The core mathematics of the actuarial profession: interest theory, life contingencies, loss models, credibility and ratemaking, plus how to plan the SOA and CAS exams.

Actuarial Mathematics

## Why it matters

Almost every number an actuary produces is a present value. A pension promise paid over 30 years, a claim that will be settled in 18 months, a life insurance benefit that might be paid next year or in 50 years: each one is money at different times, and you cannot add money at different times until you move it to the same date. The **time value of money** is the rule for doing that move. It is the first topic of the SOA Financial Mathematics (FM) exam and it runs underneath everything else in this course.

The idea is simple: 100 dollars today is worth more than 100 dollars in a year, because today's 100 can be invested and earn interest. So a payment in the future must be **discounted** to compare it with money now.

## The core formulas

Let `i` be the **effective annual interest rate** (for example 5% = 0.05).

- **Accumulation:** 1 invested today grows to `(1 + i)^t` after `t` years. So the future value of an amount `C` is `FV = C * (1 + i)^t`.
- **Discounting:** the present value of `C` paid at time `t` is `PV = C / (1 + i)^t`.

Actuaries write the one-year **discount factor** as `v = 1 / (1 + i)`, so `PV = C * v^t`. You will see `v` everywhere in life contingencies, so get used to it now.

`t` does not have to be a whole number. A payment in 6 months at 5% has `PV = C * 1.05^(-0.5)`. This is compound interest, which is the standard in actuarial work; simple interest (`C / (1 + i*t)`) only appears in short-term contexts.

## Worked example 1: one payment

You will receive 10,000 in 3 years. At 4% effective per year:

```
v = 1 / 1.04 = 0.961538
PV = 10000 * v^3 = 10000 / 1.124864 = 8889.96
```

So 8,889.96 today, invested at 4%, grows to exactly 10,000 in three years. The two amounts are **equivalent** at that rate.

## Worked example 2: several cash flows (net present value)

A project costs 5,000 now and returns 2,000 at the end of each of the next three years. At 6%:

| time | cash flow | discount factor `1.06^-t` | present value |
|---|---|---|---|
| 0 | -5,000 | 1.000000 | -5,000.00 |
| 1 | 2,000 | 0.943396 | 1,886.79 |
| 2 | 2,000 | 0.889996 | 1,779.99 |
| 3 | 2,000 | 0.839619 | 1,679.24 |
| | | **NPV** | **346.02** |

Because present values are all at the same date, they can be added. The **net present value** (NPV) is positive, so at 6% the project is worth more than it costs. The rate that makes the NPV exactly zero is the **internal rate of return** (here about 9.7%).

In Python the whole table is one line:

```python
def npv(cashflows, rate):
    return sum(amount / (1 + rate) ** t for t, amount in cashflows)

print(round(npv([(0, -5000), (1, 2000), (2, 2000), (3, 2000)], 0.06), 2))  # 346.02
```

## Nominal rates and the force of interest

Rates are often quoted **nominally**: "6% compounded monthly" means 0.5% per month, written `i^(12) = 0.06`. Convert to an effective annual rate before you discount:

```
i = (1 + i^(m) / m)^m - 1
6% compounded monthly: (1 + 0.06/12)^12 - 1 = 0.061678
```

At the limit of continuous compounding you get the **force of interest** `delta = ln(1 + i)`, and `v^t = e^(-delta * t)`. Exam FM and later life contingencies exams use all three forms, so practise switching between them.

## Common mistakes

- **Mixing periods.** A monthly cash flow needs a monthly rate, or times expressed in years with an annual effective rate. Never use a nominal annual rate as if it were effective.
- **Adding values at different dates.** Always move every cash flow to one date first.
- **Sign confusion.** Pick a side (the insurer, the borrower) and keep money out negative and money in positive.
- **Rounding too early.** Keep full precision for discount factors and round only the final answer.

## Quick recap

- `PV = C * v^t` with `v = 1 / (1 + i)`; `FV = C * (1 + i)^t`.
- Values at the same date can be added; NPV is the sum of discounted cash flows.
- Convert nominal rates to effective with `(1 + i^(m)/m)^m - 1`; the force of interest is `ln(1 + i)`.

## Try it

Open the **Do** tab and write `pv`, `fv` and `npv` functions in Python, then use them to compare two payment offers.


Price any set of future cash flows by discounting them to today with an interest rate.

Time value of money

Time value of money | Interest and debt | Finance & Capital Markets | Khan Academy

Time Value of Money - Present Value vs Future Value

## Why it matters

Most financial promises are not single payments but **series** of payments: monthly mortgage instalments, yearly pension benefits, quarterly insurance premiums. A series of payments at regular intervals is called an **annuity** (in this lesson, an *annuity-certain*: the payments happen no matter what; in module 2 the payments depend on someone being alive). Annuity formulas are the workhorse of Exam FM and of every loan, lease and pension calculation.

## Annuity-immediate and annuity-due

Take `n` payments of 1, one per period, at effective rate `i` per period, with `v = 1/(1+i)`.

- **Annuity-immediate**: payments at the *end* of each period (times 1, 2, ..., n). Its present value is

```
a_n = v + v^2 + ... + v^n = (1 - v^n) / i
```

- **Annuity-due**: payments at the *start* of each period (times 0, 1, ..., n-1). Every payment arrives one period earlier, so

```
ä_n = a_n * (1 + i) = (1 - v^n) / d,   where d = i / (1 + i)
```

The accumulated value at time `n` of an annuity-immediate is `s_n = ((1 + i)^n - 1) / i`. The formula for `a_n` is a geometric series, and it is worth deriving once by hand: write `a_n` and `v * a_n`, subtract, and almost every term cancels.

## Worked example 1: valuing a payout

A lottery pays 50,000 at the end of each year for 20 years. At 6% effective:

```
v^20 = 1.06^-20 = 0.311805
a_20 = (1 - 0.311805) / 0.06 = 11.469921
PV = 50000 * 11.469921 = 573,496.06
```

So the 1,000,000 "headline" prize is worth about 573,496 today at 6%. If the payments were at the start of each year, the value would be `573,496.06 * 1.06 = 607,905.82`.

## Loans: finding the payment

A loan of `L` repaid by `n` level end-of-period payments `P` is fair when the present value of the payments equals the amount borrowed:

```
L = P * a_n    so    P = L / a_n
```

Example: 200,000 borrowed for 30 years at 6% nominal, compounded monthly. Use monthly periods: `j = 0.005`, `n = 360`.

```
a_360 at 0.5% = (1 - 1.005^-360) / 0.005 = 166.791614
P = 200000 / 166.791614 = 1,199.10 per month
```

## The amortization schedule

Each payment first pays the interest due on the outstanding balance; the rest repays principal:

```
interest_k  = j * balance_(k-1)
principal_k = P - interest_k
balance_k   = balance_(k-1) - principal_k
```

A 4-year loan of 10,000 at 8% with annual payments (`P = 10000 / a_4 = 3,019.21`):

| year | payment | interest | principal | balance |
|---|---|---|---|---|
| 0 | | | | 10,000.00 |
| 1 | 3,019.21 | 800.00 | 2,219.21 | 7,780.79 |
| 2 | 3,019.21 | 622.46 | 2,396.74 | 5,384.05 |
| 3 | 3,019.21 | 430.72 | 2,588.48 | 2,795.56 |
| 4 | 3,019.21 | 223.65 | 2,795.56 | 0.00 |

The table shows amounts rounded to cents, but the calculation keeps full precision, so the final balance is exactly zero. Notice that interest falls and principal rises each year. A useful shortcut: the balance after `k` payments equals the present value of the payments still to come, `P * a_(n-k)` (the **prospective** method). That same idea returns as the policy reserve in module 2.

```python
def loan_payment(principal, rate, n):
    a_n = (1 - (1 + rate) ** -n) / rate
    return principal / a_n

print(round(loan_payment(10000, 0.08, 4), 2))  # 3019.21
```

## Common mistakes

- **Due versus immediate.** Read the timing of the first payment carefully; the two values differ by a factor of `1 + i`.
- **Period mismatch.** Monthly payments need the monthly effective rate and `n` in months.
- **Forgetting the zero-rate case.** If `i = 0`, `a_n = n`; the formula divides by zero, so code needs a special case.
- **Rounding the payment before building the schedule.** Lenders do round to cents, which leaves a small final adjustment; know which convention you are using.

## Quick recap

- `a_n = (1 - v^n)/i`, `ä_n = (1 + i) * a_n`, `s_n = ((1+i)^n - 1)/i`.
- Level loan payment `P = L / a_n`.
- Each payment = interest on the balance + principal; balance after `k` payments = `P * a_(n-k)`.

## Try it

In the **Do** tab, code `annuity_pv`, `loan_payment` and a full amortization schedule, and check that the loan is paid off exactly.


Value a level series of payments and build the amortization schedule for a loan.

Annuities and loans

How To Calculate The Present Value of an Annuity

Amortization Loan Formula

## Why it matters

Insurers and pension plans hold large bond portfolios to back the promises they make. An actuary needs to price those bonds, understand why their market value moves when interest rates move, and check that the assets move in step with the liabilities. The tools are a present value, a yield, and **duration**, the single most used measure of interest-rate risk. All three are on Exam FM.

## What a bond pays

A standard fixed-coupon bond has:

- a **face value** `F` (often 100 or 1,000),
- a **coupon rate** `r`: the bond pays `F * r` each period,
- a **term** of `n` periods, with the redemption amount (here equal to `F`) repaid at the end.

Its price at a **yield** `j` per period is the present value of those cash flows at `j`:

```
price = F * r * a_n + F * v^n        (a_n and v at rate j)
```

The yield to maturity is the single rate that makes this present value equal the market price. It is an internal rate of return, so it is found numerically (bisection or Newton's method), not with a closed formula.

## Worked example 1: price from yield

A 5-year bond, face 1,000, annual coupons of 6%, priced at a 5% yield:

```
coupon = 60
a_5 at 5% = 4.329477,   v^5 = 0.783526
price = 60 * 4.329477 + 1000 * 0.783526 = 259.77 + 783.53 = 1,043.29
```

The coupon rate (6%) is above the yield (5%), so the bond sells at a **premium**, above face. When coupon = yield the price equals face (**par**); when coupon < yield it sells at a **discount**. Higher yields mean lower prices: the price-yield curve slopes down and is convex.

US Treasury and most corporate bonds pay coupons **semiannually**: use half the coupon rate, half the annual nominal yield, and twice as many periods.

## Duration: how sensitive is the price?

**Macaulay duration** is the present-value-weighted average time of the cash flows:

```
D_mac = sum( t * CF_t * v^t ) / price
```

For our bond at 5%:

| t | cash flow | PV | t * PV |
|---|---|---|---|
| 1 | 60 | 57.14 | 57.14 |
| 2 | 60 | 54.42 | 108.84 |
| 3 | 60 | 51.83 | 155.49 |
| 4 | 60 | 49.36 | 197.45 |
| 5 | 1,060 | 830.54 | 4,152.69 |
| | | 1,043.29 | 4,671.61 |

```
D_mac = 4671.61 / 1043.29 = 4.478 years
D_mod = D_mac / (1 + j) = 4.478 / 1.05 = 4.265
```

**Modified duration** gives the first-order price change: `% change in price ≈ -D_mod * change in yield`. If yields rise from 5% to 5.5%, the price falls by about `4.265 * 0.005 = 2.13%`, to about 1,021.05. The exact price at 5.5% is 1,021.35; the gap is due to **convexity**, the curvature that duration ignores. Adding the convexity term `+ 0.5 * C * (change)^2` closes most of the gap.

A zero-coupon bond has Macaulay duration exactly equal to its term, because it has one cash flow. Coupons pull the duration below the term.

## Why actuaries care: matching

If an insurer's liabilities have a duration of 8 years and its bonds have a duration of 4, a fall in interest rates raises the value of the liabilities more than the value of the assets, and the surplus shrinks. **Immunization** (Redington's conditions: equal present values, equal durations, asset convexity greater than liability convexity) protects against small parallel rate moves. Asset-liability management (ALM) teams do this every quarter.

```python
def bond_price(face, coupon_rate, y, n):
    c = face * coupon_rate
    return sum(c / (1 + y) ** t for t in range(1, n + 1)) + face / (1 + y) ** n

print(round(bond_price(1000, 0.06, 0.05, 5), 2))  # 1043.29
```

## Common mistakes

- **Using the coupon rate to discount.** Discount at the yield; the coupon rate only sets the cash flows.
- **Forgetting the redemption payment** in the last period, or counting it twice.
- **Mixing Macaulay and modified duration.** Price sensitivity uses modified duration.
- **Treating duration as exact.** It is a linear approximation, good for small rate changes only.

## Quick recap

- Price = PV of coupons + PV of redemption at the yield.
- Premium when coupon > yield, par when equal, discount when coupon < yield.
- `D_mac` = PV-weighted average time; `D_mod = D_mac / (1 + j)`; `%ΔP ≈ -D_mod * Δj`.

## Try it

In the **Do** tab, write `bond_price`, `macaulay_duration` and `modified_duration`, then estimate a price change and compare it with the exact one.


Price a fixed-coupon bond from its yield and measure its interest-rate risk with duration.

Bonds and yield

Calculate Bond Price and Yield to Maturity (YTM)  |  Annual and Semi Annual Bonds

Bond Duration and Bond Convexity Explained

Price cash flows with the time value of money, value annuities and loans, and measure a bond's price and interest-rate risk.