Class 15: The Yield Curve

Financial Markets, Fall 2020, SAIF

Jun Pan

Shanghai Advanced Institute of Finance (SAIF)

Shanghai Jiao Tong University

December 16, 2020

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 1 / 22
Outline

Risk factors affecting the bond market:

▶ Level: Parallel shifts of the yield curve; duration hedging.

▶ Slope: Differential movements between the long- and short-term interest rates.

Principal component analysis (PCA):

▶ Identify common risk components in the data.

▶ Quantify the risk factors in terms of their importance.

PCA Applications: financial markets with common risk factors.

▶ Bond markets: across maturities.

▶ Domestic equity markets: SMB, HML, momentum, etc.

▶ International equity markets: across countries.

▶ Currency markets: across currencies.

▶ Commodity futures: across maturities.

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An Illustrative Example
To simplify our analysis, let’s start with zero-coupon bonds.
Coupon bonds: weighted sums of the zero-coupon bond prices.
Initially, the zero rates are:
2yr 5yr 10yr 30yr
zero rate 3.5% 4.5% 5% 5.5%
maturity 2 5 10 30
2 5 10 30
modified duration 1+3.5% 1+4.5% 1+5% 1+5.5%
1.93 4.78 9.52 28.44

Using the 2yr zero as the reference bond,

▶ 5-to-2 spread: 100 bps.

▶ 10-to-2 spread: 150 bps.

▶ 30-to-2 spread: 200 bps.

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 3 / 22
Directional Trade to Bet on a Parallel Shift in Level

The yield on the 2yr zero subsequently increases by 10 bps.

All spreads remain the same.

notional market value ($ million)

($ million) initial later change approximation
(1+3.5%)2
2yr (1 + 3.5%)2 1 (1+3.5%+10bps)2
-19.30 bps -1.93 × 10 bps
(1+4.5%)5
5yr (1 + 4.5%)5 1 (1+4.5%+10bps)5
-47.71 bps -4.78 × 10 bps
(1+5.0%)10
10yr (1 + 5.0%)10 1 (1+5.0%+10bps)10
-94.74 bps -9.52 × 10 bps
(1+5.5%)30
30yr (1 + 5.5%)30 1 (1+5.5%+10bps)30
-280.22 bps -28.44 × 10 bps

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 4 / 22
Steepener

Your view:
▶ The 10-to-2 spread will increase.

▶ No view on the overall direction of the interest rate.

Your strategy: steepener.

▶ Long $4.9286M of 2yr zero.

▶ Short $1M of 10yr zero.

initial market value notional amount

($ million) ($ million)
2yr 4.9286 4.9286 × (1 + 3.5%)2
10yr −1 −(1 + 5.0%)10

Why 4.9286? Because 4.9286=9.52/1.93.

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Two Scenarios:

1 A parallel shift of +10 bps:

initial later change approximation
($ Million) ($ Million) ($ Million ) ($ Million)
4.9286×(1+3.5%)2
2yr 4.9286 (1+3.5%+10bps)2
-95.10 bps -1.93 × 10 bps × 4.9286
(1+5.0%)10
10yr -1 − (1+5.0%+10bps) 10 94.74 bps 9.52 × 10 bps

Total -0.36 bps

2 The 2yr zero rate decreases by 15 bps and the 10yr zero rate increases by 5 bps.
initial later change approximation
($ Million) ($ Million) ($ Million ) ($ Million)
4.9286×(1+3.5%)2
2yr 4.9286 (1+3.5%−15bps)2
143.17 bps 1.93 × 15 bps × 4.9286
(1+5.0%)10
10yr -1 − (1+5.0%+5bps) 10 47.49 bps 9.52 × 5 bps

Total 190.66 bps

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 6 / 22
Curvature (or Butterfly):

Your view: the 2yr and 30yr zeros will move in the same direction while the 10yr
zero will move in opposite direction. Not sure about the overall directional of the
interest rate, nor about the slope of the yield curve.
Your strategy: butterfly trade.
Long 30yr zeros, short 10yr zeros (to hedge against parallel shifts), and long 2yr
zeros (to hedge against slope steepening or flattening).

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 7 / 22
Yield Curve Movement after 911

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 8 / 22
Three Major Factors in the Fixed-Income Market:
Closely related to the three trading strategies are the three major risk factors reported by
Litterman and Scheinkman for the fixed income market:

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 9 / 22
Treasury Yields
US Treasury Rates (%)
12
3M
2Y
10 5Y
10Y
30Y

0
1990 1995 2000 2005 2010 2015 2020

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Variance-Covariance Matrix
cov(∆y) (unit: bps2 , or equivalently, ×10−8 )
3M 1Y 2Y 5Y 10Y 30Y
3M 64.9300 40.4601 32.1459 27.1468 22.3368 17.8525
1Y 40.4601 48.3104 42.5293 39.0606 33.7733 27.6243
2Y 32.1459 42.5293 48.4517 45.1757 39.4565 31.9805
5Y 27.1468 39.0606 45.1757 51.6705 46.6458 38.8411
10Y 22.3368 33.7733 39.4565 46.6458 47.5872 40.7433
30Y 17.8525 27.6243 31.9805 38.8411 40.7433 39.7206
corr(∆y)
3M 1Y 2Y 5Y 10Y 30Y
3M 1.0000 0.7224 0.5731 0.4687 0.4018 0.3515
1Y 0.7224 1.0000 0.8790 0.7818 0.7044 0.6306
2Y 0.5731 0.8790 1.0000 0.9029 0.8217 0.7290
5Y 0.4687 0.7818 0.9029 1.0000 0.9407 0.8574
10Y 0.4018 0.7044 0.8217 0.9407 1.0000 0.9371
30Y 0.3515 0.6306 0.7290 0.8574 0.9371 1.0000
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Principal Component Analysis

Eigenvalues E
E PC1 PC2 PC3 PC4 PC5 PC6
E (bps2 ) 226.99 50.14 13.77 5.45 2.86 1.47
E/sum(E) (%) 75.49 16.68 4.58 1.81 0.95 0.49

Eigenvectors D
D PC1 PC2 PC3 PC4 PC5 PC6
3M 0.3630 -0.8017 0.4347 0.1876 -0.0365 0.0006
1Y 0.4182 -0.2371 -0.4682 -0.6806 0.2939 0.0016
2Y 0.4351 0.0257 -0.5134 0.3309 -0.6505 0.1176
5Y 0.4513 0.2493 -0.0709 0.4572 0.5076 -0.5124
10Y 0.4176 0.3430 0.2837 0.0418 0.2271 0.7577
30Y 0.3550 0.3472 0.4926 -0.4258 -0.4242 -0.3866

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The First Three PCAs

The First Three PCAs

0.5

Eigenvector or Weight

-0.5

PC1
PC2
PC3

-1
0 5 10 15 20 25 30
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Level, Slope, and Curvature

They name the first factor the level factor as it has a similar impact of the yield
curve as a parallel shift of the yield curve.
They call the second factor steepness, since shock from this factor lowers the yields
of zeros up to five years, and raises the yields for zeros of longer maturities.
They call the third factor curvature, since it increases curvature of the yield curve in
the range of maturities below twenty years.

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The Relative Importance of the Three Factors:

The first factor is by far the most important, supporting the idea that “first factor” hedging –
or its close cousin, duration hedging – takes care of most of the return risk.
Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 15 / 22
 ∆yt = a + β P C1 P C1t + β P C2 P C2t + β P C3 P C3t + ϵt .
PC1 PC2 PC3 PC1 PC2 PC3 Total
β β β R2 (%) R2 (%) R2 (%) R2 (%)
3M 0.3630 -0.8017 0.4347 46.06 49.63 4.01 99.70
1Y 0.4182 -0.2371 -0.4682 82.18 5.83 6.25 94.26
2Y 0.4351 0.0257 -0.5134 88.67 0.07 7.49 96.23
5Y 0.4513 0.2493 -0.0709 89.46 6.03 0.13 95.62
10Y 0.4176 0.3430 0.2837 83.17 12.39 2.33 97.89
30Y 0.3550 0.3472 0.4926 72.04 15.22 8.41 95.66

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 16 / 22
Daily Currency Returns from 2000 through 2016

std (%) GBP EUR AUD CAD CNY INR JPY CHF THB RUB
0.60 0.63 0.82 0.59 0.10 0.38 0.64 0.70 0.36 0.77
corr (%) GBP EUR AUD CAD CNY INR JPY CHF THB RUB
GBP 64.4 53.1 45.3 12.9 25.2 12.3 49.7 18.6 23.9
EUR 64.4 55.3 46.0 10.4 23.3 27.5 78.2 21.1 25.7
AUD 53.1 55.3 62.3 12.8 33.9 2.9 39.2 23.4 32.8
CAD 45.3 46.0 62.3 9.4 27.0 1.0 32.7 19.9 33.3
CNY 12.9 10.4 12.8 9.4 16.8 5.6 8.8 16.1 10.7
INR 25.2 23.3 33.9 27.0 16.8 -4.0 15.1 25.5 26.6
JPY 12.3 27.5 2.9 1.0 5.6 -4.0 37.5 17.6 -1.1
CHF 49.7 78.2 39.2 32.7 8.8 15.1 37.5 17.7 19.3
THB 18.6 21.1 23.4 19.9 16.1 25.5 17.6 17.7 15.9
RUB 23.9 25.7 32.8 33.3 10.7 26.6 -1.1 19.3 15.9

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The First Three PCs

The first three PC’s of currency returns

PC 1 PC 2 PC3
E/sum(E) 44.32% 17.07% 12.27%
GBP 0.3502 0.0715 0.1382
EUR 0.4191 0.2531 0.0037
AUD 0.5410 -0.2512 0.4398
CAD 0.3259 -0.1870 0.1890
CNY 0.0122 -0.0032 -0.0046
INR 0.1154 -0.1104 0.0113
JPY 0.1226 0.5569 -0.3908
CHF 0.4100 0.4488 -0.1497
THB 0.0882 0.0084 -0.0390
RUB 0.3138 -0.5558 -0.7582

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The First Three PCs

The first three PC’s of currency returns

PC 1 PC 2 PC3
E/sum(E) 54.03% 20.25% 10.10%
GBP 0.3748 -0.0312 0.2595
EUR 0.4538 0.1878 0.2925
AUD 0.5621 -0.4869 -0.4163
CAD 0.3324 -0.2946 -0.1879
JPY 0.1499 0.6699 -0.7030
CHF 0.4524 0.4371 0.3799

Financial Markets, Fall 2020, SAIF Class 15: The Yield Curve Jun Pan 19 / 22
 Fama-French 25 Portfolios
PC 1 PC 2 PC3
E/sum(E) 83.84% 4.39% 3.19%
A1 0.2874 -0.5850 -0.6434
A2 0.2456 -0.2909 0.4001
A3 0.2378 -0.1526 0.0693
A4 0.2225 -0.1552 0.1332
A5 0.2468 -0.1586 0.0062
B1 0.2048 -0.1685 0.2221
B2 0.2013 -0.0520 0.2160
B3 0.1981 -0.0093 0.1173
B4 0.2033 0.0110 0.0517
B5 0.2353 -0.0044 -0.0768
C1 0.1938 -0.0751 0.1897
C2 0.1732 0.0673 0.1030
C3 0.1765 0.1135 0.0537
C4 0.1873 0.1154 0.0207
C5 0.2283 0.1122 -0.1330
D1 0.1555 0.0565 0.1347
D2 0.1606 0.1471 0.0868
D3 0.1707 0.1824 0.0360
D4 0.1810 0.1743 -0.0947
D5 0.2289 0.2343 -0.1251
E1 0.1264 0.1461 0.0572
E2 0.1298 0.2066 0.0048
E3 0.1380 0.2427 -0.0676
E4 0.1662 0.2704 -0.1142
E5 0.2050 0.2846 -0.3807
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 Momentum 25 Portfolios
PC 1 PC 2 PC3
E/sum(E) 82.40% 5.53% 4.06%
A1 0.2791 0.0183 0.4184
A2 0.2498 -0.0775 0.3119
A3 0.2311 -0.1137 0.2692
A4 0.2267 -0.2113 0.1814
A5 0.2150 -0.3085 0.0656
B1 0.2629 0.1010 0.1670
B2 0.2212 -0.0021 0.1058
B3 0.1945 -0.0576 0.0161
B4 0.1906 -0.1614 -0.0098
B5 0.1963 -0.2603 -0.1152
C1 0.2480 0.2102 0.0695
C2 0.2057 0.0630 -0.0203
C3 0.1844 -0.0147 -0.0639
C4 0.1633 -0.0974 -0.1426
C5 0.1654 -0.2323 -0.2459
D1 0.2399 0.2932 -0.0217
D2 0.1898 0.1263 -0.1283
D3 0.1665 0.0267 -0.1472
D4 0.1569 -0.0562 -0.2211
D5 0.1499 -0.2015 -0.3146
E1 0.2049 0.6397 -0.1310
E2 0.1573 0.1926 -0.1621
E3 0.1430 0.0898 -0.2025
E4 0.1228 -0.0192 -0.2724
E5 0.1243 -0.1436 -0.3605
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Main Takeaways

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