Interest and Discounting¶

Definition¶

Under compound interest at annual effective interest rate $i>-1$$-1$, an amount $1$ invested at time $0$ grows to $(1+i)^t$ at time $t\ge0$ (in years). Equivalently, a payment of $1$ at time $t$ has present value $v^t$ at time $0$, where

Here $\log$ is the natural logarithm, so $(1+i)^t=\mathrm{e}^{\delta t}$ and $v^t=\mathrm{e}^{-\delta t}$${\mathrm{e}}^{-\delta t}$. The present value of payments $c_k$ at times $t_k$ is $\displaystyle \sum_kc_kv^{t_k}$.

Interpretation¶

The four quantities $i$, $v$, $d$ and $\delta$ describe one and the same interest basis; knowing any one of them fixes the others. They differ in what they measure:

• $i$ is interest paid at the end of the year on $1$ invested at the start;
• $d=iv=i/(1+i)$ is the same interest paid at the start of the year: borrowing $1$ repayable in a year, one receives $v=1-d$ now;
• $\delta$ is the instantaneous growth rate, $\delta=\frac{\mathrm{d}}{\mathrm{d}t}\log\big((1+i)^t\big)$$i{)}^t)$, and it is what turns discounting into an exponential function $\mathrm{e}^{-\delta t}$ for payments at any real time $t$.

The figure places the rates of the first and third examples, all for the same basis $i=0.04$$0.04$, on one number line. They are different ways of quoting the same interest, so they are close but ordered: $d<\delta<i^{(12)}<i$$i$, as in the Key Results. The nominal rate $i^{(m)}$ equals $i$ for $m=1$ and approaches $\delta$ as $m\to\infty$$\infty$; at $m=12$ it is already within $0.00007$ of $\delta$.

Discounting moves money back in time. Insurance pages multiply each possible payment by $v^t$ first and then weight it by a survival or death probability; the result is an actuarial present value. When the payments are certain, as in an annuity certain, only the discounting step remains.

The force of interest plays the same role for money as the force of mortality plays for survival. If the force varies over time as $\delta(s)$, the discount factor to time $t$ is $\exp\big(-\int_0^t\delta(s)\,\mathrm{d}s\big)$${\int }_0^t\delta (s)\mathrm{d}s)$, just as ${}_tp_x=\exp\big(-\int_0^t\mu_{x+s}\,\mathrm{d}s\big)$${\int }_0^t{\mu }_{x+s}\mathrm{d}s)$.

Key Results¶

• $d=iv=i/(1+i)$ and $i=d/(1-d)$$d)$; $\delta=-\log v=-\log(1-d)$$d)$; and $i-d=id$$id$.
• If interest is paid $m$ times a year at the nominal rate $i^{(m)}$, the rate per period is $i^{(m)}/m$, and equivalence with $i$ means $(1+i^{(m)}/m)^m=1+i$$i$, so $i^{(m)}=m\big((1+i)^{1/m}-1\big)$$1)$. As $m\to\infty$$\infty$, $i^{(m)}\to\delta$$\delta$.
• Similarly, the nominal discount rate $d^{(m)}$ deducts $d^{(m)}/m$ at the start of each $1/m$ year, with $(1-d^{(m)}/m)^m=v=1-d$$d$, so $d^{(m)}=m\big(1-v^{1/m}\big)$$v^{1/m})$. As $m\to\infty$$\infty$, $d^{(m)}\to\delta$ as well.
• For $i>0$ and $m>1$$1$: $d<d^{(m)}<\delta<i^{(m)}<i$$i$. For small rates all of them are close: $\delta=i-\tfrac{1}{2}i^2+\cdots$$\cdots$.
• Level payments of $1$ at times $0,1,\ldots,n-1$ have present value $\displaystyle \sum_{k=0}^{n-1}v^k=(1-v^n)/d$ for $d\ne0$$0$, and $n$ at $i=0$$0$, by the geometric series formula; a continuous payment stream at rate $c(t)$ has present value $\int_0^\infty c(t)\mathrm{e}^{-\delta t}\,\mathrm{d}t$.
• Rates over other periods follow by compounding, not by proportion: the effective rate for $h$ years is $(1+i)^h-1$$1$, so a half-year rate is $(1+i)^{1/2}-1$$1$, not $i/2$.

Examples¶

From $i$ to the rest. At $i=0.04$$0.04$:

A payment of $100$ in five years has present value $100v^5=100(1.04)^{-5}\approx82.19$$82.19$. Weighted by a survival probability of $0.9$, this gives the $73.97$ of the actuarial present value example.

From $\delta$ to the rest. Several Life Insurance examples use a constant force $\delta=0.03$$0.03$. Then $v=\mathrm{e}^{-0.03}\approx0.970446$$0.970446$, $d=1-\mathrm{e}^{-0.03}\approx0.029554$ and $i=\mathrm{e}^{0.03}-1\approx0.030455$$0.030455$; as expected, $d<\delta<i$$i$.

A nominal monthly rate. At $i=0.04$$0.04$, the equivalent monthly rate is $1.04^{1/12}-1\approx0.0032737$$0.0032737$, so $i^{(12)}\approx12(0.0032737)\approx0.039285$$0.039285$. It lies between $\delta\approx0.039221$ and $i=0.04$$0.04$. The ratio $i/i^{(12)}\approx1.0182$ is the UDD adjustment for monthly settlement on the page on death benefits payable at fractional-year boundaries. The matching nominal discount rate is $d^{(12)}=12\big(1-1.04^{-1/12}\big)\approx0.039157$$0.039157$, just below $\delta$.

Ten payments in advance. At $i=0.04$$0.04$, payments of $1$ at times $0,1,\ldots,9$ are worth $\ddot a_{\actuarialterm{10}}=(1-v^{10})/d=(1-1.04^{-10})/0.038462\approx8.4353$$8.4353$, compared with $10$ undiscounted.

Prerequisites¶

• Exponential Function
• Logarithm and Logarithmic Function
• Geometric Series

Related Concepts¶

• Actuarial Present Value
• Annuity Certain
• Death Benefits Payable at Fractional-Year Boundaries
• Force of Mortality
• Variance of an Insurance Present Value

Sources¶

• Life Insurance · Lecture 3 (interest-theory review, textbook §4.3)

Aliases: discount factor, force of interest, effective interest rate, discount rate, present value, compound interest, effective rate of interest, rate of discount, continuously compounded interest rate Keywords: v = (1+i)^{-1}, d = 1 - v = iv, delta = log(1+i), v^t = e^{-delta t}, annual effective interest i, nominal interest rate i(m), time value of money Symbols: delta, infinity, integral, mu, sum

Last updated: 27 October 2026 11:52