SOA Exam FM (Financial Mathematics) Glossary

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Accumulation Function: Accumulation function is a function that gives the accumulated value at time t of a single unit invested at time 0, incorporating the effect of interest over the period. $a(t) = e^{\int_0^t \delta_s\,ds}$
Amortization: Amortization is the process of repaying a loan through a series of periodic payments, each consisting of an interest component and a principal repayment component, such that the outstanding balance reaches zero at the end of the loan term.
Annuity-Due: Annuity-due is a series of equal payments made at the beginning of each period for a specified number of periods, with its present value denoted by a-double-dot. $\ddot{a}_{\overline{n}|} = \frac{1 - v^n}{d}$
Annuity-Immediate: Annuity-immediate is a series of equal payments made at the end of each period for a specified number of periods, with its present value denoted by a-angle-n. $a_{\overline{n}|} = \frac{1 - v^n}{i}$

Bond Price: Bond price is the present value of all future cash flows from a bond, including periodic coupon payments and the par value at maturity, discounted at the yield rate. $P = Fr\,a_{\overline{n}|} + Cv^n$
Book Value: Book value of a bond is the present value of remaining future cash flows (coupons and redemption value) at the original yield rate, representing the theoretical value at which the bond is carried on the investor's books.

Callable Bond: Callable bond is a bond that gives the issuer the right to redeem (call) the bond before its stated maturity date, typically at a specified call price. An investor prices a callable bond using the call date that produces the lowest price.
Convexity: Convexity is the second-order measure of the sensitivity of a bond's price to changes in interest rates, capturing the curvature of the price-yield relationship and improving duration-based price change estimates. $C = \frac{1}{P}\frac{d^2P}{di^2}$

Decreasing Annuity: Decreasing annuity is an annuity whose payments decrease by a fixed amount each period, with the first payment equal to n and subsequent payments declining by 1 each period. $(Da)_{\overline{n}|} = \frac{n - a_{\overline{n}|}}{i}$
Discount Factor: Discount factor is the present value of one monetary unit due one period in the future, representing the time value of money over a single period. $v = \frac{1}{1+i}$
Discount Rate: Discount rate (d) is the rate of interest paid at the beginning of the period as a fraction of the amount at the end of the period, related to the effective interest rate by d = i/(1+i). $d = \frac{i}{1+i} = 1 - v$
Duration (Macaulay): Macaulay duration is the weighted average time until a bond's cash flows are received, where the weights are the present values of each cash flow as a proportion of the bond's total price. $D_{\text{Mac}} = \frac{\sum_{t=1}^{n} t \cdot v^t \cdot CF_t}{P}$

Effective Annual Rate: Effective annual rate is the actual rate of interest earned or paid over a one-year period after accounting for the effect of compounding, used to compare interest rates with different compounding frequencies. $i = \left(1 + \frac{i^{(m)}}{m}\right)^m - 1$

Force of Interest: Force of interest is the instantaneous rate of interest at a given point in time, expressed as the continuous compounding rate and serving as the natural logarithm of 1 plus the effective annual rate. $\delta = \ln(1+i)$
Forward Rate: Forward rate is the interest rate applicable to a future period as implied by the current term structure of spot rates, representing the rate that equates investment over consecutive periods. $(1+s_n)^n = (1+s_{n-1})^{n-1}(1+f_{n-1,n})$
Future Value: Future value is the value at a specified future date of a present amount of money, accumulated at a given interest rate over the intervening period. $FV = PV \cdot (1+i)^n$

Immunization: Immunization is a fixed-income portfolio strategy that matches the duration and present value of assets and liabilities to protect the portfolio's surplus from small parallel shifts in the yield curve.
Increasing Annuity: Increasing annuity is an annuity whose payments increase by a fixed amount each period, with the first payment equal to 1 and subsequent payments increasing by 1 each period. $(Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|} - nv^n}{i}$
Interest Rate Swap: Interest rate swap is a derivative contract in which two parties exchange interest payment obligations, typically swapping a fixed rate for a floating rate on a notional principal amount over a specified term.
Internal Rate of Return: Internal rate of return (IRR) is the interest rate at which the net present value of all cash flows from an investment equals zero, serving as a measure of investment profitability.

Modified Duration: Modified duration is a measure of the percentage change in a bond's price for a one-unit change in yield, derived by dividing Macaulay duration by one plus the periodic yield. $D_{\text{mod}} = \frac{D_{\text{Mac}}}{1+i}$

Net Present Value: Net present value (NPV) is the sum of the present values of all cash inflows and outflows associated with a project, discounted at a specified required rate of return. $\text{NPV} = \sum_{t=0}^{n} \frac{CF_t}{(1+i)^t}$
Nominal Interest Rate: Nominal interest rate is a stated annual rate that is compounded m times per year, where the effective rate per compounding period is the nominal rate divided by m. $i^{(m)} = m\left[(1+i)^{1/m} - 1\right]$

Perpetuity: Perpetuity is an annuity that makes payments indefinitely, with its present value equal to the payment amount divided by the interest rate for a perpetuity-immediate. $a_{\overline{\infty}|} = \frac{1}{i}$
Present Value: Present value is the current worth of a future sum of money or stream of cash flows, discounted at a specified interest rate to reflect the time value of money. $PV = FV \cdot v^n = \frac{FV}{(1+i)^n}$

Redington Immunization: Redington immunization is a portfolio management technique requiring that the present value, duration, and convexity of assets match or exceed those of liabilities, protecting the surplus against small parallel yield curve shifts.

Sinking Fund: Sinking fund is a fund into which periodic deposits are made so that the accumulated value at a specified future date equals a target amount, often used to repay a loan or replace an asset. $s_{\overline{n}|} = \frac{(1+i)^n - 1}{i}$
Spot Rate: Spot rate is the yield to maturity on a zero-coupon bond, representing the annualized rate of return for a single cash flow received at a specific future date.

Term Structure of Interest Rates: Term structure of interest rates is the relationship between spot rates and their maturities, commonly depicted as a yield curve, which reflects market expectations about future interest rates, risk premiums, and liquidity preferences.

Yield Rate: Yield rate is the interest rate that equates the present value of cash inflows with the present value of cash outflows for a financial transaction, reflecting the investor's rate of return.
Yield to Maturity: Yield to maturity is the total annualized rate of return anticipated on a bond if held until it matures, accounting for all coupon payments, the par value at maturity, and the current market price.