I want to share something i learned about bootstrapping. I want you to be familiar of this concept as it can be applied in determining the discount rate to be used in valuation of pension liabilities of the company under defined benefit obligation.

Paragraph 78 of PAS 19, Employee Benefits, provides that the rate used to discount post-employment benefit obligations shall be determined by reference to market yields at the balance sheet date on **government bonds**** **in countries where there is no deep market in high quality corporate bonds (like Philippines). The currency and term of the government bonds shall be consistent with the currency and estimated term of the post-employment benefit obligations.

Philippine Interpretations Committee (PIC) issued Q&A No. 2008-1, *PAS 19.78 – Rate used in discounting post-employment benefit obligations*. According to this Q&A, the rate used to discount post-employment benefit obligations shall reflect the pattern of cash flow for the payment of retirement benefits. Employees become entitled to retirement benefits at the end of their service lives. A rate that would be reflective of a one time payment upon maturity is that of a zero coupon instrument which has a single cash flow. Although payment schemes may vary (e.g., lump sum or periodic payments over the life of the retired employees), it may be assumed for actuarial purposes that payments would be made in one lump sum.

With the effectivity of Q&A No. 2010–01, PAS 39.AG71-72 – *Rate used in determining the fair value of government securities in the Philippines*, it was clarified that the appropriate rate to be used is the PDSI-R2, in the absence of the quoted market price of government securities. Relating it to pension, of course there is no quoted price for these types of instruments so we can validly use PDSI-R2. **PDSI-R2 is a set of interpolated rates for Government Securities that uses the PDST-R2 rates as the set of “known rates” to base the interpolation, published after 4.20 pm daily.** PDSI-R2 is different from PDST-R2, and the former is the most acceptable rate to use.

If the rate reflects the yield for peso government bonds that pay out interest payments on a periodic basis over the term of the bond and the principal upon maturity, such rate may be converted to a zero coupon rate to reflect a reasonable estimate of the benefit payments. It was mentioned in the Q&A that the most common reference for the discount rates is the Philippine Dealing and Exchange Corporation (PDEX) and Bloomberg.

The preceding paragraph speaks of the **conversion of coupon rates to zero coupon rates**. This means that the yield (interest rate) of government bonds receiving periodic interest payments needs to be converted to a rate which assumes that all cash flows will be received at the end of the term, as if there were no periodic interest payments. This is important because the timing of benefit payments assumed in PAS 19 is in lump sum upon retirement of the employee. With these, we introduce you to the concept of bootstrapping which is discussed below.

Read the article below, from **www.murraystate.edu** which explains what bootstrapping is and how it is done.

**Bootstrapping**

**Glossary**

**Bootstrapping** – In a dictionary, bootstrapping is defined as “a repeating, self-sustaining operation”. As used here, bootstrapping is a procedure, repeated over and over again, to convert the yield on *coupon-bearing* bonds into yields on equivalent *zero-coupon* bonds.

**Spot rate** – A spot rate is simply the rate of return earned on a zero-coupon bond, if held to maturity.

**Yield curve** – A graph that shows the yield earned on bonds of various maturities. In short, it shows the relationship between short-term and long-term interest rates.

**The Bootstrapping Procedure**

Let’s consider Treasury bills (T-bills) for a moment. When newly issued, U.S. Treasury bills come in maturities of three months, six months, and one year. Treasury bills do not pay a stated rate of interest; they are sold at auction at a discount to their face value. In other words, they are short-term zero coupon bonds.

Since a new six-month T-bill and a new one-year T-bill trade publicly, we can look up their yields (in a database, on the Internet, in the newspaper, etc.). These yields are known quantities – we are certain about the yields on both the six-month and one-year T-bills. What we don’t know is what the yield would be on an 18-month zero-coupon Treasury. (There may not be an 18-month zero-coupon Treasury issue trading publicly at the moment.) However, we can calculate what this yield would be if such a security did trade publicly.

How do we do this? We simply take the values that we do know and create an equation with one unknown (i.e., the 18-month yield); then we solve for that unknown value. Once we solve for the 18-month yield, we can repeat the process to solve for the 24-month yield, then the 30-month yield, and so on at six month intervals. This process is known as *bootstrapping*.

Here is the information that we can look up (and therefore know):

The yield on a six-month T-bill

The yield on a one-yield T-bill

The yield on an 18-month

*coupon-bearing*Treasury security

We want to find out what the yield on the 18-month coupon-bearing Treasury security would be if it were a *zero coupon* security instead. In other words, we want to know the 18-month *spot rate*. This is the unknown value.

**A Key Fact**

To determine its fair value, a bond’s cash flows should always be discounted using spot rates. This is different that what is taught in entry-level finance classes. Those courses generally teach that you use one discount rate for all of a bond’s cash flows, i.e., the yield paid in the marketplace on bonds of similar risk and maturity. However, using one discount rate for all of the bond’s cash flows makes a crucial assumption: that the yield curve is flat. If the yield curve is flat, then it is true that investors are willing to accept the same rate of return on cash flows that are received at different maturities.

However, the yield curve typically is not flat. Therefore, investors’ required rates of return on investments vary depending on when the cash is received (i.e, the maturity of the bond). In practice, investors use several spot rates (rather than one fixed discount rate) to value a bond.

**An Illustration**

For illustration, let’s assume the following:

The yield on a six-month T-bill is 3.0% annually, or 1.5% semi-annually. This is the 6-month spot rate, or required rate of return for a cash flow to be received in six months.

The yield on a one-yield T-bill is 3.2% annually, or 1.6% semi-annually. This the one year spot rate, or required rate of return for a cash flow to be received one year from now.

The yield on an18-month coupon-bearing Treasury note is 3.50%, i.e., the bond pays interest of $35.00 per year, or $17.50 semi-annually. We will also assume that the bond sells at par ($1,000).

Since bonds pay interest semi-annually and pay the par value at maturity, the cash flows associated with the 18-month, coupon-bearing Treasury note are **$17.50** (at six months), **$17.50** (at 12 months), and**$1,017.50** (at 18 months).

If investors price the bond fairly, the bond’s market price should be equal to the present value of the bond’s cash flows (using each period’s spot rate as the discount rate). The market price is $1,000 in our illustration. Inserting the values that we know, we have an equation with one unknown: the 18-month spot rate. We can solve for this rate.

The spot rate, or zero-coupon rate, for the 18-month Treasury security is **1.7532%** semiannually, or **3.5064%** annually.

Now that we know the rate that must be paid on an 18-month zero-coupon security, we can bootstrap the results and repeat for the next period, a 24-month security. We use the information for a 24-month coupon-bearing Treasury security and insert its cash flows into the equation.

If a 24-month Treasury note pays a 3.7% yield (i.e., $37.00 annual interest or $18.50 semi-annually) and sells at par, then:

The spot rate, or zero-coupon rate, for the 24-month Treasury security is **1.8554%** semiannually, or **3.7108%** annually.

We would then repeat this process at six month intervals in order to complete our spot rate curve.

As usual, we encourage sharing your thoughts about the topic in this post.