## The yield curve part VI

##### Published: Nov 2005

Continuing our series on yield curves, this month we explain how to construct a zero-coupon yield curve. We consider two-year and three-year maturities.

Next month we will conclude our series on yield curves by demonstrating how the resulting calculations are graphically represented.

Zero-coupon yield curves can actually be constructed from a series of coupon-paying bonds. This is a technique known as ‘bootstrapping’. For example, consider the following bonds which pay annual coupons and have a maturity of two and three years respectively.

#### Two year maturity

The investor will pay 98.435 for the first bond, to receive 5 (the coupon) in one year and 105 (the coupon and repayment at par) at the end of two years.

In order to calculate the equivalent zero-coupon bond structure, we take the 98.435 that the investor pays to purchase the bond and assume the investor borrows 4.717 to offset the interim coupon payment – to make this calculation you need the one-year interest rate and we have assumed a oneyear rate of 6%. So the net cash outflow is 93.718.

At the end of the first year, the investor will use the interim coupon payment to repay the borrowing of 4.717, which with interest has become 5. At the end of the second year, the investor will receive 105.

This allows us to identify the two-year zero-coupon rate. This is the rate which discounts 105 to 93.718. To calculate this, we use the following formula:

$$i\:=\:\frac{FV}{PV}\:1 / n\:\: – \:1$$

Where,

• $$i\:= \:interest\: rate$$
• $$FV\:=\: future \:value$$
• $$PV\:= \:present \:value*$$
• $$n\:= \:number\: of\: years.$$

In this case,

$$i\:=\:\frac{105}{93.718} ^{1 / 2} \:– \:1$$

$$i\:=\:0.05848$$
So the two-year zero-coupon rate is 5.848%

*the present value must be entered as a negative value if using an HP12C or equivalent calculator because it is an expenditure.

#### Three year maturity

We can also use this information to calculate a three-year zero-coupon rate. In this case, we need to construct an artificial zero-coupon bond, by identifying how much the investor would need to borrow to offset the payment of the two interim coupons of 4.

By using the one-year rate of 6%, we can calculate the investor would need to borrow 3.774 today to match the coupon of 4 in one year.

We also need to calculate how much the investor would need to borrow today, to match the anticipated two-year coupon receipt of another 4. For this, we use the two-year rate which will be the same as the zero-coupon rate we calculated earlier – 5.848%.

The two-year discount factor is $$\frac{1}{1.058482}\:=\:0.8926\:\:$$ Using this, we can calculate that the investor would need to borrow 3.570 today to receive 4 in two years.

In order to calculate the zero-coupon rate, we assume the investor purchases the bond for 96.784 and borrows 7.344 to compensate for the two interim interest payments. The investor’s initial outlay is therefore 89.440.

Using the same formula as outlined above $$i\:=\:\frac{FV}{PV}\:1/n\:–\:1$$ , we can then identify the three-year zerocoupon rate. In other words, the rate which discounts 104 to 89.440.

In this case:

$$i\:=\:\frac{104}{89.440}\:^{1 / 3} – 1$$

$$i\:=\:0.05156$$

So the three-year zero-coupon rate is 5.156%

Using the different zero-coupon rates we have calculated, it is then possible to create a series of zero-coupon yields and then construct the yield curve.