Effective Interest Rate

The rate

i

such that an investment of 1 grows to $1+i$ at the end of one period. It reflects the actual growth per period after compounding.

Nominal Interest Rate

A rate

i^{(m)}

that is compounded

m

times per year. The effective rate per compounding period is

i^{(m)}/m

. The effective annual rate is

(1 + i^{(m)}/m)^m - 1

.

Force of Interest

The continuously compounded interest rate, denoted

\delta

. Related to the effective rate by

\delta = \ln(1+i)

. An accumulation over time

t

at constant force is

e^{\delta t}

.

Present Value

The current worth of a future cash flow, obtained by discounting at the appropriate interest rate.

PV = FV \cdot v^n

, where

v = 1/(1+i)

is the discount factor.

Annuity-Immediate

A series of equal payments made at the end of each period. The present value of an

n

-period annuity-immediate is

a_{\overline{n}|} = (1 - v^n) / i

.

Annuity-Due

A series of equal payments made at the beginning of each period. The present value is

\ddot{a}_{\overline{n}|} = (1 - v^n) / d

, where

d = i/(1+i)

is the discount rate.

Amortization

The process of repaying a loan through a series of payments that cover both interest and principal. Each payment reduces the outstanding balance until the loan is fully repaid.

Macaulay Duration

The weighted average time to receipt of a bond s cash flows, using present values as weights. Measured in time units (years). Used to assess interest rate sensitivity.

Modified Duration

Macaulay duration divided by

(1+y)

, where

y

is the yield per period. It approximates the percentage change in bond price for a small change in yield:

\Delta P/P \approx -D_{\text{mod}} \cdot \Delta y

.

Immunization

A strategy to protect a portfolio against interest rate changes by matching the duration (and sometimes convexity) of assets and liabilities. Ensures the surplus is insensitive to small yield shifts.

Put-Call Parity

For European options:

C - P = PV(F) - PV(K)

, where

C

is the call price,

P

is the put price,

F

is the forward price, and

K

is the strike price. A fundamental no-arbitrage relationship.

Forward Rate

The interest rate agreed upon today for a loan that begins at a future date. The forward rate from time

s

to time

t

can be derived from spot rates:

(1+s_t)^t = (1+s_s)^s \cdot (1+f)^{t-s}

.

FM Glossary

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