**Key words:**

*DV01, Dollar Duration, PV01/PVBP, Delta, Effective DV01, Macaulay Duration, Modified Duration, Interest rate swaps sensitivities*

The concept of interest rate risk is often confused by a lot of terminology but actually is one of the easiest and straight forward approaches: how price changes as yields change, i.e. what is the sensitivity?

There are two measures of interest rate sensitivity in fixed income: Duration and DV01.

These measures often deal with parallel shifts in the yield curve but in reality yield curves may undergo different strange twists, flattening, steepening etc.

One approach toward solving this problem is key rate duration (also called partial DV01) and bucket exposure.

For simplicity purpose we assume parallel shifting in this article.

__BONDS__**DV01**

**1.** DV01 (Dollar value of a 1bp) defines how much the clean price of the bond will change for a one basis point move in its yield to maturity.

**2.** The other definition is the change in the bond’s value for a 1bp change in market yields.

DV01 also referred to as PV01/PVBP (present value of a 1bp). Market has also developed several alternative terms such as Risk, Dollar duration or Delta. Therefore, assuming parallel shifting of the curve we define the interest rate risk as follows:

Let’s consider a bond with start date 01.01.2015 and maturity date 01.01.2018 which paying 2% coupon annually.

**1.** In the table above we calculate Risk by shifting market yields (actually Zero yield curve) by 1bp up and down.

Total resulting Risk calculated in the column “PV01”= 0.02884258. It is said to be effective DV01.

Effective DV01 takes into account both the discounting that occurs at different interest rates as well as changes in cash flows.

It is useful when a portfolio contains callable securities.

**2.** If we shift yield to maturity instead of market yields, we derive almost the same PV01 which is 0.02884479.

It is said to be DV01 or PV01/PVBP. The table below shows the calculations.

As can be seen the difference between two calculations is minor for a single bond.

Duration can be used as a measure of risk in bond market. It comes in many forms.

The most commonly used are Macaulay and Modified. The concept is the longer the duration the more sensitive a bond is to changes in interest rates.

**Macaulay Duration**

This duration is sometimes referred as the average term of the cash flows weighted by their present value.

From our calculation it follows that after 2.94 years we will have gotten $100 repaid, i.e. it takes 2.94 years to recover the true cost of the bond.

Formally, Macaulay duration is constructed by taking the derivative of the present value of each cash flow with respect to yield of a bond:

We can rewrite this as:

Taking into account that DV01=-∆P/∆Y this formula can be rewritten as:

**Modified Duration**

In order to measure an absolute change in the bond’s yield we use Modified duration that expands Macaulay duration:

We can rewrite this as: ∆P = -MD*P*∆Y. Consider our bond selling with MD = 2.8838.

For every 1% change in the bond’s yield the market value of the bond will move inversely approximately by 2.8838% without taking into account convexity effect.

The calculations are attached in the **Excel** file.

Modified duration is always smaller than Macaulay duration and is extensively used in practice.

Generally, DV01/PV01 is a more accurate and more general measure of interest rate sensitivity than duration.

__INTEREST RATE SWAPS__DV01/PV01 can be used for instruments with zero value such as Interest rate swaps where percentage changes and modified duration are less useful.

Valuation of swaps depends on a sequence of discount factors and there is no such thing as a yield on a swap.

DV01 can be used for any sequence of positive and negative cash flows.

**1.** DV01 is the change in present value of the swap for a 1bp parallel shift in the swap curve.

**2.** PV01 is the present value of a 1bp annuity with a maturity equal to the swap maturity. PV01 shows how much the value of the swap changes if the fixed coupon changed by 1bp. Note, the swap curve remains unchanged.

As we see, DV01 and PV01 are completely different things. They are not the same as in the bonds world. DV01 ≈ PV01 only if we deal with a par swap.

A swap has a positive DV01 if the swap curve shifted downward by 1bp and the swap’s value increases.

A negative DV01 is when the value of the swap decreases when the curve is shifted downwards by 1 bp.

I was looking at your spreadsheet. Where did you get YTM as 1.9947%?

You calculated PV in two different ways. They are almost identical as they should be based on the definition of YTM. However, I don’t see any calculation to find YTM.

The plus signs have gone missing in my equation!

Hi Alex

Thank you for reply. I am sorry – my question was not very clear. I understand that you are solving:

PV = C / (1 + r1) + C / (1 + r2)^2 + (C + F)/(1 + r3)^3 = C / (1 + y) + C / (1 + y)^2 + (C + F)/(1 + y)^3

I believe it can only be solved through an iterative algorithm (e.g. Newton-Raphson). Did you use one financial calculator to get this magic figure, 1.9947%? I see 1.9947% in your spreadsheet as an input. Presumably, it was computed externally.

Dear Joe,

assume you have a zero coupon curve like 1y=1,60%;2=1,80%;3=2%.

YTM = 1,9947% will give you the same result as yields above if you invest your money for 3 years.

The procedure is called “yield to maturity from spot rates”.

Thank you for your clarification. So you have used Excel itself to do the iteration.\n\nI’ll read your other articles too. I like the way you explain things that are often misrepresented or confused in books and papers!\n

Hi Joe!

On the sheet “YTM”, try to change the cell C7, so the the E12 is equal to the cell E13 (sheet “ZC”). :))))

Joe,\nthis is a main purpose of this blog. Unfortunelly, I have no time to write more…))) but sure I will