Capital, Interest & Present Value
The price of any asset — bond, share, machine — is the present value of its expected future payments. So interest rate up ⇒ price down, and expected future payoffs down ⇒ price down. The final spent a whole question on this.
What "capital" means
- Physical capital
- Machines, buildings, trucks, computers, infrastructure. The capital in MRPK.
- Financial capital
- The funds used to purchase, rent or build it.
- Investment
- The flow of new capital into the capital stock. Also a component of GDP.
- Human capital
- Education and training. Counts as capital in the growth chapters.
- On the exam
- Fall 2025's keyed answer to "in economics, the word capital means" was all of the above — physical assets, financial assets, accumulated savings investable across borders. Do not pick the narrow definition there.
Formulas
Worked
| Payment | Rate | Years out | Present value |
|---|---|---|---|
| $100 | 3% | 1 | 100/1.03 = $97.09 |
| $100 | 8% | 1 | 100/1.08 = $92.59 |
| $100 | 3% | 2 | 100/1.03² = $94.26 |
| $1,000/yr | 3% | 1–3 | 970.87 + 942.60 + 915.14 = $2,828.61 slide says 2,827.83 — see quiz note |
Stocks and bonds
- Bond
- The firm borrows, repays with interest. Price = PV of the fixed payment stream.
- Stock
- The firm sells a share of future profits, paid as dividends. Price = PV(expected dividends).
- Two reasons prices move
- Interest rates and expected dividends, for one firm or all. The pandemic is the clean experiment: Zoom's expected dividends jumped, Carnival Cruise's collapsed, prices followed.
Rules that earn points
- Write the PV formula before computing anything.
- For direction questions, ask which way the denominator or numerator moved. That is the whole answer.
- If i = 0, every denominator is 1 and PV is the sum of the payments. This exact question was on the final.
- For asset prices, write "price = PV of expected future payments," then name the input that moved.
- Defining capital as money. It is physical productive assets — except on that one multiple choice, where the key is "all of the above."
- Forgetting to raise (1+i) to the power t.
- Saying rising rates make bonds more attractive so prices rise. Existing bonds have fixed payments; prices fall.
In economics, the word "capital" means:
- AProductive physical assets like equipment and structures
- BFinancial assets such as a bank account
- CAccumulated savings investable across borders
- DAll of the above
Answer
D — all of the above. That is the keyed answer on the Fall 2025 final.
The course uses the word in all three senses: physical capital in the production function, financial capital as the funds that buy it, and accumulated savings moving across borders.
A is the trap, and it is the sense you use everywhere else — it is the capital in MRPK = MPK × MR. On this multiple choice it is incomplete. Keep the distinction in written answers: physical capital is the stuff; financial capital is the funds used to purchase, rent or build it.
The present value of $100 received two years from now, at an interest rate of 3%, is closest to:
- A$94.26
- B$97.09
- C$103.00
- D$106.09
Answer
A — $94.26.
PV = 100/(1.03)² = 100/1.0609 = 94.26.
B is the trap: 100/1.03 = 97.09 discounts only one year. Forgetting the exponent t is the most common arithmetic slip in this unit. C and D compound forward instead of discounting back.
Firms come to expect that less labor will be available in future to work alongside their machines. The price of a unit of capital today:
- ARises, because capital becomes scarcer relative to labor
- BFalls, because expected future MRPK falls
- CIs unchanged, since the interest rate has not moved
- DRises, because firms substitute capital for labor
Answer
B — it falls.
A machine is worth the present value of its future MRPK = MR × MPK. Less labor alongside it lowers its future marginal product, shrinking every numerator in the sum. Machines are worth only what they will produce.
C is the trap: present value has two moving parts. The denominator did not move, but the numerators did.
In 2020 Zoom's stock price jumped while Carnival Cruise's collapsed. The best one-line explanation is:
- AInterest rates fell for tech firms and rose for travel firms
- BA stock price is the present value of expected future dividends, and the pandemic revised those expectations in opposite directions
- CA stock price reflects the firm's past profits, which diverged in 2020
- DCarnival's physical capital was destroyed
Answer
B — expected future dividends moved in opposite directions.
Stock prices move for exactly two reasons: the interest rate used to discount, or expected future dividends. The pandemic hit the second, firm by firm.
A is the trap. The discount rate is an economy-wide price; it cannot explain two firms moving in opposite directions on the same day.
A bond pays F₁, F₂ and F₃ over the next three years. (a) The interest rate rises. What happens to the bond's price, and why? (b) Write an expression for the price if i = 0.
Model answer
(a) The price falls. The price is the present value of the payment stream, P = F₁/(1+i) + F₂/(1+i)² + F₃/(1+i)³. The payments Ft are fixed, so raising i increases every denominator and the whole right-hand side shrinks.
(b) If i = 0 then 1 + i = 1 and every denominator equals 1:
- Writing the PV formula out before answering.
- Saying the payments are fixed — that is why the price, not the payment, adjusts.
- Naming the denominator as the thing that moved.
- For (b), the sum with no discounting — not "the price is higher."
A machine will generate $1,000 of marginal revenue product at the end of each of the next three years and then be worthless. (a) Compute its present value at an interest rate of 3%. (b) Recompute at 8%. (c) Use the two answers to explain the slope of the investment demand curve.
Model answer
(c) A firm buys the machine only if its price is at or below the present value of its MRPK stream. Raising the rate from 3% to 8% cuts that value by about $250, so machines that were worth buying no longer are. Higher interest rate ⇒ less investment, which is why investment demand slopes down.
Note. The lecture slide gives the 3% total as $2,827.83, because it mistypes the first term as 970.09 instead of 970.87. The correct sum is $2,828.61. The difference never matters for credit — write the formula and your working, and either total will be accepted.
- The formula written out with exponents 1, 2, 3 before any arithmetic.
- Three separate discounted terms shown, not just a total.
- A total below $3,000 and not far below — the sanity check for three years of modest discounting.
- Connecting the fall in PV to fewer machines purchased, not just "investment falls."
- Naming the rule: firms buy until PV(MRPK) = purchase price.