Zeros
Fixed Income
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.
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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.
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