This week is about pulling everything together. Over the past four weeks, you've built a toolkit: future and present value of single cash flows (Week 1), annuities and perpetuities (Week 2), bond valuation (Week 3), and stock valuation (Week 4). Each of those tools is powerful on its own, but the real skill in finance is knowing which tool to reach for, when to combine them, and how to avoid the mistakes that trip up even careful analysts. That's what this week is about.
We'll work through three layers of review. First, we'll revisit the core valuation models with fresh examples to make sure the mechanics are solid. Second, we'll tackle problems that require combining multiple concepts — the kind of thing you'll see on the midterm. Third, we'll practice the often-overlooked skill of spotting errors and translating messy real-world situations into the clean models we've studied. If something feels shaky from a prior week, this is the time to shore it up.
Before we integrate, let's make sure each individual tool is sharp. We'll move quickly through one example from each major topic area. If any of these feel unfamiliar, go back to the relevant week's pack and review before moving on.
Recall from Week 1 that the entire field of valuation rests on a single insight: a dollar today is worth more than a dollar in the future, because today's dollar can be invested and earn a return. The future value formula captures how money grows; the present value formula captures what a future payment is worth right now. Every model we've built since — annuities, bonds, stocks — is just a structured application of this idea.
Notice that the land more than doubles — roughly $170,000 in appreciation on a $150,000 base. That's the power of compounding over a meaningful time horizon. Even at a modest 6.50%, 12 years of compounding does a lot of work.
Recall from Week 3 that a bond is just a package of cash flows: a stream of coupon payments (an annuity) plus the return of face value at maturity (a single lump sum). You price a bond by discounting both pieces at the yield to maturity. The YTM is the market's required return — and when it differs from the coupon rate, the bond trades away from par.
That last line is worth pausing on. Whenever the coupon rate is less than the YTM, the bond must trade below par to compensate investors for the below-market coupon. The discount is the market's way of making the total return (coupons plus price appreciation to par at maturity) match the required 7%.
Recall from Week 4 that a stock's value is the present value of its future dividends. The simplest case is a stock that pays a constant dividend forever — a perpetuity. This applies naturally to preferred stock, which pays a fixed dividend with no maturity date. It also connects back to Week 2, where you first learned to value perpetuities.
This is the dividend discount model at its simplest: zero growth, constant payment, valued as a perpetuity. It's the same formula you used in Week 2 for any level perpetuity — PV = PMT ÷ r. The only conceptual leap is recognizing that preferred stock is a perpetuity. That connection between a valuation model and a financial instrument is exactly the kind of thinking that shows up on exams.
Most real financial decisions don't live neatly inside one formula. An investor buying a bond cares not just about the price today, but about what happens to the coupon payments between now and maturity. A company evaluating two securities needs to compare them even though they have completely different cash flow structures. This section works through problems that require more than one model — the kind of multi-step reasoning that separates strong analysts from formula-pluggers.
When you price a bond using the YTM, you're implicitly assuming that every coupon payment gets reinvested at the YTM itself. But what if reinvestment rates differ? If coupons are reinvested at a lower rate, your actual (realized) return will fall short of the promised YTM. This is called reinvestment risk, and it's one of the most important practical complications in bond investing.
To compute the realized return, you need to think about three things: (1) what you paid for the bond, (2) what you'll receive in face value at maturity, and (3) what your coupons will have grown to by maturity when reinvested at the actual reinvestment rate. The total of items (2) and (3) is your ending wealth; compare it to item (1), and you can back out the realized return — exactly the way you solved for unknown rates in Week 1.
This example ties together three concepts from different weeks: the annuity future value formula (Week 2) to compound the coupons, the single-cash-flow rate-solving technique (Week 1) to find the realized return, and the bond valuation framework (Week 3) that provides the context. Notice the logic: lower reinvestment rates mean your coupons grow more slowly, so your ending wealth is lower than the YTM promised. The 8% YTM assumed you could reinvest at 8%; at 6%, you end up with 7.69%.
Investors frequently face choices between different types of securities. Comparing a bond to a stock requires you to understand the cash flow structure of each, value them using the appropriate model, and think critically about what the numbers mean — because a bond and a stock are fundamentally different claims, even on the same company.
The dollar figures alone don't tell you which is a "better" investment — they're apples and oranges. The stock price reflects an infinite stream of growing dividends discounted at 11%. The bond price reflects a finite set of fixed cash flows discounted at 8%. What matters for the investor is whether each security is priced correctly given its risk. The stock demands a higher return (11% vs. 8%) because equity holders bear more risk than bondholders — they're last in line if the company runs into trouble. We'll formalize this risk-return relationship after the midterm when we study the Capital Asset Pricing Model in Week 9.
Annuity valuation from Week 2 shows up constantly in real decisions — not just loan payments, but compensation packages, pension payouts, and project evaluation. When someone offers you a stream of payments versus a lump sum, you need the annuity PV formula to make the comparison.
The key insight here is that the right choice depends on the discount rate. At 6%, the annuity wins. But if the employee could earn a higher return on invested funds — say 8% — the annuity's present value would shrink and the lump sum might come out ahead. This kind of sensitivity to assumptions comes up constantly in finance, and it's worth remembering for the midterm: if a problem tells you the discount rate, trust it — but if a problem asks you to evaluate which rate matters, that's where the real thinking happens.
One of the most valuable skills in finance — and one that gets tested on exams — is the ability to look at a completed solution and identify what went wrong. This requires you to understand the models well enough to know when an answer "smells" off, and to trace the logic backward to find the mistake. Let's practice with two common error patterns.
A surprisingly common mistake in bond valuation is confusing the coupon rate with the yield to maturity. The coupon rate determines the size of the cash flows; the YTM is the discount rate you use to value them. Using the coupon rate to discount gives you a mathematically precise but financially meaningless answer.
A student is asked to price a bond with a 5% coupon, $1,000 face value, 10 years to maturity, and a YTM of 7%. Coupons are semiannual. The student writes: "C = $25, r = 0.025, n = 20. Price = $25 × [(1 − 1.025−20) ÷ 0.025] + $1,000 ÷ 1.02520 = $1,000.00." The student concludes the bond trades at par. What went wrong?
The takeaway: discounting at the coupon rate will always return par value — that's a mathematical identity, not a coincidence. If you get exactly par on a bond problem and the coupon rate doesn't equal the YTM, you've used the wrong rate. This is a mechanical check you can run on any bond calculation.
The constant growth dividend discount model is P0 = D1 ÷ (r − g). The numerator is the next dividend — one period from now. When a problem tells you the dividend "just paid" or "most recently paid," it's giving you D0, and you need to grow it by one period before plugging it in. Forgetting to do so undervalues the stock by exactly one period of growth.
A student values White Acre Technologies stock. The company just paid a dividend of $3.00 per share. Dividends grow at 5% per year. The required return is 12%. The student writes: "P0 = $3.00 ÷ (0.12 − 0.05) = $3.00 ÷ 0.07 = $42.86." What went wrong?
The $2.14 difference (about 5% of the correct value) might not seem dramatic, but in a portfolio context it adds up fast. And on an exam, the wrong answer ($42.86) is almost certainly one of the distractors. Look for language cues: "just paid" = D0 (grow it); "will pay next year" = D1 (use directly).
Real financial questions don't come with labels telling you which formula to use. A news article about rising interest rates is really a question about bond price sensitivity. A friend asking whether their retirement savings are on track is a TVM problem. The skill tested by W5.3 — and commonly on exams — is reading a situation, identifying the relevant model, and setting up the calculation correctly.
Here's a practical framework for mapping scenarios to models. When you read a problem, ask three questions in order:
1. What kind of cash flows are involved? A single lump sum in the future points to basic TVM. A series of equal payments points to an annuity. A series of payments that goes on forever points to a perpetuity. A mix of regular payments plus a lump sum at the end — that's a bond.
2. Am I valuing something today (PV) or projecting what something will be worth later (FV)? If someone asks "what is this worth right now?" or "what should I pay?", you're solving for PV. If they ask "how much will I have?" or "what will this grow to?", you're solving for FV.
3. Are there growth or sensitivity elements? If dividends are growing, it's the constant growth DDM (or two-stage DDM). If the question asks "what happens when rates change?", you're looking at price sensitivity — recompute the price at the new rate and compare.
Notice how the interpretation matters as much as the calculation. A correct answer with no context is incomplete. Telling your uncle "12.20%" without noting that this is a high hurdle would be irresponsible advice. We'll study the historical behavior of different asset classes after the midterm (Week 7), which will give you the tools to put numbers like this in perspective.
This is exactly the kind of scenario-to-model translation the midterm tests. You start with a news story (rates are rising), identify the relevant framework (bond price sensitivity), set up the right calculations (price at old yield, price at new yield), and interpret the result (long bonds take big hits when rates rise). The 20-year maturity makes this bond especially vulnerable — recall from Week 3 that longer maturity means greater interest rate risk.
Two important things happened in that example. First, we identified that "will begin paying $1.50 next year" means D1 = $1.50 — no need to grow D0. Second, the aside about changing g from 8% to 9% illustrates a point from Week 4: the DDM becomes extremely sensitive when g gets close to r. As the denominator (r − g) shrinks, the price explodes. This is one of the model's key limitations, and it's fair game for a conceptual exam question.
This collects every formula you'll need for the midterm. Nothing here is new — it's the full toolkit from Weeks 1–4.
| Concept | Formula | Week |
|---|---|---|
| Future Value (single cash flow) | FV = PV × (1 + r)t | 1 |
| Present Value (single cash flow) | PV = FV ÷ (1 + r)t | 1 |
| Solving for rate | r = (FV ÷ PV)1/t − 1 | 1 |
| Solving for time | t = ln(FV ÷ PV) ÷ ln(1 + r) | 1 |
| PV of ordinary annuity | PV = PMT × [(1 − (1 + r)−t) ÷ r] | 2 |
| FV of ordinary annuity | FV = PMT × [((1 + r)t − 1) ÷ r] | 2 |
| PV of perpetuity | PV = PMT ÷ r | 2 |
| EAR from APR | EAR = (1 + APR/m)m − 1 | 2 |
| Bond price | P = C × [(1 − (1 + r)−n) ÷ r] + F ÷ (1 + r)n | 3 |
| Current yield | Current Yield = Annual Coupon ÷ Price | 3 |
| DDM (zero growth / preferred) | P₀ = D ÷ r | 4 |
| DDM (constant growth) | P₀ = D₁ ÷ (r − g) | 4 |
| Required return from DDM | r = (D₁ ÷ P₀) + g | 4 |
Work through these with your group. Problems are ordered from direct application to multi-step integration. Solutions are in the instructor's file.
You invest $3,000 per year in a retirement account earning 7% annually. How much will you have after 25 years? (Assume end-of-year deposits.)
Gold Acre Mining has a bond with a 4.5% coupon rate, $1,000 face value, and 12 years to maturity. Coupons are paid semiannually. If the YTM is 5.5%, what is the bond's price?
Silver Acre Pharmaceuticals just paid a dividend of $1.80 per share. Dividends are expected to grow at 3.5% annually. If investors require a 10% return, what should the stock trade for today?
A classmate is solving for the present value of a 10-year ordinary annuity paying $800 per year at a 5% discount rate. They write: "PV = $800 × [((1.05)10 − 1) ÷ 0.05] = $10,062.31." They report this as the present value. What error did they make, and what is the correct answer?
Blue Acre Financial has both bonds and common stock outstanding. The bonds carry a 6.5% coupon (semiannual), 10 years to maturity, $1,000 face value, with a YTM of 7%. The stock just paid a $2.50 dividend, dividends grow at 3% annually, and the required return on equity is 9%. (a) Compute the price of each security. (b) In one or two sentences, explain why the required return on the stock is higher than the YTM on the bond, even though they're from the same company.
You borrow $25,000 at 6% annual interest for 5 years, with equal annual payments. (a) What is your annual loan payment? (b) Suppose you could invest each payment at 8% instead. What would those payments grow to over the 5-year period? (This compares the cost of borrowing with the opportunity cost of the funds.)
For each scenario below, identify the specific financial model or formula you would use, state what variable you're solving for, and set up (but do not solve) the equation. (a) A landlord offers to sell you an apartment building that generates $40,000 per year in net rent, growing at 2% per year indefinitely, and you require a 10% return. (b) You won a lawsuit and can receive $150,000 today or $12,000 per year for 20 years — you want to know which is better if your opportunity cost is 5%. (c) A zero-coupon bond with a face value of $1,000 and 8 years to maturity is currently priced at $627.
Gold Acre Mining is evaluating whether to invest $100,000 in new drilling equipment. The equipment will generate $12,000 per year in additional cash flow for 15 years. The company's cost of capital is 9%. (a) What is the present value of the cash flows from the equipment? (b) Should Gold Acre make the investment? Explain your reasoning. (Note: We haven't formally introduced NPV yet — that's Week 6 — but you have all the tools you need to answer this question. Think about what makes an investment worthwhile.)