In computing modified (or effective) duration for a portfolio of securities, we change the par interest rate (the yield to maturity) at every maturity by some small amount up and down (±Δy), and determine the percentage price change in the portfolio for 1% change in yield.  In essence, we add ±Δy to the entire par curve.  (Look here for a refresher on the par curve, and for a refresher on bootstrapping the spot curve from the par curve, which we shall be doing a bit later.)  Parallel shifts in the par curve look like this:

Modified (or effective) duration is useful for determining how the price of a security or a portfolio of securities will change when the (par) yield curve undergoes a parallel shift, but is less useful (i.e., not useful at all) when the yield curve changes in a manner other than a parallel shift (e.g., flattening or steepening).  A more versatile measure of duration is needed in those cases; key rate duration is just such a measure.

The idea of a key rate duration (also known as partial duration) is that we don’t add ±Δy to the entire par curve; we add ±Δy only to the YTM at a specific maturity on the par curve, leaving the YTM at all other maturities unchanged; that specific maturity is called the key rate.  Thus, we have the:

• ½-year key rate duration: ±Δy added to the ½-year YTM, while the YTMs at all other maturities remain unchanged
• 2-year key rate duration: ±Δy added to the 2-year YTM
• 5-year key rate duration: ±Δy added to the 5-year YTM
• 10-year key rate duration: ±Δy added to the 10-year YTM
• 20-year key rate duration: ±Δy added to the 20-year YTM
• 30-year key rate duration: ±Δy added to the 30-year YTM
• And so on

Shifts in the par curve to calculate key rate durations look like this:

Note that key rate durations can be modified (assuming no changes to the cash flows when the key rate changes) or effective (allowing that cash flows might change when the key rate changes).  There’s probably no reason to concern yourself about this as far as the exam goes, but in real world applications you might as well assume that key rate durations are effective durations.

Once the effect of the key-rate shift is incorporated into the bond price, the duration is computed using the standard (modified or effective) duration formula:

\[Dur_{key\ rate}\ =\ \frac{P_–\ -\ P_+}{2P_0\Delta y}\]

In many respects, key rate durations behave exactly like all other durations (Macaulay duration, modified duration, effective duration, etc.), but in some ways their behavior is unusual.  We’ll cover both.

Expected Behavior

Key Rate Duration of a Portfolio

As with Macaulay, modified, and effective duration (see here), any particular key rate duration of a portfolio is the weighted average of that particular key rate duration for each of its constituent bonds, where the weights are based on the market values of the constituent bonds:

\begin{align}Dur_{key\ rate_{port}}\ &=\ W_1\ ×\ Dur_{key\ rate_{bond\ 1}}\ +\ W_2\ ×\ Dur_{key\ rate_{bond\ 2}}\\
&+\ \cdots\ +\ W_n\ ×\ Dur_{key\ rate_{bond\ n}}\\
\\
&=\ \sum_{i=1}^n W_i\ ×\ Dur_{key\ rate_{bond\ i}}
\end{align}

where:

• \(W_i\): bond i‘s market value ÷ market value of the portfolio

So, for example, the 5-year key rate duration of a portfolio of bonds is the weighted average of the 5-year key rate durations of each of the constituent bonds, the 20-year key rate duration of the portfolio is the weighted average of the 20-year key rate durations of each of the constituent bonds, and so on.

Relationship of (Modified or Effective) Duration and Key Rate Durations

Another characteristic of key rate durations that is expected (or, at least, should be, once you think of it) is that the (modified or effective) duration of a security or a portfolio is the sum of all of its own (respectively, modified or effective) key rate durations:

\begin{align}Dur\ &=\ Dur_{key\ rate_1}\ +\ Dur_{key\ rate_2}\ +\ \cdots\ +\ Dur_{key\ rate_k}\ +\cdots\\
\\
&=\ \sum_{i=1}^{\infty}\ Dur_{key\ rate_i}
\end{align}

Note that this is not an infinite sum: eventually the index i will exceed the maturity of the security or portfolio; after that, all of the key rate durations will be zero (because, as we will see, all of the par rates and all of the spot rates at maturities shorter than the key rate maturity are unchanged).

Unusual Behavior

Recalling the link between the par curve and the spot curve, we can determine the effect of key rate changes on bonds of various maturities and with various coupons.  To illustrate these effects, I’ll use a simple example: a 4% flat yield curve with annual (effective) rates from 1 year to 10 years:

Suppose that we want to compute the 5-year key rate duration of a bond.  To do this, we add 50bp to the 5-year par rate (leaving all of the other par rates unchanged), and compute the effect on the spot rates, then subtract 50bp from the 5-year par rate (again, leaving all of the other par rates unchanged), and compute the effect on the spot rates.  We’ll then use those modified spot rates to compute \(P_-\) and  \(P_+\), and then use them in the duration formula.

Let’s start by adding 50bp to the 5-year par rate, leaving all other par rates unchanged at 4.0%.

The first question is: what happens to the 1-year spot yield?  The answer is easy: it remains 4.0%.  (Recall that the one-period par rate and the one-period spot rate are equal, and that the 1-year par rate hasn’t changed; only the 5-year par rate has changed.)

The 2-year spot yield is also easy: it, too, remains 4.0%, because the 2-year par yield is unchanged, and the 1-year spot yield is unchanged.  If you want to run through the algebra, it’s:

\begin{align}$1,000\ &=\ \frac{$40.00}{1.04}\ +\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\
\\
$1,000\ &=\ $38.46\ +\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\
\\
$961.54\ &=\ \frac{$1,040.00}{\left(1\ +\ s_2\right)^2}\\
\\
\left(1\ +\ s_2\right)^2\ &=\ \frac{$1,040.00}{$961.54}\ =\ 1.0816\\
\\
1\ +\ s_2\ &=\ \sqrt{1.0816}\ =\ 1.0400\\
\\
s_2\ &=\ 4.0\%
\end{align}

The 3-year spot rate and the 4-year spot rate are also easy: they’re both still 4.0%, because their par yields are unchanged, as are all of the spot yields for shorter maturities.  I’ll leave the algebra to you.

As an exciting (!) change of pace, the 5-year spot rate isn’t 4.0% (recall that we’ve added 50bp to the 5-year par rate).  A 5-year bond paying a 4.5% (= 4.0% + 50bp) will therefore sell at par, so:

\begin{align}$1,000\ &=\ \frac{$45.00}{1\ +\ s_1}\ +\ \frac{$45.00}{\left(1\ +\ s_2\right)^2}\ +\ \frac{$45.00}{\left(1\ +\ s_3\right)^3}\ +\ \frac{$45.00}{\left(1\ +\ s_4\right)^4}\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\
\\
&=\ \frac{$45.00}{1.04}\ +\ \frac{$45.00}{1.04^2}\ +\ \frac{$45.00}{1.04^3}\ +\ \frac{$45.00}{1.04^4}\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\
\\
$1,000\ &=\ $43.27\ +\ $41.61\ +\ $40.00\ +\ $38.47\ +\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\
\\
$836.65\ &=\ \frac{$1,045.00}{\left(1\ +\ s_5\right)^5}\\
\\
\left(1\ +\ s_5\right)^5\ &=\ \frac{$1,045.00}{$836.65}\ =\ 1.249022\\
\\
1\ +\ s_5\ &=\ 1.249022^{1/5}\ =\ 1.045476\\
\\
s_5\ &=\ 4.5476\%
\end{align}

This result seems reasonable: the YTM is 4.5%, so we’ll get par ($1,000) if we discount all of the cash flows at 4.5%.  Because we’ve discounted the first four payments at 4% (less than 4.5%), we have to discount the final payment at more than 4.5% to average a 4.5% discount for everything.  And because the first four payments are much smaller than the final payment, the difference on the final discount rate should be much smaller than the (50bp) difference on the first four: it’s 4.76bp.

Now comes the really interesting part (!!): the 6-year spot rate.  Let’s do the algebra (recalling that the 6-year par rate hasn’t changed: it’s still 4.0%):

\begin{align}$1,000\ &=\ \frac{$40.00}{1.04}\ +\ \frac{$40.00}{1.04^2}\ +\ \frac{$40.00}{1.04^3}\ +\ \frac{$40.00}{1.04^4}\\
\\
&+\ \frac{$40.00}{1.045476^5}\ +\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\
\\
$1,000\ &=\ $38.46\ +\ $36.98\ +\ $35.56\ +\ $34.19\ +\ $32.03\ +\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\
\\
$822.78\ &=\ \frac{$1,040.00}{\left(1\ +\ s_6\right)^6}\\
\\
\left(1\ +\ s_6\right)^6\ &=\ \frac{$1,040.00}{$822.78}\ =\ 1.264009\\
\\
1\ +\ s_6\ &=\ 1.264009^{1/6}\ =\ 1.039820\\
\\
s_6\ &=\ 3.9820\%
\end{align}

Again, this result makes sense: the 5-year spot rate has increased, so the only way that the 6-year par rate can remain unchanged is for the 6-year spot rate to decrease.  And, once again, because the final payment is much larger than the first 5 (coupon only) payments, the decrease should be small compared to the increase in the 5-year spot rate: the 5-year spot rate is 54.76bp higher than the 4% YTM, while the 6-year spot rate is 1.80bp lower than the YTM.

Note how the change in the 5-year key rate leads to some somewhat surprising results.  For example, a 6-year zero-coupon bond will have a slightly higher price (because the 6-year spot rate is lower) when the 5-year key rate is increased; in other words, a 6-year zero-coupon bond will have a negative 5-year key rate duration.  (Remember that when interest rates increase, bond prices normally decrease, and that duration is normally positive.  When the price increases with a rate increase, duration must be negative.)

Continuing in this same tedious manner, we get:

Graphically:

When we subtract 50bp from the 5-year par rate, we get:

Graphically:

Key Rate Durations of Various Bonds

Not only does the key rate duration of a bond depend on the bond’s maturity (compared to the maturity of the key rate), it also depends on the bond’s coupon rate compared to its YTM; i.e., it depends on whether the bond is priced at par, at a premium, or at a discount.  Using our 4% flat yield curve, here are the key rate durations for five 5-year, option-free bonds with varying coupon rates, along with the sum of their key rate durations, and their effective durations:

So, for example, a 5-year, zero-coupon bond has a 4-year key rate duration of −0.1633 years; if the 4-year par rate increases by 1%, the price of the bond will increase by approximately 0.1633%.  A 5-year, 8% coupon bond has a 3-year key rate duration of 0.0838 years; if the 3-year par rate decreases by 1%, the price of the bond will increase by approximately 0.0838%.

The key ideas to glean from this table are:

• Par bonds have key rate durations of zero years for any key rate maturity shorter than the bond’s maturity
• Discount bonds have negative key rate durations for key rate maturities shorter than the bond’s maturity; in particular, zero-coupon bonds have negative key rate durations for key rate maturities shorter than the bond’s maturity
• Premium bonds have positive key rate durations for key rate maturities shorter than the bond’s maturity
• All bonds have key rate durations of zero years for any key rate maturity longer than the bond’s maturity
• The sum of the key rate durations for all key rate maturities equals the bond’s effective duration

Note that these last two points was already covered under Expected Behavior, above.

Using Key Rate Durations

Key rate durations are used to determine the effect of non-parallel yield curve shifts on a portfolio of bonds.  (In fact, they can also be used to determine the effect of parallel yield curve shifts, although they’re less efficient than simply using the effective duration of each bond.)  For example, suppose that we have this par yield curve:

Suppose that we expect the yield curve to flatten: rise 50bp on the short end, fall 50bp on the long end, with a linear transition in between:

The new (par) yield curve will look like this:

To determine the effect of this flattening on the value of a fixed income portfolio, we can look at the key-rate durations of each bond in the portfolio, multiply each key rate duration by the yield change (from the table, above) and the bond’s market value, then tot them up.  Note that effective duration (of the portfolio) will not help here, as the shift is not parallel.

Similarly, if we expect the par curve to steepen, or hump, or do anything else other than a parallel shift, key rate durations will allow us to determine the first-order effect on the value of a portfolio of bonds with a range of maturities.

For whatever it’s worth, there’s also the idea of key rate convexities.  I’ve never heard of them – I just came up with the idea all on my own – but for large yield changes it would make sense to consider convexity in addition to duration.  Deep down, I’m a pioneer.

Misconceptions about Key Rate Durations

The primary misconception about key rate durations is that they correspond to a change in a single spot rate, instead of a change in a single par rate.  To be clear:

• The key rate duration of a given bond for a given maturity is the ratio of the percentage change in that bond’s price to the change in the par rate at that maturity, when the par rates at all other maturities remain unchanged.
• When the par rate at a given maturity changes, and the par rates at all other maturities remain unchanged:
  • The spot rates at maturities less than the given maturity will not change
  • The spot rate at the given maturity will change in the same direction as change in the par rate, and by an  amount greater than the change in the par rate
  • The spot rates at maturities greater than the given maturity will change in the opposite direction of the change in the par rate, and by an amount (much) less than the change in the par rate (and the change will be smaller at longer maturities)

Valuing derivatives – forwards, futures, FRAs, and swaps – is much the same as pricing them.  The value of a derivative is the amount that one party would have to pay the other if the derivative were to expire today; it depends on the price of the underlying today compared to the price at the inception of the derivative.  Therefore, valuing a derivative generally means doing the same sort of calculation you did to price the derivative, then comparing today’s price to the original price.  Finally, bear in mind that the value to one party is the negative of the value to the other party.  I encourage you to practice valuation problems always from the same party’s point of view (e.g., always from long’s point of view in valuing a forward contract); if you have to calculate it from the other party’s point of view on the exam, do it the way you’ve practiced, then change the sign.

Here are links to articles on valuing derivatives:

Also note that the formulae used for valuing derivatives will vary depending on whether the risk-free interest rate is given as an effective rate or a nominal rate; for a refresher on nominal vs. effective interest rates, look here.

Pricing currency forward contracts – determining the appropriate future exchange rate to use – is relatively straightforward; it is based on the risk-free interest rates for the currencies involved, and the no-arbitrage condition (i.e., the forward exchange rate should make arbitrage impossible).  Because the elimination of arbitrage means that the forward exchange rate has to compensate for inequality in the risk-free interest rates – it has to restore equality, or parity – and because the parity is ensured (or covered) by the forward contract, the approach in known as covered interest rate parity (covered IRP, or CIRP).  The formula is:

\[F_{PC/BC}\ =\ S_{PC/BC}\ ×\ \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}\]

where:

• \(F_{PC/BC}\): forward (future) exchange rate, quoted as price currency / base currency
• \(S_{PC/BC}\): spot (current) exchange rate, quoted as price currency / base currency
• \(r_{PC}\): risk-free rate for the price currency
• \(r_{BC}\): risk-free rate for the base currency

An easy way to remember which rate goes in the numerator and which rate goes in the denominator is to think of the formula this way:

\begin{align}F_{PC/BC}\ &=\ \frac{PC_{future}}{BC_{future}}\ =\ \frac{PC_{today}\ ×\ \left(1\ +\ r_{PC}\right)}{BC_{today}\ ×\ \left(1\ +\ r_{BC}\right)}\\
\\
&= \frac{PC_{today}}{BC_{today}}\ ×\ \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}\ =\ S_{PC/BC} × \frac{1\ +\ r_{PC}}{1\ +\ r_{BC}}
\end{align}

Writing it this way shows that the future price currency is today’s price currency grown at the price currency risk-free rate (the numerators) and the future base currency is today’s base currency grown at the base currency risk-free rate (the denominators).

Note that the forward exchange rate is calculated for a certain amount of time t in the future, so the risk-free interest rates have to be the rates applicable for that amount of time.  For example, suppose you are given that the:

• spot exchange rate for US dollars and British pounds is USD/GBP 1.6453
• 180-day USD LIBOR rate is 2.4%
• 180-day GBP LIBOR rate is 3.0%

You are asked to calculate the 180-day forward USD/GBP exchange rate.  The calculation is:

\begin{align}F_{USD/GBP}\ &=\ S_{USD/GBP}\ ×\ \frac{1\ +\ r_{USD}}{1\ +\ r_{GBP}}\\
\\
&=\ 1.6453\ ×\ \frac{1\ +\ 2.4\%\left(\frac{180}{360}\right)}{1\ +\ 3.0\%\left(\frac{180}{360}\right)}\\
\\
&=\ 1.6453\ ×\ \frac{1.012}{1.015}\\
\\
&=\ 1.6404
\end{align}

Thus, the 180-day forward exchange rate is USD/GBP 1.6404.

(Note that the quoted LIBOR rates are annual, nominal rates.)

If, instead, you are given that the:

• spot exchange rate for US dollars and British pounds is USD/GBP 1.6453
• effective (annual) 180-day USD risk-free rate is 2.4%
• effective (annual) 180-day GBP risk-free rate is 3.0%

then the calculation of the 180-day forward USD/GBP exchange rate is:

\begin{align}F_{USD/GBP}\ &=\ S_{USD/GBP} × \frac{1\ +\ r_{USD}}{1\ +\ r_{GBP}}\\
\\
&=\ 1.6453\ ×\ \frac{\left(1\ +\ 2.4\%\right)^{180/365}}{\left(1\ +\ 3.0\%\right)^{180/365}}\\
\\
&=\ 1.6453\ ×\ \frac{1.011764}{1.014684}\\
\\
&=\ 1.6406
\end{align}

Thus, the 180-day forward exchange rate is USD/GBP 1.6406.

An Approximation

As a quick and dirty check, the percentage increase/decrease of the forward rate over the spot rate is approximately equal to the difference between the price currency’s and base currency’s risk-free rates:

\[\frac{F_{PC/BC}}{S_{PC/BC}}\ -\ 1\ ≈\ r_{PC}\ -\ r_{BC}\]

To illustrate this, consider the first example, above:

\[\frac{1.6406}{1.6453}\ -\ 1\ =\ -0.286\% ≈\ -0.300\%\ =\ \ 2.4\%\left(\frac{180}{360}\right)\ -\ 3.0\%\left(\frac{180}{360}\right)\]

Why?  In a Word: Arbitrage

The reason that the forward exchange rate must satisfy the formula given above is that any other forward rate will create an arbitrage opportunity.  The arbitrage transaction is fairly simple:

• Borrow currency A at its risk-free rate
• Convert currency A to currency B at the spot exchange rate
• Enter into a forward contract to exchange currency B for currency A at the forward exchange rate (note: the amount in the forward contract will be current amount of currency B plus the interest you will earn on currency B)
• Invest currency B at its risk-free rate
• Wait (until the forward contract matures)
• Convert currency B (principal plus interest) to currency A at the agreed forward rate
• Pay off the currency A loan (principal and interest)
• Enjoy the profit

Note that this series of transactions is risk-free (hence, arbitrage): the money is borrowed (so the arbitrageur has no money at risk), the interest rates are risk-free, and the future exchange rate is locked in at the outset by the forward contract.

To see how this works, imagine that in the first example (LIBOR rates), the quoted forward exchange rate were USD/GBP 1.6420 (higher than the required 1.6406).  In that case, the future GBP would be exchanged for more future USD than the formula specifies; this tells us that we need to start the transaction by borrowing USD (because we’ll end up with more USD than we should: we’re earning a higher interest rate on USD than we should).  If we borrow USD1,000,000, the transactions would look like this:

• Borrow USD1,000,000 at 2.4% USD LIBOR for 180 days
• Convert USD1,000,000 to GBP607,792 at USD/GBP 1.6453
• Enter into a forward contract to convert GBP616,909 (= GBP607,792 × (1 + 3%(180/360))) at USD/GBP 1.6420 in 180 days; the amount is GBP607,792 plus interest
• Invest GBP607,792 at 3% GBP LIBOR  for 180 days
• Wait 180 days; the GBP607,792 investment grows to GBP616,909
• Convert GBP616,909 to USD1,012,964 at USD/GBP 1.6420
• Pay off the USD loan for USD1,012,000 (= USD1,000,000 × (1 + 2.4%(180/360)))
• Enjoy a profit of USD964

If, instead, the quoted forward exchange rate were USD/GBP 1.6391 (lower than the required 1.6406), then the future GBP would be exchanged for fewer USD than the formula specifies; this tells us that we need to start the transaction by borrowing GBP (the reverse of the previous case: if we borrowed USD we’d end up with fewer USD than we should, earning a lower interest rate on USD than we should).  If we borrow GBP1,000,000, the transactions would look like this:

• Borrow GBP1,000,000 at 3.0% GBP LIBOR for 180 days
• Convert GBP1,000,000 to USD1,645,300 at USD/GBP 1.6453
• Enter into a forward contract to convert USD1,665,044 (= USD1,645,300 × (1 + 2.4%(180/360))) at USD/GBP 1.6391 in 180 days; the amount is USD1,645,300 plus interest
• Invest USD1,645,300 at 2.4% USD LIBOR for 180 days
• Wait 180 days; the USD1,645,300 investment grows to USD1,665,044
• Convert USD1,665,044 to GBP1,015,828 at USD/GBP 1.6391
• Pay off the GBP loan for GBP1,015,000 (=GBP1,000,000 × (1 + 3%(180/360)))
• Enjoy a profit of GBP828

There are four articles on IS/LM:

• Synopsis
• Part 1: the IS curve
• Part 2: the LM curve
• Part 3: combining the IS and LM curves
  

I’m still working on Level III Equity Portfolio Management articles; as I finish them I’ll list them here.  If you’d like to see specific articles on Level III Equity Portfolio Management, just ask.

I’m still working on Level III GIPS® articles; as I finish them I’ll list them here.  If you’d like to see specific articles on Level III GIPS®, just ask.

Here are some articles on Level III Evaluation and Attribution. If you’d like to see specific articles on Level III Evaluation and Attribution, just ask.

Here are some articles on Level III Trade Execution, Monitoring, and Rebalancing.  If you’d like to see specific articles on Level III Trade Execution, Monitoring, and Rebalancing, just ask.

Here are some articles on Level III Derivatives.  If you’d like to see specific articles on Level III Derivatives, just ask.

I’m still working on Level III Risk Management articles; as I finish them I’ll list them here.  If you’d like to see specific articles on Level III Risk Management, just ask.