Introduction to Actuarial Mathematics

Chapter 3
Theory of Interest and Financial Mathematics

Post Graduate Program in Insurance Management
Introduction to Actuarial Mathematics • Post Graduate

Theory of Interest and Financial Mathematics

Introduction and context

Insurance is an intertemporal business. Premiums are received before claims are paid, reserves are invested over many years, and the value of liabilities depends on the timing, amount and uncertainty of future cash flows. Theory of interest provides the mathematical language for comparing monetary amounts at different dates. Financial mathematics extends that language to loans, bonds, projects, inflation-linked obligations and asset–liability management.

The central principle is that a monetary amount has a time-dependent value. One unit available today can be invested and become more than one unit in the future; conversely, a future payment must be discounted to determine its equivalent value today. Actuaries therefore require a disciplined distinction between accumulation, discounting, interest-rate conventions, yield curves and cash-flow timing.

Learning objectives

By the end of this chapter, the reader should be able to:

calculate present and accumulated values under effective, nominal, discount and force-of-interest rates;
convert consistently between equivalent interest-rate conventions;
value level and non-level cash-flow streams;
interpret yield curves and select appropriate discount rates;
incorporate inflation and changing interest rates;
analyse the duration and sensitivity of asset cash flows;
identify assumptions, limitations and governance issues in financial valuation.

1. Time value of money and the accumulation function

Let time be measured in years, with time 0 as the valuation date. An accumulation function, a(t), gives the value at time t of one monetary unit invested at time 0. It normally satisfies a(0)=1 and, under positive investment returns, is increasing.

If an amount C is invested at time 0, its accumulated value at time t is:

C a(t)

The corresponding discount factor is:

v(t) = 1/a(t)

Thus, a payment X due at time t has present value:

PV = X v(t) = X/a(t)

These relationships assume a deterministic rate structure and that the payment is certain. If payment is uncertain, expected present value requires probabilities or risk-adjusted cash flows in addition to discounting.

One-period effective interest rate

The effective annual interest rate, denoted i, is the interest earned during one year per unit invested at the beginning of that year. Therefore:

1+i = a(1)

For a constant effective annual rate over n years:

a(n) = (1+i)ⁿ

and:

PV = X(1+i)^-n

The exponent n must be expressed in the same period unit as the rate. For example, a three-year payment discounted at an annual effective rate uses n=3, not 36, unless the rate has first been converted to a monthly rate.

Periodic rates and compounding

If interest is credited m times per year at a rate j^(m) nominal per annum convertible mthly, the rate per conversion period is j^(m)/m. Under a constant rate:

a(t) = (1 + j^(m)/m)^(mt)

where t is measured in years and mt is the number of conversion periods. The associated annual effective rate is:

i = (1 + j^(m)/m)^m - 1

A nominal rate is not itself an annual growth rate. It is a quotation convention and must not be compared directly with an effective annual rate without conversion.

2. Discount rates and the force of interest

Effective discount rate

The effective annual discount rate, d, is defined as the proportion of the amount due at the end of a year that is deducted at the beginning of the year. If the future value is one, the present value is 1-d. Since one unit invested grows to 1+i,

1-d = 1/(1+i)

Consequently:

d = i/(1+i)

and:

i = d/(1-d)

For n years at a constant effective discount rate:

vⁿ = (1-d)ⁿ

A nominal discount rate, d^(m), convertible mthly, gives a discount per conversion period of d^(m)/m. The present value of one due at the end of t years is:

v(t) = (1 - d^(m)/m)^(mt)

The equivalent annual effective interest rate is therefore:

i = (1 - d^(m)/m)^(-m) - 1

Force of interest

The instantaneous force of interest, δ(t), describes the continuously compounded rate at time t. If a(t) is differentiable and positive:

δ(t) = a'(t)/a(t) = d[ln a(t)]/dt

Integrating gives the general accumulation function:

a(t) = exp(∫₀ᵗ δ(s) ds)

If the force is constant, δ(t)=δ, then:

a(t)=e^(δt)

The equivalent annual effective rate is:

i=e^δ-1

and the equivalent force corresponding to an effective annual rate is:

δ=ln(1+i)

The force of interest is especially useful for continuously varying rates, theoretical work and models in which the yield curve is represented as a smooth function. It should not be confused with a nominal rate convertible continuously, although the limiting relationships coincide under appropriate assumptions.

Rate-conversion summary

Quotation	Accumulation over one year	Equivalent effective annual rate
Effective interest i	1+i	i
Nominal interest j^(m)	(1+j^(m)/m)^m	(1+j^(m)/m)^m-1
Effective discount d	1/(1-d)	d/(1-d)
Nominal discount d^(m)	(1-d^(m)/m)^-m	(1-d^(m)/m)^-m-1
Constant force δ	e^δ	e^δ-1

3. Present value of cash flows

Suppose payments C_k occur at times t_k, for k=1,…,n. Under a deterministic accumulation function, the present value is:

PV = ∑ₖ₌₁ⁿ Cₖ v(tₖ)

If a single constant effective annual rate i applies and all times are measured in years:

PV = ∑ₖ₌₁ⁿ Cₖ(1+i)^(-tₖ)

The accumulated value at time T is:

AV(T)=∑ₖ₌₁ⁿ Cₖ a(T-tₖ)

for cash flows received no later than T. A payment at time t_k must therefore be accumulated only for T-t_k years.

Annuities

An annuity-immediate pays one unit at the end of each of n periods. At an effective rate i, its present value is:

a-angle-n = (1-vⁿ)/i

where v=(1+i)^(-1). An annuity-due pays at the beginning of each period, so:

a-double-dot-angle-n = (1+i)a-angle-n

These formulas assume equal payments, equal-length periods and a constant effective rate per period. For irregular dates or changing rates, direct cash-flow discounting is safer than applying an annuity shortcut.

Worked example 1: effective, nominal and force rates

An insurer invests £100,000 for three years. Compare the accumulated value under: (a) an effective annual rate of 6%; (b) a nominal rate of 5.85% convertible monthly; and (c) a constant force of interest of 5.8%.

(a) Effective rate:

AV = 100000(1.06)³ = £119,101.60 approximately

(b) Nominal monthly rate:

AV = 100000(1+0.0585/12)³6 ≈ £119,120.00

AV = 100000e^(0.058×3) ≈ £119,015.10

The results differ because the quoted rates represent different annual growth factors. The comparison is meaningful only after identifying the compounding convention.

Worked example 2: present value of an insurance cash-flow stream

An insurer expects to receive £500,000 at the end of year 1, pay claims of £300,000 at the end of year 2, and pay expenses of £100,000 at the end of year 4. If the constant effective annual discount rate is 4%, the net present value is:

PV = 500000(1.04)^(-1) - 300000(1.04)^(-2) - 100000(1.04)^(-4)

Using (1.04)^(-1)≈0.961538, (1.04)^(-2)≈0.924556 and (1.04)^(-4)≈0.854804:

PV ≈ 480,769 - 277,367 - 85,480 = £117,922

The sign convention treats receipts as positive and payments as negative. A different convention is acceptable if applied consistently.

4. Yield curves and term structure

A yield curve shows interest rates or discount factors by maturity. In practice, the rate for a one-year cash flow need not equal the rate for a ten-year cash flow. A single flat rate is therefore a simplification, not a universal law.

The discount factor for maturity t is denoted P(0,t), where:

P(0,t)=the present value at time 0 of one unit paid at time t

The continuously compounded spot yield is:

y(0,t) = -ln P(0,t)/t

The effective annual spot yield is:

i(0,t)=P(0,t)^(-1/t)-1

For a cash-flow stream, the correct valuation is:

PV=∑ₖ CₖP(0,tₖ)

rather than discounting every cash flow at an arbitrarily selected average rate.

A forward rate describes the implied rate applying between two future dates. Under annual effective notation, the one-period forward rate from t to t+1 satisfies:

1+f(t,t+1) = P(0,t)/P(0,t+1)

More generally, for dates s<t:

(1+f(s,t))^(t-s)=P(0,s)/P(0,t)

These are no-arbitrage relations in a deterministic, frictionless framework. Actual forward rates are expectations only under additional assumptions; they may include risk premia, liquidity effects and market imperfections.

Constructing and interpreting a curve

Bond prices can be used to infer discount factors through bootstrapping. A security with known cash flows gives an equation in unknown discount factors. Starting with the shortest maturity, one solves sequentially for longer maturities. The resulting curve should be checked for data quality, liquidity, interpolation assumptions and consistency with observable prices.

Common curve shapes include upward-sloping, downward-sloping, flat and humped curves. An upward curve may reflect expected economic growth, inflation or term premia; it does not provide a unique explanation. Insurers use curves for liability valuation, investment strategy, solvency analysis and product pricing.

5. Inflation and real versus nominal values

Inflation changes the purchasing power of money. Let π be the effective inflation rate over a period. If a nominal cash amount is X_nom, its real value measured in base-period purchasing power is:

X_real = X_nom/(1+π)^t

If the nominal effective interest rate is i and inflation is π, the exact real interest rate r satisfies the Fisher relation:

1+r=(1+i)/(1+π)

Thus:

r=(i-π)/(1+π)

The approximation r≈i-π is useful only when rates are relatively small. Nominal cash flows should be discounted with nominal rates; real cash flows should be discounted with real rates. Mixing the two creates systematic valuation error.

For inflation-linked insurance liabilities, inflation may be stochastic, claim-specific and correlated with interest rates. Medical inflation, wage inflation and general consumer-price inflation can differ materially. A professional model must specify the inflation index, lag, caps, floors and contractual adjustment rules.

6. Changing interest rates and reinvestment

If the effective rate in period k is i_k, the accumulation from time 0 to time n is:

a(n)=∏ₖ₌₁ⁿ(1+iₖ)

The present value of a cash flow C_k at time k is:

PV=∑ₖ Cₖ / ∏ⱼ₌₁ᵏ(1+iⱼ)

This formula assumes rates are known and apply to the relevant periods. If rates are uncertain, valuation may require scenario analysis, stochastic interest-rate models or market-consistent techniques.

A key distinction is between the valuation yield and the realised reinvestment return. A bond purchased at a price reflecting a yield curve may not earn that yield over its actual holding period if rates change or coupons are reinvested at different rates. This is known as reinvestment risk. An insurer must therefore distinguish:

market value at a valuation date;
expected holding-period return;
yield to maturity under stated assumptions;
realised return after future rate movements.

7. Asset cash flows, duration and immunisation

For an asset with cash flows C_k at times t_k, discounted at a flat effective annual yield i, price is:

P=∑ₖ Cₖ(1+i)^(-tₖ)

The Macaulay duration is the present-value-weighted average payment time:

D_M = [∑ₖ tₖCₖ(1+i)^(-tₖ)]/P

The modified duration under annual effective yield is:

D_mod = D_M/(1+i)

For a small parallel change ∆i in yield:

∆P/P ≈ -D_mod∆i

The approximation ignores convexity and is most reliable for small yield changes. A second-order improvement is:

∆P/P ≈ -D_mod∆i + ½ Convexity(∆i)²

where convexity is the appropriately scaled second derivative of price with respect to yield.

Immunisation seeks to align asset cash flows and liability cash flows so that changes in interest rates have limited effect on surplus. Basic duration matching is not sufficient in all circumstances. Effective immunisation also considers present-value matching, convexity, non-parallel curve movements, liquidity, transaction costs, credit risk and the possibility that asset and liability discount curves differ.

Worked example 3: duration interpretation

Suppose an asset portfolio has market value £20 million and modified duration 6.2 years. If the relevant yield increases by 0.50 percentage points, ∆i=0.005. The first-order price effect is:

∆P/P≈-6.2×0.005=-0.031

The estimated fall in value is 3.1%, or:

∆P≈£20,000,000×(-0.031)=-£620,000

This is an estimate, not a guaranteed result. Convexity, cash-flow changes, credit spreads and non-parallel shifts can materially alter the outcome.

8. Applied case: pricing and managing a deferred annuity portfolio

Background

This illustrative case concerns Meridian Life, an insurer selling ten-year deferred annuities. Premiums are received immediately, while guaranteed benefits begin after a deferral period. The chief risk officer must determine whether the proposed asset portfolio is suitable for the liabilities.

Problem

The liability cash flows are £4 million at the end of years 6, 7, 8, 9 and 10. The market provides the following continuously compounded spot yields:

Maturity	Spot yield
6 years	3.20%
7 years	3.30%
8 years	3.40%
9 years	3.50%
10 years	3.60%

The insurer must calculate the liability value and assess sensitivity to a parallel 1% increase in yields.

Analysis and approach

For continuous spot yields, the discount factor is P(0,t)=e^-y(0,t)t. Therefore:

L=4,000,000∑ₜ₌₆¹⁰e^[-y(0,t)t]

Approximate discount factors are 0.825, 0.794, 0.762, 0.730 and 0.698. Their sum is approximately 3.809, giving:

L≈£4,000,000×3.809=£15.236 million

The liability duration is approximately:

D_L = [∑ₜ₌₆¹⁰ t×4,000,000×P(0,t)]/L ≈ 7.9 years

Under a small parallel increase of 1% in continuously compounded yields, the approximate proportional change in value is -D_L×0.01, or about −7.9%. The liability value would fall by approximately £1.20 million. If assets have materially shorter duration, their value may fall by less, but the resulting change in surplus depends on both asset and liability responses.

Outcome and managerial decision

Meridian Life should select assets whose market value, duration and convexity are consistent with the liability profile, while retaining sufficient liquidity for expenses and unexpected claims. It should stress-test steepening, flattening and credit-spread shocks rather than relying solely on the parallel-shift estimate.

Learning points

Each liability cash flow should be discounted using the rate applicable to its maturity.
Duration is a sensitivity measure, not a complete risk measure.
Asset–liability management requires consideration of liquidity, credit quality, reinvestment and regulatory capital.
Model outputs must be accompanied by assumptions, validation and scenario analysis.

9. Limitations, governance and professional practice

Financial mathematics is precise, but precision in calculation does not remove uncertainty in assumptions. Important limitations include:

Rate-model risk: the selected curve or stochastic model may not represent future markets.
Basis risk: asset and liability values may respond to different reference curves or inflation indices.
Liquidity risk: quoted prices may not be achievable for large transactions.
Credit risk: a promised cash flow may not be received in full or on time.
Optionality: policyholder lapses, guarantees, calls and embedded options make cash flows path-dependent.
Data and implementation risk: date conventions, rounding, missing observations and coding errors can produce material misvaluation.

Professional governance requires documented assumptions, independent review, version control, reproducible calculations, sensitivity testing and clear communication of uncertainty. Models should be validated against known cases and reconciled to accounting, actuarial and investment records. Automated and AI-assisted tools may help detect anomalies, generate scenarios or explain model outputs, but they do not replace actuarial judgement, data governance, human oversight or regulatory accountability. Sensitive policyholder and market data must be protected, and models must be monitored for bias and inappropriate extrapolation.

Contemporary practice increasingly uses market-consistent discounting, scenario-based asset–liability management, stochastic interest-rate models, climate-related stress testing and integrated economic capital analysis. These developments improve risk understanding but also increase model complexity. Complexity is justified only when it improves decisions sufficiently to outweigh additional data, validation and governance burdens.

Best-practice checklist

Before approving a valuation or investment analysis, the practitioner should:

define the valuation date, cash-flow dates and sign convention;
identify whether rates are effective, nominal, discount or continuously compounded;
convert all rates to compatible periods and units;
use maturity-specific discount factors when a yield curve is available;
distinguish nominal from real cash flows and rates;
test changing rates, inflation, credit spreads and liquidity assumptions;
quantify duration and, where material, convexity and key-rate sensitivities;
document assumptions, data sources and interpolation methods;
independently verify calculations and reconcile outputs;
explain limitations clearly to management, policyholders and regulators.

Chapter Summary

Theory of interest measures how monetary values change through time. The accumulation function converts present values into future values, while discount factors convert future cash flows into present values. Effective rates describe growth over a stated period; nominal rates require a compounding frequency; discount rates describe deduction from a future amount; and the force of interest provides a continuous-time representation.

Financial valuation is fundamentally a cash-flow problem. Under a deterministic curve, each cash flow is multiplied by its maturity-specific discount factor. Yield curves, spot rates and forward rates support no-arbitrage valuation, although practical interpretation must recognise liquidity, credit and risk premia. Inflation requires consistent matching of nominal cash flows with nominal rates or real cash flows with real rates.

Changing rates affect both valuation and realised investment returns. Duration and convexity provide useful sensitivity measures, while immunisation links asset cash flows to liability cash flows. In insurance, sound practice combines mathematical accuracy with model governance, scenario analysis, professional judgement and regulatory awareness.

Key Terms

Accumulation function: Function giving the value at a future time of one unit invested at a reference date.

Accumulated value: Value of a present amount or cash-flow stream at a specified future date.

Annuity-immediate: Equal payments made at the ends of successive periods.

Annuity-due: Equal payments made at the beginnings of successive periods.

Basis risk: Risk that related assets and liabilities respond differently to market movements.

Convexity: A second-order measure of the curvature of a security’s price–yield relationship.

Discount factor: Present value of one unit payable at a specified future time.

Effective interest rate: Interest earned over a stated period relative to the amount invested at its beginning.

Effective discount rate: Proportion of a maturity amount deducted at the beginning of a period.

Force of interest: Instantaneous rate of accumulation, represented by the derivative of log accumulation.

Forward rate: Rate implied for borrowing or investing between two future dates.

Inflation: General increase in prices that reduces purchasing power.

Macaulay duration: Present-value-weighted average time of an asset’s cash flows.

Modified duration: Approximate proportional price sensitivity to a change in yield.

Nominal rate: Quoted annual rate associated with a stated compounding or discount frequency.

Present value: Value at the valuation date of future cash flows after discounting.

Real interest rate: Interest rate measured after allowing for inflation.

Spot rate: Rate applying to a single payment at a particular maturity.

Yield curve: Relationship between interest rates or yields and maturities.

Review Questions

Short Answer Questions

Distinguish an effective annual interest rate from a nominal annual rate convertible monthly.
Define the force of interest and state its relationship with the accumulation function.
Why must nominal cash flows normally be discounted using nominal rates?
What is the difference between an annuity-immediate and an annuity-due?
State two limitations of duration as a risk measure.

Long Answer Questions

Derive the relationship between an effective interest rate and an effective discount rate, and illustrate it numerically.
Explain how spot discount factors and forward rates are related under no-arbitrage conditions.
Discuss the valuation consequences of using a flat discount rate instead of a maturity-specific yield curve.
Explain the distinction between nominal and real interest rates and evaluate the approximation r≈i-π.
Discuss the role of asset–liability immunisation in life insurance and its principal limitations.

Analytical / Application Questions

An insurer receives £2 million at the end of year 2 and pays £1.4 million at the end of year 5. Using a constant effective annual rate of 5%, calculate the net present value.
A bond portfolio has market value £50 million and modified duration 4.8. Estimate the first-order change in value if yields rise by 0.75 percentage points. Explain why the estimate may be inaccurate.
An annuity liability is indexed to annual inflation while its supporting assets pay fixed nominal coupons. Analyse the principal risks and recommend two hedging or asset–liability management actions.

Multiple Choice Questions with Answers

Question 1

An effective annual interest rate of 6% is equivalent to which annual effective discount rate?

A. 5.66%

B. 6.00%

C. 6.38%

D. 4.00%

Correct Answer: A. 5.66%

Question 2

What is the accumulation factor over three years at a constant effective annual rate i?

A. 1+3i

B. (1+i)^3

C. 3(1+i)

D. (1+i)/3

Correct Answer: B. (1+i)^3

Question 3

If the constant force of interest is δ, the accumulation function over t years is:

A. 1+δt

B. (1+δ)^t

C. e^(δt)

D. δ^t

Correct Answer: C. e^(δt)

Question 4

A nominal annual interest rate j^(12) is convertible monthly. What is the monthly interest rate?

A. 12j^(12)

B. j^(12)/12

C. (1+j^(12))^12−1

D. j^(12)−12

Correct Answer: B. j^(12)/12

Question 5

Which expression gives the present value of cash flows C_k at times t_k using discount factors P(0,t_k)?

A. ∑ C_k/P(0,t_k)

B. ∑ C_kP(0,t_k)

C. ∏ C_kP(0,t_k)

D. ∑ C_kt_kP(0,t_k)

Correct Answer: B. ∑ C_kP(0,t_k)

Question 6

If a nominal annual interest rate is 8% and annual inflation is 3%, the exact real rate is approximately:

A. 5.00%

B. 4.85%

C. 11.00%

D. 2.91%

Correct Answer: B. 4.85%

Question 7

If the modified duration of an asset is 5 and yield rises by 0.2%, the first-order proportional price change is approximately:

A. +1.0%

B. −1.0%

C. −0.2%

D. +5.2%

Correct Answer: B. −1.0%

Question 8

Under annual effective notation, the forward rate from s to t satisfies which relationship?

A. (1+f(s,t))^(t−s)=P(0,s)/P(0,t)

B. 1+f(s,t)=P(0,t)/P(0,s)

C. f(s,t)=P(0,s)+P(0,t)

D. f(s,t)=P(0,s)P(0,t)

Correct Answer: A. (1+f(s,t))^(t−s)=P(0,s)/P(0,t)

Question 9

An annuity-due differs from an annuity-immediate because its payments occur:

A. Only at maturity

B. At the beginning of each period

C. At random times

D. Only after interest is paid

Correct Answer: B. At the beginning of each period

Question 10

Which risk arises when coupons must be reinvested at rates different from those assumed at purchase?

A. Reinvestment risk

B. Longevity risk

C. Operational risk

D. Moral hazard

Correct Answer: A. Reinvestment risk

Introduction to Actuarial Mathematics
Chapter 3 of 14

Chapter 3 Theory of Interest and Financial Mathematics