CL’s Handy
Formula Sheet
(Useful formulas from Marcel Finan’s FM/2 Book)
Compiled by Charles Lee
8/19/2010
Interest
Interest
Simple
Compound
Discount
Simple
Compound
The Method of Equated Time
a(t)
Period
when
greater
Interest Formulas
Force of Interest
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”
o Uses exact days
o 365 days in a nonleap year
o 366 days in a leap year (divisible by 4)
 Ordinary
o “30/360”
o All months have 30 days
o Every year has 360 days
o
 Banker’s Rule
o “actual/360”
o Uses exact days
o Every year has 360 days
Basic Formulas
Annuities
Basic Equations
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
Perpetuity
Immediate
Due
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
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
Perpetuity
Due
General
P, P+Q,…,
P+(n-1)Q
Increasing
P=Q=1
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 .
Decreasing
P=n
Q = -1
Under compound interest, i.e.
Perpetuity
, 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
Invest 1 at the beginning of each period for n periods at rate i, with
interest reinvested at rate j
Uniqueness of IRR
Theorem 1

Theorem 2
 Let Bt be the outstanding balance at time t, i.e.
o
o
 Then
o
o
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+∑ct(1-t)
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
If we assume uniform cash
flow, then
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
o
 Makeham Formula
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 tth coupon has just been paid, then
Let It denote the interest earned after the tth coupon has been made
Let Pt denote the corresponding principal adjustment portion
Date
Coupon
Interest
earned
Amount for
Amortization
of Premium
Book Value
June 1, 1996
Dec 1, 1996
June 1, 1997
Approximation Methods of Bonds’ Yield Rates
Exact
Where
Approximation
Power series
expansion
Bond Salesman’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
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)


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.



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
o The outstanding loan balance at any time is equal to the
present value at that time of the remaining payments
Retrospective Method
o The outstanding loan balance at any time is equal to the
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
Interest paid
Principal
repaid
Outstanding loan
balance
…
…
…
…
…
…
…
…
…
…
Total
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
80
80
80
80
Sinking
fund
deposit
221.92
221.92
221.92
221.92
Interest
earned
on
sinking
fund
0
17.75
36.93
57.64
Amount
in
sinking
fund
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
Duration
 Method of Equated Time (average term-to-maturity)
o
where R1,R2,…,Rn are a series of payments

made at times 1,2,…,n
Macaulay Duration
o

, where
o
is a decreasing function of i
Volatility (modified duration)
o
o
o if P(i) is the current price of a bond, then
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 rate of short sale

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
Inflation
Given i' = real rate, i = nominal rate, r = inflation rate,
Fischer Equation
A common approximation for the real interest rate:
of the portfolio is
Similarly, the convexity 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 A1,A2,…,An and cash outflows L1,L2,…,Ln.
Then the net cash flow at time t is
Immunization conditions
 We need a local minimum at i

o The present value of cash inflows (assets) should be
equal to the present value of cash outflows (liabilities)

o The modified duration of the assets is equal to the
modified duration of the liabilities

o The convexity of PV(Assets) should be greater than the
convexity of 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 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 forward n years from now satisfied
where it is the t-year spot rate
Option Styles
European option – Holder can exercise the option only on the
expiration date
American option – Holder can exercise the option anytime during the
life of the option
Bermuda option – 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 + buy put
Cap – short + buy call
Covered Call – stock + write call = write put
Covered Put – short +write put = write call
Profit
Price at Maturity
Cash-and-Carry – buy asset + short forward contract
Synthetic Forward – a 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)
-Forward Price
Guaranteed price
PT-K
Unlimited
Forward Price
Short(sell)
Guaranteed price
K-PT
-FV(Premium)
Unlimited
-FV(Premium)
FV(Premium) – Strike Price
Unlimited
FV(Premium)
Strike Price – FV(Premium)
FV(Premium)
Long(buy)
Short(sell)
Short(sell)
Long(buy)
Insures against high price
Sells insurance against high price
Insures against low price
Sells insurance against low price
max{0,PT-K}
-max{0,PT-K}
max{0,K-PT}
-max{0,K-PT}
(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 K1, sell call with strike
price K2>K1 and same expiration date
o Bull Put
 Buy put with strike price K1, sell put with strike
price K2>K1 and same expiration date
o Two profits are equivalent  (Buy K1 call) + (Sell K2
call) = (Buy K1 put) + (Sell K2 put)
o Profit function

Bear Spread
o Investor speculates stock price will decrease
o Exact opposite of a bull spread
o Bear Call
 Sell K1 call, buy K2 call, where 0<K1<K2
o Bear Put
 Sell K1 put, buy K2 put, where 1<K1<K2
Long Box Spread
Bull Call Spread Bear Put Spread
Synthetic Long Forward
Buy call at K1
Sell put at K1
Synthetic Short Forward
Sell call at K2
Buy put at K2
Regardless of spot price at expiration, the box spread guarantees a cash
flow of K2-K1 in the future.
Net premium of acquiring this position is PV(K2-K1)
If K1<K2, then lending money
Invest PV(K2-K1), get K2-K1
If K1>K2, then borrow money
Get PV(K1-K2), pay K1-K2
Butterfly Spread
An insured written straddle
 Let K1<K2<K3
o Written straddle
 Sell K2 call, sell K2 put
o Long strangle
 Buy K1 call, buy K3 put
 Profit
o Let F
o
Profit
Payoff
K2-K1
K2-K1-FV[…
PT
PT
K1
K2
-FV[…
Asymmetric Butterfly Spread
Collar
Used to speculate on the decrease of the price of an asset
 Buy K1-strike at-the-money put
 Sell K2-strike out-of-the-money call
 K2>K1
 K2-K1 = collar width
Profit Function
Collared Stock
Collars can be used to insure assets we own
 Buy index
 Buy at-the-money K1 put
 Buy out-of-the-money K2 call
 K1<K2
Profit Function
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
Strangle
A straddle with lower premium cost
 Buy K1-strike call
 Buy K2 strike put
 K1<K2
Profit Function
Profit Function
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
 Let FP0,T denote the prepaid forward price for an asset bought
at time 0 and delivered at time T
 If no dividends, then FP0,T = S0, otherwise arbitrage
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 eδT shares at
time T
Forward Contracts
 Discrete dividends
o
 Continuous dividends
o
 Forward premium = F0,T / S0
 The annualized forward premium α satisfies
o


If no dividends, then α=r
If continuous dividends, then α=r-δ
Financial Engineering of Synthetics
 (Forward) = (Stock) – (Zero-coupon bond)
o Buy e-δT shares of stock
o Borrow S0e-δT to pay for stock
o Payoff = PT – F0,T
 (Stock) = (Forward) + (Zero-coupon bond)
o Buy forward with price F0,T = S0e(r-δ)T
o Lend S0e-δT
o Payoff = PT
 (Zero-coupon bond) = (Stock) – (Forward)
o Buy e-δT shares
o Short one forward contract with price F0,T
o Payoff = F0,T
o If the rate of return on the synthetic bond is i, then
 S0e(i-δ)T = F0,T or
 Implied repo rate