will earn interest on both the coupon and the face value. WHat this means is:
The investor receives return on the bond, and the return is part cpupon and part face value. In comparison, zero coupon bonds would only give investors return on the face value. With coupon bonds however, the investor will be given the face value, and the coupon. Since the bond is trading at a discount, it trades at a price that is lower than the face value.
ALSO: When a coupon bond is trading at a discount, we can basically guarantee that the coupon rate is smaller than the discount rate.
What happens with the YTM if the bond is trading at a discount? (coupon bond)
If the bond is a coupon bond and is trading at a discount, we know that the YTM must exceed the coupon rate.
What values of YTM is “legal” when we know that the bond is a coupon bond and is trading at a premium?
If the bond is trading at a premium, it means that the price of the bond is greater than the face value.
This means that the YTM must be less than the coupon rate. This is because we will loose money on the price vs face value trade. Since we only gain money from the coupons, but loose some additionally, the YTM (which is the total yield of the bond) cannot be larger than the coupon rate.
What can we say about a bond that has YTM same as coupon?
It will trade exactly at par.
There is no gain from the price-faceValue deal, so all cash flows stems from the coupons. Therefore, the coupon rate (given as APR of course) is the same as the YTM.
A $1000 one year coupon bond with a coupon rate of 5% trades at what price…?
Depends on a whole lotta shit. we need the risk free discount rate IF the bond is risk free etc.
However, in general terms, the bonds price will be equal to the present value of its cash flows.
-price + coupon + face Value
price = 50/r + FV/r
Given a discount rate of 5 percent:
price = 50/1.05 + 1000/1.05 = 1000
What can we say about the issuer of bonds in regards to what the coupons are typically set to be?
The coupons are typically set so that the bond will initially trade very close to par. Then after this, the market price typically fluctuate as a result of two things:
1) Time
2) Changes in market interest rate
How does the US treasury set bond coupons?
Us treasury sets bond coupons so that the bonds initially trade at par.
What factors influence the price action/movement of a bond?
Time & changes in market interest rate.
Shortly on how time affect bond prices
As time move by, and we become closer to the maturity date, the present value of the bond converge towards the face value.
Shortly on how market interest affect bond prices
It will affect the present value of the cash flows, which will affect price
Elaborate on how time affect bond price more deeply
If everything were to remain the same, except for time, if we have a zero coupon bond it will see a price increase equivalent with the yield to maturity. This is because since everything else remains the same, we have to account for the time that pass by (time value of money).
For instance, say a 30 year zero $1000 coupon bond with a 5% ytm. Originally, it will trade at 1000/1.o5^30 = 231.
one year later, we will have the same bond with the same face value, but now with 29 time periods to discount. Gives us 243. if we take 243 / 231 we get 5%, which is equal to the yield.
So, the bond will loose value each time period equal to the discounting rate.
What can we say about a bond investment if the YTM does not change in the holding period?
The IRR of the bond investment will be equal to the YTM. This also applies if you sell the bond early. And it holds for coupon bonds.
What can we say about the price of a coupon bond right before and right after a coupon paymenmt?
The price right before and right after the payment will have a difference of exactly equal the coupon.
What is the point of convergence in terms of bond prices?
Typically face value of the bond. However, it the last coupon is payed at maturity, then the convergence is towards FV + coupon, and not just the FV.
Elaborate on whats going on ehre
Shows how the bond prices change with the passage of time for a variety of coupon rates.
The YTM remains the same: 5%. Recall that if the bonds are US treasury, then they are risk free, and a risk free investment cannot earn a larger return one place compared to the other place, so they need to have the same ROI. Therefore, all of these bonds have the same YTM.
As we can see, the zero coupon bond shows a smooth line, indicating that time is the only thing affecting its price.
FOr the couupon bonds, there are soemthing causing the zig zags, whcih is the coupon effect.
The upper bond has a coupon of 10%, which is higher than the YTM of 5%. Because of this larger coupon, the bond price will trade at a premium. Therefore, its price is higher than the others, but will converge towards the par face value (and final coupon value) as time pass by.
The bond with YTM = Coupon rate has no time effect, in terms of percentage of face value.
Do not confuse zero coupon bonds and coupon bonds. It is very important to understand that the coupons cause a bond to trade at a price higher than its equivalent zero coupon bond. This is because coupon bonds produce more cash flows, which naturally makes its price higher.
Elaborate on clean price vs dirty price
The dirty price is the actual cash price of the bond. However, as we know, the bond price tend to move in this sawtooth pattern. However, traders are typically more interested in price changes of the bond that occur as a result of changing YTM. Therefore, the “clean price” is an adjusted price.
Clean price = Dirty price - Accrued interest
What is accrued itnerest?
Accrued interest is equal to the amount of the next coupon oayment that has already been accrued.
Accrued interest = Coupon Amount x (Days since last coupon / Days in current coupon period)
Accrued interest will rise and fall with the sawtooth pattern.
What types of bonds are more sensitive to changes in interest rates?
Long term bonds
What can cause bond prices to rise?
If interest rates rise and bond yield rise, the price will FALL. And vice versa.
How does the coupons affect the duration of the bond?
Recall that duration refers to the sensitiity of the bond.
Large coupons have the impact of making more of the cash flow happen early. Therefore, larger coupons will make bonds less sensitive towards changes in for instance market interest rates
Recall YTM
YTM is the internal rate of return that equates price and the present value of the cash flows of the bond
How do we use the law of one price to determine the price and yield of any default free bond given the spot interest rates?
The key is to replicate the specific bond in question.
Since we want ot find the deatils of a default free bond, either zero coupon or coupon, we make use of the fact that the risk level is the same as with the zero coupons (spot interest rates) default free bonds .
We identify the cash flow of the specific coupon, and then re-create it using the spot interest rates. The sum of individual prices of the bonds that together create the replicated cash flow, MUST be equal to the price of the specific bond, because of the law of one price.
For instance, consider a $1000 bond that pays 10% coupons over 3 years. It will have the following cash flow:
100
100
100 + 1000
We replicate this cash flow using zero coupons
What do we need to determine the price and yield of any default-free bonds?
we need the spot interest rates.
Recall that the spot interest rates refers to the yields of zero coupon default free bonds.
We need the spot interest rates, and then we need the law of one price to make the compuationsl
What is important to remember when doign the replication of cash flows?
The zero coupon bonds MUST be discounted using their own individual discount rate. This is because these investments WILL DIFFER with differnet time frames. A 2 year zero coupon will not offer the same yield as the 1 year zero coupon.
So, we use those interest rates to discount, and the final result is the price of the zero coupon bond portfolio. Since the law of one price applies, we know that the specific bond in question is priced the same.
Most important part about the corporate bonds
There is a risk of default. Promised payments are promised, not guaranteed. The promised payments serves as an upper limit of what investors will receive. which means that the expected value is less than this.
What is credit risk?
Risk of default
How does credit risk affect prices of corporate bonds?
Since credit risk implies that the expected return is less than the promised return, investors will not be willing to pay as much for these bonds, as they would pay for the equiaveltn treasury bond.
what can we say about the YTM of a bond that is not risk free?
It will be higher then the risk free counterpart. this is to compensate for the fact that there is a risk of default.
In the case of certain default, YTM is higher then that of risk free. However, since certainty imples no risk, we discount using the same risk free rate.
Since we know how much we would get from the defaulting firm, and this sum is lower than face value, the price of the bond must be lower as well.
YTM is computed using face value. This is why YTM is higher for risky bonds. Their prices are lower, but YTM is computed on the PROMISED face value payment.
We could use expected value though, which would make less impact on YTM.
For a corporation, what can we say about its bonds expected return?
The expected return of the bond (from the perspective of the investor) is equal to the firm’s DEBT COST OF CAPITAL.
What are the most important classes of bonds in terms of credit rating?
Investment grade
Speculative/junk/high yield
What does the bond rating depend on?
The risk of default.
Also depends on the bond buyers ability to lay a claim on assets in the event of a bankruptcy
Define credit spread
Credit spread, also commonly referred to as the “default spread” is the difference in yield between the specific risky bond and the equivalent risk free bond.
Elaborate on the important aspects of sovereign bonds
First of all, many states are actually risky. This makes many sovereign bonds behave like corporate bonds.
However, there is also the risk of how the company might behave in the event of finanacial struggle. FOr instance, printing more money to repay debt. This will essentially inflate away gains. This would make the bonds highly useless.
what can we say about the coupons of a coupon bond that is currently trading at a discount?
The coupon rate is less than the yield.
what happens if we have a negtive yield?
Negative yield is basically arbitrage.
We just sell the bill and receive free money.
Very unlikely scenario, typically the result of extreme cases like 2008 etc
