Fixed Income Securities

FIXED INCOME SECURITIES

LECTURE NOTE: CURVE

PROFESSOR DAVID T. BROWN

UNIVERSITY OF FLORIDA

WARRINGTON COLLEGE OF BUSINESS

Overview.

The duration of a portfolio measures the sensitivity of the value of a portfolio to a change in interest rates. In particular, duration measures the change in the value of a portfolio resulting from a parallel shift in the term structure of interest rates. However, in reality, interest rate changes are often not parallel shifts in the term structure of interest rates. The following materials and examples discuss how to measure the exposure of a portfolio to non-parallel shifts in the term structure of interest rates: “curve risk”.[1] In addition, this material introduces the concept of dollar duration.

It is useful to consider the following examples to motivate this material. Consider three portfolios that have the same duration, say five years. Portfolio A is a bullet portfolio that is entirely invested in five-year zero-coupon bonds. Portfolio B is a barbell portfolio that is invested half invested in ten year zeros and half invested in very short-term bonds. Portfolio C is a ladder portfolio that has small investments in several maturities. If interest rates rise in a manner where long term rates rise more than short-term rates, then the Portfolio B will have the worst performance, followed by Portfolio C. The bullet will have the best performance.

Portfolio Curve Exposure.

An example from the portfolio curve exposure calculator is shown on the following page. The example shows a ladder style portfolio. The portfolio positions in all of the bonds are collapsed into the cash flows due in each year in the future.

The portfolio has market value of $137.65 and a duration of 11.342 years. The dollar duration of each cash flow is the present value of the cash flow times the maturity of the cash flow. The portfolio dollar duration is the sum of the individual dollar durations. The dollar duration of 55.839 in the ten-year maturity cell tells you that the portfolio will increase (decrease) in value by .55839 dollars for every 1% decrease (increase) in the 10 year rate.

The contribution to dollar duration is just the dollar duration of a particular maturity divided by the total dollar duration. The contributions to dollar duration add to one. The highest contribution to dollar duration is in the 17-year bucket. That means that a change in the 17-year rate will have the largest effect on the portfolio value.

KEY RATE DURATION EXAMPLE: CURVE RISK
Treasury / Contribution
Strip / Portfolio / Portfolio / Dollar / to Dollar
Year / Curve / Cash Flow / PV / Duration / Duration
1 / 6% / 10 / $ 9.43 / 9.434 / 0.604%
2 / 6% / 10 / $ 8.90 / 17.800 / 1.140%
3 / 6% / 10 / $ 8.40 / 25.189 / 1.613%
4 / 6% / 10 / $ 7.92 / 31.684 / 2.029%
5 / 6% / 10 / $ 7.47 / 37.363 / 2.393%
6 / 6% / 10 / $ 7.05 / 42.298 / 2.709%
7 / 6% / 10 / $ 6.65 / 46.554 / 2.982%
8 / 6% / 10 / $ 6.27 / 50.193 / 3.215%
9 / 6% / 10 / $ 5.92 / 53.271 / 3.412%
10 / 6% / 10 / $ 5.58 / 55.839 / 3.577%
11 / 6% / 10 / $ 5.27 / 57.947 / 3.712%
12 / 6% / 10 / $ 4.97 / 59.636 / 3.820%
13 / 6% / 10 / $ 4.69 / 60.949 / 3.904%
14 / 6% / 10 / $ 4.42 / 61.922 / 3.966%
15 / 6% / 10 / $ 4.17 / 62.590 / 4.009%
16 / 6% / 10 / $ 3.94 / 62.983 / 4.034%
17 / 6% / 10 / $ 3.71 / 63.132 / 4.044%
18 / 6% / 10 / $ 3.50 / 63.062 / 4.039%
19 / 6% / 10 / $ 3.31 / 62.797 / 4.022%
20 / 6% / 10 / $ 3.12 / 62.361 / 3.994%
21 / 6% / 10 / $ 2.94 / 61.773 / 3.957%
22 / 6% / 10 / $ 2.78 / 61.051 / 3.910%
23 / 6% / 10 / $ 2.62 / 60.213 / 3.857%
24 / 6% / 10 / $ 2.47 / 59.275 / 3.797%
25 / 6% / 10 / $ 2.33 / 58.250 / 3.731%
26 / 6% / 10 / $ 2.20 / 57.151 / 3.661%
27 / 6% / 10 / $ 2.07 / 55.989 / 3.586%
28 / 6% / 10 / $ 1.96 / 54.776 / 3.509%
29 / 6% / 10 / $ 1.85 / 53.521 / 3.428%
30 / 6% / 10 / $ 1.74 / 52.233 / 3.346%
Market Value / $ 137.65
Dollar Duration / 1561.236
Portfolio Duration / 11.342

The following page shows a different portfolio that also has market value of $137.65 and very similar duration: 11.331 years. This portfolio has very different exposures to different interest rate changes.

Treasury / Contribution
Strip / Portfolio / Portfolio / Dollar / to Dollar
Year / Curve / Cash Flow / PV / Duration / Duration
1 / 6% / 20 / $ 18.87 / 18.868 / 1.210%
2 / 6% / 8 / $ 7.12 / 14.240 / 0.913%
3 / 6% / 8 / $ 6.72 / 20.151 / 1.292%
4 / 6% / 8 / $ 6.34 / 25.347 / 1.625%
5 / 6% / 8 / $ 5.98 / 29.890 / 1.916%
6 / 6% / 8 / $ 5.64 / 33.838 / 2.169%
7 / 6% / 8 / $ 5.32 / 37.243 / 2.388%
8 / 6% / 8 / $ 5.02 / 40.154 / 2.574%
9 / 6% / 8 / $ 4.74 / 42.617 / 2.732%
10 / 6% / 8 / $ 4.47 / 44.672 / 2.864%
11 / 6% / 26 / $ 13.70 / 150.661 / 9.659%
12 / 6% / 8 / $ 3.98 / 47.709 / 3.059%
13 / 6% / 8 / $ 3.75 / 48.759 / 3.126%
14 / 6% / 8 / $ 3.54 / 49.538 / 3.176%
15 / 6% / 8 / $ 3.34 / 50.072 / 3.210%
16 / 6% / 8 / $ 3.15 / 50.387 / 3.230%
17 / 6% / 8 / $ 2.97 / 50.506 / 3.238%
18 / 6% / 8 / $ 2.80 / 50.450 / 3.234%
19 / 6% / 8 / $ 2.64 / 50.238 / 3.221%
20 / 6% / 8 / $ 2.49 / 49.889 / 3.198%
21 / 6% / 8 / $ 2.35 / 49.418 / 3.168%
22 / 6% / 8 / $ 2.22 / 48.841 / 3.131%
23 / 6% / 8 / $ 2.09 / 48.171 / 3.088%
24 / 6% / 8 / $ 1.98 / 47.420 / 3.040%
25 / 6% / 8 / $ 1.86 / 46.600 / 2.988%
26 / 6% / 8 / $ 1.76 / 45.720 / 2.931%
27 / 6% / 8 / $ 1.66 / 44.791 / 2.872%
28 / 6% / 20 / $ 3.91 / 109.553 / 7.024%
29 / 6% / 19.5 / $ 3.60 / 104.367 / 6.691%
30 / 6% / 21 / $ 3.66 / 109.689 / 7.032%
Market Value / $ 137.65
Dollar Duration / 1559.798
Portfolio Duration / 11.331

Using dollar duration to assess a position against and index.

The idea here is very similar. In this example, a portfolio is managed against and index or a liability. The market value of the index is always equal to the market value of the portfolio. In this case, the portfolio has a very similar duration to the index. However, the portfolio has considerable curve risk against the index.

The curve risk is the dollar duration under or over weights. The dollar duration under or over weights sum to zero in this case because the durations are the same. If the portfolio had a greater (lower) duration than the index, then the over or under weights would sum to a positive (negative) number. A dollar-duration overweight in a particular maturity means that you will out perform the index if that interest rate rises by the product of the rate change and the dollar duration.

Dollar
Treasury / Present / Duration
Strip / Portfolio / Index / Value / Over or Under
Year / Curve / Cash Flow / Cash Flow / Difference / Weight
1 / 6% / 20 / $ 10.00 / $ 9.43 / 9.434
2 / 6% / 8 / $ 10.00 / $ (1.78) / -3.560
3 / 6% / 8 / $ 10.00 / $ (1.68) / -5.038
4 / 6% / 8 / $ 10.00 / $ (1.58) / -6.337
5 / 6% / 8 / $ 10.00 / $ (1.49) / -7.473
6 / 6% / 8 / $ 10.00 / $ (1.41) / -8.460
7 / 6% / 8 / $ 10.00 / $ (1.33) / -9.311
8 / 6% / 8 / $ 10.00 / $ (1.25) / -10.039
9 / 6% / 8 / $ 10.00 / $ (1.18) / -10.654
10 / 6% / 8 / $ 10.00 / $ (1.12) / -11.168
11 / 6% / 26 / $ 10.00 / $ 8.43 / 92.715
12 / 6% / 8 / $ 10.00 / $ (0.99) / -11.927
13 / 6% / 8 / $ 10.00 / $ (0.94) / -12.190
14 / 6% / 8 / $ 10.00 / $ (0.88) / -12.384
15 / 6% / 8 / $ 10.00 / $ (0.83) / -12.518
16 / 6% / 8 / $ 10.00 / $ (0.79) / -12.597
17 / 6% / 8 / $ 10.00 / $ (0.74) / -12.626
18 / 6% / 8 / $ 10.00 / $ (0.70) / -12.612
19 / 6% / 8 / $ 10.00 / $ (0.66) / -12.559
20 / 6% / 8 / $ 10.00 / $ (0.62) / -12.472
21 / 6% / 8 / $ 10.00 / $ (0.59) / -12.355
22 / 6% / 8 / $ 10.00 / $ (0.56) / -12.210
23 / 6% / 8 / $ 10.00 / $ (0.52) / -12.043
24 / 6% / 8 / $ 10.00 / $ (0.49) / -11.855
25 / 6% / 8 / $ 10.00 / $ (0.47) / -11.650
26 / 6% / 8 / $ 10.00 / $ (0.44) / -11.430
27 / 6% / 8 / $ 10.00 / $ (0.41) / -11.198
28 / 6% / 20 / $ 10.00 / $ 1.96 / 54.776
29 / 6% / 19.5 / $ 10.00 / $ 1.75 / 50.845
30 / 6% / 21 / $ 10.00 / $ 1.92 / 57.456
Portfolio Market Value / $ 137.65
Dollar Duration / 1559.798
Portfolio Duration / 11.331
Index Market Value / $ 137.65
Dollar Duration / 1561.2362
Index Duration / 11.342

[1]The material on key rate durations on pages 80-83 of Fabozzi is related.