Zero coupon bonds

Professor Anh Le

1 – Zero coupon bond and zero yields

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

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Yields

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 ...

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