MATLAB code empirical analysis (table 1, table 2 and figure 6)

1. Import data and set risk-free rate of return

R_M=xlsread('Data_Empirical Analysis',1,'B3:BV235'); % 73 mutual fund returns
r=0.001; % risk-free rate of return

2. Find five best and five worst performing mutual funds according to the Sharpe ratio

mean=mean(R_M);
vola=std(R_M);
SR=(mean-r)./vola; % Sharpe ratio SR
SR_rank=tiedrank(-SR) % SR ranking in descending order

3. Import data of the 10 mutual funds and market index

clear
R_M(:,1)=xlsread('Data_Empirical Analysis',2,'B3:B235'); % S&P500
R_M(:,2)=xlsread('Data_Empirical Analysis',1,'V3:V235'); % Franklin Growth Series Class A
R_M(:,3)=xlsread('Data_Empirical Analysis',1,'BJ3:BJ235'); % T. Rowe Price Blue Chip Growth Fund
R_M(:,4)=xlsread('Data_Empirical Analysis',1,'U3:U235'); % Franklin DynaTech Fund Class A
R_M(:,5)=xlsread('Data_Empirical Analysis',1,'H3:H235'); % Brown Capital Management Small Company Fund
R_M(:,6)=xlsread('Data_Empirical Analysis',1,'BA3:BA235'); % T. Rowe Price Institutional Mid-Cap Equity
R_M(:,7)=xlsread('Data_Empirical Analysis',1,'AR3:AR235'); % BMO Large-Cap Growth Fund Class Y
R_M(:,8)=xlsread('Data_Empirical Analysis',1,'K3:K235'); % Commerce Growth Fund
R_M(:,9)=xlsread('Data_Empirical Analysis',1,'AV3:AV235'); % MainStay Large Cap Growth Fund Class A
R_M(:,10)=xlsread('Data_Empirical Analysis',1,'B3:B235'); % AB Growth Fund Class B
R_M(:,11)=xlsread('Data_Empirical Analysis',1,'AA3:AA235'); % Goldman Sachs Capital Growth Fund Class

4. Set parameters, range of x and compute c and lower partial moment with target c

n=2; % lower partial moment order
t=0.002; % target t
r=0.001;
x=[-2:0.00001:3];

for i=1:11
    R_P.mean(:,i)=mean(x.*R_M(:,i)+(1-x)*r); % mean of portfolio return R_P
    R_P.LPM(:,i)=nthroot(lpm(x.*R_M(:,i)+(1-x)*r,t,n),n); % n-th root of lower partial moment 
    for portfolio return R_P [LPM_n,t(R_P)]^(1/n)
    x_MLPM.ind(i)=find(R_P.LPM(:,i)==min(R_P.LPM(:,i)),1); % find index number of MLPM 
    portfolios
    x_MLPM.value(i)=x(x_MLPM.ind(i)); % retrieve value of x at the MLPM portfolio
    c(i)=(t-(1-x_MLPM.value(i))*r)/(x_MLPM.value(i)); % compute c
    LPM2c(i)=nthroot(lpm(R_M(:,i),c(i),2),2) % compute lower partial moment with target c
end

5. Return and downside risk of the 10 mutual funds and market index

mean=mean(R_M)
vola=std(R_M)

LPM2t=nthroot(lpm(R_M(:,1:11),t,2),2) % lower partial moment of order 2 with target t
MLPM_slope=(mean-r)./LPM2c % slope of MLPM portfolios

6. Plot portfolio lines of the 10 mutual funds and market index

figure('position', [200 0 1600 850])
plot(R_P.LPM(:,1),R_P.mean(:,1),'color','k','LineWidth',1) % plot portfolio line of the index
hold on
plot(t-r,r,'.k','MarkerSize',16)
text(t-r-0.0010,r-0.0002,'$r$','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(LPM2t(1),mean(1),'.k','MarkerSize',12)
text(LPM2t(1)+0.00015,mean(1)-0.0002,'S\&P500','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,2),R_P.mean(:,2),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 1
hold on
plot(LPM2t(2),mean(2),'.k','MarkerSize',12)
text(LPM2t(2)+0.00015,mean(2)-0.0002,'1','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,3),R_P.mean(:,3),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 2
hold on
plot(LPM2t(3),mean(3),'.k','MarkerSize',12)
text(LPM2t(3)+0.00015,mean(3)-0.0002,'2','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,4),R_P.mean(:,4),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 3
hold on
plot(LPM2t(4),mean(4),'.k','MarkerSize',12)
text(LPM2t(4)+0.00015,mean(4)-0.0002,'3','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,5),R_P.mean(:,5),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 4
hold on
plot(LPM2t(5),mean(5),'.k','MarkerSize',12)
text(LPM2t(5)+0.00015,mean(5)-0.00020,'4','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,6),R_P.mean(:,6),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 5
hold on
plot(LPM2t(6),mean(6),'.k','MarkerSize',12)
text(LPM2t(6)+0.00015,mean(6)-0.0002,'5','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,7),R_P.mean(:,7),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 69
hold on
plot(LPM2t(7),mean(7),'.k','MarkerSize',12)
text(LPM2t(7)+0.00015,mean(7)-0.0003,'6','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,8),R_P.mean(:,8),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 70
hold on
plot(LPM2t(8),mean(8),'.k','MarkerSize',12)
text(LPM2t(8)+0.00015,mean(8)+0.00015,'7','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,9),R_P.mean(:,9),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 71
hold on
plot(LPM2t(9),mean(9),'.k','MarkerSize',12)
text(LPM2t(9)+0.00015,mean(9)-0.0002,'8','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,10),R_P.mean(:,10),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 72
hold on
plot(LPM2t(10),mean(10),'.k','MarkerSize',12)
text(LPM2t(10)+0.00015,mean(10)-0.0003,'9','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
hold on
plot(R_P.LPM(:,11),R_P.mean(:,11),'color','k','LineWidth',1) % plot portfolio line of mutual fund with SR rank 73
hold on
plot(LPM2t(11),mean(11),'.k','MarkerSize',12)
text(LPM2t(11)+0.00015,mean(11)+0.00015,'10','Interpreter','latex','FontSize',20,'FontName','Times New Roman')
set(gca,'xLim',[0 0.05],'ylim',[-0.002 0.009], 'FontSize', 28,'FontName','Times New Roman','xcolor','k')
ax = gca; ax.XAxis.Exponent = 0; ax.YAxis.Exponent = 0; % prevent scientific format
xtickformat('%.2f') % format with 2 decimal places
ytickformat('%.3f')
xticks([0.01 0.02 0.03 0.04 0.05]) % selected ticks on x axes
yticks([0.000 0.002 0.004 0.006 0.008])
xlabel('$$\sqrt[2]{\textrm{LPM}_{2,\tau}(R)}$$','Interpreter','latex','FontSize',28,'FontName','Times New Roman')
ylabel('$$\textrm{E}(R)$$','Interpreter','latex','FontSize',28,'FontName','Times New Roman')
box off % prevent box format

7. Import data 73 mutual funds and set parameters

clear
R_M=xlsread('Data_Empirical Analysis',1,'B3:BV235');
n=2;
t=0.002;
r=0.001;
x=[-0.50:0.00001:1];

8. Compute c, lower partial moment with target c and Omega ratio

for i=1:73
    R_P.LPM(:,i)=nthroot(lpm(x.*R_M(:,i)+(1-x)*r,t,n),n); % n-th root of lower partial moment
    for portfolio return R_P [LPM_n,t(R_P)]^(1/n)
    x_MLPM.ind(i)=find(R_P.LPM(:,i)==min(R_P.LPM(:,i)),1); % find index number of MLPM
    portfolios
    x_MLPM.value(i)=x(x_MLPM.ind(i)); % retrieve value of x at the MLPM portfolio
    c(i)=(t-(1-x_MLPM.value(i))*r)/(x_MLPM.value(i)); % compute c
    LPM2c(i)=nthroot(lpm(R_M(:,i),c(i),2),2); % compute lower partial moment with target c
    OM1t(i)=lpm(-R_M(:,i), -t, 1) / lpm(R_M(:,i), t, 1) % compute Omega ratio
end

9. Rank correlation of downside performance measures

mean=mean(R_M);
vola=std(R_M);
SR=(mean-r)./vola;

LPM2t=nthroot(lpm(R_M,t,2),2);
KR2t=(mean-t)./LPM2t; % Kappa ratio of order 2 with target t

LPM2r=nthroot(lpm(R_M,r,2),2);
KR2r=(mean-r)./LPM2r; % Kappa ratio of order 2 with target r

UPM2r=-r+mean+LPM2r;
OM2r=nthroot(UPM2r./LPM2r,2); % generalalized Omega ratio of order 2 with target r
MLPM_slope=(mean-r)./LPM2c; % slope of MLPM portfolios

SR_rank=tiedrank(SR);
KR2t_rank=tiedrank(KR2t);
KR2r_rank=tiedrank(KR2r);
OM1t_rank=tiedrank(OM1t);
OM2r_rank=tiedrank(OM2r);
MLPM_slope_rank=tiedrank(MLPM_slope);

RC=corr([SR_rank',KR2t_rank', KR2r_rank', OM1t_rank',OM2r_rank',MLPM_slope_rank'],'rows','pairwise') % rank correlations