Receive $100 now and leave in a bank at 1% interest (APR) then it will be worth $101 in a year.
If we only receive the $100 a year from now then we lose $1.
So money now is worth more than money in the future,
this is the time value of money.
Inflation reduces the value of money, deflation increases it.
If inflation is 1% each year (per annum) then our $100 will only be worth $100/1.01≈$.99 a year from now.
To work out what a future payment is now we have to discount it using the interest rate.
Multiply payments by a discount factor, v:
$$v=\frac{1}{1+i}$$
Effect of compounding period on interest rate
Interest rate stated
Semi-annual
Quarterly
Monthly
Daily
{{ i | percent2 }}
{{ toNormalizedPeriodRate(i, 2) }}
{{ toNormalizedPeriodRate(i, 4) }}
{{ toNormalizedPeriodRate(i, 12) }}
{{ toNormalizedPeriodRate(i, 365) }}
$100 paid in December is not exactly the same as $50 paid in June and $50 paid in December.
By the time value of money, $50 is paid in June it is worth more $50 paid in December.
So the actual or effective interest rate (EAR) depends on the period between payments,
which is called the compounding period.
1% compounded quarterly is not the same as 1% compounded annually.
So 'stated interest rate' is not the same as the the effective interest rate.
1% compounded over 365 periods is \((1.01)^{365}\) = {{ Math.pow(1.01, 365) | percent }}.
Mortgage schedule
Monthly payment: {{ periodicPayment }}
Period
Year
Month
Remaining principal
Interest payment
Principal payment
% Interest
{{ row.period }}
{{ row.year }}
{{ row.month }}
{{ row.percentInterest }}%
Annuity
PV={{ pv }}
About
In the abstract, an annunity is a series of regular payments,
we can discount them to find out what they are worth now
to get their individual present values (PV).
Summing these gives the present value of the annutity.
An annuity-certain is where a fixed number of payments is known in advance.
An annuity-immediate is where the payemsns are mad at the end of each period,
whereas with an annuity-due the payments are made at the start of each period.
...
...
Life annuity
Present value (PV) - how much a future payment is worth now. The sum of a discounted set of future cashflows.
Net present value (NPV) - the present value of a future income and costs.
Contingent cashflow - a future cashflow that might not be paid.
Interest rate parity
About
Typically each central bank sets a base rate, a interest rate used to lend to other banks and also
used as a reference or benchmark for the currency in question.
Interest rate parity relates foreign exchange (FX) with interest rates (IR).
The basic idea being that investing in different currencies ought to provide the
same return even with different interest rates.
Typically currencies with higher interest rates are riskier than others.
The spot exchange rate is what the bank will exchange at today.
The forward exchange rate is what the bank will agree today for an exchange in the future.
One can in theory use this to estimate forward rates.
...
Covered interest rate parity is where forward contracts are used to eliminate exchange rate risk.
$$(1 + i_{domestic}) = \frac{Forward}{Spot} \times (1+i_{overseas})$$
Uncovered interest rate parity is a condition,
interest rate differences will equal the relative change in FX rates.
In theory, not in practice, if one country's interest rate in 2% higher than anothers then
one would expect it's currency to deprecate 2% against the other.
$$Forward = Spot \times \frac{1+i_{overseas}}{1 + i_{domestic}}$$
complete
Options pricing
Call/Put price per asset price(S)
Calculates the price of european options using Black-Scholes.
.
What is an option?
We might like the option of building some property someplace in the future,
we might be happy to pay for the privillege now and decide later whether it is worth
spending the money at the later date.
We might like the option to buy or sell something in the future at a given price.
An option can be seen as an insurance, we pay now and then depending on future prices
we might make no payment or a payment later (improve).
A call option is an option to buy an asset at a certain price in the future,
we don't have to buy it but if we do we have a fixed price.
If the asset is worth much more in the future then we can make a profit
by exercising the option to buy at the lower price and immediately selling it on the market at the higher price.
We have minimal risk since the worst we can lose is the premium we paid.
The issuer of the call option however ...
A put option is an option to sell an asset at a certain price in the future.
If the asset is worth much more than it is now then,
if we don't have one already, we would have to buy it at high market prices
and sell to you at the much lower exercise price of the option.
In theory since prices can theoretically go up for ever so can our loss.
The issuer ...
....
Spot price (S) is the current market price of the asset. How much it would cost to buy now.
Strike price (K) is the future price of the option ...
PV(K) the present value of the strike price. How much it would be worth now.
Risk free interest rate (r) - ...
Time to maturity (T-t) ...
Volatility (σ) ...
Black-scholes-merton
A formula for calculating the price of options,
that assumes that returns follow a normal distribution.
We can make that assumption and I understand it holds a lot in practice apart from when it does not
There is a joke about ...
A differential equation is what it says on the packet, an equation with differentials.
A differential such as \(\frac{dy}{dx}\) refers to how much \(y\) changes for every change in \(x\).
With "Normal equations" one can plug in values and get an answer with,
one can't really do that with differential equations.
So one has to solve them, which means finding an equation without any differentials that means the same,
which is far from easy and often not possible.
The black scholes differential equation is:
$$\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} -rV = 0$$
The solutions, used for pricing are:
$$call(S_t,t) = N(d_1)S_t - N(d_2)PV(K)$$
$$put(S_t,t) = N(-d_2)PV(K) - N(-d_1)S_t$$
$$d_1 = \frac{1}{\sigma\sqrt{T-t}}(\ln(\frac{S_t}{K} + (r+\frac{\sigma^2}{2})(T-t))$$
$$d_2 = d_1 = \sigma\sqrt{T-t}$$
$$PV(K)=Ke^{r(T-t)}$$
...
Assume a stock price moves by random coin tosses, randomly left and right like a drunk.
Think of a stock price (S) on average going up at a fixed rate \(\mu\) but also moving randomly up and down (with a standard deviation \(\sigma\)),
then the change in S is:
$$\frac{dS}{S} = \mu dT + \sigma dW (1)$$
Now the price of an option (V) depends on the stock price (S) and the time until maturity (T), so V is a function of S and T: V(S,T).
Take the taylor series expansion of V;
$$dV=\frac{\partial V}{\partial T}dT + \frac{\partial V}{\partial S}dS + \frac{\partial^2 V}{\partial S^2}dS^2 + ... (2)$$
Replace \(dW^2\) with \(dt\) and set \(dT^2=0\) and \(dt dW = 0\):
$$dV=(\frac{\partial V}{\partial T} + \mu\frac{\partial V}{\partial S} + \frac{\sigma^2}{2}\frac{\partial^2 V}{\partial S^2})dT + \sigma^2 dW$$
...
Volatility
The volatility gives the randomness of the asset price,
with zero volatility the future price is entirely predictable,
with more volatility the uncertain the future price.
Since Black-scholes assumes that assets returns follow a normal probability distribition,
(and asset prices a lognormal probability distribition),
and because the normal probability distribition function takes two parameters: a mean (μ) and a standard deviation (σ).
The current asset price (S) is the mean (μ) [check r-rf] and the volatility is the standard deviation (σ).
This model assumes that volatility is a constant for the whole time period and for all options, which is unrealistic.
We can estimate the volatility by taking the standard deviation of historical returns.
We can also look the price of options on the market and work backwards to work out what volatility is implied by their prices.
In theory they should all have the same implied volatility for all strike prices,
but if we plot them we might see a curve called a volatility smile.
graph volatility smile and expand text
Put-call parity
In theory the price of an call and a put are intimately related,
so much so that given a put one can work out the price of the call and vice versus.
This is called put-call parity:
$$C + PV(K) = P + S$$
Or the price of a call plus that of a bond equals the price of the put and that of the stock.
\(PV(K)\) is the present value of the strike price K,
one can think of this as a non-interest bearing bond with par-value K.
One pays for a bond now and at some point in the future you get paid the par-value
(For interest bearing bonds there would also be regular interest payments).
Since the bond pays in the future and because future payments are worth less than
payments now (the time value of money), one has to discount the future value of par-value
to it's present value.
In practice if you get real prices for puts and calls from the market,
put-call parity is unlikely to hold (expand ...).
Greeks
$$Δ call = N(d1)$$ - also Delta hedging: Holding stock and cash
Financial statements
Under construction
Balance sheet
period 1
Liabilities {{ round2(totalLiabilities) }}
Assets {{ round2(totalAssets) }}
Liquidity Ratios
Solvency Ratios
Cash flow statement
period 1-2
Operations {{cashFlowFromOperations }}
Investing {{cashFlowFromInvesting }}
Financing {{cashFlowFromFinancing }}
Ratios
Balance sheet
period 2
Liabilities {{ totalLiabilities }}
Assets {{ totalAssets }}
Liquidity Ratios
Solvency Ratios
Investment Ratios
Statements
There are three basic financial statements: the balance sheet, the income statement (profit and loss) and the cashflow statement.
Thinking a bit like a computer scientist one can think of a enterprise as a state machine,
One can think of the balance sheet as the state of an enterprise at a point in time,
and the income statement describing the transitions between balance statements.
A balance sheet is quite an abstract concept, a table of assets and liabilities
I could create a balance statement for myself, for my family as a group or for any group or individuals.
Normally though companies publish balance sheets on a regular basis.
A cashflow statement is like an income statement in that it describes changes in the period between two balance sheets.
However, it just shows how much cash went in and out and to what purpose.
If an income statement shows great sales and profits and that is not reflected in the cashflow,
one might be concerned, I believe this occurred in the Enron scandal.
A cashflow statement should be less easy to manipulate since the auditors can just check the bank statements,
though I believe Wirecard managed to do it.
One normally analyses statements using financial ratios which can be are hard to remember if you are not using them everyday.
The idea of this app is get a feel for what happens to the ratios when one changes the inputs.
The share price is determined by those who buy and sell the company,
there are variious to compare this with numbers given in financial statements.
To show what the market thinks the company is worth against what is indicated in the financial statements.
The Price/Earnings ratio says how many years it would take for the earnings to pay the price.
There are a lot of dual words and antonyms in finance.
Asset/Liability
Put/Call Option
Contango/Backwardation
Accounts Receivable (A/R)/Accounts Payable(A/P)
Credit/Debit
Profit/Loss
Alpha/Beta
Deposit/Widthdrawl
Knock-In/Knock-Out Option
Inflation/Deflation
Earned/Unearned income
...
Cashflow forecast
Years forecast:
Interest rate:
Cash:
Show cash:
Simulations:
Growth volatility:
Date volatility:
A simple cashflow forecast that works by repeating last years payment pattern over successive years.
for general cashflows or particular cases like dividend forecasting.
Working Capital Metrics
Traditional Metrics using the top down method
The Traditional Metrics and DSO, DPO, DIO and CCC are calculated at the enterprise level,
top down method using numbers from the latest Financial Statements.
The figures for AP and AR taken from the Balance Sheet with Sales and COGS taken from the Income Statement.
$$\begin{aligned}
DSO &= AR / Sales / days \\
DPO &= AP / COGS / days \\
DIO &= Inventory / COGS * days \\
CCC &= DSO + DIO - DPO
\\
days &= 5 * 52 & \text{For one year of working days.} \\
\end{aligned}$$
Transaction based metrics using the bottom up "Rolling" method
"Rolling" metrics can be calculated for invoices over any given period (yearly, monthly)
with COGS and Sales estimated using the sum all Invoices raised within the period.
$$\begin{aligned}
AR &= \sum{Receivable Invoices Outstanding } \\
AP &= \sum{Payable Invoices Outstanding } \\
Sales &= \sum{Receivable Invoices \land Invoice Date \in Period} \\
COGS &= \sum{Payable Invoices \land Invoice Date \in Period} \\
Invoices Outstanding &= \{ Invoice Date \in Period \land CloseDate \not \in Period\}
\end{aligned}$$
This can be used for Year to Date (YTD) metrics be setting the start of the period to be the current date minus 365 and the end date for the period to be today's date.
To calculate a value for the previous month one would set the start date for the period to be the current date minus 365 and the end date for the period to be 30 days before today's date.
To calculate calendar month metrics for the last year: one might use the first day of each month of the period and the last day of each month as the end of each period.
Unlike the traditional metrics, WAT, WADC, and WADP are calculated at the level of individual invoices.
So each invoice would have it's own values for WAT, WADC, and WADP.
Weighted Average Terms (WAT) measures how long the counterparty is allowed to delay payment
(weighted by the relative amount of the invoice).
Weighted Average Days to Collect (WADC) measures the time to it took to collect for each invoice (weighted by the relative amount of the invoice).
$$\begin{aligned}
WAT &= (Due Date – Invoice Date) \times Invoice Amount/Total Invoice Amounts \\
WADC &= (Collection Date – Invoice Date) \times Invoice Amount/Total Invoice Amounts \\
WADP &= (Payment Date – Invoice Date) \times Invoice Amount/Total Invoice Amounts \\
\\
WAD &= (Closed Date – Invoice Date) \times Invoice Amount/Total Invoice Amounts \\
\\
Total Invoice Amounts &= \sum_{Invoice \in Selected Invoices}{Invoice Amount}
\end{aligned}$$
The "Selected Invoices" might be per counter party, per sector or any other grouping.
So rather than a single WAT, WAD or WADP value for each invoice, there might be a WAT per counterparty, WAT per sector, WAT per enterprise.
Note that WADC and WADP use the formula with Collection Date being the Closed Date for Payables and Payment Date being the Closed Date for Receivables.
In this document we refer to this as WAD.
See book "The Stress Test Every Business Needs: A Capital Agenda for Confidently" for discussion of WADC, WADP and WAT.
AP
Accounts Payable (from balance sheet)
AR
Accounts Receivable (from balance sheet)
CCC
Cash Conversion Cycle: \(CCC = DSO + DIO - DPO\)
COGS
Cost of Goods Sold (from income statement)
DIO
Days Inventory Outstanding: \(DIO = Inventory / COGS * days\)
DPO
Days Payables Outstanding: \(DPO = AP / COGS / days\)
DSO
Days Sales Outstanding: \(DSO = AR / Sales / days\)