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# Zero Coupon Discount Factors

Introduction

Present value (PV) calculations are commonly used in financial markets. They are particularly relevant to over-the-counter derivatives. Their use includes pricing and marking-to-market transactions.

PV uses a discount factor to convert future money into today’s money. The sum of any deal’s cash flows in present value terms is referred to as the net present value (NPV). From a dealer’s perspective this is important. Transactions with positive NPVs equate to profit and those with negative NPVs losses.

The following is an introductory explanation to zero coupon discount factors. The examples use USD rates where the instruments pay interest on an actual/360 day count basis.

Zero Coupon Discount Factors

Bank deposits shorter than a year in maturity pay interest at maturity. This is known as simple interest. Libor rates from the interbank market are regularly used to derive short term discount factors using a simple formula:

Discount factor = 1/ (1 + rate x days / 360)

Here is a example using 1 year and a 6.00% interest rate.

1/ (1 + 0.06 x 365 / 360) = 0.942655

1 Dollar in one year's time is worth 94.2655 cents today.

But when we try and calculate a 2 year discount factor this approach does not work. Why?

Because the interest rates used to derive the discount factors are taken from the swap market. Swaps pay regular interest therefore swap rates are not simple interest rates. They are internal rates of return.

Suppose the two year swap rate is 6.07%.What is the two year discount factor?

1. The two year swap rate is 6.07 %, based on a notional amount of \$100, the annual interest is \$100 x 6.07% x 365/360 =  \$6.1543

2. We know the one year discount factor is 0.942655 (see above) so the present value of the first swap interest payment is \$6.1543 x 0.942655 = \$5.80138

3. The swap NPV is zero and the notional is \$100, deduct \$5.80138 and you are left with \$94.1986. What is this?

4. \$94.1986 is the present value of the interest and notional amount payable in 2 years time. So what is the discount factor?

5. \$94.1986 today is equal to \$106.1543 in two years.

94.1986 / 106.1543 =  0.887374

\$100 payable in three years is worth \$88.73 today.

These 5 steps look like this:

[100-(6.07 x  (365/360) x Df1Yr) ] / [100 + (6.07 x 365/360) ] = 0.887374

The process is repeated to find subsequent discount factors.

Here is the third year.

Assume the swap rate is 6.12%

[ 100 - ((0.942655 + 0.887374)  x  (6.12  x  365 / 360)) ]  /    [ 1 + (6.12  x  365 / 360) ]

= 0.834656 the three year discount factor

\$100 payable in three years is worth \$83.46 today.

The importance of the zero coupon approach

Why go to such trouble to calculate discount factors? Because when you are pricing and valuing large transactions even small discrepancies can lead to costly mistakes. In fact the above example is simplified and trading systems go to much greater lengths to determine the appropriate discount factors.

Our example highlights the crucial issue. Note how in order to find the two year discount factor you need the one year discount factor and so on. (This is sometimes referred to as “boot strapping the yield curve”). No assumptions have been made about the rate of interest that has been used to compound or discount future cash flows in order to determine the discount factor

Next time you are valuing transactions by using discount factors spare a thought for the process involved.

An example for you

If you have a spread sheet you may wish to calculate the discount factors for 1 to 10 years using the following yield curve.

USD Annual Actual/360

1 Week                       5.50

1 Month                      5.50

3 Months                    5.75

6 Months                    5.75

1 Year                          6.00

2 Years                        6.07

3 Years                        6.12

4 Years                        6.15

5 Years                        6.18

6 Years                        6.22

7 Years                        6.23

8 Years                        6.27

9 Years                        6.28

10 Years                      6.29

 Rate Days Discount factor 1 Week 5.50% 7 0.99893 1 Month 5.50% 30 0.99544 3 Months 5.75% 60 0.99051 6 Months 5.75% 182 0.97175 1 Year 6.00% 365 0.94266 2 Years 6.07% 730 0.88737 3 Years 6.12% 1095 0.83466 4 Years 6.15% 1460 0.78490 5 Years 6.18% 1825 0.73764 6 Years 6.22% 2190 0.69228 7 Years 6.23% 2555 0.65068 8 Years 6.27% 2920 0.60968 9 Years 6.28% 3285 0.57260 10 Years 6.29% 3650 0.53763