Everything about Swaps : Part 1

Valuation of swap is one of the most important areas of CFA level II curriculum. It looks complex but if understood clearly,  it is high scoring and there are fewer chances for committing a mistake.

There are mainly three types of swaps. (i) Interest rate swaps (ii) Currency swap and (iii) Equity swap.

Out of these, equity swap is the easiest to value and calculate. Let’s take an example: equity to equity swap. Suppose there are two parties: Markos and Kathy. Markos has one portfolio of stocks and Kathy has other. They entered into a swap of notional value $1 million with the understanding that they will swap the returns of portfolios of each other. Suppose during the swap terms, the portfolio of Markos rises up by 12% and the portfolio of Kathy falls down by 8%. Now, Kathy will receive 12% of the Markos’ portfolio and should pay Markos -8% return on her portfolio. Overall she will be gaining 20% on the notional value which comes out to be 0.2*$1million = $200,000.

Another type can be equity to fixed income security swap (where fixed income security can have either fixed rate coupons or floating rate coupons). The value of equity side of the swap can be calculated simply by calculating rise or fall in the index value. The valuation of the fixed income security is a little complex. Let us try to understand it in detail.

Let us first take a fixed paying coupon bond. Now suppose the coupon rate (C% semi-annually) is such that the bond is trading at face value and the face value is $100 and the time to maturity as 2 years. Assume that the bond is a semi-annual coupon paying bond. Assume that the annual spot rates for 6 month, 1-year, 18 months and 2-years are 2.5%, 3.0%, 3.5% and 4% respectively. Then the value of bond will be 100 = (C/1.025) + (C/1.03^2) + (C/1.035^3) + (C/1.04^4) + (100/1.04^4). Now, we can see that any coupon payment can be termed as a zero-coupon bond. For example (C/1.035^3) is nothing but a zero-coupon bond of face value C and period interest as 3.5% and number of periods as 3. (C/1.035^3) can also be written as [C/(1+R3)] or C*Z3 where R3=discount rate for the 3rd period coupon and Z3=[1/(1+R3)] – price of 3rd period zero coupon bond per $ of face value.

I hope you are not lost. Let’s calculate 2-3 values to regain the concept as it is very vital to understand it. Value of 2nd period zero-coupon bond per $ of face value = [1/(1+0.035*540/360)] = 1/1.0525 = 0.9501. Calculate yourself the values of 1st, 2nd and 4th-period zero-coupon bond per $ of face value. (0.98765, 0.9709 and 0.9259)

Now coming back to the value of the coupon C.

100 = C*(Z1+Z2+Z3+Z4) + 100*Z4

C =(1-Z4)/(Z1+Z2+Z3+Z4)

Now you guys must be thinking why this kolaveri? Why have we calculated all this? What’s the use of calculating the coupon rate? Well, when we swap a fixed rate or floating rate coupon bond with some other security, we do it for a particular notional value and since the value of the swap at inception is zero, the bond should trade at the notional value. So, by calculating the coupon rate, we basically calculated the swap rate. Do not forget that in our exam it is the semi-annual rate. To make it annual just multiply it by 360/180. It can be quarterly or monthly or of any other period. Now, suppose after few days (say 200 days) market coupon rates are (remember that there are 720-200=520 number of days left) 3.2% for 160 days (360-200=160 days until next coupon date), 3.8% for 340 days (540-200=340 days for the 2nd coupon date from now) and 4.3% for 520 days (maturity date). Now, to calculate the value of fixed coupon bond, we know that we will receive the fixed coupon of C at each of those dates. Just discount the future payments with the new rates to get the value of the bond.

Value of bond = [C/(1+0.032*160/360)] + [C/(1+0.038*340/360)] + [(100+C)/(1+0.043*520/360)]

Now this is the value of the bond and similarly, you can calculate the value of other security and calculate the value of swap by subtracting those values. Now let us look at the case of valuation of floating rate bond. The valuation of floating rate is relatively easy as compared to the fixed rate bond. We just need to remind ourselves of the concept from the fixed income securities that when coupon rate is equal to the yield, then the bond trades at par. Now in our previous example, the bond will trade at par i.e. $100 in the beginning. Now, let us calculate its value after 200 days. Now, we are going to receive the coupon after 160 days and after that coupon rate will adjust to the market rate and again bond will trade at par. So, after 160 days, the value of the bond will be $100 and the coupon received at the end of 160 days will depend on the coupon rate set at the previous coupon payment i.e. 180 days from the beginning or say 20 days before today (today is the 200th day). Suppose the 180-day interest rate at that time was 3.1%. So, the coupon will be 0.031*(180/360)*100(=y). Now, after 160 days we can sell the bond at $100 after receiving coupon y. So, the value of bond = (y+100)/(1+0.032*160/180). Please note carefully that on discounting these payments we have used the current market rate and not the interest rate set at the last coupon date. If this is clear, then congratulations, you have mastered the swap value calculation problem. You just need a little practice now.

Now, let us look at another type of swaps. Interest rate swaps- plain vanilla swap. In a plain vanilla swap, one party pays fixed coupon and receives floating and it is opposite for the other party. We just need to calculate the value of fixed coupon bond and floating coupon bond and accordingly calculate the value of swap after few days. If say after few days, the value of the fixed-rate bond is $98.5 and the value of floating rate bond is $101.5. Now if you are floating rate receiver, you will gain from the swap. The swap will have a positive 3 value for you while it will be negative $3 for the fixed rate receiver.

Coming to the most complex swaps i.e. currency swaps. No need to worry. They are also not as complex if the above-mentioned process is clear. The only thing is that in that case you will have different interest rates for both currencies and you just need to be careful to put the correct values. During calculation of the swap value, we just need to convert those to one currency (either of the one) and then calculate accordingly. It can be of four types.

(i)Pay fixed, received fixed (say for currencies USD and EUR)

(ii)Pay fixed, receive floating

(iii)Pay-floating, receive fixed

(iv)Pay-floating, receive floating

In the second case, we need to calculate the value of fixed rate coupon bond for USD and floating rate coupon bond for EUR. Suppose, the notional money is EUR1 million and the exchange rate at the beginning is 1.5 USD per EUR. That means we have a notional value of EUR bond as 1 million and the notional value of USD bond as 1.5 million. Now, suppose after 50 days we want to calculate the value of swap and the value of USD bond is 1.4 million and that of EUR bond as 0.95 million and exchange rate as 1.4 USD per EUR. Then,we can convert 1.4 million USD in EUR and that would be 1 million EUR (by dividing by the exchange rate). So, the value of swap will be more for USD receiver and that party will be gaining 0.05 million EUR on the swap.

I hope that this is clear. I will cover rest of the topics related to swaps (very few) and will provide some problem in the next article. All the best.

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A blog of the NYU Colloquium on Market Institutions and Economic Processes

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Thoughts of a teacher & practitioner of value investing and behavioral economics

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