Zero Rates & Bootstrapping

As discussed in the video, £100 deposited today, results in £103.41 in one years time.

So £100 is the present value of £103.41 (the future value) in a years time when invested at a rate of 3.41%.

So the discount factor for a period of one year can be determined as shown.

This discount factor converts the future value of £103.41, to its present value of £100.

You can try this with a calculator (0.96702447 x 103.41)

For the two year deposit, we have the intermediate cash flow of £3.87 after a one year period.

Using the above discount factor of 0.96702447 (which is the one year discount factor), we can bring that cash flow to now (t=0) which gives us the present value of that cash flow.

This discounted cashflow, which is positive, reduces to £3.7423847

This reduces the net present value to £-96.2576153 (-£100 + £3.7423847)

We have no intermediate cash flows anymore and can therefore caculate the two year discount factor.

The discount factor for a two year period is obtained by stripping out the intermediate one year cash flow.

The two year discount factor is therefore 0.92671238

This technique of stripping out intermediate cash flows by using the respective discount factor from an interest rate curve is known as bootstrapping.

For annually compounded interest, the zero rate is determined by this equation. We will introduce the concept of continuous compunding later, which is the convention used in quantitative finance (and involves natural logarithms).

But we keep it simple here.

Here T is the number of years and z is the zero rate. A zero rate is when there are no intermediate cash flows.

Plugging the discount factor of 0.92671238 and a value of T=2, the two year zero rate is 3.878942%. As you may have noticed the zero rate is slightly greater than the interest rate. 

Did you work that out?

Let us repeat this exercise for the three year deposit.

Remember, here we will use two separate discount factors.

• • One for year one, which is 0.96702447

  • The other is for year two, which is 0.92671238

The cash flows for year one and year two are 4.20 as determined by the three year rate of the Rate Curve.

The year one cash flow movement reduced the present value to -£95.93849723. We use the year one discount factor of 0.96702447

For year two cash flow we use the second year discount factor of 0.92671238. This further reduces the net present value to -£92.21561599

Having stripped out the two intermediate cash flows, we can now determine the discount factor for year three.

As before, having stripped out year 1 and year 2 cash flows our net present value reduces to 92.21561599

This gives is the discount factor for year three as 0.88336185.

The three year zero rate is %4.220652