Fixed Income 1 – Zero coupon bond and zero yields
Zero coupon bonds
Professor Anh Le
A zero coupon bond (or zero for short), as its name suggests, is a bond that pays no coupons. It only pays the face value on the maturity date. Not surprisingly, sellers of zero coupon bonds have to offer them at a deep discount in order to sell them to the public. For example, a 30‐yr zero, face value $1,000 could be selling for as little as $53.54. One question you may ask right now is: if you only get back the face value on the maturity date and no coupons between now and then, isn’t it weird since you don’t earn any interest? The answer is: every bond earns interest and zeros are no exception. Let’s think about it this way. Instead of buying the above 30‐yr zero, you put $53.54 in a bank account that pays an interest rate of 10% p.a. semi‐annually compounding – how much would you end up after 30 years? By now, you should be quite proficient with this: 10% semi‐annually compounding really means 5% per six months, therefore, after 30 years, $53.54 would grow to $53.54(1.0560) = $1,000. So, the interest rate that you earn from a 30‐yr zero is implicit in the discount that you receive. From our calculations above, a 30‐year zero face value $1,000 selling for $53.54 is implicitly offering you a rate of interest of 10% p.a. semi‐annual compounding. This rate of interest rate has a name. It is called zero yield. Knowing about the zero yield for a maturity is very useful because we can price zero for that corresponding maturity. In the above example, the zero yield for the 30‐year maturity is 10%p.a., therefore, the price for the 30‐yr zero, face value $1000, must be $1000/1.0560 = $53.54. Should the 30‐ yr zero yield be 5%, the price of the 30‐yr zero must be $1000/1.02560 = $227.28. In short, zero yields are the interest rates that we earn on zeros AND we can use such zero yields to price zeros of the corresponding maturities. 2 – The zero yield curve It is common understanding that when you lend your money for a short period, you will get a lower rate of interest than when you lend money for a longer period of time. The rate that you will get from your term deposits tends to be higher than what banks pay on your money market savings account. Likewise, if bond sellers issue a 1‐year zero, the public will generally require a lower interest rate/zero yield than if they issue a 10‐year zero. Here is an example of the borrowing rates for different maturities: Maturities Yields 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
3.00% 3.60% 4.20% 4.70% 5.20% 5.60% 6.00% 6.30% 6.50% 6.60%
Often, people (myself included) don’t like tables and they would put the numbers above into a graph like the one below where the x‐axis corresponds to maturities and the y‐axis corresponds to interest rates or yields. This curve is called the zero yield curve. It is also called the term structure of interest rates.
Fixed Income
Zero coupon bonds
Professor Anh Le
Zero yield curve
7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% 0 1 2 3 4 5 6
Historically speaking, the zero yield curve can take quite a variety of shapes. The most common shape is upward sloping as in the graph above. However, there have been times when we had hump‐shaped yield curves where interest rates are highest for medium maturities and low for very short and very long maturities. More recently, we have an interesting situation where the yield curve is inverted. Effectively, lending long will earn a lower rate of interest than lending short! How strange! I often wonder how banks can survive with an inverted yield curve. If they are in the business of borrowing short and lending long, doesn’t it mean that their borrowing cost is higher than their interest revenues? More interestingly, people believe that the presence of an inverted yield curve suggests that a recession is eminent. Statistically speaking, an inverted yield curve tends to appear 2 years before a recession comes upon us. 3 – The zero yield curve and bond pricing Regardless of the shape, the important point to take away from a zero yield curve is that interest rates or yields are different from maturity to maturity. How would we price a 1‐yr T‐note, face value $1000, paying a coupon rate of 6% p.a. if we are given such a zero yield curve? First, let’s work out the cash flows from this bond. It’s quite straightforward: • • $30 in 6 months $1030 in 12 months
Second, let’s ask the questions that we asked before: • • How much would we have to put into a bank account to generate $30 in 6 months? How much would we have to put into a bank account to generate $1030 in 12 months?
Fixed Income
Zero coupon bonds
Professor Anh Le
To answer the first question, we would need to put some money into a bank account for 6 months and earn the rate of interest corresponding to that maturity, which, as implied by the table/graph above, is 3.00% p.a. This suggests that we would have to put into a bank account an amount equal to $30/1.015. Similarly, to answer the second question, we would need to lend for 12 months at the rate of interest of 3.60%. This suggests that we would need: $1030/1.0182. Together, these calculations suggest that the price of the 1‐yr T‐note must be the sum of the 2 amounts: $30 1.015 $1030 1.018 $1,023.45
In essence, in the presence of the zero yield curve, to price the T‐note, we need to discount each of its cash flows by the correct zero yields. For the first cash flow that comes in 6 months, we need to use the zero‐yield corresponding to the 6‐month maturity. For the second cash flow that comes in 12 months, we need to use the zero‐yield corresponding to the 12‐month maturity. And so on. Let’s do a different example to make sure we understand what is going on: Let’s price a 1.5‐yr T‐note, face $100, paying coupon 10% p.a. The price of this T‐note will be: $5 $5 $105 $108.40 1.015 1.018 1.021 Let me explain. The first coupon payment of $5 comes in 6 months therefore being discounted at the 6 month rate of 3%. The second coupon payment of $5 comes in 12 months therefore being discounted at the 12 month rate of 3.6%. The third payment of $105 comes in 18 months therefore being discounted at the 18 month rate of 4.2%. Let’s do another one: Let’s price a 1.5‐yr T‐note, face $100, paying coupon 20% p.a. The price of this T‐ note will be: $10 $10 $110 $122.85 1.015 1.018 1.021 Let me explain again. The first coupon payment of $10 comes in 6 months therefore being discounted at the 6 month rate of 3%. The second coupon payment of $10 comes in 12 months therefore being discounted at the 12 month rate of 3.6%. The third payment of $110 comes in 18 months therefore being discounted at the 18 month rate of 4.2%. Alright, you got it. I will stop boring you now. But generally, given the above zero yield curve, we can price any bond. If a T‐note pay CF0.5 in 6 months, CF1 in 12 months and CF1.5 in 18 months, we simply discount CF0.5 at the 6 month rate of 3%, CF1 at the 12‐month rate of 3.6% and CF1.5 at the 18‐month rate of 4.2%. Therefore, the price of the bond must be:
. .
1.015
1.018
1.021
Fixed Income
Zero coupon bonds
Professor Anh Le
whatever the cash flows are. If the bond has cash flows that go beyond 1.5 years, it is not a big deal: our zero yield curve goes beyond the maturity of 1.5 years too. The most important thing is to make sure that you discount cash flows using the correct zero yields corresponding to their respective maturities. 4 – The discount factors Let’s look at the last pricing equation we have from the last section:
. .
1.015
1.018
1.021
As I said, this equation allows us to price any 1.5‐yr bond that pays CF0.5 in 6 months, CF1 in 12 months and CF1.5 in 18 months regardless of what the cash flows CF0.5, CF1 and CF1.5 are. As we price bond after bond after bond, however, we realize that we always discount the cash flows by the same zero yields. For example, we always discount CF1.5 by dividing CF1.5 by 1.021 to the third every time. Punching in 1.025 to the third every time hurts our finger – therefore, we can save ourselves some work by pre‐computing the
.
which is 0.939556. Next time, to discount a cash flow that comes in 18
months, all we need to do is to multiply the cash flow CF1.5 by 0.939556. Similarly, for cash flows that come in at 1‐yr, since
.
0.964949, to discount a cash flow that
comes in 12 months, we multiply the cash flow CF1 by 0.964949. Finally, to discount a cash flow that comes in 6 months, we multiply the cash flow CF0.5 by 0.985222. So the price for any 1.5‐yr bond that pays cash flows CF0.5, CF1.0, CF1.5 would be: P = CF0.5 x 0.985222 + CF1.0 x 0.964949 + CF1.5 x 0.939556 Let’s double check, and compute the price of the 1.5‐yr T‐note, face $100, paying coupon 10% p.a. that we priced before. For this bond, CF0.5 = CF1.0 = $5 and CF1.5 = $105, according to the equation above, the price of this bond should be: P = 5 x 0.985222 + 5 x 0.964949 + 105 x 0.939556 = 108.40 which coincides with our number before. These numbers (0.985222, 0.964949, 0.939556) have an intuitive name: discount factors. Instead of discounting the cash flows at the appropriate zero yields, we multiply the cash flows by these numbers. Let’s review how we compute our discount factors, which from now on I will denote by d0.5, d1, d1.5. • • • d0.5, the discount factor for the 0.5‐yr maturity: d0.5 = 1/1.015 = 0.985222 d1.0, the discount factor for the 1.0‐yr maturity: d1.0 = 1/1.0182 = 0.964949 d1.5, the discount factor for the 1.5‐yr maturity: d1.5 = 1/1.0213 = 0.939556
Fixed Income
Zero coupon bonds
Professor Anh Le
Generally, when we price a cash flow of, say, $100, that comes in at year n at the zero yield for the n‐ year maturity of yn: , the discount factor for the n‐year maturity is dn = . This way, to , we can simply do the
discount the $100 backwards n years, instead of doing the bulky thing multiplication: 100 x dn.
Finally, before concluding this section, let’s look at the relationship between the discount factor and its
corresponding zero yield for the n‐year maturity: dn =
We realize that one way to interpret the discount factor dn is to think of it as the price of a zero coupon bond that pays $1 at year n. The reason is that looking at the right hand side of the above equation, it is essentially the same as discounting $1 at the zero yield yn for n years. 5 – Ytm Let’s come back to the 1.5‐yr T‐note, face $100, paying coupon 10% p.a. that we priced using the zero yield curve before. We have showed that it should be: P = 5 x 0.985222 + 5 x 0.964949 + 105 x 0.939556 = 108.40 In the financial press or elsewhere, people often report ytm’s instead of prices. To go from price to ytm is not too hard, we just need an Excel function called IRR. But before getting to know exactly how to use the IRR function, let’s review what a ytm is. Ytm, or yield to maturity is some discount rate that when used to discount all the cash flows from a bond, it gives us the price of that bond. In other words, for the bond above, its ytm, denoted by y, must be such that: 108.40 5 1 2 1 5 2 1 105 2
Solving the above equation for y, that would be the bond’s ytm. We can use the IRR function in Excel to solve for y as follows: Enter ‐108.40, 5, 5, 105 in cell A1, A2, A3 and A4 respectively. In cell B1, enter the following: = IRR(A1:A4) That would give you an internal rate of return, R, that solves: 108.40 5 1 1 5 1 105
In this case R = 2.0826%. Double this value, that would be the yield to maturity of the bond y = 4.1652%. 6 – The STRIPS zero markets
Fixed Income
Zero coupon bonds
Professor Anh Le
Examples of zero coupon bonds are T‐bills whose maturities are less than 1 year. The government doesn’t issue zero coupon bonds with one year or longer maturities. However, the government, through a program called STRIPS, allows dealers to buy their coupon bonds and sell the cash flows separately as independent zeros. Just imagine someone buying a happy meal from McDonald, and then separately selling the burger, the soda, the fries and the toy. A 10‐year T‐note, for example, can be split into 21 individual securities, consisting of 20 coupon payments and one principal payment. Why would you need something like the STRIPS program to create zero coupon bonds? You would imagine that it is a simple task. However, it is not quite that simple. Physically created zeros were created in the early days but it involved setting up a custodian bank or a trust company to hold the securities and a transfer agent to track ownership of each of the zeros. More importantly, if the trust company defaults on the payments, the government has no obligation to honor them. This arrangement therefore was both costly and risky. With the STRIPS program, both of these issues are resolved. First, the government tracks the ownership details of each zeros created under the program. Second, STRIPS zeros are direct obligations of the government. On the maturity date of each of the zeros, instead of going to the trust to claim payments, you go straight to the government. Since the government keeps your details in their book‐entry system, they have no problem in honoring the zeros. With a similar program from McDonald, if someone buys the sodas stripped from a happy meal and got a stomach‐ ache, McDonald will be responsible for it. This would help people feel safe to buy the stripped burger, soda, fries and toy even if they are not sold directly from McDonald. Understandably, the STRIPS program helps greatly with the liquidity of the zero‐coupon bond market. 7 – The Bootstrap method Despite the STRIPS program, as a matter of fact, STRIPS zeros are not as liquid as T‐bills, T‐notes and T‐ bonds. Therefore, using zero yields implied from STRIPS zeros may introduce market noises and errors into the zero yield curve and hence the discount factors. These errors could potentially result in mispricing if we use such a zero yield curve to price bonds. It would be ideal if we can use the prices of the more liquid T‐bills, T‐notes and T‐bonds that are free from microstructure noises to construct a zero yield curve. It turns out that this is entirely feasible thanks to a procedure called the Bootstrap method. Essentially, this method is a way to imply what the discount factors for different maturities must be given a set of existing bond prices. Let’s work on the following simple example, assuming that we have the following two coupon bonds: • • Bond 1: 0.5‐yr maturity, 4.25% p.a. coupon, face value = $100 and price = $99.40625 Bond 2: 1‐yr maturity, 4.375% p.a. coupon, face value = $100 and price = $98.96875
I will show how we can use the bootstrap method to work out the discount factors d0.5, d1.0 for maturities 0.5 year and 1 year. As I have shown you above, given the discount factors d0.5, d1.0, any bond that pays cash flows CF0.5 and CF1.0 at 0.5 year and 1 year, respectively, must be priced as:
Fixed Income
Zero coupon bonds CF0.5 x d0.5 + CF1.0 x d1.0
Professor Anh Le
First, bond 1 has one cash flow of (100+
.
) = $102.125 in 6 months times, its price must be: 102.125 x d0.5
Second, bond 2 has two cash flows: a coupon payment of
.
.
2.1875 and another cash flow of (100+
) = $102.1875 2.1875 x d0.5 + 102.1875 x d1.0
Since bond 1 costs $99.40625, this suggests that 102.125 x d0.5 = $99.40625. This means the discount factor for 6 months d0.5 =
$ . .
$0.973378.
Now, since bond 2 costs 98.96875: 2.1875 x d0.5 + 102.1875 x d1.0 = 98.96875. Plugging in d0.5 = $0.973378, we have: 2.1875 x 0.973378 + 102.1875 x d1.0 = 98.96875, therefore d1.0 = (98.96875 – 2.1875 x 0.973378) / 102.1875 = $0.947665 So, from the prices of the two coupon bonds, bond 1 and bond 2, I have showed that those prices imply that the discount factors for the 0.5‐yr and 1‐yr maturities must be: d0.5 = $0.973378 d1.0 = $0.947665 Using this formula: dn = y0.5 = 5.47%, y1.0 = 5.448%. It is important to note that without directly observing the zeros prices in the markets, we can infer what the zero‐yields must be by applying the Bootstrap method to prices of coupon‐paying bonds. To imply zero yields for maturities longer than 1 year, we again can apply the Bootstrap method to other coupon paying bonds with longer maturities. 8 – The replicating portfolio approach Assume that we desperately want to buy a 1‐yr zero, face value $100. From our calculations above d1.0 = 0.947665, therefore the price of such a bond must be 100 x 0.947665 = $94.7665. Knowing how much such a zero should sell for is good but we can actually do better. We can indeed combine the two coupon bonds that we have in such a smart way that the resulting combination of bonds will exactly generate cash flows that come from a 1‐yr zero, face value $100. Just to remind you that we have the following two bonds: • Bond 1: 0.5‐yr maturity, 4.25% p.a. coupon, face value = $100 and price = $99.40625 to solve for yn, the zero yields for 0.5‐yr and 1‐yr are as follows:
Fixed Income •
Zero coupon bonds
Professor Anh Le
Bond 2: 1‐yr maturity, 4.375% p.a. coupon, face value = $100 and price = $98.96875
We will need to combine N1 bond1 with N2 bond 2 in a way that replicates a 1‐year zero. • • At time 0.5, the portfolio will generate a cash flow of: N1 x (1+0.0425/2) + N2 x 0.04375/2 At time 1, the portfolio with generate a cash flow of: N1 x 0 + N2 x (1+0.044375/2)
We need N1 and N2 to solve the following: • • N1 x (1+0.0425/2) + N2 x 0.04375/2 = 0 N1 x 0 + N2 x (1+0.04375/2) = 1
Solve this, and we have: N2 = .9786, N1 = ‐0.021. This suggests that we can long .9786 x Bond 1 and short ‐0.021 bond 2. The price of this replicating portfolio is: (0.9786 x $98.96875 – 0.021 x $99.40625) = $94.7665. In absence of arbitrage, with no market frictions, this must be the price of the 1‐year zero with face value = $100, consistent with our calculations above. 9 – Par yields and the par yield curve In this final section, I would like to mention briefly the concept of par yields and most importantly to differentiate par yields from zero yields. Par yields are simply ytms of par bonds. Similar to zero yields, people often put par yields into chart. Not surprisingly, this chart is called a par yield curve. The yield curves reported in the Wall Street Journal are par yield curves. Note the difference between the par yield and the zero yield curve: • A zero yield curve is a set of zero yields plotted against maturities. For example, a 1‐yr zero yield of 5% p.a. corresponds to a lending rate implied by a 1‐yr zero. In other words, lending $1 for one year would yield $1.025^2 = $1.050625 at the end of the one year. No coupon is paid in between. A par yield curve is a set of par yields plotted against maturities. For example, a 1‐yr par yield of 5% p.a. corresponds to a lending rate implied by a 1‐yr par bond, face value $100. Two things we know about par bonds regard their prices and their coupon rate. Their prices are always equal to par. Their coupon rates are always equal to their ytm. For our par bond here, therefore, the par bond sells for $100 in return for a cash flow of $2.5 in 6 months and another $102.5 in 12 months.
•
Where do par bonds come from? Most often, they come fresh from treasury auctions. We know that the Fed chooses to set coupon rates of new T‐notes/T‐bonds right after their auctions in a way that makes their prices closest to par. The most important thing to remember is we can always construct a zero yield curve given a set of par yields using the bootstrap method (as we did in the mini‐case) – and only the zero yield curves are really useful in pricing other fixed income securities.