Bootstrapping Zero Curves

What is a yield curve

A yield curve is a representation of what interest rates you could lock in today for investments over different periods. It’s effectively a set of yields for securities of different maturities (typically cash rates at the short end, futures and then swaps at the longer maturities – see the wikipedia entry). These yields are typically calculated from market prices using standard price/yield formulas.

Why yield curves can’t directly be used in PV calculations

The problem is that the quoted rates are from coupon paying securities and tell us the value of a series of cash flows, but they don’t tell us the rate for a cash flow at the maturity point independently of the other cash flows. So these rates don’t directly imply the present value of a dollar to be received at the maturity points – that is, they cant be used directly as discount rates.

The Zero curve to the rescue

A zero curve represents the set of interest rates assuming that there are no periodic cash flows – as though the rate reflects a single payment of interest and principal made at that maturity point on the curve. As such, the rate can directly provide the present value of a dollar received at these maturity points.

Let’s take a very simple example:

A yield curve with 3 points:
Year 1: 5%
Year 2: 6%
Year 3: 7%

This means if you invest $1:

• For 1 year, you’ll receive 5% interest plus your principal – $1.05 – at the end of year 1.
• For 2 years, you’ll get 6% annually, with your principal included at the end of year 2. So you will get $.06 at the end of year 1, and $1.06 at the end of year 2.
• For 3 years, you’ll get $.07 at the end of year 1, $.07 at the end of year 2 and $1.07 at the end of year 3.

The key point here is that the 2 and 3 year rates effectively represent a coupon paying security. In the 2 year rate, $1 invested now for two years represents the combined PV of two cash flows. But what is the PV of just the second cash flow? If we can work this out, we’ll have a zero (or discount) rate at 2 years, enabling us to PV any money received 2 years in the future.

Deriving Zero curves

We know the PV of $1 received at the end of the first year using our basic time value of money formula:

PV = \frac{FV}{(1 + r)^n} = \frac{1}{(1 + .05)^1} = \frac{1}{1.05}

This is effectively the 1 year discount factor. So any monies received at the end of 1 year can be multiplied by this to get the PV. Building (or “bootstrapping”) the zero curve is obtaining these discount factors for all maturity points.

To get the zero rate, or discount factor, for the 2 year point, we can say that $1 invested today will equal the PV of 1st payment plus the PV of 2nd payment. So:

PV = 1 = (\frac{.06}{1.05}) + (\frac{1.06}{(1+r)^2})

(where r equals the 2 year discount factor)

Working this out, we end up with:

r_{\tiny{Z2}}=.0603

or 6.03%

For the 3rd year zero rate, we build on the above:

PV = 1 = (\frac{.07}{1.05}) + (\frac{.07}{(1+.0603)^2}) + (\frac{1.07}{(1+r)^3})

(where r equals the 3 year discount factor)

This gives us:

r_{\tiny{Z3}}=.07097

or 7.097%

Now we have our zero curve that we can used as discount rates for cash flows at the corresponding
maturities.

So the corresponding zero curve:
Year 1: 5%
Year 2: 6.03%
Year 3: 7.097%

An observation

It’s interesting to note that the zero rates are slightly higher than the yield curve rates when the curve is sloping upwards. If you have trouble seeing why think of an extreme case of a yield curve with 0% rate in year 1 and 100% rate in year 2.

Implied forward rates

Another concept worth touching on here are the implied forward rates from these zero rates. Let’s say we invest $1 for 2 years under our zero rate, which is 6.03%. This would give us a return of:

FV = 1 * (1 + .0603)^2 = 1.124236

We could also have invested for 1 year, then take that return and lock in a forward rate for the second year (the rate at the end of year 1 for investing for year 2). Under the rules of arbitrage, this forward rate for year 2 needs to give us the same return as investing for 2 years using the 2 year zero rate. So:

FV = 1.124236 = 1.05 * (1 + r_{\tiny{1,2}})

where r_{\tiny{1,2}} is the forward rate for the second year

This gives an implied forward rate r_{\tiny{1,2}} = .070707 = 7.07%

You can keep building these implied forward rates for all future years using this technique.