## Constructing a zero-coupon yield curve

##### Published: Nov 2008

The zero-coupon yield curve can be constructed using a series of coupon-paying bonds using an iterative technique known as ‘bootstrapping’. This works on the premise that the investor ‘borrows’ money today, the day that the bond is purchased, to compensate for not receiving any coupons over the life of the bond. In this example we use the following bonds which pay annual coupons and assume that the one year interest rate is 6%.

#### Two year maturity

The investor pays $98.435 for$100 face value of the two year bond. They will receive $5 (the coupon) in one year and$105 (the coupon and repayment at par) at the end of two years. By discounting the cash flows we can create a synthetic two year zero-coupon bond. In order to calculate how much the investor will need to ‘borrow’, we will need to work out the present value of the coupon today, using the following formula:

$$\mathrm{Present\: value\:PV}=\frac{Future \:Value\: FV}{1\:+\:interest\: rate\: i^n \:number\: of\: years}$$

In our instance the first interest payment is discounted as: 5/(1+0.06)1 = 4.717. The initial outlay is therefore $93.718 and the$4.717 is notionally repaid with interest (making $5) after one year: The rate which discounts$105 to $93.718 is the two year zero-coupon rate. To calculate this, we use the following formula: $$\mathrm{Interest\: rate \:i}=\frac{Future\: Value \:FV}{Present \:value \:PV}\mathrm{1\:/\: number \:of\: years\: (n)\:-\:1}$$ So the two year zero-coupon rate in our instance is 5.848%, calculated from: $$\frac{105}{93.718}\mathrm{1/2\:-\:1}=\mathrm{0.05848}$$ #### Three year maturity We can calculate a three year zero-coupon rate in the same way. First we identify how much the investor would need to borrow to eliminate the impact of the two interim coupons of$4. By using the PV method outlined above, we can calculate that 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%. Therefore, the investor would need to borrow$3.570 today to receive $4 in two years. That makes a total borrowing of$7.344 and the initial outlay is therefore \$89.440 (96.784 – 7.344).

Using the i = (FV/PV)1/n – 1 formula , we can then derive the three year zero-coupon rate – in this instance 5.16%: (104/89.440)1/3 – 1 = 0.0516. By extending the above method to different maturities, it will be possible to determine a series of zero-coupon yields which can be plotted against the term (in years) to construct the yield curve.