# ret2price

Convert returns to prices

## Description

example

[Prices,ticks] = ret2price(Returns) returns the matrix of numVars price series Prices, and corresponding observation times ticks, computed from a matrix of continuously compounded return series Returns.

example

PriceTbl = ret2price(ReturnTbl) returns the table or timetable of price series PriceTbl computed from converting each variable in the table or timetable of continuously compounded return series ReturnTbl. To select different variables in Tbl from which to compute prices, use the DataVariables name-value argument.

example

[___] = ret2price(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. ret2price returns the output argument combination for the corresponding input arguments. For example, ret2price(Tbl,Method="periodic",DataVariables=1:5) computes prices from the simple periodic returns in the first five variables in the input table Tbl.

## Examples

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Load the Schwert Stock data st Data_SchwertStock.mat, which contains monthly returns of the NYSE index from 1871 through 2008 DataTableMth.Return, among other variables (enter Description for more details).

returns = DataTableMth.Return;
numObs = numel(returns)
numObs = 1656
dates = datetime(datesMth,ConvertFrom="datenum");

Set missing (NaN) returns to 0.

returns(isnan(returns)) = 0;

Convert the monthly NYSE returns to prices.

prices = ret2price(returns);

prices is a 1657-by-1 vector of monthly NYSE prices from the continuously compounded returns. ret2price sets the starting price to 1 by default; specify the StartPrice name-value argument to set an appropriate starting price.

r10 = returns(9)
r10 = 0.0114
p9_10 = [prices(9) prices(10)]
p9_10 = 1×2

1.1274    1.1403

returns(10) = 0.0114 is the monthly return of the prices in the interval [1.1274, 1.1403].

plot(dates,DataTableMth.Return)
ylabel("Return")
yyaxis right
plot([dates(1) - month(1); dates],prices)
ylabel("Price")
title("NYSE Index Prices and Returns")

Compute price series from return series, which are variables in a table. This example also shows how to convert prices in a timetable.

Load the Schwert Stock data st Data_SchwertStock.mat, which contains monthly returns of several series in the table DataTableMth. Replace all missing values (NaN) with 0.

DTM = fillmissing(DataTableMth,"constant",0);

Convert Returns in Table to Prices

Convert all return series in the table to prices.

PriceTbl = ret2price(DTM);
tail(PriceTbl)
Tick    Interval    Return    DivYld    CapGain    CapGainA
____    ________    ______    ______    _______    ________

1649       1        46004      16.9     5.3445      5.4712
1650       1        42215      16.9     5.3445      5.4712
1651       1        41801      16.9     5.3445      5.4712
1652       1        42313      16.9     5.3445      5.4712
1653       1        38592      16.9     5.3445      5.4712
1654       1        32615      16.9     5.3445      5.4712
1655       1        30263      16.9     5.3445      5.4712
1656       1        30501      16.9     5.3445      5.4712

Because DTM is a table, PriceTbl is a table. PriceTbl contains the price series computed from the continuously compounded returns (variable names match the input variable names), observation times Tick, and time intervals Interval.

You can choose a subset of variables to convert to prices by using the DataVariables name-value argument.

Convert Returns in Timetable to Prices

Convert the table DTM to a timetable.

dates = datetime(datesMth,ConvertFrom="datenum");
TT = table2timetable(DTM,RowTimes=dates);
TT.Dates = [];

When the input series is a timetable, ret2price requires a starting time, which is specified by the StartTime name-value argument.

Convert only the NYSE and capital gain returns to prices, and specify the following options:

• The starting time is exactly one month before the first observed return.

• The starting prices of the NYSE and capital gains returns are 100 and 10, respectively.

• The returns are periodic.

startTime = TT.Time(1) - month(1);
varnames = ["Return" "CapGain"];
startPrice = [100 10];
PriceTT = ret2price(TT,DataVariables=varnames,Method="periodic", ...
StartTime=startTime,StartPrice=startPrice);
Time        Interval    Return    CapGain
___________    ________    ______    _______

31-Dec-1870      NaN          100        10
01-Jan-1871        1       101.24    10.023
01-Feb-1871       31       164.61    14.374
01-Mar-1871       28       301.26    24.381
01-Apr-1871       31       619.29     46.79
01-May-1871       30       1148.5    79.862
01-Jun-1871       31       599.29    60.866
01-Jul-1871       30       443.99    27.923

Because TT is a timetable, PriceTT is a timetable. PriceTT contains only those requested price series. Because the input data is time aware, the time-related variables in PriceTT contain more information about the observation times than returned when the input data is a matrix or table. For example, rather than tick times, the output timetable contains observation dates for the prices and the interval is in days.

Create two stock price series from continuously compounded returns that have the following characteristics:

• Series 1 grows at a 10 percent rate at each observation time.

• Series 2 changes at a random uniform rate in the interval [-0.1, 0.1] at each observation time.

• Each series starts at price 100 and is 10 observations in length.

rng(1); % For reproducibility
numObs = 10;
p1 = 100;
r1 = 0.10;
r2 = [0; unifrnd(-0.10,0.10,numObs - 1,1)];

s1 = 100*exp(r1*(0:(numObs - 1))');
cr2 = cumsum(r2);
s2 = 100*exp(cr2);
S = [s1 s2];

Convert each price series to a return series, and return the observation intervals.

[R,intervals] = price2ret(S);

Prepend the return series so that the input and output elements are of the same length and correspond.

[[NaN; intervals] S [[NaN NaN]; R] r2]
ans = 10×6

NaN  100.0000  100.0000       NaN       NaN         0
1.0000  110.5171   98.3541    0.1000   -0.0166   -0.0166
1.0000  122.1403  102.7850    0.1000    0.0441    0.0441
1.0000  134.9859   93.0058    0.1000   -0.1000   -0.1000
1.0000  149.1825   89.4007    0.1000   -0.0395   -0.0395
1.0000  164.8721   83.3026    0.1000   -0.0706   -0.0706
1.0000  182.2119   76.7803    0.1000   -0.0815   -0.0815
1.0000  201.3753   72.1105    0.1000   -0.0627   -0.0627
1.0000  222.5541   69.9172    0.1000   -0.0309   -0.0309
1.0000  245.9603   68.4885    0.1000   -0.0206   -0.0206

price2ret returns rates matching the rates from the simulated series. price2ret assumes prices are recorded in a regular time base, therefore all durations between prices are 1.

Convert the prices to returns again, but associate the prices with years starting from August 1, 2010.

tau1 = datetime(2010,08,01);
dates = tau1 + years((0:(numObs-1))');
[Ry,intervalsy] = price2ret(S,Ticks=dates);
[[NaN; intervalsy] S [[NaN NaN]; Ry] r2]
ans = 10×6

NaN  100.0000  100.0000       NaN       NaN         0
365.2425  110.5171   98.3541    0.0003   -0.0000   -0.0166
365.2425  122.1403  102.7850    0.0003    0.0001    0.0441
365.2425  134.9859   93.0058    0.0003   -0.0003   -0.1000
365.2425  149.1825   89.4007    0.0003   -0.0001   -0.0395
365.2425  164.8721   83.3026    0.0003   -0.0002   -0.0706
365.2425  182.2119   76.7803    0.0003   -0.0002   -0.0815
365.2425  201.3753   72.1105    0.0003   -0.0002   -0.0627
365.2425  222.5541   69.9172    0.0003   -0.0001   -0.0309
365.2425  245.9603   68.4885    0.0003   -0.0001   -0.0206

price2ret assumes time units are days. Therefore, all durations are approximately 365 and the returns are normalized for that time unit.

Compute returns again, but specify that the observation times are years.

[Ryy,intervalsyy] = price2ret(S,Ticks=dates,Units="years");
[[NaN; intervalsyy] S [[NaN NaN]; Ryy] r2]
ans = 10×6

NaN  100.0000  100.0000       NaN       NaN         0
1.0000  110.5171   98.3541    0.1000   -0.0166   -0.0166
1.0000  122.1403  102.7850    0.1000    0.0441    0.0441
1.0000  134.9859   93.0058    0.1000   -0.1000   -0.1000
1.0000  149.1825   89.4007    0.1000   -0.0395   -0.0395
1.0000  164.8721   83.3026    0.1000   -0.0706   -0.0706
1.0000  182.2119   76.7803    0.1000   -0.0815   -0.0815
1.0000  201.3753   72.1105    0.1000   -0.0627   -0.0627
1.0000  222.5541   69.9172    0.1000   -0.0309   -0.0309
1.0000  245.9603   68.4885    0.1000   -0.0206   -0.0206

price2ret normalizes the returns relative to years, and now the returned rates match the simulated rates.

Complete the roundtrip by converting all returns series back to prices. Specify the returned observation intervals and the starting prices p1.

[P,ticks] = ret2price(R,Intervals=intervals,StartPrice=p1);
[Py,ticksy] = ret2price(Ry,Intervals=intervalsy,StartPrice=p1);
[Pyy,ticksyy] = ret2price(Ryy,Intervals=intervalsyy,StartPrice=p1,Units="years");

[ticks ticksy ticksyy]
ans = 10×3
103 ×

0         0         0
0.0010    0.3652    0.0010
0.0020    0.7305    0.0020
0.0030    1.0957    0.0030
0.0040    1.4610    0.0040
0.0050    1.8262    0.0050
0.0060    2.1915    0.0060
0.0070    2.5567    0.0070
0.0080    2.9219    0.0080
0.0090    3.2872    0.0090

[S P Py Pyy]
ans = 10×8

100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000
110.5171   98.3541  110.5171   98.3541  110.5171   98.3541  110.5171   98.3541
122.1403  102.7850  122.1403  102.7850  122.1403  102.7850  122.1403  102.7850
134.9859   93.0058  134.9859   93.0058  134.9859   93.0058  134.9859   93.0058
149.1825   89.4007  149.1825   89.4007  149.1825   89.4007  149.1825   89.4007
164.8721   83.3026  164.8721   83.3026  164.8721   83.3026  164.8721   83.3026
182.2119   76.7803  182.2119   76.7803  182.2119   76.7803  182.2119   76.7803
201.3753   72.1105  201.3753   72.1105  201.3753   72.1105  201.3753   72.1105
222.5541   69.9172  222.5541   69.9172  222.5541   69.9172  222.5541   69.9172
245.9603   68.4885  245.9603   68.4885  245.9603   68.4885  245.9603   68.4885

## Input Arguments

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Time series of returns, specified as a numObs-by-numVars numeric matrix. Each row of Returns corresponds to a sampling time specified by the optional Ticks name-value argument. Each column of Returns corresponds to an individual return series.

Data Types: double

Time series of returns, specified as a table or timetable with numObs rows. Each row of Tbl is a sampling time. For a table, the optional Ticks name-value argument specifies sampling times. For a timetable, ReturnTbl.Time specifies sampling times and it must be a datetime vector.

Specify numVars variables, from which to compute prices, by using the DataVariables argument. The selected variables must be numeric.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: ret2price(Tbl,Method="periodic",DataVariables=1:5) computes prices from the simple periodic returns in the first five variables in the input table Tbl.

Start prices to use for each series, specified as a nonnegative scalar or vector.

For a scalar, all series have the start price StartPrices.

For a vector, StartPrices(j) is the start price for series j in the input data.

Data Types: double

Start time for start prices, specified as a numeric or datetime scalar that ret2price applies to all price series.

The default is 0 when the input data is a matrix or table.

StartTime is required when the input data ReturnTbl is a timetable.

Data Types: double | datetime

Time intervals corresponding to the returns, specified as a positive scalar, or a length numObs positive or duration vector. ret2price applies scalar Intervals to all input series.

When the input return series are in a matrix or table, the default is 1.

When the input return series are in a timetable, ret2price computes time intervals from the start time StartTime and the row times in ReturnTbl.Time, and ignores Intervals. ReturnTbl.Time must be a datetime vector.

Example: Intervals=years(ones(100,1)) specifies 100 1-year durations for the input return series.

Data Types: double | duration

Time units to use when time intervals Intervals are durations, specified as a value in this table.

ValueDescription
"milliseconds"Milliseconds
"seconds"Seconds
"minutes"Minutes
"hours"Hours
"days"Days
"years"Years

ret2price requires time units to convert duration intervals to numeric values for denormalizing returns.

When the value of the Intervals name-value argument is a numeric vector, ret2price ignores the value of Units.

Example: Units="years"

Data Types: char | string

Compounding method, specified as a value in this table.

ValueDescription
"continuous"Compute prices from continuously compounded returns
"periodic"Compute prices from simple periodic returns

Example: Method="periodic"

Data Types: char | string

Variables in ReturnTbl, from which ret2price computes prices, specified as a string vector or cell vector of character vectors containing variable names in ReturnTbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.

Example: DataVariables=["GDP" "CPI"]

Example: DataVariables=[true true false false] or DataVariables=[1 2] selects the first and second table variables.

Data Types: double | logical | char | cell | string

## Output Arguments

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Price series, returned as a (numObs + 1)-by-numVars numeric matrix. ret2price returns Prices when you supply the input Returns.

Returns in row i ri are associated with price interval [pi,pi+1], i = 1:numObs, according to the compounding method Method and using the start price p(1):

• When Method is "periodic",

${r}_{i}=\frac{\left({p}_{i+1}/{p}_{i}\right)-1}{{\tau }_{i+1}-{\tau }_{i}}.$

• When Method is "continuous",

${r}_{i}=\frac{\mathrm{log}\left({p}_{i+1}/{p}_{i}\right)}{{\tau }_{i+1}-{\tau }_{i}}.$

When time intervals (see Intervals) are durations, the magnitude of the normalizing interval τi+1τi depends on the specified time units (see Units).

Observation times τ of the Price series, returned as a length numObs + 1 numeric vector, beginning with the value of the StartTime name-value argument. ret2price returns ticks when you supply the input Returns.

Price series and observation times, returned as a table or timetable, the same data type as ReturnTbl, with numObs + 1 rows. ret2price returns PriceTbl when you supply the input ReturnTbl.

PriceTbl contains the outputs Prices and ticks.

PriceTbl associates observation time τi+1 with the end of the interval for the returns in row i ri.

## Algorithms

Consider the following variables:

• p is a price series (Prices), with p1 as the starting price (StartPrice).

• r is the corresponding return series (Returns).

• τ is a vector of the observation times (Ticks), with τ1 as the starting time (StartTime).

• δ is the series of lengths between observation times (Intervals).

The following figure shows how the inputs and outputs are associated.

## Version History

Introduced before R2006a

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