The Exercise of Options Embedded in Norwegian Student Loans

Transcription

1 The Exercise of Options Embedded in Frode Brevik and Axel Kind University of St. Gallen Swiss Society of Economics and Statistics 2007 Annual Meeting, St. Gallen March 23, 2007 Brevik and Kind (2007) - 1

2 Brevik and Kind (2007) - 2

3 Outline Norwegian State Educational Loan Fund (SL) Loan Provisions Option s Inner Value The Value of Early Exercise Interest Rate Results Observed vs. Optimal Option Exercise Behavior Heuristics Explaining Observed Exercise Behavior Costs of Suboptimal Exercise Brevik and Kind (2007) - 3

4 Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Brevik and Kind (2007) - 4

5 Norwegian State Educational Loan Fund Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Inception in 1947 Total customers: (support phase), 6% of Norwegian population (repayment phase), 10% of Norwegian population Students graduating in 2004: - average debt NOK (USD ) Sum of nominals per year (Mio real NOK) Number of new loans per year (1000) Brevik and Kind (2007) - 5

6 Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Monthly base loan of NOK (ca. USD 1 210) Various other grants available for special needs (transportation costs, parenthood, illness, study fees) tuition-fee loan: NOK (ca. USD 3 160) p.a. travel grant: NOK (ca. USD 1 060) p.a. Upon graduation 40 percent of this is converted to a grant are interest free for the duration of the studies Insurance provisions Brevik and Kind (2007) - 6

7 Repayment Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Quarterly interest payments Default: floating interest rates Option to commit for three years to fixed interest rate payments Bermudan-style option: exercise four times per year Multiple options: option renewed after three-year fixed interest period Brevik and Kind (2007) - 7

8 Time Line of Exercise Dates Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Support Period Quarterly { exercise dates }} { t 0 t 1 t 2 t n 1 t n t n+n Start of payback period Last exercise date Maturity of SL loan Brevik and Kind (2007) - 8

9 Costs of Exercising the Option Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Just two mouse clicks... Forside Personlige opplysninger Status søknader Omgjøring etter eksamen Samlet gjeld Kontobevegelser Neste terminbeløp Beløp til selvangivelsen Søke om betalingsutsettelse Søke om fast rente Bestillinger BREVIK FRODE Kundenummer: Søke om fast rente Ved å inngå en fastrenteavtale, binder du deg til å ha samme rentesats i tre eller fem år. Avtalen kan ikke sies opp. Du kan inngå fastrenteavtale fra 1. januar, 1. april, 1. juli eller 1. oktober. Søknadsfristen er den 10. i den måneden du søker fastrente fra. Sjekk rentesatser her. Fast rente Fristen for å inngå avtale om fast rente fra og med 01. januar 2006 er nå gått ut. Neste tilbud om fastrenteavtale i Lånekassen er klart i midten av mars Lånekassen kan ennå ikke oppgi rentesatsene for denne avtalen. Du kan benytte Kassafonen, telefon eller for å få opplysning om rentesatser og for å bestille avtale om fast rente. Once logged in, two mouse clicks are enough for exercising the option Brevik and Kind (2007) - 9

10 SL Swaps vs. Traditional Swaps Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line 1. Floating rate: averages of lagged three-month yields on treasury bills from second to last quarter bp 2. Fixed rate: averages of lagged three-year interest rates from the mid month of the last quarter bp 3. Decision lag: 10 days retroactive option exercise 4. Annuity loan: notional underlying changes over time Brevik and Kind (2007) - 10

11 Time Line Norwegian State Educational Loan Fund Repayment Time Line of Exercise Dates Costs of Exercising the Option SL Swaps vs. Traditional Swaps Time Line Q3 Jul 1, 99 Q2 Apr 1, 99 Q1 Jan 1, 99 Jul 10, 99 Exercise date for committing to fixed interest rate payments in Q3. Fixed interest rate: Daily average of three-year government bond yields in the mid month of the last quarter +100 bp. Flexible interest rate: Daily average of short-term government bond yields in the second to last quarter +100 bp. Brevik and Kind (2007) - 11

12 Option Inner Value - Value of SL Swap Pricing Bermudan SL Swaptions Interest Rate and Calibration Interest Rate Data Brevik and Kind (2007) - 12

13 Option Inner Value - Value of SL Swap Option Inner Value - Value of SL Swap Pricing Bermudan SL Swaptions Interest Rate and Calibration Interest Rate Data V t = where, B(t i, t j ) = exp N k=t+1 ) B t,t+j L t+j (Ẽt [r t+j ] R t /4, (1) ( ) j 1 k=i f(t i, t k ) h Historical evolution of the SL swap Brevik and Kind (2007) - 13

14 Pricing Bermudan SL Swaptions Option Inner Value - Value of SL Swap Pricing Bermudan SL Swaptions Interest Rate and Calibration Interest Rate Data Dynamic programming formulation Terminal Condition: At any other time: W(t n ) = Max(V (t n ); 0) (2) W(t i ) = Max V (t i )+B(t i, t i+n )Ẽt i [W(t i+n )];B(t i, t i+1 )Ẽt } {{ } i [W(t i+1 )] Multiple Exercise (3) where, for i > n, W(t i ) = 0 and ( B(t i, t j ) = exp ) j 1 k=i f(t i, t k ) h Brevik and Kind (2007) - 14

15 Interest Rate and Calibration Option Inner Value - Value of SL Swap Pricing Bermudan SL Swaptions Interest Rate and Calibration Interest Rate Data Interest Rate Gaussian three-factor HJM model. Calibration: Principal Component Analysis (PCA) on historical variances of changes in forward rates. Parametric function: fit each of the three eigenvectors with the following parametric form: σ k (t i,t j ) = [α k + β k (t j t i )]e γ k(t j t i ) + δ k (4) Brevik and Kind (2007) - 15

16 Interest Rate Data Option Inner Value - Value of SL Swap Pricing Bermudan SL Swaptions Interest Rate and Calibration Interest Rate Data Daily yields on treasury bills and bonds ranging back to 1996 obtained from the Norwegian central bank Cross check: Norwegian Interbank Offered Rates (NIBOR) for maturities shorter than one year and swaprates for longer maturities Interest Rate (in %) Months 1 Year 3 Years 10 Years Time Brevik and Kind (2007) - 16

17 Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Brevik and Kind (2007) - 17

18 Data on Preliminary Results Data on Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Data on the exercise behavior of ca. half a million students between July 1, 1999, and October 1, economically relevant decisions (of almost half a million individuals) 9 maturity classes at 22 exercise dates. Brevik and Kind (2007) - 18

19 Optimal Exercise Boundary Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Longstaff and Schwartz (2001) Andersen (2000) Inner value of swap (in CHF) Exercise boundary (in % of notional) Exercise boundary (in % of notional) 18Y 15Y 12Y 9Y 6Y Time until final repayment Inner value of swap (in CHF) Y 15Y 12Y 9Y 6Y Time until final repayment Brevik and Kind (2007) - 19

20 Optimal vs. Observed Exercise Behavior Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II 1. Total exercise decisions = (6.03%) 2. Two types of sub-optimal behavior: (a) not exercising when exercising is optimal: (b) exercising when non-exercising is optimal: 3. Results: (a) 7.91% out of non-exercise decisions were suboptimal (b) 40.84% out of exercise decisions were suboptimal (c) 11% out of exercise occurred with negative values of the SL swap Brevik and Kind (2007) - 20

21 What affects the Exercise Behavior? Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II 1. Difference between the SL fix rate and the 3-year market rate (-) 2. Difference between the SL float rate and the 3-month market rate (+) 3. Difference between the SL fix rate and the SL float rate (-) 4. Maturity (-) 5. Media (+), Dine Penger, Verdens Gang 6. Exercise value (++) Brevik and Kind (2007) - 21

22 Univariate with P = log P 1 P P i,t = c + Σβ kx (k) i,t + ǫ i,t SL Fix SL Float SL Fix Media: Exercise Regression Constant minus minus minus Maturity Value Adj. R 2 BIC Market (3y) Market (3m) SL Float (-11.07) (-1.19) (-29.76) ( 9.50) (-42.91) (-12.16) (-17.68) ( 4.28) (-42.09) ( 4.58) (12.45) (-32.30) ( 3.66) Brevik and Kind (2007) - 22

23 Multivariate Continued... SL Fix SL Float SL Fix Media: Exercise Regression Constant minus minus minus Maturity Value Adj. R 2 BIC Market (3y) Market (3m) SL Float (-16.63) (-8.90) (14.23) (-18.16) (-9.78) (15.63) ( 6.43) (-18.08) (-2.50) ( 6.79) ( 2.69) ( 6.03) (-10.09) (-1.95) (12.46) ( 2.49) (-25.63) (-13.15) ( 5.87) (-45.85) (-6.02) ( 3.74) ( 6.50) (-42.80) (-11.20) (-0.48) (-29.77) (-6.66) ( 6.66) ( 4.13) ( 7.18) (-30.61) (-7.25) ( 6.87) ( 4.54) ( 7.96) (-3.16) (-32.26) ( 7.13) ( 7.13) ( 3.01) ( 7.07) ( 3.55) (-15.07) (-0.04) ( 7.05) ( 7.11) ( 2.95) ( 6.58) ( 2.14) Brevik and Kind (2007) - 23

24 The Costs of Suboptimal Exercise Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Value of SL option in percent of notional using as exercise rule: the optimal L&S(2001): 5.49% regression model (8): 3.53% regression model (11): 3.32% Brevik and Kind (2007) - 24

25 Can Expert-Advice Help? Part I Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Q1:2003: Major Norwegian newspaper Dagbladet ran an article with the header Lock in your interest rate now (Bind renta naa; Expert view quoted in the article: 6.9 percent is a good interest rate, because the floating interest rate is 8.1 percent, and, while it is predicted to fall a bit, it will not go down much. For you to loose on this it would have to go down to 5.4 percent, but it will not go down so far Inner value given an initial loan of 40,000 USD Offical student loan interest rates Floating rate fixed rate Value lags/averaging in floating rate lags/averaging in fixed rate remainder Q1:2000 Q1:2001 Q1:2002 Q1:2003 Q1:2004 Q1: Q1:2003 Q3:2003 Q1:2004 Q3:2004 Q1:2005 Q3:2005 Q1:2006 Brevik and Kind (2007) - 25

26 Good Advice? Part II Preliminary Results Data on Norwegian Student Optimal Exercise Boundary Optimal vs. Observed Exercise Behavior What affects the Exercise Behavior? Univariate Multivariate The Costs of Suboptimal Exercise Can Expert-Advice Help? Part I Good Advice? Part II Q3:2003: Largest online newspaper runs story with title Lock in your interest rate Ex ante Inner value given an initial loan of 40,000 USD Value lags/averaging in floating rate lags/averaging in fixed rate remainder Q1:2000 Q1:2001 Q1:2002 Q1:2003 Q1:2004 Q1: Ex post Offical student loan interest rates Floating rate fixed rate Q1:2003 Q3:2003 Q1:2004 Q3:2004 Q1:2005 Q3:2005 Q1:2006 Brevik and Kind (2007) - 26