Difficulty: Intermediate to Advanced | Prerequisites: Consumer Theory study notes; comfort with present value arithmetic and basic probability.
This unit extends consumer choice into two new dimensions: time and uncertainty. Intertemporal choice applies the same utility-maximisation framework to decisions about when to consume (now vs. later), while the risk section asks how people behave when outcomes are uncertain. Both connect back to the demand analysis you have already covered, and forward to how firms evaluate investment projects. If you are comfortable with indifference curves and budget constraints, the mechanics here will feel familiar; the new ingredient is the interest rate, which plays the role of a price, converting future consumption into present-value terms.
The interest rate is the price of consuming now rather than later. Consumers choose saving or borrowing to smooth consumption over their lifetime, and the same discounting logic lets firms evaluate investment projects via NPV and IRR. When outcomes are uncertain, risk-averse consumers will pay a premium for certainty, which explains why insurance markets exist and why diversification reduces portfolio risk.
Compound interest
Interest earned on previously accumulated interest, producing exponential growth over time.
Present discounted value (PDV)
The current value of a stream of future cash flows, each discounted by the interest rate. Think of it as "what those future pounds are worth to you today."
Nominal interest rate
The stated, unadjusted rate. What the bank quotes you.
Real interest rate
The nominal rate adjusted for inflation. Reflects actual purchasing power.
Fisher equation
Links nominal and real rates: (1 + R_nominal) = (1 + R_real)(1 + inflation). In simple terms, the real rate is roughly the nominal rate minus inflation.
Intertemporal budget constraint
The budget line for consumption today vs. consumption tomorrow. Its slope depends on the interest rate.
Life cycle hypothesis
Consumers plan to smooth consumption over their lifetime, saving during working years and spending down savings in retirement.
Net present value (NPV)
The sum of discounted future cash flows minus the initial investment cost. A positive NPV means the project is worth undertaking.
Internal rate of return (IRR)
The discount rate that makes NPV exactly zero. If IRR exceeds the market interest rate, the project is profitable.
Human capital
The economic value of a worker's education, training, and skills. Evaluated as an investment with costs now and returns (higher earnings) later.
Expected payoff
The probability-weighted average of all possible outcomes.
Variance / standard deviation
Measures of how spread out outcomes are around the expected value. Higher variance means more risk.
Risk-averse
A person who prefers a certain outcome to a gamble with the same expected value. Their utility function is concave.
Risk-neutral
A person who cares only about expected value, indifferent to the spread of outcomes.
Risk-loving
A person who prefers the gamble. Convex utility function.
Certainty equivalent
The guaranteed amount that gives a risk-averse person the same utility as a risky prospect.
Risk premium
The difference between the expected value and the certainty equivalent. The maximum a risk-averse person would pay to eliminate uncertainty.
Fair insurance
Insurance whose premium equals the expected loss. Zero expected profit for the insurer.
Diversification
Spreading investments across assets to reduce overall portfolio risk without sacrificing expected return.
Interest rates and compound interest
The interest rate is the cost of borrowing and the return on saving. It converts future consumption into present-value terms.
Compound interest: FV = PV x (1 + R)^T. Growth is exponential, which is why even small rate differences matter over long horizons.
Present discounted value
PDV = F_1/(1+R) + F_2/(1+R)^2 + ... + F_T/(1+R)^T.
Future money is worth less today because of the opportunity cost of waiting. The higher the interest rate, the more steeply future cash flows are discounted.
Bond prices and interest rates
Bond prices and interest rates move inversely. When rates rise, the present value of a bond's fixed future payments falls, so the bond's price drops.
Real vs. nominal interest rates
The Fisher equation: (1 + R_nominal) = (1 + R_real)(1 + inflation).
Approximate version: R_real is roughly R_nominal minus inflation.
Always use the real rate when comparing consumption across periods, because inflation erodes purchasing power.
Intertemporal budget constraint and consumer choice
Two-period model: I = P_1 * C_1 + P_2 * C_2 / (1 + R).
The consumer chooses current consumption C_1 and future consumption C_2 to maximise utility subject to this constraint.
The slope of the intertemporal budget line is -(1 + R): giving up one unit of consumption today yields (1 + R) units tomorrow.
Substitution and income effects of interest rate changes
A higher interest rate makes saving more attractive (substitution effect: consume less today, more tomorrow).
The income effect depends on whether the consumer is a net saver or borrower. For a saver, a higher rate increases wealth, which may increase current consumption.
The net effect is ambiguous for savers; for borrowers, both effects reduce current consumption.
Intertemporal optimisation
At the optimum: MU_1 / P_1 = MU_2 / [P_2 x (1 + R)]. Marginal utility per pound is equalised across periods.
Life cycle hypothesis
Consumers smooth consumption over their lifetime: save during high-earning years, draw down in retirement.
Implies that consumption depends on lifetime wealth (not just current income), and that people's saving rate changes predictably with age.
NPV and IRR
NPV = sum of discounted future cash flows minus the upfront cost. Invest if NPV > 0.
IRR = the discount rate at which NPV = 0. Invest if IRR > the market interest rate (cost of capital).
When NPV and IRR conflict (rare, but possible with non-standard cash flow patterns), NPV is the more reliable criterion.
Human capital investment
Education and training are investments: costs now (tuition, foregone earnings), returns later (higher wages).
Evaluated with the same NPV/IRR framework. The decision depends on expected future earnings, the cost of education, and the discount rate.
Risk and expected payoff
Expected payoff = sum of (probability x payoff) across all states.
Variance = sum of [probability x (payoff minus expected)^2]. Higher variance = more risk.
Risk preferences
Risk-averse: concave utility function (U'' < 0). Prefers the expected value for certain over the gamble.
Risk-neutral: linear utility. Indifferent between the gamble and its expected value.
Risk-loving: convex utility. Prefers the gamble.
Certainty equivalent and risk premium
Certainty equivalent: the guaranteed amount giving the same utility as the risky prospect.
Risk premium = expected value minus certainty equivalent. It is what a risk-averse person would pay to eliminate the risk.
Insurance
Fair insurance: premium = expected loss. Zero expected profit for the insurer. A risk-averse person always buys full fair insurance.
Partial insurance: covers only a portion of the loss.
Risk-averse consumers are willing to pay premiums above expected losses to obtain certainty.
Risk management strategies
Risk sharing: spreading risk across a group (e.g., mutual insurance).
Hedging: holding assets with negatively correlated payoffs.
Diversification: investing in many assets to reduce unsystematic risk.
Information acquisition: research that reduces uncertainty about outcomes.
Present discounted value
PDV = F_1/(1+R) + F_2/(1+R)^2 + ... + F_T/(1+R)^T
Fisher equation
(1 + R_nominal) = (1 + R_real)(1 + inflation)
Intertemporal budget constraint (two periods)
C_1 + C_2/(1+R) = I_1 + I_2/(1+R)
Intertemporal optimum
MU_1 / P_1 = MU_2 / [P_2 x (1+R)]
Net present value
NPV = -Cost_0 + F_1/(1+R) + F_2/(1+R)^2 + ... + F_T/(1+R)^T
Internal rate of return
The rate R* such that NPV = 0 when R = R*.
Expected payoff
E[X] = sum of p_i x X_i
Variance
Var(X) = sum of p_i x (X_i - E[X])^2
Risk premium
RP = E[X] - CE, where CE is the certainty equivalent.
The PDV framework is how firms, governments, and individuals evaluate any long-lived asset: a rental property, a government bond, a university degree. The life cycle hypothesis explains why young professionals often have negative savings rates while retirees draw down wealth. Diversification is the logic behind index funds: by holding many stocks, investors eliminate company-specific risk while retaining the market return.
Students often confuse nominal and real interest rates in present-value calculations. If cash flows are in real terms, discount with the real rate; if in nominal terms, use the nominal rate. Mixing them produces wrong answers.
A common error is to assume that a positive IRR always means "invest." What matters is whether IRR exceeds the cost of capital, not whether it is positive.
Students sometimes think risk-averse means "avoids all risk." Risk-averse people take risky positions all the time; they simply demand a higher expected return to compensate.
Many students apply the variance formula without squaring the deviations. Variance uses squared differences; the standard deviation is the square root of variance.
⚠️ Be comfortable computing PDV for a stream of cash flows, and know what happens to PDV when the interest rate rises.
⚠️ Expect to calculate NPV and IRR for a simple investment project and state the investment rule.
⚠️ Questions on risk often ask you to compute expected payoff and variance, then rank options for a risk-averse consumer using certainty equivalents.
⚠️ The Fisher equation and the distinction between real and nominal rates appear on most exams. Know how to convert between them.
True or false: When interest rates rise, bond prices fall. (True.)
Fill in the blank: A project should be accepted if its NPV is _______ . (positive / greater than zero)
True or false: A risk-neutral person would never buy insurance at a premium above the expected loss. (True.)
Fill in the blank: The certainty equivalent for a risk-averse person is _______ than the expected value. (less / lower)
True or false: Diversification eliminates all risk in a portfolio. (False. It eliminates unsystematic risk, but market-wide/systematic risk remains.)
Q: You can invest in a project that costs 1,000 today and returns 600 in one year and 600 in two years. The interest rate is 10%. Should you invest?
A: NPV = -1000 + 600/1.1 + 600/1.21 = -1000 + 545.45 + 495.87 = 41.32. NPV is positive, so yes, invest.
Q: A consumer earns 50,000 this year and 55,000 next year. The interest rate is 5%. What is the present value of lifetime income?
A: PV = 50,000 + 55,000/1.05 = 50,000 + 52,380.95 = 102,380.95.
Q: A gamble pays 100 with probability 0.5 and 0 with probability 0.5. A risk-averse consumer has a certainty equivalent of 40 for this gamble. What is the risk premium?
A: Expected value = 0.5(100) + 0.5(0) = 50. Risk premium = 50 - 40 = 10.
Q: Explain why a risk-averse consumer always buys full insurance at a fair premium.
A: Fair insurance has a premium equal to the expected loss, so buying full coverage gives the consumer a certain wealth level equal to their expected wealth minus the expected loss (which they would face anyway in expectation). Because the consumer is risk-averse, certainty at the same expected wealth is strictly preferred to bearing the risk.
Intertemporal choice extends consumer theory from a static setting to a dynamic one, using the same tangency-condition logic. The NPV framework reappears in firm investment decisions and cost-benefit analysis of government policy. Risk preferences connect to insurance markets, financial economics, and the analysis of moral hazard and adverse selection (topics in information economics).
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