Boiling Point Elevation:Freezing Point Depression Worksheet.pdf ... - Free Printable

Let’s solve each problem step by step. We’ll go one at a time, carefully checking our work.

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Problem 1:

> A bond has a face value of $1000 and pays annual coupons of $80. It matures in 5 years. If the yield to maturity is 6%, what is the current price of the bond?

We need to find the present value (PV) of all future cash flows from the bond:
- Annual coupon payments: $80 per year for 5 years
- Face value repayment: $1000 at end of year 5

Yield to maturity (YTM) = 6% → discount rate = 0.06

Use the bond pricing formula:

Price = PV(coupons) + PV(face value)

PV(coupons) = C × [1 - (1 + r)^(-n)] / r
= 80 × [1 - (1.06)^(-5)] / 0.06

Calculate (1.06)^(-5):

1.06^5 ≈ 1.3382255776 → so (1.06)^(-5) ≈ 1 / 1.3382255776 ≈ 0.747258

Then:

[1 - 0.747258] = 0.252742

Divide by 0.06: 0.252742 / 0.06 ≈ 4.212367

Multiply by 80: 80 × 4.212367 ≈ 336.99

Now PV(face value) = 1000 / (1.06)^5 ≈ 1000 × 0.747258 ≈ 747.26

Total Price = 336.99 + 747.26 = 1084.25

✔ Check: Since coupon rate (8%) > YTM (6%), bond should trade above par — yes, $1084.25 > $1000. Good.

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Problem 2:

> A stock just paid a dividend of $2.00. Dividends are expected to grow at 5% per year forever. The required return is 10%. What is the current stock price?

This is a constant growth dividend model (Gordon Growth Model):

P₀ = D₁ / (r - g)

D₀ = $2.00 → D₁ = D₀ × (1 + g) = 2.00 × 1.05 = $2.10

r = 10% = 0.10, g = 5% = 0.05

P₀ = 2.10 / (0.10 - 0.05) = 2.10 / 0.05 = $42.00

✔ Makes sense: growing dividends, reasonable return → price around 20x next dividend.

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Problem 3:

> You invest $1000 today. In 3 years, you get back $1300. What is the annual rate of return?

We use the compound interest formula:

FV = PV × (1 + r)^n

1300 = 1000 × (1 + r)^3

Divide both sides by 1000:

1.3 = (1 + r)^3

Take cube root of both sides:

(1.3)^(1/3) = ?

Try 1.09: 1.09³ = 1.09×1.09=1.1881 ×1.09≈1.295 → close!

Try 1.091: 1.091² = 1.190281; ×1.091 ≈ 1.2986 → still low

Try 1.092: 1.092² = 1.192464; ×1.092 ≈ 1.302 → too high

So between 1.091 and 1.092. Let's interpolate or use calculator-style:

Actually, 1.3^(1/3) ≈ 1.09139...

So r ≈ 0.09139 → 9.14%

✔ Check: 1000 × (1.0914)^3 ≈ 1000 × 1.300 → yes.

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Problem 4:

> A project costs $5000 now and returns $2000 per year for 3 years. Discount rate is 8%. What is NPV?

NPV = Sum of discounted cash flows - initial investment

Cash flows: Year 0: -$5000; Years 1–3: +$2000 each

Discount factor for year t: 1/(1.08)^t

Year 1: 2000 / 1.08 ≈ 1851.85
Year 2: 2000 / (1.08)^2 = 2000 / 1.1664 ≈ 1714.68
Year 3: 2000 / (1.08)^3 = 2000 / 1.259712 ≈ 1587.66

Sum of PVs = 1851.85 + 1714.68 + 1587.66 = 5154.19

NPV = 5154.19 - 5000 = $154.19

✔ Positive NPV → good project.

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Problem 5:

> A company has debt with market value $4 million, equity $6 million. Cost of debt = 5%, cost of equity = 12%. Tax rate = 30%. What is WACC?

WACC = (E/V) × Re + (D/V) × Rd × (1 - Tc)

Where:
E = equity = 6M
D = debt = 4M
V = total = 10M
Re = 12% = 0.12
Rd = 5% = 0.05
Tc = 30% = 0.30

Equity weight = 6/10 = 0.6
Debt weight = 4/10 = 0.4

After-tax cost of debt = 0.05 × (1 - 0.30) = 0.05 × 0.70 = 0.035

WACC = (0.6 × 0.12) + (0.4 × 0.035)
= 0.072 + 0.014 = 0.086 → 8.6%

✔ Standard calculation — weights sum to 1, tax shield applied correctly.

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Problem 6:

> A portfolio has two assets: Asset A (weight 40%, return 10%), Asset B (weight 60%, return 15%). What is expected portfolio return?

Expected return = wA × RA + wB × RB
= 0.4 × 10% + 0.6 × 15%
= 4% + 9% = 13%

✔ Simple weighted average — no risk here, just return.

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Problem 7:

> A firm has EBIT = $1 million, interest expense = $200,000, tax rate = 30%. What is net income?

Net Income = (EBIT - Interest) × (1 - Tax Rate)

= (1,000,000 - 200,000) × (1 - 0.30)
= 800,000 × 0.70 = $560,000

✔ Straightforward income statement logic.

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Problem 8:

> A machine costs $10,000, lasts 5 years, salvage value $2,000. Straight-line depreciation. What is annual depreciation?

Depreciation = (Cost - Salvage) / Life
= (10,000 - 2,000) / 5 = 8,000 / 5 = $1,600 per year

✔ Basic straight-line method.

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Problem 9:

> You borrow $10,000 at 6% annual interest, compounded monthly. How much do you owe after 1 year if no payments?

Formula: FV = PV × (1 + r/n)^(n×t)

r = 6% = 0.06, n = 12 (monthly), t = 1

FV = 10,000 × (1 + 0.06/12)^(12)
= 10,000 × (1.005)^12

Calculate (1.005)^12:

1.005^2 = 1.010025
1.005^4 = (1.010025)^2 ≈ 1.020150
1.005^8 ≈ (1.020150)^2 ≈ 1.040703
Then 1.005^12 = 1.005^8 × 1.005^4 ≈ 1.040703 × 1.020150 ≈ 1.061678

So FV ≈ 10,000 × 1.061678 = $10,616.78

✔ Matches standard compound interest tables.

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Problem 10:

> A call option has strike price $50, stock price $55, premium $6. What is intrinsic value? Time value?

Intrinsic value = max(0, Stock Price - Strike) = max(0, 55 - 50) = $5

Time value = Premium - Intrinsic Value = 6 - 5 = $1

✔ Basic options terminology.

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Final Answer:

1. $1084.25
2. $42.00
3. 9.14%
4. $154.19
5. 8.6%
6. 13%
7. $560,000
8. $1,600
9. $10,616.78
10. Intrinsic: $5, Time: $1