​​​ zero-coupon yield curve, zero-coupon bond, CAGR
  1. The use and abuse of derivatives

The use and abuse of derivatives

While the zero-coupon pricing technology has made it possible to price any combination of cash flows...

By: | Published: May 19, 2015 12:11 AM

While the zero-coupon pricing technology has made it possible to price any combination of cash flows, this instrument has been ‘used’ as well as ‘abused’.

A zero-coupon bond (ZCB) is exactly what it says — it pays no periodic interest, or coupon. As the ZCB does not pay any interest, an investor would only buy it if it would be sold at a discount, and the discount would be expected to be equal to the rate of return he expects for the time horizon.

Let’s take an example (see chart):  If an investor expects a return of 8% CAGR for five years, he would discount it at 8% and pay R68.06 for it.

By investing R68.06 and receiving R100 after five years, the investor would have earned  a CAGR of 8% — and the beauty about a ZCB is that the ‘final’ return on this investment would be known today.

Suppose the investor also had the choice of investing R68.06 in a normal five-year bond or a fixed deposit, where the precise return is inherently unknown until the end because he will not know the rate at which interest payments will be invested for the year 5.  The jargon for this is ‘reinvestment risk’.


The elegance of a zero-coupon instrument is that there is no coupon and, therefore, no reinvestment risk — and the return on investment is known immediately.  In the example above, by investing R68.05 and getting R100 in five years, the investor earns 8% annually compounded — and this is known on the first day.

You don’t have to wait for five years to find out.  Conversely, if you were promised R100 after five years, you could pay out R68.06 today (by discounting at 8%). Thus, in the world of interest rates, there would be no difference in value between R68.06 today and  R100 in five years, assuming there is no credit risk.

It is this feature of ZCB that stands at the heart of pricing interest swaps. Until ZCB maths was developed in the the 1980s, pricing of swaps was done using IRR — the internal rate of return.  But IRR is not a ‘return’ or a ‘yield’, but a measure of value to indicate if one investment proposition is better than another.

So, the problem was: Where do you get a zero-coupon yield curve (ZCYC) in a world of illiquid coupon bearing bonds? The solutions were ingenious, from bootstrapping, curve fitting (Nelson-Siegel) and other interpolation methods to create a ZCYC. This curve lies at the heart of the rich structures that swap teams were able to provide.

Once you have a ZCYC, it is possible to calculate the value today of R100 after one year, two  years or even two years and 43 days! The ZCY allowed practitioners to equate any cash flow along the yield curve.

For a start ,ZCYC allows you to calculate precisely the forward rate of interest.  If the one-year ZCY is 8% and the two- year ZCY is 9%, the implied one-year rate after a year is 10% (intuitively, if we are required to pay someone a 9% CAGR for two years and are able to earn only 8% in the first year, we will need to earn 10% between the first and second years  —  which is the forward rate).

Using the ZCYC, the bank can derive the forward rate every six months and, then, arrive at the fixed rate X — such that the implied value of the forward rates equals the fixed rate.  This is a simple fixed-to-floating swap rate and if you think about it, this innovation is amazing.  It permits a derivatives team to, at a point in time, express a floating rate (which is an unknowable unknown) into a fixed rate.   And it is useful for a corporate because it transforms a floating rate loan into a fixed rate loan, or the other way round.

A currency swap transaction is as follows.  A corporate which has a five-year dollar loan has to pay Libor every six months and the principal amount at the end of five years.  How does this corporate protect itself from changes in $ Libor and also changes in the dollar rupee rate (INR)?

By entering into a currency swap the bank says — okay, we will fix the rate at which you need to repay those dollars at today’s dollar rate of R60.  We will also pay you the $ Libor every six months, so that you can pay that to your lender. In exchange, you pay us 7% in rupees.

How did the bank calculate this?  It is easy once you have a ZCYC for INR and the dollar. The bank calculates what the $ Libor payments will be by using forward rates.  It also calculates the cost of giving $ to the company five years later but at today’s rate of R60. It, then, arrives at the fixed rate of interest that the company will have to pay every six months so that the value of what is paid and what is received is equal – which say, works out to 7%.  This enables a transformation of a floating rate USD loan into a fixed rate rupee loan.

This article is the first in a two-part series. The writer is managing director & CEO of Tata Asset Management. Views are persona

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