financial_mathematics_20150928225714 Flashcards
Simple Interest
1+it
Real Rate of Interest Formula
(i - r) / (1 + r)
Force of Interest for Simple Interest
i / (1 + it)
Force of Interest for Compound Interest
ln( 1 + i)
Net Present Value
total value of all cashflows either in or out divided by a discount rate, so basically present value for many different payments.
Compound Interest
(1 + i) ^ t
a(t)
accumulation function -> how much an investment of 1 dollar would grow to in a certain amount of time
A(t)
a(t) * A(0) FYI: (A(0) is the principal)
Present and Future Value
Using current interest rates, the value that the current money would grow to.
Convertible monthly.
i^(12)
i^(m)
( 1 + (i/m) ) ^ m
Equivalent Rates of Interest and Discount
i = d / (1 - d)d = i / (1 + i)
Rate of Discount.
It is used to determine the current value by reducing future value. i.e. RoD = 0.07 and Money at time 1 is 100Then money at time 0 is 93. Or if we had time 0, then time 1 is 93 * ( 1 / 0.93)
Equation of Value
Equates the present value of all payments disbursed and received.
Effective Annual Rate of Interest
( A(t) - A(t - 1) ) / A(t - 1)
Effective Annual Rate of Discount
( A(t) - A(t - 1) ) / A(t)
Force of Interest - 2 Formulas
- Derivative of a(t) over a(t)2. exp(integral with respect to t of the equation)
a n | i (there should be a line over the n) Formula and PV or AV
( 1 - v^n ) / i PV Bonus: v + v^2 + … + v^n
s n | i (there should be a line over the n) Formula and PV or AV
( ( 1 + i )^(n) - 1) / iAVBonus: 1 + (1 + i) + (1 + i)^2 + … + (1 + i)^(n - 1)
Definition difference between annuity-immediate and annuity-due?
Payments made under an ordinary annuity occur at the end of the period while payments made under an annuity due occur at the beginning of the period.
Formula difference between annuity-immediate and annuity-due
Annuity immediate = annuity due * (1 + i)
PV of m-year deferred n-year annuity-immediate
v^m * a n | i
PV of m-year deferred n-year annuity-due
v^m * (a n | i) * (1 + i)
PV of a perpetuity-immediate
1 / i
PV of a perpetuity-due
1 / d
What to do if payment period doesn’t equal interest period?
change interest period to an equivalent rate.
PV of a annuity-immediate convertible m-nthly.
( i / i^(m) ) * a n|
AV of a annuity-immediate convertible m-nthly.
( i / i^(m) ) * s n|
PV of a annuity-due convertible m-nthly.
( d / d^(m) ) * a n|
AV of a annuity-due convertible m-nthly.
( d / d^(m) ) * s n|
PV of a annuity-immediate perpetuity convertible monthly
1 / ( i ^ (m) )
PV of a annuity-due perpetuity convertible monthly
1 / (d ^ (m) )
PV of an annuity-immediate arithmetic progression with payment P, P+Q, P+2Q, … P + (n-1)Q
( P * (a|n) ) + Q * (a|n - nv^n) / i
PV of a perpetuity-immediate withpayments 1, 2, 3, …
1 / (id) or (1/i) + (1/i^2)
PV of a perpetuity-due withpayments 1, 2, 3, …
1 / d^2
PV of a annuity-immediate with payments n, n-1, n-2, …, 1
( n - an|i ) / i
PV of an n-year annuity-immediate with payments 1, (1 + k), (1 + k)^2, …, (1 + k)^n-1
( 1 - ( (1 + k) / (1 + i) ) ^ n ) / (i - k)
Level Continuous Annuity
integral with respect to n of v^tor(i / (force of interest) ) * a n|i
Increasing Continuous Annuity
Prospective Outstanding Balance
Present Value of all future paymentsR = Level PaymentsR * a (n - t)|i
Retrospective Outstanding Balance
Current Value of all payments minus accumulated value of all past paymentsR * (ani * (1 + i) ^ t - st|i)
Total Interest for a loan with n payments of 1
n - an|i
Interest Period T for a loan with n payments of 1
1 - v^n - t + 1
Total Principal Repaid for a loan with n payments of 1
a n|i
Principal Repaid during Period T for a loan with n payments of 1
v^n - t + i
balloon paymen
An oversized payment due at the end of a mortgage, commercial loan or other amortized loan. Because the entire loan amount is not amortized over the life of the loan, the remaining balance is due as a final repayment to the lender.
drop payment
Probably if it is less than the others i think.
Amortization
The paying off of debt with a fixed repayment schedule in regular installments over a period of time. Consumers are most likely to encounter amortization with a mortgage or car loan.
Sinking Fund (Formula for Interest and Sinking Fund Deposits)
I = B * i *interest rate on loanSFD = B / (sn|i) *interest on sinking fundwhere B is loan balance
Purpose of a Sinking Fund
A sinking fund allows the investor to accumulate some of the principal in an interest bearing savings account which helps offset some of the cost involved. Note the interest in the savings account is usually lower than that on the bond.
Principal paid in t = Principal paid in t + 10 by what.
(Principal paid in t) * (1 + i) ^ (t + 10 - t)= Principal paid in t + 10
book value
The value at which an asset is carried on a balance sheet. To calculate, take the cost of an asset minus the accumulated depreciation.
amortization of premium
The amount of principal that is repaid when a bond is bought at a premium. AKA writing down a bond.
accumulation of discount
The negative portion of the principal paid when a bond is bought at a discount
redemption value
Redemption value is the price at which the issuing company may choose to repurchase a security before its maturity date.
par value/face value
The nominal value or dollar value of a security stated by the issuer. For stocks, it is the original cost of the stock shown on the certificate. For bonds, it is the amount paid to the holder at maturity (generally $1,000). Also known as “par value” or simply “par.”
yield rate
The amount of return an investor will realize on a bond. Though several types of bond yields can be calculated, nominal yield is the most common. This is calculated by dividing amount of interest paid by the face value.
callable/ non - callable
Callable - A bond that can be redeemed by the issuer prior to its maturity. Usually a premium is paid to the bond owner when the bond is called.
coupon rate
sum of coupons / face value of bond
Basic Price of Bond Formula
((Face Value) * (Coupon Rate) * a n|i) + Face Value * v^n
If Price of Bond > Redemption Value of Bond
Premium
If Price of Bond
Discount
Premium/Discount Formula
Price = C + (Fr - Ci) * an|i
Full or Dirty Price Formula
price including accrued interest (price that is actually paid)If the price for the period is B, then the price in between coupons is:B (1 + i) ^ (t) where t is the fraction of the coupon payment that we are looking at
Clean Price
price including accrued interest minus the portion of coupon to next periodIf the price for the period is B, then the price in between coupons is:B (1 + i) ^ (t) - tFr where t = fraction of the coupon payment that we are looking atF = Face valuer = coupon rate
Increase in Book Value at time t if the bond is bought at a discount.
(Ci - Fr) * v^n-t+1
Decrease in Book Value at time t if the bond is bought at a premium
(Fr - Ci) * v^n-t+1
Lowest Price of a Premium Bond
Earliest
Lowest Price of a Discount Bond
Lastest
Rate of Return
The gain or loss on an investment over a specified period, expressed as a percentage increase over the initial investment cost. Gains on investments are considered to be any income received from the security plus realized capital gains.(Final sale cost - Initial Cost) / Initial CostBasically, it is yield for anything.
Dollar Weighted Rate of Return
it is the discount rate on which the NPV = 0 or the present value of inflows = present value of outflows - Same as Internal Rate of ReturnCalculated using Financial Calculator - http://www.actuarialoutpost.com/downloads/baiiplus.pdf
Time-Weighted Rate of Return
How much a dollar would grow to in a period. Uses time instead of money as the weight.HPR = ((MV1 - MV0 + D1 - CF1)/MV0)Where: MV0 = beginning market value, MV1 = ending market value D1 = dividend/interest inflows, CF1 = cash flow received at period end (deposits subtracted, withdrawals added back)Compounded Version =[ (1 + HPR1) * (1 + HPR2) and so forth) ] - 1Annualized = (( 1 + compounded version ) ^ (1 / y)) - 1
Portfolio
A grouping of financial assets such as stocks, bonds and cash equivalents, as well as their mutual, exchange-traded and closed-fund counterparts. Portfolios are held directly by investors and/or managed by financial professionals.
spot rate
The price quoted for immediate settlement on a commodity, a security or a currency. The spot rate, also called “spot price,” is based on the value of an asset at the moment of the quote. This value is in turn based on how much buyers are willing to pay and how much sellers are willing to accept, which depends on factors such as current market value and expected future market value. As a result, spot rates change frequently and sometimes dramatically.
forward rate
A rate applicable to a financial transaction that will take place in the future. Forward rates are based on the spot rate, adjusted for the cost of carry and refer to the rate that will be used to deliver a currency, bond or commodity at some future time. It may also refer to the rate fixed for a future financial obligation, such as the interest rate on a loan payment.An agreement to pay or buy at a certain rate.
stock dividend
A dividend payment made in the form of additional shares, rather than a cash payout. A dividend is a distribution of a portion of a company’s earnings, decided by the board of directors, to a class of its shareholders. Dividends can be issued as cash payments, as shares of stock, or other property.
yield curve
A line that plots the interest rates, at a set point in time, of bonds having equal credit quality, but differing maturity dates. The most frequently reported yield curve compares the three-month, two-year, five-year and 30-year U.S. Treasury debt. This yield curve is used as a benchmark for other debt in the market, such as mortgage rates or bank lending rates. The curve is also used to predict changes in economic output and growth.
Macaulay Duration
The weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price, and is a measure of bond price volatility with respect to interest rates.higher duration means more volatile and lower duration means less volatile.Σ( t * CF(t) * v^t ) All present current value of bond.
modified duration
Modified duration follows the concept that interest rates and bond prices move in opposite directions. This formula is used to determine the effect that a 100-basis-point (1%) change in interest rates will have on the price of a bond.Modified Duration = Macaulay Duration * v
Macaulay Duration of a n-year zero coupon bond
n
Macaulay Convexity
Σ( t^2 * CF(t) * v^t ) All over present current value of bond.
Macaulay Convexity of a n-year zero coupon bond
n^2
Convexity
Σ t * (t + 1) * v^(t + 2) * CF(t) All over present current value of bond
Estimated Change in Price due to change in interest Rates
P(i) * (- (delta i) * ModD + (1/2)((delta i)^2) * Conv)
Cash Flow Matching
As before, there is a stream of liabilities to be funded at specified time intervals. To achieve this, a cash flow matching strategy makes use of cash flows from principal and coupon payments on various bonds that are chosen so that the total cash flows exactly match the liability amounts.
Reddington Immunization
Meets three conditions1. PV Assets = PV Liabilities2. Duration of Assets = Duration of Liabilities3. Convexity of Assets >= Convexity of LiablilitiesTo solve use 1 and 2 as a system of equations in two variables.
Full Immunization
Same as above, but Condition # 3 needs to be satisfied for all i, not just I nought.
derivative
A derivative is a security with a price that is dependent upon or derived from one or more underlying assets.
over the counter market
sold through decentralized trading markets, less fees and regulations. Brokers deal with each other and means less transparency. AKA unlisted stock. Usually smaller companies do this.In general, the reason for which a stock is traded over-the-counter is usually because the company is small, making it unable to meet exchange listing requirements. Also known as “unlisted stock”, these securities are traded by broker-dealers who negotiate directly with one another over computer networks and by phone.
short selling
Borrowing an asset and promising to return it in a different time. Immediately sell the object and get current market price. Make money if the value of the asset decreases.
short position
make money if the price decreases
long position
make money if the price increase
lease rate for a short sell
The amount of money paid over a specified time period for the rental of an asset, such as real property or an automobile. The lease rate that the lender earns from allowing someone else to use his property compensates him for not being able to put that property to another use during the term of the lease.
maintenance margin
Maintenance margin is the minimum amount of equity that must be maintained in a margin account. In the context of the NYSE and FINRA, after an investor has bought securities on margin, the minimum required level of margin is 25% of the total market value of the securities in the margin account. Keep in mind that this level is a minimum, and many brokerages have higher maintenance requirements of 30-40%.
margin
Borrowed money that is used to purchase securities. This practice is referred to as “buying on margin”.
margin call
A broker’s demand on an investor using margin to deposit additional money or securities so that the margin account is brought up to the minimum maintenance margin. Margin calls occur when your account value depresses to a value calculated by the broker’s particular formula.
mark to market
A measure of the fair value of accounts that can change over time, such as assets and liabilities. Mark to market aims to provide a realistic appraisal of an institution’s or company’s current financial situation.Check the current value of an asset based on market conditions.
arbitrage
The simultaneous purchase and sale of an asset in order to profit from a difference in the price. It is a trade that profits by exploiting price differences of identical or similar financial instruments, on different markets or in different forms. Arbitrage exists as a result of market inefficiencies; it provides a mechanism to ensure prices do not deviate substantially from fair value for long periods of time.
risk averse
Losing an amount of money is worse than gaining the same amount of money.A description of an investor who, when faced with two investments with a similar expected return (but different risks), will prefer the one with the lower risk.
Long Call (Definition)
Right, but not obligaion to buy at a strike price
Long Call (Max Profit and Loss)
infinite and FV(Premium)
Long Call (Position in Underlying Asset)
Long
Short Call (Definition)
Obligation to sell at strike price
Short Call (Max Profit and Loss)
FV(Premium) and infinite
Short Call (Position in Underlying Asset)
Short
Long Put (Definition)
Right to sell at the strike price
Long Put (Max Profit and Loss)
K - FV(Premium) and FV(Premium)
Long Put (Position in Underlying Asset)
Short
Short Put (Definition)
Obligation to buy at strike price
Short Put (Max Profit and Loss)
FV(Premium) and - K + FV(Premium)
Short Put (Position in Underlying Asset)
Long
European Option
Exercise option only at expiration date
American Option
Exercise anytime during the contract
Bermuda option
Exercise during specified periods
Option Writer
Seller of an option
in-the-money
would have a positive payoff if the option was exercised immediately
at-the-money
would have no payoff if the option was exercised immediately
out-of-the-money
would have a negative payoff if the option was exercised immediately
covered call
Buying the underlying asset and selling a call, so long in the asset and short in the call.For example, let’s say that you own shares of the TSJ Sports Conglomerate and like its long-term prospects as well as its share price but feel in the shorter term the stock will likely trade relatively flat, perhaps within a few dollars of its current price of, say, $25. If you sell a call option on TSJ for $26, you earn the premium from the option sale but cap your upside
naked writing
Selling an option, but not holding the underlying asset.Naked writers are exposed to additional risk, since they hold no position with which to hedge against the adverse movement of the underlying security’s price. If the options contract is exercised, the naked writer would be forced to buy or sell a certain number of shares at a potentially undesirable price. Naked writers try to profit by receiving premiums for writing the contracts without the need to purchase share lots.
put-call parity
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Outright Purchase
Price: S0, Payment and item are received at time 0
Fully Leverage Purchase or Deferred Purchase
Price S0 * interest; Payment at time T and item at time 0
Prepaid Forward Contract
F(t) -> Pay at time 0, but receive item at time T
Forward Contract
Price accumulated value of F(t) -> pay and receive item at time T
Cost of Carry
Costs incurred as a result of an investment position. These costs can include financial costs, such as the interest costs on bonds, interest expenses on margin accounts and interest on loans used to purchase a security, and economic costs, such as the opportunity costs associated with taking the initial position.
Dividend Discount Model
Used to value stocks.Formula isDividend / (Discount Rate - Dividend Growth Rate)
Future Value of a Contract
AV (Stock) - AV (Dividends)
Future Value of a Contract (exponential)
Original Amount * e ^ (i - dividend rate) * t
swap
Traditionally, the exchange of one security for another to change the maturity (bonds), quality of issues (stocks or bonds), or because investment objectives have changed. Recently, swaps have grown to include currency swaps and interest rate swaps.
notional amount
amount of the item controlled by the contract.
swap spread
The difference between the negotiated and fixed rate of a swap. The spread is determined by characteristics of market supply and creditor worthiness.
deferred swap
Only the payment is deferred, so the goods come today.
Hedging
Making an investment to reduce the risk of adverse price movements in an asset. Normally, a hedge consists of taking an offsetting position in a related security, such as a futures contract.
Non-Diversifiable Risk
The risk inherent to the entire market or an entire market segment. Systematic risk, also known as “undiversifiable risk,” “volatility” or “market risk,” affects the overall market, not just a particular stock or industry. This type of risk is both unpredictable and impossible to completely avoid. It cannot be mitigated through diversification, only through hedging or by using the right asset allocation strategy.
Diversifiable Risk
Company- or industry-specific hazard that is inherent in each investment. Unsystematic risk, also known as “nonsystematic risk,” “specific risk,” “diversifiable risk” or “residual risk,” can be reduced through diversification. By owning stocks in different companies and in different industries, as well as by owning other types of securities such as Treasuries and municipal securities, investors will be less affected by an event or decision that has a strong impact on one company, industry or investment type. Examples of unsystematic risk include a new competitor, a regulatory change, a management change and a product recall.
bull spread
Long Call (k1) + Short Call (k2) K1
bear spread
Short Call (k1) + Long Call (k2) K1
Short Synthetic Forward
Long Put + Short Call
Long Synthetic Forward
Short Put + Long Call
Collar
Long Put + Short CallThe effect is a collar around a certain price, but if the stock moves quite a bit down then you make money and if it moves up then you lose money.Insurance on stock
Strangle
Long Put + Long Call , but the strike prices differYou make money whether the stock moves right or left.
Straddle
Long Put + Long Call and the strike prices are the same.
Written Straddle
Short Put + Short Call, basically betting against any change in the stock price.
Collar Width
The difference between the strike prices of the two options in a collar.
Collared Stock
A collar provide insurance on a stock. If the price goes down then the collar provides the difference and if the price goes up then the you lose some if it goes above the short call.
zero-cost collar
A collar whose premium on the long put is offset by the premium received by the premium short call.
ratio spread
Long and short call and puts, so there is little payment and if the stock moves only a little up, then you get money, else you end up losing money on the deal and it can be a lot of money because you are holding multiple short positions.An options strategy in which an investor simultaneously holds an unequal number of long and short positions. A commonly used ratio is two short options for every option purchased.
box spread
a strategy that attempts to exploit an arbitrage situation and buy a bearish and bullish position to create an automatic gain after a certain amount of time. Only for advanced traders.
vertical spread
An options trading strategy with which a trader makes a simultaneous purchase and sale of two options of the same type that have the same expiration dates but different strike prices.So it is a bear or bull spread where you make money on a small change but sell off a large change to receive short term money.
symmetric butterfly spread
benefit from a neutral market, with limited downside risk. purchase one option in the money and one out of the money. Also purchase two at the money and difference between all three need to be the same, so 1 45s, 2 50s and 1 55s.
asymmetric butterfly spread
lambda = (k3 - k2) / (k3 - k1)Buy lambda of low positionBuy one of middle groundBuy (1 - lambda) of high position
