Value Over Time, Risk, and Insurance – Study Notes

Core idea: A dollar today > a dollar tomorrow because today’s dollar can be spent or invested immediately.
Interest rate (\% per dollar per unit of time)
- Measures the opportunity cost of having to wait for money.
- Lender’s perspective: compensation for giving up alternative uses of the funds.
- Borrower’s perspective: price paid for earlier access to funds.
Simple future-value formula (one period)
- $\text{FV}=x\times(1+r)$
  where $x$ = current amount, $r$ = interest rate per period.
Immediate vs. delayed pay-offs
- $100{,}000$ now vs. $100{,}000$ in 10 yrs → most choose the immediate sum.
- Increase delayed amount to $1{,}000{,}000$ and many will wait → illustrates how interest rate bridges the gap.
Opportunity-cost framing
- What else could you earn with the money between now and the future date?
- Banks embody this via the interest they charge or pay.
Common mistake alert: Real-world rates also include inflation & default risk; chapter analysis abstracts from these to focus purely on time value.

Compounding = interest on previously earned interest.
Multi-period future value formula (Eq. 11-3)
- $FV = PV \times (1+r)^n$
  $PV$ = present value, $n$ = number of periods.
Example @5 %:
- Year 1 on $1{,}000$ → $1{,}050$.
- Year 2 interest charged on $1{,}050$ → final $1{,}102.50$ (not $1{,}100$).
Long-run impact
- After 20 yrs: $1{,}000\times(1.05)^{20}=2{,}653.29$
- Compounding doubles money in ~14.2 yrs at 5 %, versus 20 yrs with simple addition of $50/yr$.
Rule of 70 (back-of-envelope)
- Doubling time $\approx \frac{70}{r(\%)}$ .
- 2 % → ~35 yrs; 5 % → ~14 yrs.
- Works for growth of savings and growth of debt (e.g.
  credit-card balances at 16 % double in ≈4.4 yrs).
Compounding can be friend (long-term saving) or foe (high-rate debt, e.g. credit cards 16–20 %, private student loans 18 %).

Converts a future sum into its equivalent today.
Formula (Eq. 11-4):
$PV = \frac{FV}{(1+r)^n}$
Rearranged from the compounding formula; interest rate & time are the only two inputs needed for translation.
College example
- Extra $20{,}000/yr for 30 yrs (total $600{,}000$), first payment in 5 yrs, $r=5\%$.
- PV of first $20k$ payment: $\frac{20{,}000}{(1.05)^5}=15{,}670.52$
- PV of whole stream (\$600,000) ≈ \$252,939.
- Verdict: so long as total college costs < \$252,939 today, the degree pays off at 5 % discount rate.
Practical uses
- Retirement planning: "How much must I save now for a given future income?"
- Business capital budgeting: compare PV of future sales vs. present cost of machinery.

Risk: future outcomes unknown; probabilities may (risk) or may not (uncertainty) be measurable.
Everyday illustrations
- Flight delays, used-car repairs, stock performance, job market after graduation.
Distinction (used loosely in text):
- Risk (known probabilities); uncertainty (unknown/immeasurable probabilities).

Definition (Eq. 11-5):
$EV = \sum{i=1}^{n} pi \times si$ where $pi$ = probability, $s_i$ = payoff of outcome $i$.
College income example
- No degree: 50 % × $1.5 M +$ 50 % × $0.9 M$ = $1.2 M$.
- With degree: 50 % × $2.4 M +$ 25 % × $1.5 M +$ 25 % × $0.9 M$ = $1.8 M$.
- EV gain = $0.6 M$ despite possibility of worse outcome.
EV aids decision-making under uncertainty (retirement portfolios, insurance pricing, etc.) but must be paired with risk attitude.

Risk-averse individuals
- Prefer lower variance when EV is identical.
- Example prize: Option A (certain $99{,}999–$100,001) vs.
  Option B (50 % $200k$, 50 % $0$). Both $EV=100k$; most pick A.
Risk-seeking individuals
- Accept higher variance for same or even lower EV (casino patrons, speculative investors).
Degree of risk aversion varies; affects required premium for engaging in risky choice (e.g.
how much extra you need in Option B to switch).

Insurance policy: pay premium to transfer large uncertain cost to insurer.
Why pay more than EV?
- Utility from peace of mind; risk aversion means $\text{Premium} > EV$ is acceptable.
Premium vs. expected payout example – auto insurance
- Accident probabilities 1.5 % @ \$10k and 0.2 % @ \$200k.
- EV of coverage $=0.015\times10k+0.002\times200k = 550$.
- If premium = \$1,000, excess \$450 = value of reduced anxiety + insurer’s profit & admin costs.

Risk pooling
- Group collectively absorbs individual shocks.
- Insurer with 1 M clients uses many premiums to cover few claims.
- UK student-loan system: government pools earnings risk; repayment only if income exceeds threshold.
Risk diversification
- Spread resources across uncorrelated risks.
- Stock example: 50 shares of X vs. 25 X + 25 Y; EV same (\$820) but diversified portfolio reduces chance of big loss (4 % versus 20 %).
- Insurance firms diversify products & geography (earthquake, hurricane, car insurance) to avoid correlated payouts.

Adverse selection
- High-risk individuals disproportionately seek insurance.
- Information asymmetry: customers know their own risk better than insurer → premiums rise → low-risk customers exit.
Moral hazard
- Behavior changes after coverage (park in unsafe area, request unnecessary medical tests).
- Raises cost to insurer; necessitates deductibles, co-pays, monitoring.

Rule of 70 for doubling.
Check failure rates & repair costs before buying extended warranties – often overpriced relative to EV; may still yield utility via peace of mind.
When comparing alternatives across time:
1. Identify timing of each cash flow.
2. Choose appropriate discount rate $r$ (opportunity cost).
3. Convert all flows to PV (or FV) and sum.
4. Incorporate risk via EV and consider worst-case outcomes.

Who should bear the earnings risk of higher education?
- Taxpayers (UK model), graduates via income-driven repayment (IDR) plans, or pooled graduate-tax schemes.
- Implications for access, social equity, and encouraging pursuit of lower-paying social-value careers.