Chapter 6 - Bonds

Question 1

why do we need valuation of bonds?

Answer 1

There are several reasons.

First, the prices of risk free treasury bonds can be used to determine the risk free interest rates that produce the yield curve. The yield curve helps us predict future market movements.

Second, firms often issue bonds to fund their investments, and the returns that investors receive on those bonds is one factor determining the firm’s cost of capital.

Question 2

What is a bond?

Answer 2

A bond is a way for someone to raise money by receiving money today against the promise of returning the money in the future. Can be coupon bond or zero-coupon bond.

Question 3

The terms of a bond is called..?

Answer 3

Bond certificate.

The bond certificate includes things like time to maturity, and the dates of all payments, and the amount to be payed.

The maturity date is the final payment date.

Question 4

What is a “term”?

Answer 4

Term is used to describe the time remaining until the final repayment date, also known as the maturity date.

Question 5

The promised interest payments of a bond is called …?

Answer 5

Coupons

Question 6

What is the face value of a bond?

Answer 6

The face value of a bond is the notional amount we use to compute the interest payments. Usually, the face value is repayed at the maturity date.

Question 7

The amount of each coupon payment is called?

Answer 7

The coupon rate.

By convention, the coupon rate is set as APR. therefore, the amount of each coupon payment, CPN, is computed as:

CPN = (CouponRate x FV) / (Number of coupon payments in a year)

Question 8

Why do we use APR and not EAR on bonds?

Answer 8

using EAR with bonds would be a little weird. If we have a bond that does quarterly payments, then it would be a little difficult for the issuer to express the EAR, as they have no clue what the investor will do with the money. They could make EAR based on their interest rate, but that would not be very good since the investor is not getting that rate on his coupon payments (since they are given to him, and not accumulated in the face value of the bond to grow). There is no compounding on a bond, so using EAR makes little since

Question 9

What is the principal of the bond?

Answer 9

The same as face value.

It is NOT the price or the value of the bond.

Question 10

What are zero-coupon bonds?

Answer 10

Zero coupon bonds are bonds that do not pay any coupons. no interest payment.

However, they will trade at a “discount”, which means that they are trading for less than their face value, or the amount that will be returned to the buyer at the maturity.

Zero-coupon bonds trade at a discount because the present value of a future cash flow is always smaller than the future value. As long as regular conditions apply at least. In other words, the bond trades lower than future value because it simply is not worth as much today as in the future. So, the buyer will be compensated for the time value of money.

US treasury bills (T-bills) are example of zero-coupon bonds.

Question 11

What are T-bills

Answer 11

T-bills, short for Treasury bills, are zero-coupon bonds that lasts up to a year. They are issued by the US government.

People often talk about interest rates of the T-bill, but it is actually the difference between the discount value and the face value. So it is not an explicitly given interest rate, but rather implicit.

Question 12

What is YTM?

Answer 12

Yield to maturity.
YTM is a special name given to bonds, and it specifically refers to the internal rate of return IRR that the investors get from buying the bond at the discount, and holding it until maturity.

IRR is the rate that ivnestors get that make the NPV of the cash flow equal to 0.
For zero-coupon bonds, this IRR is basically just the return they get.

But YTM, or sometimes just the yield, is more generalizable. we take the price the bond sells for, and equals it to its promised cash flow, and find the IRR. This IRR is the yield to maturity. REMEMBER: We only use present values.

Question 13

Why are zero coupon bonds with N periods easy to calcualte?

Answer 13

No interest payment. Only face value, original discount price, and IRR, which we call the YTM.

Question 14

What are coupon bonds?

Answer 14

Coupon bonds are bonds that pay interest in addition to the face value.

There are 2 types of US treasury securities trading today:
1) Treasury notes: one to 10 year original maturity.
2) Treasury bonds: 10 to upwards original maturity.

Question 15

Precisely define YTM

Answer 15

Yield to maturity is the single discount rate that would equate the current value of the bond with the present value of the cash flows of the remaining bond.

Question 16

Formula for YTM of a coupon bond

Answer 16

The annuity formula seems important af.

BUT, here we reach the IRR problem. We cannot solve it analytically.

Question 17

What is important to remember about YTM

Answer 17

The YTM, or IRR, is a per-coupon rate. Not necessarily APR. It is common to convert the result to APR when done.

Question 18

coupon bonds can trade at a discount and …

Answer 18

a premium, and a par.

The value is dynamic.

Question 19

How can YTM exceed the coupon rate?

Answer 19

the bond must be a coupon bond, and trade at a discount

Question 20

Define a bond that trades at a par

Answer 20

its yield to maturity is the same as the coupon rate. Its face value will be the same as its price

Question 21

Elaborate on the effect of time on bonds

Answer 21

If the YTM remains unchanged, the less time there is to maturity, the smaller the discount will be. This means that as time move by, the price of the bond will most closer to the face value. This is because, since the YTM remains unchanged, and we have less time, the price of the bond must increase to balance the time vs constant YTM.

YTM remains unchanged is a sort of “lightspeed in vacuum” approach. The only goal here is to understand that as time pass by, and all other factors remain constant, the price/value of the bond will move closer to its face value.

Question 22

Consider a coupon bond of 30 years with face value of 100. It has a coupon rate of 10%. If this bond has a YTM of 5%, what will it trade at?

How about right before the first coupon payment?

how about right after the first coupon payment?

Answer 22

We compute it using the present value of annuity, but we need to use the YTM as the interest.

P = (Coupon/YTM) x (1 - 1/(1+YTM)^30) + 100/(1+YTM)^30

To find the price right before/immediately before the first coupon payment, we can consider the payment as date 0, and add the coupon as the initial “investment”.

P = Coupon + (Coupon/YTM)x(1- …^29) + 100/(…)^29

To find immediately after the first payment, we can subtract the coupon, and keep the 29.

There are some intuition here as well. Consider the price for the bond immediately before the first coupon payment. We know that we skip the time value of money for one year, get the coupon, continue with the bond. We can use the IRR, or the YTM to find this value since the YTM tells us the interest we get EVERY DAMN PERIOD. Since we are about to value it one period forward in time, we multiple the original price by the YTM. NB: this is a method that actually only works when we ignore the cash payout times. It works if we have continuous coupon payments. It will give us the value of the bond one year (one period) forward in time.

Question 23

elaborate on bond prices and itnerest rate changes

Answer 23

As interest rates and demand for bond yield rise, bond prices will fall, and vice versa.

Question 24

elaborate on changes in interest rate and its impact on bond price sensititivty

Answer 24

Bonds that have maturity in the close future are not much impacted. This is because there are fewer years to see the interest change impact on.

However, bonds with longer time to maturity can have substantial changes in the price of the bond.

the same principle applies to bonds with higher frequency coupon payments. Because they pay more of the cash earlier, they will be less sensitive to changes in interest rate.

Question 25

elaborate on the zero coupon yield curve and the law of one price

Answer 25

In short: the zero coupon yield curve is all we need to price all other risk free bonds.

We can use the zero coupon yield curve to build a portfolio that emulate any coupon bond cash flow. The price of the portfolio will be equal to the sum of its parts. Then, because of the law of one price, we know that this portfolio price is the same as the coupon bond that offers the same cash flow.

Now, we can value a coupon bond by using zero coupon bonds. The price of the bond should be equal to the present value of the cash flow from the bond, when discounted (obviously).
HOWEVER: We must discount using the interest rates that are given in the yield curve for the zero coupon bonds, as they represent the alternative investment that makes our money have a time value. Therefore, the discount interest we use in year 2 is likely not the same as the interest we use to discount year 3.
The final value we get is the price of the bond. This price should be equal to the price of the portfolio of the zero coupon bonds we used earlier, because of the law of one price.

Question 26

elaborate on bonds

Answer 26

Bonds have a maturity (date), a face value, a price, and potentially coupons. Based on these, they will essentially work together to create a cash flow. A bond can be regarded as a cash flow. This cash flow can be used to find the IRR, which we call yield to maturity.
When working with risk-free bonds, also known as default-free bonds, if we are given the yield curve of zero-coupon bonds, we can use this yield curve to find the price of all risk-free coupon bonds as well, and also then find their corresponding yield to maturity. The law of one price will be the driving force here, so it assumes an arbitrage-free market.
There will be bonds of various term structures: We have different coupon rates and different face values and different payout policy and different time to maturity. The great “equator” will be the price. The price will be adjusted so that the present value of all the risk-free bonds are the same.

Question 27

how do we go from the zero-coupon yield curve to finding the YTM for coupon bonds?

Answer 27

step 1) Find the price (present value) of the coupon bond. We do this by constructing it from the zero coupon bond.

step 2) equate the price to the cash flow, and solve for the interest rate. This is the IRR. IRR is the same as YTM.

Question 28

how can bonds with same risk and same maturity have differnet yields?

Answer 28

Different coupons will carry different yields.

If the coupon is small, most of the “weight” of the bond-value lies in the final payment, since the face value is payed.

if the coupon is large, the payment is more evenly spread out.

This “spread” in combinaiton with the various YTM for the zero-coupon bonds of various time frames will determine the yield.

In general, as the coupon increases, earlier cash flows become more important. As the coupon decreases, later cash flows becomes more important.

Question 29

What is the “credit risk” of a bond?

Answer 29

Credit risk is the risk for default of the issuer.

Question 30

What can we say about the YTM of a “risky” bond?

Answer 30

It will be higher than the YTM of a corresponding treasury bond that is risk free. This is because we use the PROMISED and not the actual cash payment/coupon to compute the YTM. Since the corporation might default on the bond, the expected cash flow will be smaller than the promised cash flow. It will never be larger than it.

Question 31

What do we call the difference in yield between the treasury bond and corporate bond?

Answer 31

Credit spread, or default spread.

Credit spread will fluctuate based on the perceptions regarding the probability of default.

Question 32

Elaborate on certain default as a risk free investment

Answer 32

The point is that if we can say for certain that some corporation will default, and we know for certain that they will then only be able to pay say 90% of the promised face value, then we can treat this investment as a risk-free investment with a face value of 1000*0.9 = 900 usd. Because of the law of one price, we can then use the risk free yield curve to do the rest.

When studying the prices, we will quickly find out that having the certain default will obviously lower profit, which will decrease the price of the bond.

And the case of uncertain probability of default lies somewhere between. The price of the bond WILL be lower, but not necessarily as low as the certain default case.

Question 33

How is YTM impacted of certain default case?

Answer 33

YTM is calculated using PROMISED face value. Therefore, the yield will be much higher with risky investments than the actual yield. This is a sort of fake yield. It is promised, but not necessarily the yield you will get.

Question 34

elaborate on yield curve and its relationship with bond prices in general

Answer 34

if people assume that interest rates will decrease, it means that there will be more difficult for them to earn risk-free cash. Therefore, they will be willing to pay more/accept lower yield from a bond investment, regardless of the type of bond.Likewise, if interest rates increase/people believe they will increase, there are better opportunities in the market to earn risk free profits from the rates, and therefore the prices of bonds become smaller as a result of decreased demand