Equity Duration as a Risk Factor in the Norwegian Stock Market

Dato: 02.06.2020 Totalt antall sider: 78

0

Emnekode: BE305E

Navn: Sigurd Bakke Jensen & Gjermund Odden Bergerengen

Equity Duration as a Risk Factor in the

Norwegian Stock Market

E(Rp) =^!

i=1

w_i·E(Ri)

E(Rp) w_i E(Ri)

σ²_p =^!n

i=1

!n j=1

w_iw_jCov(rir_j)

σ_p² Cov(rir_j)

E(r) σ

E(R_p) =r_f+β_i∗[E(r_m)−r_f]

E(R_p) r_f

[E(r_m)−r_f] β_i

(rm−r_f)

(rf)

SM L

β r

i

i

i

E(Ri,t+1−R_f,t+1) =γCov(Ri,t+1, RM_t+1) +γ_zCov(Ri,t+1,z˜_t+1)

R_f,t E(Ri,t+1) R_t+1

RM_t+1 z_t+1 k

Cov(R_i,t+1, RM_t+1) γ_z R_t+1 RM_t+1

˜

z z γ_z

γ_z ≡ −Jw_z(W, z, t) Jw(W, z, t)

Jw Jwz

γ_z

Jw Jw_z

z

ra=rf +β₁f₁+β₂f₂+...+βnfn

r_a r_f n

β_n f_n

β_n

R_it=α_i+β_iMR_{M t}+β_{iSM B}SM B_t +β_{iHM L}HM L_t+e_it

R_i,t t α_i SM B

HM L

W_i = q_i·p_i

q₁·p₁+q₂·p₂+...+q_n·p_n W_i= M CAP_i

"_n

j=1M CAP_j

M CAPi i ^"n_j=1

q_i i p_i i

D_t^k ≡ −∂P_t^k/P_t^k

∂d^k_t/d^k_t

D^k_t k t

P_t^k k t CF_i^k

P_t^k = ^!∞

i=t+1

E_t(CF_i^k)/d(d^k_t)^i−t

∆P_t^k

P_t^k −∂P_t^k

∂t · ∆t

P_t^k =β_{M CF}^k ·M CF_t+F CF_t^k−D^k_t ·∆d^k_t d^k_t

β_{M CF}^k k

∆d^k_t d^k_t

∂P_t^k

∂d^k_t = ^!∞

i=t+1

E_t(CF_t^k)·(d^k_t)^t−i−1·(t−i)·∂E_t(CF_i^k)

∂d^k_t ·(d^k_t)^t−i

D^k_t = ^!∞

i=t+1

E_t(CF_i^k)·(d^k_t)^t−i·(i−t) P_t^k − ^!∞

i=t+1

∂E_t(CF i^k)

∂d^k_t · (d^k_t)^t−i+1 P_t^k

t

r_i,t−r_f =α_i+β_i(rm,t−r_f,t)εi,t

r_i,t−r_f,t β_i

r_m,t r_f,t

T_p = rp−rf

β_p

r_p r_f β_p

S_p = r_p−r_f σ_p

S_p r_p r_f σ_p

IR= r_p−r_m

σ_p−m → α_p σ_p−m

r_p−r_m σ_p−m

β F_j,t

m n

R_1,t= α₁+β_1,F1F_1,t+β_1,F2F_2,t+· · ·+β_{1,F m}F_m,t+ε_1,t R_2,t= α₂+β_2,F1F_1,t+β_2,F2F_2,t+· · ·+β_{2,F m}F_m,t+ε_2,t

R_n,t =α_n+β_n,F1F_n,t+β_{n,F n}F_n,t+· · ·+β_{n,F m}F_m,t+ε_n,t

R_i,t i t n

β j m F_j,t t

T

R_i,1=γ_1,0+γ_1,1βˆ_i,F1+γ_1,2βˆ_i,F2+· · ·+γ_1,mβˆ_{i,F m}+ε_i,1 R_i,2=γ_2,0+γ_2,1βˆ_i,F1+γ_2,2βˆ_i,F2+· · ·+γ_2,mβˆ_{i,F m}+ε_i,2

R_i,T =γ_T,0+γ_n,1βˆ_i,F1+γ_n,2βˆ_i,F2+· · ·+γ_n,mβˆ_{i,F m}+ε_i,n

R_i,T γ

λˆ= 1 T

!

t=1

Tλˆ_t εˆ_i = 1 T

!T t=1

ˆ ε_i,t

Y_t

Y_t−p Y

Y_t Y_t−1 p^th Y_t−p

Y_t−1 Y_t

Y_t =β₀+β₁Y_t−1+u_t

β u_t

p^th p

p

Y_t p

Y_t =β₀+β₁Y_t−1+β₂Y_t−2+· · ·+β_pY_t−p+u_t

t

Y

p

q p, q

Y_t =β₀+β₁Y_t−1+· · ·+β_pY_t−p+u_t+θ₁u_t−1· · ·+θ_qu_t−q

β θ

p, d, q

k k

p

p Y_t X_t p

Y_t =β₁₀+β₁₁Y_t−1+· · ·+β_1pY_t−p+γ₁₁X_t−1+· · ·+γ_1pX_t−p+u_ut X_t =β₂₀+β₂₁Y_t−1+· · ·+β_2pY_t−p+γ₂₁X_t−1+· · ·+γ_2pX_t−p+u_2t

β γ u_1t u_2t

u_1t u_2t Y X

E(u_t |Y_t−1, Y_t−2, ..., X_1t−1, X_1t−2, ..., X_kt−1, X_kt−2, ...) = 0 Y X Y_t, X_1t, ..., X_kt Y_t−j, X_1t−j, ..., X_kt−j

j

X_1t X_kt Y_i

t t+1

D(BE) =D(A)· A

BE −D(L)· L BE

D(A) =D(CS)· CS

A +D(CA)·CA

A +D(F A)· F A A

D(F A) = F A

Depreciation and Amortization· N et P P E Gross P P E

D(L) =D(CL)·CL

L +D(LL)·LL L

R_i,T =α_i+β_iM KT_i,t+s_iSM B_i,t+h_iHM L_i,t+m_iM OM_i,t+ε_i,t

i,t α _i,t

ε

R_i,T =α_i+β_iT ERM! _i,t+s_iRF^#_i,t+h_iOIL"_i,t+ε_i,t

i,t α

ε

n

R_i,T = α_i+β_iM KT_i,t+s_iSM B_i,t+h_iHM L_i,t+m_iM OM_i,t+l_iLDM HD_i,t+ε_i,t

β_i s_i h_i m_i l_i

T

R_i,1=γ_1,0+γ_1,1βˆ_i,F1+γ_1,2βˆ_i,F2+· · ·+γ_1,mβˆ_{i,F m}+ε_i,1 R_i,2=γ_2,0+γ_2,1βˆ_i,F1+γ_2,2βˆ_i,F2+· · ·+γ_2,mβˆ_{i,F m}+ε_i,2

R_i,T =γ_T,0+γ_n,1βˆ_i,F1+γ_n,2βˆ_i,F2+· · ·+γ_n,mβˆ_{i,F m}+ε_i,n

R_i,T β

γ ε

λˆ= 1 T

!

t=1

Tλˆ_t εˆ_i = 1 T

!T

t=1εˆ_i,t σˆ_j =

$%

%&1 T

!T

t=1(ˆλ_,t−ˆλ_j)² t_λj =√ Tλˆ_j

ˆ σ_j

t t+1

D(BE) =D(A)· A

BE −D(L)· L BE

D(A) =D(CS)· CS

A +D(CA)·CA

A +D(F A)· F A A

D(F A) = F A

Depreciation and Amortization· N et P P E Gross P P E

D(L) =D(CL)·CL

L +D(LL)·LL L

R_i,T =α_i+β_iM KT_i,t+s_iSM B_i,t+h_iHM L_i,t+m_iM OM_i,t+ε_i,t

i,t α _i,t

ε

R_i,T =α_i+β_iT ERM! _i,t+s_iRF^#_i,t+h_iOIL"_i,t+ε_i,t

i,t α

ε

n

R_i,T =α_i+β_iM KT_i,t+s_iSM B_i,t+h_iHM L_i,t+m_iM OM_i,t+l_iLDM HD_i,t+ε_i,t

β_i s_i h_i m_i l_i

T

R_i,1=γ_1,0+γ_1,1βˆ_i,F1+γ_1,2βˆ_i,F2+· · ·+γ_1,mβˆ_{i,F m}+ε_i,1 R_i,2=γ_2,0+γ_2,1βˆ_i,F1+γ_2,2βˆ_i,F2+· · ·+γ_2,mβˆ_{i,F m}+ε_i,2

R_i,T =γ_T,0+γ_n,1βˆ_i,F1+γ_n,2βˆ_i,F2+· · ·+γ_n,mβˆ_{i,F m}+ε_i,n

R_i,T β

γ ε

λˆ= 1 T

!

t=1

Tλˆ_t εˆ_i = 1 T

!T

t=1εˆ_i,t σˆ_j =

$%

%&1 T

!T

t=1(ˆλ_,t−λˆ_j)² t_λj =√ Tλˆ_j

ˆ σ_j

α

t t+1

α

Return α β_{M KT} β_{SM B} β_{HM L} β_{M OM} D(BE) SharpeRatio

RF#

RF#

"

OIL

T ERM!

T ERM!

RF#

"

OIL

R²

T ERM! RF^#

RF#

T ERM!

"

OIL

RF# T ERM!

! T ERM

RF#

"

OIL

R²

Return α β_{M KT} β_{SM B} β_{HM L} β_{M OM} D(BE) SharpeRatio

Return α β_{M KT} β_{SM B} β_{HM L} β_{M OM} D(BE) SharpeRatio

Consumer.discretionary Consumer.staples Energy Health.care Industrials Information.technology Materials Telecom

Low−Duration Sector Distribution

0.0 0.1 0.2 0.3 0.4

Consumer.discretionary Consumer.staples Energy Health.care Industrials Information.technology Materials Utilities

High−Duration Sector Distribution

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Jul 2003

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Jun 2018

Oil price development 2003−07−31 / 2018−06−30

40 60 80 100 120

Jul 2003

Jan 2005

Jul 2006

Jan 2008

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Jan 2011

Jul 2012

Jan 2014

Jul 2015

Jan 2017

Jun 2018

Risk−free rate development 2003−07−31 / 2018−06−30

0.01 0.02 0.03 0.04 0.05 0.06