A RUBE GOLDBERG & HEAT ROBINSON APPROACH TO PRICE CORPORATE BONDS: ALTMAN Z-SCORE

A quick and dirty formula to price corporate bonds/notes might be given by the following formula:

rf / PS  (it looks like a KLUDGE or some sort of MacGyverism)

rf is the market risk free rate as read from the Temporary Structure Of Interest Rate or Yield Curve.  PS is the probability of success for the Corporation or (1-Probability Of Default)

This is the catch!   Probability of Default or Probability Of Success is not easy to get.  You will need to run an algorithm entitled: Altman Z-Score (by Professor Edward I. Altman) with the following figures belonging to a "Columbian" Corporation listed on BVC:

Working Capital=333,060,600

Total Assets=1312,650,600

EBIT=117,550,800

Sales=2546,934,000

Market value of equity is :744,488,400

Total Liabilites:  470,203,200

Retained Earnings: 587,754,000

T₁ = Working Capital / Total Assets

T₂ = Retained Earnings / Total Assets

T₃ = Earnings Before Interest and Taxes / Total Assets

T₄ = Market Value of Equity / Total Liabilities

T₅ = Sales/ Total Assets

Z score bankruptcy model:

Z = 1.2T₁ + 1.4T₂ + 3.3T₃ + 0.6T₄ + .999T₅

Zones of Discrimination:

Z > 2.99 -“Safe” Zones

1.81 < Z < 2.99 -“Gray” Zones

Z < 1.81 -“Distress” Zones

The Altman Z-Score is a Gaussian statistic and for this corporation  =  4.11522388059702.  This means a probability of success = 0.999980659810466  (you can use NORMSDIST function embedded on any spreadsheet to get this value)

The very same day these Financial Reports were issued, the "Columbian" yield curve for a 10 year Bond is 7.84%   after applying our formula and if we want to issue our Corporate Bond our cost would be 7.8402%

Is this cost a straight jacket or alluring to investors?  maybe not!  you have to compare this one to the cost you would face if you were taking a loan from a bank or an ensemble of them.  So thus, the coupon rate of your bond could be defined by the following interval [rf, ActiveRate]

After all, other than full control and a lower cost of corporate debts you are looking for an arbitrage opportunity.  The cheapest bank in this country is paying on a 10 year CD 8% and for a 10 year loan is collecting 16% a year...Our new coupon rate would be 0.08/0.999980659810466 = 8.000154725%

Either a bank or the yield curve.  The day this debt is issued to be market traded, the market's discount rate will be higher than your coupon rate...Beware of not selling your bonds at a discount rate near or above the cost of a loan from any Bank....

WHAT IF?   By using the yield curve the key reference rate is negative? No problem, at least one of the rates per maturity will be positive....We can use this one to build our Yield curve by applying calculation of the short rate curve

Acknowledgements: Edward I. Altman from NYU & Pasquale Cirillo from TU Delft
Sources: RiskCenter, Investopedia, Wikipedia, EdX.org, Coursera.org

COMPUTATIONAL FINANCE: Internal Rate Of Return for uneven cash flows (The Time Lags).

You wanted to know what was the return of a share last year, but you can not compute it due to dividends?

This is basically a problem of computational finance not one of Economics Engineering nor Financial Mathematics.

There is R, an excellent programming language if you are dealing with statistics and data science....

The following algorithm and source code is a courtesy of Professor Pasquale Cirillo:

#f=function(y) D1*exp(-t0.5*y)+D2*exp(-t1*y)+D3*exp(-t1.5*y)+PT*exp(-T*y)-P0

#uniroot(f,lower=j,upper=k)

This uniroot function is crucial and of capital importance when you are writing the upper and lower values

The following example will clear this up :

On december 18, 2013.  You bought a share of Stock Exchange.  The next 3 dates are dividends and the last date is when you are closing your position...

21.6    18/12/2013
0.78    30/04/2014
0.32    27/06/2014
0.32    31/10/2014
19.2     17/12/2014

Computer work:

f=function(y) 0.78*exp(-133*y)+0.32*exp(-191*y)+0.32*exp(-317*y)+19.2*exp(-365*y)-21.6
uniroot(f,lower=-1,upper=1) 

Your Work:

y=-0.000152587.  This is a daily rate in continuous time .  Year rate= y*365, exp^(-0.055694255)=0.9458283   =>  -1 = -0.0541717

effective rate: -5.41717%

What if...   uniroot(f,lower=-2,upper=2) or -1 and 0 ??

y=-0.0001220703.  This is a daily rate in continuous time. Annual = -0.0445556595
Converting to discrete time =0.9564224-1 = -0.0435776  => -4.35776%

By using a year scale instead of a day scale, the yearly continuous rate is: -0.048047, so thus:
0.953089-1 =-0.046911.  This is -4.6911%

By using interpolation.  Our daily continuous rate is: -0.0001316 or -4.8034% per year but in discrete time this translates to:  -4.68986189% per year


Your effective IRR will be the third computation....These algorithms can be deployed to compute rates for ventures and other financial vehicles with not regular events

Sources: Pasquale Cirillo at TU Delft and R
              http://pbil.univ-lyon1.fr/Rweb/Rweb.general.html  (online computations)

FINCA RAIZ: VISIÓN ECONÓMICO-CONTABLE, VISIÓN MONETARIA Y VISIÓN FINANCIERA

Tasar finca raíz, en particular propiedad horizontal, en su precio justo, al menos en teoría; es y será historia de nunca acabar.

En las siguientes líneas se desarrollarán 3 modelos para valorar inmuebles, que casi siempre arrojan resultados irreconciliables, dados los criterios para determinar el valor presente de tales activos.

Los 3 modelos aquí expuestos se fundamentan en pasado, presente o futuro.  Sin híbridos o sin tocar límites de los otros.

->Desde una perspectiva económico-contable, es decir, el pasado.  El valor de un inmueble vendría determinado por el último valor de transacción ajustado por las valorizaciones y correciones efectuadas por el instituto catastral u oficina de predios.

Ejemplo: si se compra un inmueble directamente al constructor y el valor de transacción fue de 1000 y han pasado 5 años en los que las valorizaciones oficiales a objeto de determinar el impuesto de bienes inmuebles o de predios, han evolucionado así: 5%, 20%, 30%, 10% y 8%.  La valorización acumulada sería: 1.05*1.2*1.3*1.1*1.08=1.945944 o 94.5944%  y el precio teórico justo: (1000-depreciaciones)*1.945944 < 1945,944

->Desde una perspectiva financiera, es decir, el futuro.  El valor de un inmueble vendría determinado por los alquileres que podría generar a futuro y las valorizaciones futuras en dinero de hoy. Incluyendo el valor residual o de remate de la propiedad, según se pueda deducir de la información fiscal.

En el caso anterior, suponga que las rentas anuales del inmueble son de 60 anuales y las valorizaciones de 10 anuales .  Sin incrementos por inflación u oferta/demanda. Aplicando la fórmula de Myron Gordon (GGM) y asumiendo un costo del capital del 12% anual, tendríamos:

70/0.12= 583,333333   a este valor hay que sumarle el valor residual/de remate del inmueble hoy día, que puede ser el mismo valor comercial catastral o el determinado por la oficina de predios y que es de 1000

Lo cual nos arroja un total de 1583,33333

->Desde una perspectiva monetaria, se esperaría una conciliación de las dos posturas anteriores.  El presente determina el aquí y el ahora. No hubo un antes, ni va a haber un después.  El valor de un inmueble queda determinado por la oferta y demanda: inventarios disponibles, número de posibles compradores, cantidad de dinero en la economía  y capacidad de endeudamiento.

Ejemplo no numérico:  suponga que en el microcosmos económico de la ciudad R, la cantidad de dinero es M, la cantidad de inmuebles para la venta es I y el nivel de precios es P.

Si por emisiones de la autoridad monetaria o un aumento en la inversión directa la cantidad de dinero aumenta a M+,  al igual que el número de demandantes y la cantidad de inmuebles para la venta sigue siendo I , es más que claro que P tendría que aumentar.

Por el contrario, si hay fuga de capitales o contracción de la base monetaria y ahora hay M- e igualmente baja el número de demandantes, e I sigue en la misma cantidad, es más que claro que P ha de bajar.

Fuentes: Perry Mehrling de Barnard College y Walter Bagehot: Lombard Street.

STANDARD DEVIATION (DESVIACIÓN TÍPICA) : FROM CONTINOUS TIME TO DISCRETE TIME (DE TIEMPO CONTINUO A TIEMPO DISCRETO)

Heteroskedacity is a problem when we are handling time series data. La discontinuidad en la periodicidad de los datos en las series temporales resulta ser un problema de manipulación. Continous time transformation is needed. Transformación en tiempo continuo es necesaria.  Example (Ejemplo) :

Given the following 10 intraday share prices, what is the standard deviation of the return rates? Dados los siguientes 10 precios intradía, cuál es la desviación típica de las tasas de rentabilidad ?

From last to first, Del último al primero: 3700 , 3740 , 3735 , 3800 , 3735 , 3790 , 3695 , 3705 , 3875 , 3865

AS IS (rates) , COMO TAL (tasas):  0.9893 , 1.0013 , 0.9829 , 1.0174 , 0.9855 , 1.0257 , 0.9973 , 0.9561 , 1.0026

These rates must be turned into continous time rates by using natural logarithm: ln.  Estas tasas deben ser convertidas a tasa en tiempo continuo, usando logaritmo natural: ln.  

-0.0107 , 0.0013 , -0.0172 , 0.0172 , -0.0146 , 0.0254 , -0.0027 , -0.0449 ,  0.0026

We can now apply the classic formula for a sample.  Podemos aplicar ahora la fórmula clásica para una muestra.

STD = 0.0205730233503667  This is a value in continuos time... But reality is nearer to discrete time.  Este es un valor en tiempo continuo... Pero la realidad  es más cercana al tiempo discreto.

e^μ * (e^σ^2 -1)^(1/2)  1 period transformation (fórmula de transformación para un periodo)  

 If (si) μ = -0.00484765137657851 , e = Euler's number (número de Euler) ,σ = standard deviation in continous time (desviación típica en tiempo continuo)

STD in discrete time (en tiempo discreto) = 0.0204757003886505

What is the conceptual difference? Cuál es la diferencia conceptual ?While under continous time, we calculate a standard deviation for a n number of data; under discrete time we have got a standard deviation for an elapsed number of time units.

Mientras que bajo tiempo continuo, calculamos una desviación típica para un número n de datos; bajo tiempo discreto conseguimos una desviación típica para un número transcurrido de unidades temporales. 

Here is the formula for a cumulated, discrete time, standard deviation.  Aquí la fórmula para una desviación típica, acumulada y en tiempo discreto.
e^μ*T * (e^σ^2*T -1)^(1/2) multiperiod transformation

Help yourselves by solving for a 15 day cumulated value.  Sírvanse ustedes mismos, resolviendo para un valor acumulado a 15 días.


Sources/Fuentes: John Cochrane. The University Of Chicago. Asset Pricing.

TASAS DE INTERÉS 7 (INTEREST RATES): TASA AL CONTADO (SPOT RATE), TASA INSTANTÁNEA (SHORT RATE), TASA A FUTURO (FORWARD RATE), SWAP RATE

Warning: spot rates are calculated by employing Central Bank Treasuries only. So therefore, the first spot rate is the Internal Rate Of Return of such a theoretical zero coupon treasury, the following spot rates -on the yield curve- undergo a bootstrapping process.  Precaución: las tasas de interés al contado se calculan empleando títulos del Banco Central únicamente.  Entonces y por lo tanto, la primera tasa al contado es la Tasa Interna De Retorno de un teórico título sin cupones, las tasas al contado siguiente -sobre la curva de rendimientos- se someten un proceso de iteración/recálculo.


La siguiente es la información después de un día de operaciones en el mercado de renta fija en términos anuales.  The following information is given by the fixed rent market after an annualized trading day (Oct. 11, 2012):

Fecha de vencimiento/                  Spot Rate/Tasa al contado (annual effective):
Maturity Date:

          Feb. 15, 2013                                                        0.1250782%

          Aug. 15, 2013                                                       0.3967851%

         Feb. 15, 2014                                                        0.4751252%

         Aug. 15, 2014                                                       0.5253753%

         Feb. 15, 2015                                                        0.5163284%

         Aug. 15, 2015                                                       0.4922074%

         Feb. 15, 2016                                                        0.3917654%

         Aug. 15, 2016                                                       0.537439%

         Feb. 15, 2017                                                        0.8889277%

Para títulos emitidos por el BANCO CENTRAL.  For "treasuries" issued by the Central Bank.

Cuál es la curva de Tasas Instantáneas?  What is the Short Rate curve ?

Time to maturity in years/Plazo al vencimiento en años: 0.3479452, 0.8438356, 1.3479452, 1.8438356, 2.3479452, 2.8438356, 3.3479452, 3.8465753, 4.3506849 

Habrá que reescribir las tasas de descuento en semestre efectivo y no año efectivo/ You will have to rewrite the rates in effective semesters and not years

La primera tasa instantánea es igual a la primera tasa al contado/The first short rate is equal to the first spot rate:  0.0625195% per semester or 0.0435026% until maturity.  r(0,1)

r(1,2) =  (1.001981961^2 / 1.000625195)-1  =   0.334057%  per semester or 0.2910886% per time fraction

r(2,3) = 0.305232% per time fraction or 0.315497% per semester 

r(3,4) = 0.3277799% per time fraction or 0.337568% per semester

r(4,5) = 0.2433175% per time fraction or 0.239787% per semester

r(5,6) = 0.1873067% per time fraction or 0.185672% per semester

r(6,7) = -0.087239% per time fraction or -0.104445% per semester

r(7,8) = 0.7555578% per time fraction or 0.7785148% per semester

r(8,9) =  1.804683% per time fraction or 1.855504% per semester
The short rate curve tells us the market expectation and the expected value for the 3 month rate in 3, 6, 9, 12 months from now and so on OR the annual effective rate in 1, 2, 3, 4 years and so on...../La curva para tasas instantáneas nos dice la expectativa del mercado de un periodo a otro y el valor de la tasa a 3 meses en 3, 6, 9, 12 meses desde ahora y en adelante O la tasa efectiva anual en 1, 2, 3, 4 años y en adelante ... Ahora la curva de tasas a futuro.  Now the forward rate curve.

f(1,2) = 0.3340567% semester ;  f(1,3) = 0.324776% semester ; f(1,4) = 0.329040% semester

f(1,5) = 0.306719% semester ;  f(1,6) = 0.282498% semester ;  f(1,7) = 0.217904% semester

f(1,8) = 0.2978% semester ;  f(1,9) = 0.491202% semester

f(1,9) = (1.004434805^9/1.000625195)^(1/8)  -1

These are the expected spot rates in 0.6958904 semesters, there is no forward rate for nine semesters.  Estas son las tasas al contado esperadas en 0.6958904 semestres, no hay tasa futura para nueve semestres.

Swap rates or spreads. Tasa swap o márgenes diferenciales:

Given an 8 semester forward rate and an 8 semester spot rate.  (0.491201 - 0.2683594)*100  =  22.28416 bps.  Dada una tasa a futuro para 8 semestres y una tasa al contado para 8 semestres....


Sources/Fuentes:  www.oup.com/us/ppt/derivatives/DMCH13.ppt (David Dubofsky & Thomas Miller)
-Finance By Kay Giesecke from Stanford University
-Financial Engineering & Risk Management By Martin Haugh & Garud Iyengar from Columbia University

BUDGETS & OTHER FORECASTS: DCF ANALYSIS TO APPRAISE DIFFERENT PROJECTS (BANG FOR THE BUCK)

Net Present Value is widely deployed to perform contrast and comparison among projects, proven the discount rate, the life expectancy and the initial investment value are all the same.

Could I use discount cash flow (DCF) analysis to appraise projects with different discount rates, amounts and life expectancies??? YES! But with a twist....

Remember? COST-->BENEFIT or the old fashioned ratio: BENEFIT/COST??

Read the following projects:

year/project	A	B	C	D	E	F	G	H
0	-2000000	-10000000	-10000000	-10000	-110000	-17200	-1000000	-350000
1	1000000	4000000	0	300	25000	10000	200000	35000
2	1200000	4000000	0	500	25000	10000	206000	35000
3	1400000	5000000	14000000	1200	25000	10000	212180	35000
4				2000	25000	10000	218545,4	42000
5				2000	25000	10000	225101,76	42000
6					25000	10000	231854,81	42000
7					25000	10000	238810,46	42000
8					25000	10000	245974,77	42000
9					25000	10000	253354,02	42000
10					25000	10000	260954,64	42000
11						10000	268783,28	42000
12						10000	276846,77	42000
13						10000	285152,18	42000
14						10000	293706,74	42000
15						10000	302517,94	42000
16						10000	311593,48	42000
17						10000	320941,29	42000
18						10000	330569,53	42000
19						10000	340486,61	42000
20						10000	350701,21	42000
21								42000
22								42000
23								42000
24								42000
25								42000

These are the free cash flows for every venture. Each of them has a unique market discount rate above the Cost of Capital related to our Investment Bank.....What is/was the best project overall??

"The Bang For The Buck" is the answer.....The discount rates are as follow (%): 4, 4.1, 4.2, 4.5, 5, 10, 8, 10

We know the present value for CapEx, Costs and Investment. Now we need the present value for the net cash income.....They must be read as :

A= 3315600.82 , B= 11965766.65, C= 12374420.65, D= 14061.80, E= 193043.37 , F= 85135.64 , G= 2450008.29     H= 2144175.17 and now: the ratios of efficiency:

A= 1.65780041, B= 1.196576665, C= 1.237442065, D= 1.40618, E= 1.7549397273, F= 4.949746349, G= 2.450008290, H= 6.126214771 Why investment F has the same discount rate than investment H? Although, F life is shorter than H, F is a high tech venture while H is a monopolic public transport service
**************************************************************************************************************************************

NOW, HOW TO DRAW A BUDGET: In most countries, forecasting the last financial statements by using Customer's Price Index rate of change is bylaw.....There is a major flaw: this rate is an average of the economy as a whole, so it might no mirror the change rates of your business....The alternative: Ground Zero Budget

The author's favorite method:

1. You should calculate the historical geometric average for the change rate of your incomes (both operational and non operational)

2.Apply down analysis to calculate the share for costs, expenses, profit/loss

....Now you can forecast these variables and to draw an income statement with an expected profit/loss

3. The expected profit/loss goes to the EQUITY section on the Balance sheet and again ....based on historical calculations you must apply down analysis to draw the pending values for the accounts on the equity section and the accounts on the LIABILITIES SECTION....The share for each account and subaccount are defined by the arithmetical average of previous years

For example: total income for your company last fiscal year was $150 million.....The historical geometric average for the change is 15% a year

Total operational costs, CapEx and non operational Expenditure. Down Analysis tells us that the historical average weight of them is 80%

150*1.15 = 172 .5 expected total income for next year and 172.5 * 0.8 = 138 the total expected costs, CapEx and non operational expenditure (taxes have been discounted, already)

expected net profit = 34.5 Down analysis tells us that the historical average weight for profits is 50% of the equity

expected equity = $69 millions down analysis tells us that the historical average weight of equity on the financial structure for this company is 70%

liabilites are 30% = expected $29.57 millions

total asset : $98.57 millions

See you around!

Sources: Kay Giesecke, Dmitry Smelov, David Luenberger, Jorge E. Burbano, Alberto Ortiz & Stanford U.