(Useful formulas from Marcel Finan’s FM/2 Book) Compiled by Charles Lee 8/19/2010
Interest Interest Simple Compound a(t) Period when greater
Discount Simple Compound
Interest Formulas
Force of Interest
The Method of Equated Time
The Rule of 72 The time it takes an investment of 1 to double is given by
Date Conventions Recall knuckle memory device. (February has 28/29 days) Exact o “actual/actual” Uses exact days o 365 days in a nonleap year o 366 days in a leap year (divisible by 4) o Ordinary o “30/360” All months have 30 days o Every year has 360 days o
o
Banker’s Rule o “actual/360” Uses exact days o Every year has 360 days o
Basic Formulas
Annuities Basic Equations Immediate
Due
Perpetuity
Annuities Payable Less Frequently than Interest is Convertible Let = number of interest conversion periods in one payment period Let = total number of conversion periods Hence the total number of annuity payments is
Immediate
Due
Perpetuity
Annuities Payable More Frequently than Interest is Convertible Let = the number of payments per interest conversion period Let = total number of conversion periods Hence the total number of annuity payments is
Coefficient of
is the total amount paid during on interest
conversion period
Immediate
Due
Perpetuity
Continuous Annuities
Geometric
a=1 r = 1+k k≠i If k = i a=1 r = 1-k k≠i If k=i
Varying Annuities Arithmetic Immediate
General P, P+Q,…, P+(n-1)Q Increasing P=Q=1
Decreasing P=n Q = -1
Due
Perpetuity
Perpetuity
Continuously Varying Annuities Consider an annuity for n interest conversion periods in which payments are being made continuously at the rate and the interest rate is variable with force of interest .
Under compound interest, i.e. , the above becomes
Rate of Return of an Investment Rate of Return of an Investment Yield rate, or IRR, is the interest rate at which
Interest Reinvested at a Different Rate Invest 1 for n periods at rate i, with interest reinvested at rate j
Hence yield rates are solutions to NPV(i)=0
Invest 1 at the end of each period for n periods at rate i, with interest reinvested at rate j
Discounted Cash Flow Technique
Uniqueness of IRR Theorem 1
Theorem 2 Let Bt be the outstanding balance at time t, i.e.
o o
Then
o o
Invest 1 at the beginning of each period for n periods at rate i, with interest reinvested at rate j
Dollar-Weighted Interest Rate A = the amount in the fund at the beginning of the period, i.e. t=0 B = the amount in the fund at the end of the period, i.e. t=1 I = the amount of interest earned during the period ct = the net amount of principal contributed at time t C = ∑ct = total net amount of principal contributed during the period i = the dollar-weighted rate of interest Note: B = A+C+I Exact Equation
Simple Interest Approximation
Summation Approximation The summation term is tedious.
Define
“Exposure associated with i"= A+∑c t(1-t)
If we assume uniform cash
flow, then
Time-Weighted Interest Rate Does not depend on the size or timing of cash flows. Suppose n-1 transactions are made during a year at times t1,t2,…,tn-1. Let jk = the yield rate over the kth subinterval Ct = the net contribution at exact time t Bt = the value of the fund before the contribution at time t Then
The overall yield rate i for the entire year is given by
Bonds Notation P = the price paid for a bond F = the par value or face value C = the redemption value r = the coupon rate Fr = the amount of a coupon payment g = the modified coupon rate, defined by Fr/C i = the yield rate n = the number of coupons payment periods K = the present value, compute at the yield rate, of the redemption value at maturity, i.e. K=Cvn G = the base amount of a bond, defined as G=Fr/i. Thus, G is the amount which, if invested at the yield rate i, would produce periodic interest payments equal to the coupons on the bond Quoted yields associated with a bond 1) Nominal Yield a. Ratio of annualized coupon rate to par value 2) Current Yield a. Ratio of annualized coupon rate to original price of the bond 3) Yield to maturity a. Actual annualized yield rate, or IRR Pricing Formulas Basic Formula
o
Premium/Discount Formula o
Base Amount Formula
Makeham Formula
o
o
Yield rate and Coupon rate of Different Frequencies Let n be the total number of yield rate conversion periods. Case 1: Each coupon period contains k yield rate periods
o
Case 2: Each yield period contains m coupon periods o
Amortization of Premium or Discount Let Bt be the book value after the t th th coupon has just been paid, then
Let It denote the interest earned after the t th th coupon has been made Let Pt denote the corresponding principal adjustment portion
Date
June 1, 1996 Dec 1, 1996 June 1, 1997
Coupon
Interest earned
Amount for Amortization of Premium
Book Value
Approximation Methods of Bonds’ Yield Rates Exact
Where
Approximation
Power series expansion
Bond Salesma Salesman’s n’s Method
Equivalently
Valuation of Bonds between Coupon Payment Dates
Premium or Discount between Coupon Payment Dates
The purchase price for the bond is called the flat price and is denoted by The price for the bond is the book value, or market price, and is denoted by The part of the coupon the current holder would expect to receive as interest for the period is called the accrued interest or accrued coupon and is denoted by From the above definitions, it is clear that
$
Flat price
Book value
Theoretical Method 1 The flat price should be the book value Bt after the preceding coupon accumulated by (1+i)k
2
3
4
Practical Method Uses the linear approximation
Semi-theoretical Method Standard method of calculation by the securities industry. The flat price is determined as in the theoretical method, and the accrued coupon is determined as in the practical method.
Callable Bonds The investor should assume that the issuer will redeem the bond to the disadvantage of the investor. If the redemption value is the same at any call date, including the maturity date, then the following general principle will hold: 1) The call date will be at the earliest date possible if the bond was sold at a premium, which occurs when the yield rate is smaller than the coupon rate (issuer would like to stop repaying the premium via the coupon payments as soon as possible) 2) The call date will be at the latest date possible if the bond was sold at a discount, which occurs when the yield rate is larger than the coupon rate (issuer is in no rush to pay out the redemption value)
Serial Bonds Serial bonds are bonds issued at the same time but with different maturity dates. Consider an issue of serial bonds with m different redemption dates. By Makeham’s formula , where
Loan Repayment Methods Amortization Method
Prospective Method The outstanding loan balance at any time is equal to the o present value at that time of the remaining payments Retrospective Method The outstanding loan balance at any time is equal to the o original amount of the loan accumulated to that time less the accumulated value at that time of all payments previously made
Consider a loan of at interest rate i per period being repaid with payments of 1 at the end of each period for n periods. Period
Payment amount
…
…
…
…
Total
Interest paid
Principal repaid
…
…
…
…
Outstanding loan balance
…
…
Sinking Fund Method Whereas with the amortization method the payment at the end of each period is
, in the sinking fund method, the borrower both deposits
into the sinking fund and pays interest i per period to the lender.
Example Create a sinking fund schedule for a loan of $1000 repaid over four years with i = 8%. If R is the sinking fund deposit, then Period
0 1 2 3 4
Interest paid
Sinking fund deposit
Interest earned on sinking fund
Amount in sinking fund
80 80 80 80
221.92 221.92 221.92 221.92
0 17.75 36.93 57.64
221.92 461.59 720.44 1000
Net amount of loan
1000 778.08 538.41 279.56 0
Measures of Interest Rate Sensitivity Stock
Preferred Stock o Provides a fixed rate of return o Price is the present value of future dividends of a perpetuity o
Common Stock o Does not earn a fixed dividend rate o Dividend Discount Model o Value of a share is the present value of all future dividends o
Duration Method of Equated Time (average term-to-maturity)
Inflation Given i' = i' = real rate, i = nominal rate, r = r = inflation rate,
Fischer Equation A common approximation for the real interest rate:
where R 1,R 2,…,R n are a series of payments
made at times 1,2,…,n Macaulay Duration
, where
o
is a decreasing function of i Volatility (modified duration) o
o
Short Sales In order to find the yield rate on a short sale, we introduce the following notation: M = Margin deposit at t=0 S0 = Proceeds from short sale St = Cost to repurchase stock at time t dt = Dividend at time t i = Periodic interest rate of margin account j = Periodic yield yield rate of short sale sale
o
o o
if P(i) is t he current price of a bond, then
Convexity o
Modified Duration and Convexity of a Portfolio Consider a portfolio consisting of n bonds. Let bond K have a current price , modified duration , and convexity . Then the current value of the portfolio is The modified duration
Similarly, the convexity
of of the portfolio is
of of the portfolio is
Thus, the modified duration and convexity of a portfolio is the weighted average of the bonds’ modified durations and convexities respectively, using the market values of the bonds as weights.
Redington Immunization Effective for small changes in interest rate i Consider cash inflows A 1,A2,…,An and cash outflows L 1,L2,…,Ln. Then the net cash flow at time t is
Immunization conditions We need a local minimum at i
o
o
o
The present value of cash inflows (assets) should be equal to the present value of cash outflows (liabilities) The modified duration of the assets is equal to the modified duration of the liabilities
The convexity of PV(Assets) should be greater than the convexity of PV(Liabilities), PV(Liabilities), i.e. asset growth > liability growth
Full Immunization Effective for all changes in interest rate i A portfolio is fully immunized if
Full immunization conditions conditions for a single liability cash flow 1) 2) 3) Conditions (1) and (2) lead to the system
where δ=ln(1+i) and k=time of liability
Dedication Also called “absolute matching” In this approach, a company structures an asset portfolio so that the cash inflow generated from assets will exactly match the cash outflow from liabilities.
Interest Yield Curves The k -year -year forward n years from now satisfied
where it is the t -year -year spot rate
Option Styles European option – option – Holder Holder can exercise the option only on the expiration date American option – option – Holder Holder can exercise the option anytime during the life of the option Bermuda option – option – Holder Holder can exercise the option during certain prespecified dates before or at the expiration date
Call Put
Buy
Write
↑ ↓
↓ ↑
Long Forward
Short Forward
Long Call
Short Call
Payoff
Floor – own – own + buy put Cap – Cap – short short + buy call Covered Call – Call – stock stock + write call = write put Covered Put – Put – short short +write put = write call
Profit Price at Maturity
Cash-and-Carry – Cash-and-Carry – buy buy asset + short forward contract Synthetic Forward – Forward – aa combination of a long call and a short put with the same expiration date and strike price
Long Put
Short Put
Fo,T = no arbitrage forward price Call(K,T) = premium of call
Put-Call Parity
Derivative Position Long Forward Short Forward Long Call Short Call Long Put Short Put
Maximum Loss
Maximum Gain
Strategy
Payoff
Unlimited
Position wrt Underlying Asset Long(buy) Long(buy)
-Forward Price
Guaranteed Guaranteed price
P T-K
Unlimited
Forward Price
Short(sell)
Guaranteed Guaranteed price
K-PT
-FV(Premium) Unlimited -FV(Premium) FV(Premium) – FV(Premium) – Strike Strike Price
Unlimited FV(Premium) Strike Price – Price – FV(Premium) FV(Premium) FV(Premium)
Long(buy) Long(buy) Short(sell) Short(sell) Long(buy) Long(buy)
Insures against high price Sells insurance against high price Insures against low price Sells insurance against low price
max{0,P T-K} -max{0,PT-K} max{0,K-PT} -max{0,K-P -max{0,K-P T}
(Buy index) + (Buy put option with strike K) = (Buy call option with strike K) + (Buy zero-coupon bond with par value K) (Short index) + (Buy call option with strike K) = (Buy put option with strike K) + (Take loan with maturity of K)
Spread Strategy Creating a position consisting of only calls or only puts, in which some options are purchased and some are sold Bull Spread o Investor speculates stock price will increase o Bull Call Buy call with strike price K 1, sell call with strike price K 2>K 1 and same expiration date o Bull Put Buy put with strike price K 1, sell put with strike price K 2>K 1 and same expiration date o Two profits are equivalent (Buy K 1 call) + (Sell K 2 call) = (Buy K 1 put) + (Sell K 2 put) o Profit function
PT K 1
K 2
-FV[…
o
K 2-K 1-FV[… PT
Butterfly Spread An insured written straddle Let K 1
Profit
Payoff
Bull Call Spread Bear Put Spread Spread Synthetic Long Forward Buy call at K 1 Sell put at K 1 Synthetic Short Forward Sell call at K 2 Buy put at K 2 Regardless of spot price at expiration, the box spread guarantees a cash flow of K 2-K 1 in the future. Net premium of acquiring this position is PV(K 2-K 1) If K 1K 2, then borrow money Get PV(K 1-K 2), pay K 1-K 2
Bear Spread o Investor speculates stock price will decrease o Exact opposite of a bull spread o Bear Call Sell K 1 call, buy K 2 call, where 0
K 2-K 1
Long Box Spread
Asymmetric Butterfly Spread
Collar Used to speculate on the decrease of the price of an asset Buy K 1-strike at-the-money put Sell K 2-strike out-of-the-money call K 2>K 1 K 2-K 1 = collar width
Profit Function
Collared Stock Collars can be used to insure assets we own Buy index Buy at-the-money K 1 put Buy out-of-the-money K 2 call K 1
Profit Function
Zero-cost Zero-cost Collar A collar with zero cost at time 0, i.e. zero net premium
Straddle A bet on market volatility Buy K-strike call Buy K-strike put
Profit Function
Strangle A straddle with lower premium cost Buy K 1-strike call Buy K 2 strike put K 1
Profit Function
Equity-linked Equity-linked CD (ELCD)
Can financially engineer an equivalent by Buy zero-coupon bond at discount Use the difference to pay for an at-the-money call option
Prepaid Forward Contracts on Stock P Let F 0,T denote the prepaid forward price for an asset bought at time 0 and delivered at time T P If no dividends, then F 0,T = S0, otherwise arbitrage opportunities opportunities exist If discrete dividends, then
o
If continuous dividends, then o Let δ=yield rate, then the
and 1 share at time 0 grows to e
δT
time T
Forward Contracts Discrete dividends
o
Continuous dividends o
Forward premium = F 0,T / S0 The annualized forward premium α satisfies o
Financial Engineering of Synthetics (Forward) = (Stock) – (Stock) – (Zero-coupon (Zero-coupon bond) -δT o Buy e shares of stock -δT o Borrow S0e to pay for stock o Payoff = P T – F – F0,T (Stock) = (Forward) + (Zero-coupon bond) (r-δ)T (r-δ)T o Buy forward with price F 0,T = S0e -δT o Lend S0e o Payoff = P T (Zero-coupon bond) = (Stock) – (Stock) – (Forward) (Forward) -δT o Buy e shares o Short one forward contract with price F 0,T o Payoff = F0,T o If the rate of return on the synthetic bond is i, then (i-δ)T (i-δ)T = F0,T or S0e
If no dividends, then α=r If continuous dividends, then α=r α=r -δ
shares at
Implied repo rate
