26 Duration

26.1 Duration of a Single Cash Flow

Consider an example of two zero-coupon bond, each with maturity value of $\$1000$. The first bond matures in 10 years while the second one matures in 20 years. Both bonds are purchased today to yield $6\%$ effective.

The price of the 10-year bond is $1000\times(1.06)^{-10} = 558.39$
The price of the 20-year bond is $1000\times (1.06)^{-20} = 311.80$.

Now suppose that the effective rate of interest suddenly increased by 0.1% to 6.1%. Then, that change in the interest rate operating over a 20-year period would have a relatively greater impact on the present value than the same change operating over a 10-year period. Indeed:

The price of the 10-year bond at $6.1\%$ yield: $1000\times(1.061)^{-10} = 553.15$
The price of the 20-year bond at $6.1\%$ yield: $1000\times(1.061)^{-20} = 305.98$

Then, we can calculate the percentage change of the bonds:

$\%\Delta(\text{10-year bond}) = \frac{553.15-558.39}{558.39} = -0.94\%$
$\%\Delta(\text{20-year bond}) = \frac{305.98-311.80}{311.80} = -1.87\%$ We call the percentage change in price the price sensitivity of a bond to a change in the interest rate. The time remaining to a single cash inflow or cash outflow is called its duration. We will expand on the concept to multiple cash flow.

26.2 Duration of a General Cash Flow

For single cash flow, the greater the duration, the more sensitive the PV is to changes in the interest rate. To generalize this concept, we can weigh the time of each cash flow by the present value of the cash flow.

Let us take the example of a bond with $5\%$ annual coupons maturing at $\$1000$ in 10 years. Then, the duration of the cash flow:

\[\frac{(50v)(1) + (50v^2)(2) + .... + (50v^{10})(10) + (1,000v^{10})(10)}{50(v+v^2+...+v^{10}) + 1,000v^{10}} = \frac{50(Ia)_{\overline{10|}} + 10,000v^{10}}{50a_{\overline{10|}} + 1,000v^{10}} = 8.02\] This duration is called the Macaulay duration

26.2.1 Macaulay Duration

For a set of cash flow $C_1, C_2,..., C_n$ at time $1,2,...,n$. The Macaulay duration of the set of cash flow under a flat yield is denoted $D_{mac}(i)$ and is defined to be: \[D_{mac}(i) = \frac{\sum_{t > 0} t \times C_t \times (1+i)^{-t}}{\sum_{t>0} C_t \times (1+i)^{-t}}\]

Or, under a general term structure:

\[ D_{mac} = \frac{\sum_{t > 0}t \times C_t \times v^t_{spot}}{\sum_{t>0} C_t \times v_{spot}^t} \] Recall that, $i = e^{\delta} - 1 \implies di = e^\delta d\delta \implies d\delta = v \times di$. Hence, we can also express Macaulay duration as change in present value with respect to change in the force of interest:

\[D_{mac}(i) = -\frac{\frac{d}{d\delta}P(i)}{P(i)}\]

As interest rate volatility and risk grew, the finance industry developed an alternative to Macaulay duration to measure the sensitivity of bond prices to changes in yield rates. The concept of modified duration was introduced, defined in terms of the derivative of bond price with respect to change in yield. ### Modified Duration

For a set of cash flow $C_1, C_2,..., C_n$ at time $1,2,...,n$ with present value $P(i) = \sum_{t > 0} C_t \times v^t$ under a flat yield. The modified duration of the set of cash flow is denoted $D_{mod}(i)$ and is defined to be:

\[D_{mod}(i) = -\frac{\frac{d}{di}P(i)}{P(i)} = -\frac{d}{di}\ln P(i) = \sum_{t > 0}t v^{t+1}\times C_t\]

The modified duration is a price sensitivity measure, defined as the percentage derivative of price with respect to yield. The modified duration for periodically compounded yields is:

\[D^{(m)}_{mod}(i) = -\frac{\frac{d}{di^{(m)}}P(i)}{P(i)} = \frac{1+i}{1+(i^{(m)}/ m)}\times D_{mod}(i)\] As $m \to \infty$, $D^{(m)}_{mod}(i) \to D_{mac}(i)$. The relationship between Macaulay and modified duration can be expressed as:

\[D_{mod} (i) = \frac{1+(i^{(m)}/m)}{1+i} D^{(m)}_{mod}(i)= \frac{1}{1+i}\times D_{mac}(i)\]

Even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield.

Note that, when the text refers to duration, Macaulay Duration is implied.

26.3 Duration of a Portfolio

Suppose we are considering $m$ cash flows in our portfolio. Denote $X_k$ to be the present value, and $D_{\text{mac, k}}(i); D_{\text{mod, k}}(i)$ to be the Macaulay and modified duration of the k-th cash flow.

The Macaulay duration of the portfolio is:

\[D_{\text{mac, total}(i)} = \frac{\sum_{k=1}^m -(1+i)\frac{d}{di} X_k}{\sum_{k=1}^m X_k} = \frac{\sum_{k=1}^m D_{\text{mac, k}}\times X_k}{\sum_{k=1}^m X_k}\] and the modified duration of the portfolio is:

\[D_{\text{mod, total}(i)} = \frac{\sum_{k=1}^m D_{\text{mod, k}}\times X_k}{\sum_{k=1}^m X_k}\] Example: A portfolio consists of four bonds, each of which has annual coupons:

2-year bond with face amount $\$100,000$, and $5\%$ coupon rate,
10-year bond with face amount $\$80,000$, and $10\%$ coupon rate,
30-year bond with face amount $\$120,000$, and $5\%$ coupon rate, and
60-year bond with face amount $\$75,000$, and $15\%$ coupon rate.

Find the Macaulay duration of this portfolio of bonds if the term structure is flat with effective annual interest rate $10\%$.

Solution: The bond prices are: (i) $91,322$, (ii) $80,000$, (iii) $63,439$, and (iv) $112,377$. The combined price of all bonds in the portfolio is $347,138$.
The Macaulay durations for the bonds can be calculated to be: (i) 1.950, (ii) 6.759, (iii) 11.434, and (iv) 10.91.

The Macaulay duration of the portfolio is: \[\begin{align}D_{\text{mac, total}}(0.1) &= \frac{(1.950)(91,322) + (6.759)(80,000)+(11.434)(63,439) + (10.910)(112,377)}{347,138}\\ &= 7.60\end{align}\]

26.4 First-Order Approximation of Present Value

The first-order modified approximation of the present value function is: \[P(i) \approx P(i_0) (1 - (i-i_0) D_{mod}(i_0))\] This can be derived from the Taylor Series for $P(i)$, as $D_{mod}(i_0) = -\frac{P'(i_0)}{P(i_0)}$.

The first-order Macaulay approximation of the present value function is:

\[P(i) \approx P(i_0) \bigg(\frac{1+i_0}{1+i}\bigg)^{D_{mac}(i_0)}\] To get this formula, we can apply the linear approximation for the current value of the cash flow at time $T = D_{mac}(i_0)$, i.e. when the current value is neither increasing nor decreasing: \[V(i) = P(i) \times (1+i)^{T}\] and $V'(i_0) = 0$. Applying the first-order Taylor approximation to $V(i)$ about $i_0$ we have:

\[\begin{align}V(i)&\approx V(i_0) + (i-i_0) V'(i_0) = V(i_0) \\ P(i) \times (1+i)^{T}&\approx P(i_0) \times (1+i_0)^{T}\\ P(i) &\approx P(i_0)\times \bigg(\frac{1+i_0}{1+i}\bigg)^{T}\end{align}\]