Pnl Explain

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P&L Explain – Bonds and Swaps
Tony Morris antony.morris@db.com
MICS – DKS Manila

Contents

1. Bond Pricing – basic concepts

2. P&L sensitivities of a bond

i. PV01
ii. CS01
iii. Theta
iv. Carry

3. Extension to interest rate swaps

1. Bond Pricing – basic concepts

Let’s say you have a 4 year 10% annual coupon bond, with a yield (‘yield to maturity’ or ‘yield to redemption’) of 12%. From this information, the price can be calculated as 93.93%. The price is calculated by pricing each of the bond’s cash flows using the yield to maturity (YTM) as a discount rate. Why? Because the YTM is defined as the rate which, if used to discount the bond’s cash flows, gives its price. We could picture it like this:

Bond Cash Flows on a Time Scale

Each fixed coupon of 10% is discounted back to today by the yield to maturity of 12%: 93.93% = 10 + 10 + 10 + 110 (1.12)1 (1.12)2 (1.12)3 (1.12)4

All we are doing is observing the yield in the market and solving for the price. Alternatively, we could work out the yield if we have the price from the market. Bond price calculators work by iteratively solving for the yield to maturity. For a bond trading at par, the yield to maturity and coupon will be the same, e.g. a four year bond with a fixed coupon of 10% and a yield of 10% would be trading at 100%. Note that bond prices go down as yields go up and bond prices go up as yields go down. This inverse relationship between bond prices and yields is fairly intuitive. For our par bond above, if four year market yields fall to 9% investors will be willing to pay more than par to buy the above market coupons of 10%. This will force its price up until it, too, yields 9%. If yields rise to, say, 11% investors will only be willing to pay less than par for the bond because its coupon is below the market. For a detailed example of the bond pricing process, see Appendix 3. For now, note that the dirty price of a bond is the sum of the present values of the cash flows in the bond. The price quoted in the market, the so-called “clean” price or market price, is in fact not the present value of anything. It is only an accountants’ convention. The market price, or clean price, is the present value less accrued interest according to the market convention.

2. P&L sensitivities of a bond

As we saw above, the price of a bond can be determined if we know its cash flows and the discount rate (i.e. YTM) at which to present value them.

The yield curve from which are derived the discount factors for a bond can itself be considered as the sum of two curves: 1. the “underlying” yield curve (normally Libor), and 2. the “credit” curve i.e. the spread over the underlying curve

The sensitivity of the bond price to a change in these two curves is called: i. PV01, and
ii. CS01
respectively.

In terms of the example above, the discount rate of 12% might be broken down into, say, a Libor rate of 7% together with a credit spread of 5%.

(Note, in the following, it is important not to confuse the discount rate, which is an annualised yield, and the discount factor, which is the result of compounding the discount rate over the maturity in question.)

In addition to the sensitivities described above, we can also consider the impact on the price of the bond of a one day reduction in maturity. Such a reduction affects the price for two reasons:

a) assuming the yield curve isn’t flat, the discount rates will alter because, in general, the discount rate for time “t” is not the same as that for time “t-1” b) since one day has elapsed, whatever the discount rate, we will compound it based on a time interval that is shorter by one day

The names given to these two sensitivities are, respectively: iii. Theta, and
iv. Carry

Note that, of these four sensitivities, only...
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