Case Assignment: Valuing bonds, stocks and a firm (discounted cash flow)
1. You buy a bond that pays $1,000 in 10 years plus an annual coupon of $50. The interest rate is 4%.
- What is the value of this bond?
- If the bond is a zero-coupon bond that still pays $1,000 in 10 years, what is its value?
- What would happen to the value of the bond if the inflation rate unexpectedly goes up?
- If the bond still pays a $50 coupon but the interest rate drops to 2%, what is its new value?
2. XYZ Corporation pays a dividend of $1 per share, and its required rate of return is 8%.
- Assuming zero growth in dividends, what is the value of each share?
- With 4% annual dividend growth, what is the value of each share?
- With 4% growth and a required return of 6%, what is the new value?
3. Acme Medical Corp. expects cash flows of $500,000 (2018), $550,000 (2019), $620,000 (2020), $700,000 (2021) and $800,000 (2022), then 3% annual growth. Its WACC is 7%.
- What is the present value of Acme’s future cash flows using the DCF model?
- With 200,000 common shares, no preferred shares and debt with a market value of $10,000,000, what is the value of each share?
- How would your answers change if the discount rate rose to 10%?
PV of coupons + PV of face value
PV of dividends (perpetuity or Gordon growth)
PV of forecast cash flows + terminal value
Worked solutions
Original example for study purposes (about 900 words). Calculations were checked in Python; show your own work and cite your course readings.
Question 1: Bond Valuation
A bond’s price equals the present value of its coupon payments plus the present value of its face (par) value, discounted at the market interest rate:
P = C × [1 − (1 + r)−n] / r + F / (1 + r)n
A. Coupon bond at 4%. C = $50, F = $1,000, r = 0.04, n = 10.
- PV of coupons = 50 × [1 − 1.04−10] / 0.04 = 50 × 8.1109 = $405.54
- PV of face value = 1,000 / 1.0410 = 1,000 / 1.4802 = $675.56
- Bond value = $405.54 + $675.56 = $1,081.11
The bond sells at a premium (above $1,000) because its coupon rate (5%) is higher than the market rate (4%).
B. Zero-coupon bond. With no coupons, only the face value is discounted: 1,000 / 1.0410 = $675.56. The zero sells at a deep discount because the investor’s entire return comes from the difference between the price paid and the $1,000 received at maturity.
C. Unexpected rise in inflation. The bond’s cash flows are fixed in nominal dollars, so higher-than-expected inflation erodes their purchasing power. Lenders demand compensation for inflation, so nominal interest rates rise (the Fisher effect: nominal rate ≈ real rate + expected inflation). Because bond prices move inversely to interest rates, the bond’s value falls. For example, if the market rate rose from 4% to 6%, the coupon bond would fall from $1,081.11 to about $926.40, below par. The zero-coupon bond would fall proportionally more because all of its cash flow arrives at the end, giving it a longer duration (Financial Industry Regulatory Authority [FINRA], n.d.). The reverse also holds: if inflation unexpectedly fell and rates declined, bond values would increase.
D. Interest rate drops to 2%.
- PV of coupons = 50 × [1 − 1.02−10] / 0.02 = 50 × 8.9826 = $449.13
- PV of face value = 1,000 / 1.0210 = $820.35
- New bond value = $1,269.48
Lower interest rates raise the bond’s value from $1,081.11 to $1,269.48, an increase of about 17%, confirming the inverse price–yield relationship.
Question 2: Stock Valuation (Dividend Discount Model)
A. Zero growth. A constant dividend forever is a perpetuity: P = D / r = $1 / 0.08 = $12.50 per share.
B. 4% constant growth. The Gordon growth model values a dividend that grows at a constant rate g: P0 = D1 / (r − g). Treating the $1 as the dividend just paid (D0), next year’s dividend is D1 = 1 × 1.04 = $1.04.
P0 = 1.04 / (0.08 − 0.04) = $26.00 per share
C. Required return falls to 6%. P0 = 1.04 / (0.06 − 0.04) = $52.00 per share. A two-point drop in the required return doubles the share price because the denominator (r − g) shrinks from 4% to 2%. This shows how sensitive growth-stock valuations are to discount rates, especially when r is close to g.
Note: if your course treats the $1 as next year’s dividend (D1), the answers are $25.00 for Part B and $50.00 for Part C. State your assumption clearly.
Question 3: Acme Medical Corp. (Discounted Cash Flow)
The DCF model values a firm as the present value of forecast free cash flows plus a terminal value representing all cash flows after the forecast period (Corporate Finance Institute, n.d.). I treat 2018 to 2022 as Years 1 to 5 and discount to the start of 2018.
| Year | t | Cash flow | Discount factor at 7% | Present value at 7% | Present value at 10% |
|---|---|---|---|---|---|
| 2018 | 1 | $500,000 | 0.9346 | $467,290 | $454,545 |
| 2019 | 2 | $550,000 | 0.8734 | $480,391 | $454,545 |
| 2020 | 3 | $620,000 | 0.8163 | $506,105 | $465,815 |
| 2021 | 4 | $700,000 | 0.7629 | $534,027 | $478,109 |
| 2022 | 5 | $800,000 | 0.7130 | $570,389 | $496,737 |
| PV of forecast cash flows | $2,558,201 | $2,349,753 | |||
A. Present value at 7%.
- Terminal value at end of 2022 = CF2022 × (1 + g) / (r − g) = 800,000 × 1.03 / (0.07 − 0.03) = 824,000 / 0.04 = $20,600,000
- PV of terminal value = 20,600,000 / 1.075 = $14,687,515
- Total present value (enterprise value) = $2,558,201 + $14,687,515 = $17,245,717
About 85% of Acme’s value comes from the terminal value, which is typical of DCF models and explains why the growth and discount rate assumptions matter so much.
B. Value per share at 7%. Equity value = enterprise value − debt − preferred stock = $17,245,717 − $10,000,000 − $0 = $7,245,717. Dividing by 200,000 shares gives $36.23 per share.
C. Discount rate rises to 10%.
- PV of forecast cash flows = $2,349,753
- Terminal value = 824,000 / (0.10 − 0.03) = $11,771,429; PV = 11,771,429 / 1.105 = $7,309,131
- Enterprise value = $9,658,884, a drop of about 44%
- Equity value = $9,658,884 − $10,000,000 = −$341,116
At a 10% discount rate, the firm’s operations are worth less than its debt, so the calculated equity value is negative (about −$1.71 per share). Because shareholders have limited liability, the shares cannot be worth less than zero; in practice, the stock would trade at a small positive value reflecting the option-like chance that results improve. The lesson is that a higher cost of capital reduces the present value of every future cash flow, and the effect is magnified for long-lived terminal cash flows and for highly leveraged firms, where debt claims are paid before shareholders.
Check your work: in Excel, Part 1A is =PV(4%,10,-50,-1000) and Part 3A’s forecast portion is =NPV(7%,500000,550000,620000,700000,800000). Add the discounted terminal value separately.
References
Corporate Finance Institute. (n.d.). DCF terminal value formula. https://corporatefinanceinstitute.com/resources/financial-modeling/dcf-terminal-value-formula/
Financial Industry Regulatory Authority. (n.d.). Brush up on bonds: Interest rate changes and duration. https://www.finra.org/investors/insights/bonds-interest-rate-changes-duration
[Your course background readings, in APA format.]
Frequently asked questions
Why do bond prices fall when interest rates rise?
A bond’s coupons are fixed. When new bonds offer higher rates, investors will only buy the older bond at a lower price that gives them a competitive yield.
Why can the Gordon growth model not be used when g is greater than or equal to r?
The formula divides by r − g. If growth equals or exceeds the required return, the denominator is zero or negative and the model gives no meaningful value.
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