Johanna's Café

Contents

1. Business Scenario
2. Time Value of Money
3. Interest Measurement
4. Bonds and Bond Types
5. Annuities
6. Loans and Borrowing
7. Yield Rates and IRR
8. Risk Management
9. Capital Budgeting
10. Term Structure
11. Recap

Notation Glossary

The Business Scenario

Johanna is opening a specialty café that requires $100,000 in startup capital, with expected business lifetime of 20 years. Her business plan includes:

1. Initial Financing Options:
   • Commercial bank loan: $100,000 at 6% annual interest for 5 years
   • Bond issuance: 100 bonds at $1,000 each with various structures
   • Angel investor: $100,000 for 30% ownership with various cash flow arrangements
   • Equipment financing: $40,000 with different payment structures
   • Line of credit: $25,000 with variable interest rates
2. Expansion Plans (Year 5):
   • Second location requiring additional $150,000
   • Upgrade equipment for $75,000
   • Refinance original debt
3. Long-term Financial Planning:
   • Retirement fund for herself
   • Business succession planning
   • Protection against interest rate and inflation risks

Time Value of Money

Money available today is worth more than the same amount in the future

This explains why all financial relationships in Johanna’s business require compensation for the time dimension of money.

Decision Point: Johanna must decide between:

• Immediate discount: $5,000 off equipment if purchased today
• Future benefit: 12 months of free maintenance worth $6,000 received over the next year

Using time value principles, she calculates that the future benefit is worth less than $5,000 today at her cost of capital and chooses the immediate discount.

Question 1: Johanna is offered two payment options for new café furniture: $8,500 now or $9,000 in 6 months. If her business earns 8% annually on investments, which option should she choose?

Solution: To compare the options, we need to find the future value of $8,500 after 6 months: \(FV = 8,500 \times (1 + \frac{0.08}{2}) = 8,500 \times 1.04 = \$8,840\)

Since $8,840 < $9,000, Johanna should pay $8,500 now and invest the $500 difference, which would grow to: \(500 \times 1.04 = \$520\)

This gives her a net advantage of $520 - ($9,000 - $8,840) = $360.

Interest Measurement

Simple Interest

Formula: \(I = P \times r \times t\)

Where:

• \(I\) = Interest amount
• \(P\) = Principal
• \(r\) = Interest rate (decimal)
• \(t\) = Time (in years)

To find simple interest, multiply the principal by the rate by the time.

Example: Johanna’s line of credit uses simple interest at 8%: \(25,000 \times 0.08 \times 1 = \$2,000\) interest for one year

Question 2: Johanna loans $15,000 to her friend Alex to help start a bakery. They agree on 6% simple interest for 30 months. How much will Alex repay Johanna at the end of the term?

Solution: Interest amount: \(I = 15,000 \times 0.06 \times \frac{30}{12} = 15,000 \times 0.06 \times 2.5 = \$2,250\)

Total repayment: \(15,000 + 2,250 = \$17,250\)

Compound Interest

Formula: \(FV = PV(1+r)^t\)

Where:

• \(FV\) = Future value (accumulated amount)
• \(PV\) = Present value (principal)
• \(r\) = Interest rate per period
• \(t\) = Number of periods

To find the accumulated amount, multiply the present value by one plus the interest rate raised to the power of time.

Example: Johanna’s bank loan after 5 years without payments: \(100,000(1+0.06)^5 = \$133,823\)

Key Insight: Compound interest operates like a snowball rolling downhill - as Johanna’s debt grows, it collects more interest, which makes it collect even more interest in the next period.

Question 3: Johanna invests $20,000 of her café’s profits in a 5-year CD with 4.5% interest compounded quarterly. How much will she have at maturity?

Solution: Quarterly rate: \(r = \frac{0.045}{4} = 0.01125\)

Number of periods: \(t = 5 \times 4 = 20\)

Accumulated amount: \(FV = 20,000 \times (1 + 0.01125)^{20} = 20,000 \times 1.2508 = \$25,016\)

Present Value (Discounting)

Formula: \(PV = \frac{FV}{(1+r)^t} = FV \times v^t\)

Where:

• \(PV\) = Present value
• \(FV\) = Future amount
• \(r\) = Interest rate per period
• \(t\) = Number of periods
• \(v\) = Discount factor \((1+r)^{-1}\)

To find what a future amount is worth today, divide that amount by one plus the interest rate raised to the power of time.

Example: The present value of Johanna’s bank loan repayment in 5 years: \(\frac{133,823}{(1+0.06)^5} = \$100,000\)

Decision Point: Johanna compares two loans:

• 6.1% compounded annually
• 5.95% compounded monthly (effective annual rate: 6.12%)

Despite the lower nominal rate on the second loan, she chooses the first loan due to its lower effective cost.

Question 4: Johanna’s café will need a new espresso machine in 3 years that will cost $12,000. If she can earn 5% interest compounded semi-annually, how much should she set aside today?

Solution: Semi-annual rate: \(r = \frac{0.05}{2} = 0.025\)

Number of periods: \(t = 3 \times 2 = 6\)

Present value: \(PV = \frac{12,000}{(1+0.025)^6} = \frac{12,000}{1.1596} = \$10,348.40\)

Discount Factor

Formula: \(v = \frac{1}{1+r}\)

Where:

• \(v\) = Discount factor
• \(r\) = Interest rate per period

To find the discount factor, divide 1 by one plus the interest rate.

Example: Bank’s annual discount factor: \(\frac{1}{1.06} = 0.9434\)

Question 5: Johanna calculated that a supplier payment of $2,000 due in 18 months has a present value of $1,820. What annual interest rate is she using?

Solution: Using the present value formula: \(PV = \frac{FV}{(1+r)^t}\)

Rearranging: \((1+r)^t = \frac{FV}{PV}\)

\[(1+r)^{1.5} = \frac{2,000}{1,820} = 1.0989\]

Taking the root: \(1+r = (1.0989)^{1/1.5} = 1.0989^{0.6667} = 1.0648\)

\[r = 0.0648 = 6.48\%\]

Force of Interest (Continuous Compounding)

Formula: \(FV = PV \cdot e^{\delta t}\)

Where:

• \(FV\) = Future value
• \(PV\) = Present value
• \(\delta\) = Force of interest
• \(t\) = Time
• \(e\) = Mathematical constant (≈ 2.71828)

To find the accumulated amount with continuous compounding, multiply the present value by e raised to the power of the force of interest times time.

Example: If Johanna’s loan used continuous compounding with force of interest \(\delta = 0.058\): \(100,000 \times e^{0.058 \times 5} \approx \$133,750\)

Question 6: Johanna invests $50,000 in a fund that uses continuous compounding with a force of interest of 5.2%. How much will she have after 4 years?

Solution: \(FV = 50,000 \times e^{0.052 \times 4} = 50,000 \times e^{0.208} = 50,000 \times 1.2312 = \$61,560\)

Force of Interest as a Derivative

Formula: \(\delta_t = \frac{a'(t)}{a(t)}\)

Where:

• \(\delta_t\) = Force of interest at time \(t\)
• \(a(t)\) = Accumulation function
• \(a'(t)\) = Derivative of accumulation function

To find the instantaneous growth rate, take the derivative of the accumulation function divided by the accumulation function itself.

Example: Johanna’s 5% annually compounded investment has a force of interest \(\delta = \ln(1.05) \approx 4.88\%\). Marcus, her accountant, explains this means at any moment in time, her investment is growing at exactly 4.88% of its current value.

Question 7: Johanna has an investment that grows according to the accumulation function \(a(t) = (1+t)^2\) where \(t\) is measured in years. Find the force of interest after 3 years.

Solution: First, find the derivative of the accumulation function: \(a'(t) = 2(1+t)\)

Then calculate the force of interest at \(t = 3\): \(\delta_3 = \frac{a'(3)}{a(3)} = \frac{2(1+3)}{(1+3)^2} = \frac{8}{16} = 0.5 = 50\%\)

Real Rate of Interest

Formula: \(r_{real} = \frac{r_{nom}-i}{1+i}\)

Where:

• \(r_{real}\) = Real interest rate
• \(r_{nom}\) = Nominal interest rate
• \(i\) = Inflation rate

To find the real interest rate after inflation, subtract the inflation rate from the nominal rate, then divide by one plus the inflation rate.

Example: If Johanna’s loan has a 6% nominal rate and inflation is 2%: \(r_{real} = \frac{0.06-0.02}{1+0.02} = \frac{0.04}{1.02} = 0.0392 = 3.92\%\)

Elena, Johanna’s banker, explains this means high-interest investments during high-inflation periods aren’t as profitable as they appear.

Question 8: Johanna has a supplier contract with fixed annual payments increasing at 3% per year. If inflation is currently 4.5%, what is the real annual rate of change in her payments?

Solution: Using the real rate formula with \(r_{nom} = 3\%\) and \(i = 4.5\%\): \(r_{real} = \frac{0.03-0.045}{1+0.045} = \frac{-0.015}{1.045} = -0.0144 = -1.44\%\)

This means her payments are actually decreasing by about 1.44% per year in real terms.

Equivalent Rates Across Compounding Periods

Formula: \(\left(1+\frac{r^{(m)}}{m}\right)^m = (1+r_{eff})\)

Where:

• \(r^{(m)}\) = Nominal rate compounded \(m\) times per year
• \(r_{eff}\) = Effective annual rate
• \(m\) = Number of compounding periods per year

To find the equivalent annual rate for a rate compounded multiple times per year, take one plus the rate divided by the number of compounding periods, raise it to that number, and subtract one.

Example: Johanna’s 6% annual rate compounded monthly: \(\left(1+\frac{0.06}{12}\right)^{12} - 1 = (1.005)^{12} - 1 = 0.0617 = 6.17\%\) effective annual rate

Question 9: Johanna has two investment options for her café’s cash reserves:

• Option A: 5.8% compounded semi-annually
• Option B: 5.7% compounded continuously Which option provides the better return?

Solution: For Option A (semi-annual compounding):

\[r_{eff} = \left(1+\frac{0.058}{2}\right)^2 - 1 = (1.029)^2 - 1 = 0.0594 = 5.94\%\]

For Option B (continuous compounding): \(r_{eff} = e^{0.057} - 1 = 1.0586 - 1 = 0.0586 = 5.86\%\)

Since 5.94% > 5.86%, Option A provides the better return.

Question 10: Johanna needs to accumulate $40,000 for café renovations in 4 years. If she currently has $30,000 to invest, what annual interest rate (compounded quarterly) does she need to achieve her goal?

Solution: Using the compound interest formula: \(FV = PV(1+r/m)^{mt}\)

Substituting the values: \(40,000 = 30,000 \times \left(1+\frac{r}{4}\right)^{4 \times 4}\)

\[\frac{40,000}{30,000} = \left(1+\frac{r}{4}\right)^{16}\] \[1.3333 = \left(1+\frac{r}{4}\right)^{16}\] \[\left(1+\frac{r}{4}\right) = 1.3333^{1/16} = 1.3333^{0.0625} = 1.0182\] \[\frac{r}{4} = 0.0182\] \[r = 0.0182 \times 4 = 0.0728 = 7.28\%\]

Johanna needs an annual interest rate of 7.28% compounded quarterly to reach her goal.

Bonds and Bond Types

A bond is essentially a loan that Johanna’s business takes from multiple lenders (bondholders).

Zero-Coupon Bonds

Formula: \(PV = \frac{F}{(1+r_m)^n}\)

Where:

• \(PV\) = Price (present value)
• \(F\) = Face value (par value)
• \(r_m\) = Yield rate
• \(n\) = Term to maturity

To find the price of a zero-coupon bond, divide the face value by one plus the interest rate raised to the power of the time until maturity.

Example: If Johanna issues 5-year zero-coupon bonds with 7% yield: \(PV = \frac{1,000}{(1+0.07)^5} = \$712.99\)

Johanna receives $712.99 today and pays back $1,000 in 5 years.

Question 11: Johanna issues zero-coupon bonds to finance her kitchen equipment. If the bonds have a face value of $1,000 each, mature in 4 years, and investors require an 8% yield, how much will Johanna receive for each bond today?

Solution: \(PV = \frac{1,000}{(1+0.08)^4} = \frac{1,000}{1.3605} = \$735.02\)

Johanna will receive $735.02 per bond issued.

Regular Coupon Bonds

Formula: \(PV = C \times PV(A,n,r_m) + F \times v^n\)

Where:

• \(PV\) = Bond price
• \(C\) = Coupon payment
• \(PV(A,n,r_m)\) = Present value of annuity for \(n\) periods at rate \(r_m\)
• \(F\) = Face value
• \(v^n\) = Discount factor \((1+r_m)^{-n}\) for \(n\) periods

To find the price of a coupon bond, add the present value of all coupon payments (coupon amount times the present value of an annuity) plus the present value of the face value (face value times the discount factor).

Example: If Johanna issues 5-year bonds with 5% annual coupons (paying $50/year per $1,000 bond):

Question 12: Johanna is considering issuing 3-year bonds with a face value of $1,000 each and annual coupon payments of 6%. If the market yield is 7%, what should be the issue price of each bond?

Solution: Annual coupon payment = $1,000 × 0.06 = $60

\[PV = 60 \times PV(A,3,0.07) + 1,000 \times (1.07)^{-3}\] \[PV = 60 \times \frac{1-(1.07)^{-3}}{0.07} + 1,000 \times (1.07)^{-3}\] \[PV = 60 \times 2.6243 + 1,000 \times 0.8163\] \[PV = 157.46 + 816.30 = \$973.76\]

Each bond should be issued at $973.76.

Premium/Discount Formula

Formula: \(PV = F + (F \times r_c - F \times r_m) \times PV(A,n,r_m)\)

Where:

• \(PV\) = Bond price
• \(F\) = Face value
• \(r_c\) = Coupon rate
• \(r_m\) = Yield rate
• \(PV(A,n,r_m)\) = Present value of annuity for \(n\) periods at rate \(r_m\)

To find a bond’s price, add the face value (redemption value) to the present value of the difference between coupon payments and yield payments.

If \(C = F \cdot r_c\) (coupon payment), then the bondholders compare:

• What they receive: actual coupon payment \(F \cdot r_c\) per period
• What they expect: market yield payment \(F \cdot j\) per period (the interest the market expects). The “yield payments” are not real payments — they are the theoretical payments the market would require from a bond of that risk and maturity.

The term \((F r_c - F j)a_n\) in the formula captures the present value of this difference:

• If \(F \cdot r_c > F \cdot j\): bond pays more than market demands → premium bond
• If \(F \cdot r_c < F \cdot j\): bond pays less than market demands → discount bond

Example: For Johanna’s 5% coupon bonds when market yield is 6%: \(PV = 1,000 + (1,000 \times 0.05 - 1,000 \times 0.06) \times PV(A,5,0.06)\) \(PV = 1,000 + (-0.01 \times 1,000) \times 4.2124 = 1,000 - 42.12 = \$957.88\)

Question 13: Johanna issues 4-year bonds with a face value of $1,000 and a coupon rate of 7.5%. If the market yield is 6%, calculate the bond price using the premium/discount formula.

Solution: \(PV = 1,000 + (1,000 \times 0.075 - 1,000 \times 0.06) \times PV(A,4,0.06)\)

\[PV = 1,000 + (0.015 \times 1,000) \times \frac{1-(1.06)^{-4}}{0.06}\] \[PV = 1,000 + 0.015 \times 1,000 \times 3.4651\] \[PV = 1,000 + 15 \times 3.4651\] \[PV = 1,000 + 51.98 = \$1,051.98\]

The bond would sell at a premium of $51.98 above face value.

Makeham’s Formula

Formula: \(PV = K + \frac{G}{r_m}(F-K)\)

Where:

• \(PV\) = Bond price
• \(K\) = \(F \times v^n\) (present value of face value at maturity)
• \(G\) = Coupon rate per dollar of face value
• \(r_m\) = Yield rate
• \(F\) = Face value

To calculate bond price with Makeham’s formula, add the present value of the redemption amount to a fraction of the difference between the face value and this present value.

Example: For Johanna’s 5% coupon bonds with 6% yield: \(K = 1,000 \times (1.06)^{-5} = \$747.26\) \(G = 0.05\) \(PV = 747.26 + \frac{0.05}{0.06}(1,000-747.26) = 747.26 + 0.833 \times 252.74 = \$957.88\)

Question 14: Using Makeham’s formula, calculate the price of a $5,000 face value bond with a coupon rate of 4%, yield of 5%, and 10 years to maturity.

Solution: First, calculate \(K\): \(K = 5,000 \times (1.05)^{-10} = 5,000 \times 0.6139 = \$3,069.50\)

Then apply Makeham’s formula: \(PV = 3,069.50 + \frac{0.04}{0.05}(5,000-3,069.50)\)

\[PV = 3,069.50 + 0.8 \times 1,930.50\] \[PV = 3,069.50 + 1,544.40 = \$4,613.90\]

Bond Interpolation

Formula: \(PV_{t+k} = PV_t(1+r_m)^k - F \times r_c \cdot k\)

Where:

• \(PV_{t+k}\) = Bond price \(k\) periods after a coupon date
• \(PV_t\) = Bond price at the last coupon date
• \(r_m\) = Yield rate
• \(F\) = Face value
• \(r_c\) = Coupon rate
• \(k\) = Fraction of period elapsed

To find a bond’s price between coupon dates, take the price at the last coupon date, grow it at the yield rate for the fraction of the period elapsed, and subtract the accrued coupon interest for that fraction of time.

Example: If Johanna’s bond is priced at $957.88 at a coupon date, and 3 months (0.25 year) have passed: \(PV_{0.25} = 957.88(1.06)^{0.25} - 1,000 \times 0.05 \times 0.25 = 957.88 \times 1.0147 - 12.50 = 972.20 - 12.50 = \$959.70\)

Question 15: Johanna’s café issued bonds with a face value of $2,000, coupon rate of 6% paid annually, and yield of 7%. If the bonds were worth $1,900 at the last coupon date and 8 months have passed, calculate the current bond price.

Solution: \(PV_{8/12} = 1,900 \times (1.07)^{8/12} - 2,000 \times 0.06 \times 8/12\)

\[PV_{8/12} = 1,900 \times (1.07)^{0.6667} - 2,000 \times 0.06 \times 0.6667\] \[PV_{8/12} = 1,900 \times 1.0458 - 120 \times 0.6667\] \[PV_{8/12} = 1,987.02 - 80.00 = \$1,907.02\]

Clean vs. Dirty Price

• Clean price: Price excluding accrued interest
• Dirty price: Price including accrued interest
• Relationship: Dirty price = Clean price + Accrued interest

Example: In the bond above, $959.70 is the dirty price, $947.20 is the clean price, and $12.50 is the accrued interest.

Question 16: Johanna’s bonds with $1,000 face value and 8% annual coupon are quoted at a clean price of $1,050 in the market. If it’s been 3 months since the last coupon payment, what is the dirty price a buyer would actually pay?

Solution: Accrued interest = $1,000 × 0.08 × 3/12 = $1,000 × 0.08 × 0.25 = $20

Dirty price = Clean price + Accrued interest = $1,050 + $20 = $1,070

The buyer would pay $1,070 per bond.

Callable Bonds

Feature: Johanna can redeem them early (useful if interest rates drop)

Valuation: \(PV_{callable} = PV_{regular} - \text{Call option value}\)

Example: If Johanna’s regular 5-year bonds are worth $957.92 and the call option is valued at $25: \(PV_{callable} = \$957.92 - \$25 = \$932.92\)

Raj, Johanna’s financial advisor, explains investors pay less for callable bonds because Johanna could repay early if rates fall, denying them high returns in a low-rate environment.

Question 17: Johanna issues 7-year bonds with a face value of $1,000, coupon rate of 7% paid annually, and yield of 6.5%. The call option is valued at $35 per bond. Calculate the price of the callable bond.

Solution: First, calculate the price of the regular bond: \(PV_{regular} = 70 \times PV(A,7,0.065) + 1,000 \times (1.065)^{-7}\)

\[PV_{regular} = 70 \times \frac{1-(1.065)^{-7}}{0.065} + 1,000 \times 0.6364\] \[PV_{regular} = 70 \times 5.3137 + 1,000 \times 0.6364\] \[PV_{regular} = 371.96 + 636.40 = \$1,008.36\]

Now calculate the callable bond price: \(PV_{callable} = 1,008.36 - 35 = \$973.36\)

Convertible Bonds

Feature: Bondholders can convert to ownership shares

Valuation: \(PV_{convertible} = PV_{regular} + \text{Conversion option value}\)

Example: If Johanna’s regular bonds are worth $957.92 and the conversion option is valued at $50: \(PV_{convertible} = \$957.92 + \$50 = \$1,007.92\)

Decision Point: Johanna must choose between issuing regular bonds at 7% interest or convertible bonds at 5.5%. Despite the lower interest cost, she chooses regular bonds to avoid potential ownership dilution if her business succeeds dramatically.

Question 18: Johanna is considering issuing convertible bonds with a face value of $1,000, coupon rate of 5%, and 6 years to maturity. The market yield for similar non-convertible bonds is 6.5%, and the conversion option is valued at $80. Calculate the market price of the convertible bond.

Solution: First, calculate the price of a regular bond: \(PV_{regular} = 50 \times PV(A,6,0.065) + 1,000 \times (1.065)^{-6}\)

\[PV_{regular} = 50 \times \frac{1-(1.065)^{-6}}{0.065} + 1,000 \times 0.6778\] \[PV_{regular} = 50 \times 4.7472 + 1,000 \times 0.6778\] \[PV_{regular} = 237.36 + 677.80 = \$915.16\]

Now calculate the convertible bond price: \(PV_{convertible} = 915.16 + 80 = \$995.16\)

Annuities (Regular Payment Streams)

Present Value of Annuity Immediate

Formula: \(PV(A) = \frac{1-(1+r)^{-n}}{r}\)

Where:

• \(PV(A)\) = Present value of annuity for \(n\) periods
• \(r\) = Interest rate per period
• \(n\) = Number of periods

To find the present value of a series of equal payments made at the end of each period, divide the difference between 1 and the discount factor raised to the number of periods by the interest rate.

Example: Johanna’s bank loan with 60 monthly payments:

Monthly rate: \(i = \frac{0.06}{12} = 0.005\)

Present value: \(a_{60} = \frac{1-(1.005)^{-60}}{0.005} = 51.2473\)

Question 19: Johanna wants to offer a coffee subscription service where customers pay $30 at the end of each month for unlimited coffee. If she values this revenue stream at 8% annual interest and expects customers to subscribe for 2 years on average, what is the present value of each subscription?

Solution: Monthly interest rate: \(r = \frac{0.08}{12} = 0.0067\)

Number of payments: \(n = 2 \times 12 = 24\)

Present value: \(PV = 30 \times PV(A,24,0.0067) = 30 \times \frac{1-(1.0067)^{-24}}{0.0067}\)

\[PV = 30 \times \frac{1-0.8528}{0.0067} = 30 \times \frac{0.1472}{0.0067} = 30 \times 21.97 = \$659.10\]

Future Value of Annuity

Formula: \(FV(A) = \frac{(1+r)^n-1}{r}\)

Where:

• \(FV(A)\) = Future value of annuity for \(n\) periods
• \(r\) = Interest rate per period
• \(n\) = Number of periods

To find the future value of a series of equal payments, divide the difference between one plus the interest rate raised to the number of periods and 1 by the interest rate.

Example: If Johanna saves $500 monthly at 4% annual interest:

Monthly rate: \(i = \frac{0.04}{12} = 0.0033\)

After 5 years: \(s_{60} = \frac{(1.0033)^{60}-1}{0.0033} = 73.5746\)

Total accumulated: \(500 \times 73.5746 = \$36,787.30\)

Question 20: Johanna sets aside $800 at the end of each quarter to save for new café furniture. If she invests at 5% annual interest compounded quarterly, how much will she have accumulated after 3 years?

Solution: Quarterly interest rate: \(r = \frac{0.05}{4} = 0.0125\)

Number of quarters: \(n = 3 \times 4 = 12\)

Accumulated value: \(FV = 800 \times FV(A,12,0.0125) = 800 \times \frac{(1.0125)^{12}-1}{0.0125}\)

\[FV = 800 \times \frac{1.1607-1}{0.0125} = 800 \times \frac{0.1607}{0.0125} = 800 \times 12.856 = \$10,284.80\]

Annuity Due (Payments at Beginning)

Formula: \(PV(A_{due}) = PV(A) \times (1+r)\)

Where:

• \(PV(A_{due})\) = Present value of annuity due for \(n\) periods
• \(PV(A)\) = Present value of ordinary annuity for \(n\) periods
• \(r\) = Interest rate per period

To find the present value of a series of equal payments made at the beginning of each period, multiply the present value of an equivalent ordinary annuity by one plus the interest rate.

Example: If Johanna’s equipment lease requires payments at the beginning of each month:

Question 21: Johanna’s supplier requires payments of $2,000 at the beginning of each quarter for specialty coffee beans. If the contract runs for 2 years and Johanna’s cost of capital is 6% per annum, what is the present value of this payment stream?

Solution: Quarterly interest rate: \(r = \frac{0.06}{4} = 0.015\)

Number of quarters: \(n = 2 \times 4 = 8\)

Present value of ordinary annuity: \(PV(A,8,0.015) = \frac{1-(1.015)^{-8}}{0.015} = \frac{1-0.8873}{0.015} = \frac{0.1127}{0.015} = 7.5133\)

Present value of annuity due: \(PV(A_{due},8,0.015) = 7.5133 \times 1.015 = 7.6260\)

Total present value: \(PV = 2,000 \times 7.6260 = \$15,252\)

Perpetuities (Payments Forever)

Formula: \(PV(P) = \frac{PMT}{r}\)

Where:

• \(PV(P)\) = Present value of perpetuity
• \(PMT\) = Regular payment amount
• \(r\) = Interest rate per period

To find the present value of an infinite series of equal payments, divide the payment amount by the interest rate.

Example: If Johanna’s investor wanted $5,000 annual payments indefinitely at 5% yield:

Lisa, Johanna’s investment partner, explains this is like buying an income stream that never stops.

Question 22: As part of her business succession plan, Johanna wants to establish a scholarship fund that will provide $3,000 annually forever to train future baristas. If the fund can earn 4.5% annually, how much capital does Johanna need to set aside?

Solution: \(PV = \frac{3,000}{0.045} = \$66,667\)

Johanna needs to set aside $66,667 to establish the perpetual scholarship fund.

Geometrically Increasing Annuities

Formula: \(PV(G) = PMT \times \frac{1-(1+g)^n(1+r)^{-n}}{r-g}\)

Where:

• \(PV(G)\) = Present value of geometric annuity
• \(PMT\) = Initial payment
• \(g\) = Growth rate of payments
• \(r\) = Interest rate
• \(n\) = Number of periods

To find the present value of payments that grow at a constant rate, multiply the initial payment by the difference between 1 and the growth factor raised to \(n\) multiplied by the discount factor raised to \(n\), divided by the difference between the interest rate and the growth rate.

Example: If Johanna agrees to pay her investor $10,000 the first year with 3% annual increases for 10 years, at 6% discount rate:

Question 23: Johanna’s lease agreement for her café space starts at $3,500 per month and increases by 2% each year for 5 years. If her discount rate is 7% per annum, what is the present value of all lease payments?

Solution: Monthly discount rate: \(i = \frac{0.07}{12} = 0.0058\)

Monthly growth rate: \(g = (1.02)^{1/12} - 1 = 1.0017 - 1 = 0.0017\)

Number of monthly payments: \(n = 5 \times 12 = 60\)

Present value: \(PV = 3,500 \times \frac{1-(1+0.0017)^{60}(1+0.0058)^{-60}}{0.0058-0.0017}\)

\[PV = 3,500 \times \frac{1-(1.0017)^{60}(1.0058)^{-60}}{0.0041}\] \[PV = 3,500 \times \frac{1-1.1069 \times 0.7079}{0.0041}\] \[PV = 3,500 \times \frac{1-0.7836}{0.0041} = 3,500 \times \frac{0.2164}{0.0041} = 3,500 \times 52.78 = \$184,730\]

Arithmetically Increasing Annuities

Formula: \(PV(I) = d \times \frac{PV(A) - n \times v^n}{r}\)

Where:

• \(PV(I)\) = Present value of an annuity increasing by constant amount \(d\) per period
• \(d\) = Increment amount
• \(PV(A)\) = Present value of level annuity for \(n\) periods
• \(n\) = Number of periods
• \(v\) = Discount factor \((1+r)^{-1}\)
• \(r\) = Interest rate

To find the present value of payments that increase by a constant amount each period, multiply the increment amount by the result of subtracting the number of periods times the discount factor raised to that power from the present value of a level annuity, then divide by the interest rate.

Example: Johanna pays a marketing bonus that starts at $1,000 and increases by $1,000 each year for 5 years, at 5% discount rate:

Total present value: $1,000 × 8.24 = $8,240

Question 24: Johanna hires a consultant to improve operations. The consultant will be paid $5,000 for the first year, with payments increasing by $1,000 each year for a total of 4 years. If Johanna’s discount rate is 6%, what is the present value of this payment stream?

Solution: First, calculate the present value of a level annuity: \(PV(A,4,0.06) = \frac{1-(1.06)^{-4}}{0.06} = \frac{1-0.7921}{0.06} = \frac{0.2079}{0.06} = 3.4650\)

Then calculate the increasing annuity formula: \(PV(I,4,0.06,1) = \frac{PV(A,4,0.06) - 4 \times (1.06)^{-4}}{0.06}\)

\[PV(I,4,0.06,1) = \frac{3.4650 - 4 \times 0.7921}{0.06}\] \[PV(I,4,0.06,1) = \frac{3.4650 - 3.1684}{0.06} = \frac{0.2966}{0.06} = 4.9433\]

Initial payment = $5,000 Increment = $1,000

Total present value: \(PV = 5,000 \times PV(A,4,0.06) + 1,000 \times PV(I,4,0.06,1)\)

\[PV = 5,000 \times 3.4650 + 1,000 \times 4.9433\] \[PV = 17,325 + 4,943.30 = \$22,268.30\]

Arithmetically Decreasing Annuities

Formula: \(D\bar{a}_{\angln}(1) = n - i \cdot I\bar{a}_{\angln}(1)\)

Where:

• \(D\bar{a}_{\angln}(1)\) = Present value of an annuity decreasing by 1 per period
• \(n\) = Number of periods
• \(i\) = Interest rate
• \(I\bar{a}_{\angln}(1)\) = Present value of an increasing annuity

To find the present value of payments that decrease by a constant amount each period, subtract the interest rate times the present value of an increasing annuity from the number of periods.

Example: Johanna offers a signing bonus that pays $5,000 initially, decreasing by $1,000 each year for 5 years:

Total present value: $1,000 × 4.588 = $4,588

Key Insight: Think of Johanna’s annuity payments as stones dropped in a pond - each creating ripples that spread over time. Earlier stones have larger total ripple effects because they have more time to spread.

Question 25: To manage cash flow during startup, Johanna arranges a supplier agreement where she’ll pay $8,000 in the first year, decreasing by $1,500 annually for 4 years. Using a 5.5% discount rate, calculate the present value of this arrangement.

Solution: First, calculate the increasing annuity value: \(PV(I,4,0.055,1) = \frac{PV(A,4,0.055) - 4 \times (1.055)^{-4}}{0.055}\)

\[PV(I,4,0.055,1) = \frac{3.5160 - 4 \times 0.8112}{0.055}\] \[PV(I,4,0.055,1) = \frac{3.5160 - 3.2448}{0.055} = \frac{0.2712}{0.055} = 4.9309\]

Now calculate the decreasing annuity value: \(PV(D,4,0.055,1) = 4 - 0.055 \times PV(I,4,0.055,1) = 4 - 0.055 \times 4.9309 = 4 - 0.2712 = 3.7288\)

With initial payment = $8,000 and decrement = $1,500: \(PV = 8,000 \times PV(A,4,0.055) - 1,500 \times PV(I,4,0.055,1)\)

Or using the decreasing annuity formula: \(PV = (8,000 + 1,500(4-1)) \times PV(A,4,0.055) - 1,500 \times PV(D,4,0.055,1)\)

\[PV = (8,000 + 4,500) \times 3.5160 - 1,500 \times 3.7288\] \[PV = 12,500 \times 3.5160 - 1,500 \times 3.7288\] \[PV = 43,950 - 5,593.20 = \$38,356.80\]

Continuous Annuities

Formula for present value: \(\overline{PV(A)} = \frac{1-e^{-\delta n}}{\delta}\)

Where:

• \(\overline{PV(A)}\) = Present value of continuous annuity
• \(\delta\) = Force of interest
• \(n\) = Time period

To find the present value of a continuous stream of payments, divide the difference between 1 and e raised to the negative force of interest times the time by the force of interest.

Formula for future value: \(\overline{FV(A)} = \frac{e^{\delta n}-1}{\delta}\)

Where:

• \(\overline{FV(A)}\) = Future value of continuous annuity
• \(\delta\) = Force of interest
• \(n\) = Time period

To find the future value of a continuous stream of payments, divide the difference between e raised to the force of interest times the time and 1 by the force of interest.

Example: Johanna’s café generates continuous revenue stream equivalent to $10,000 annually for 5 years, discounted at 6% force of interest:

Present value: $10,000 × 4.32 = $43,200

Question 26: Johanna’s drive-thru window generates revenue continuously throughout the day, equivalent to $50,000 annually. If she expects this revenue stream to continue for 7 years and uses a 5.5% force of interest, what is the present value of this revenue stream?

Solution: \(\overline{PV(A,7,0.055)} = \frac{1-e^{-0.055 \times 7}}{0.055} = \frac{1-e^{-0.385}}{0.055} = \frac{1-0.6804}{0.055} = \frac{0.3196}{0.055} = 5.8109\)

Present value: \(PV = 50,000 \times 5.8109 = \$290,545\)

Contingent Payments

Formula: \(a_x = \sum_{t=0}^{\infty} v^t \cdot {}_{t}p_x\)

Where:

• \(a_x\) = Present value of contingent payments
• \(v\) = Discount factor
• \({}_{t}p_x\) = Probability of payment at time \(t\)

To find the present value of payments that depend on an uncertain event, sum the products of the discount factor for each period times the probability the payment will be made in that period.

Example: Johanna considering a business insurance policy that pays $5,000 annually as long as her café remains open, with 90% probability of surviving each year, at 5% interest:

Decision Point: Johanna must choose between a fixed equipment maintenance contract at $1,200 annually or a variable one starting at $900 and increasing by 10% each year. She calculates the present value of each over 7 years and selects the fixed payment option.

Question 27: Johanna is offered a business opportunity that will pay $4,000 at the end of each year as long as a particular coffee supplier stays in business. If the annual probability of the supplier remaining in business is 85% and Johanna’s discount rate is 6%, what is the present value of this opportunity?

Solution: We can use the formula for contingent payments:

\[a_x = 4,000 \times \sum_{t=1}^{\infty} (1.06)^{-t} \cdot (0.85)^t\]

This is a geometric series with first term \(\frac{0.85}{1.06} = 0.8019\) and ratio \(\frac{0.85}{1.06} = 0.8019\):

\[a_x = 4,000 \times \frac{0.8019}{1-0.8019} = 4,000 \times \frac{0.8019}{0.1981} = 4,000 \times 4.0479 = \$16,191.60\]

Johanna should value this opportunity at $16,191.60.

Question 28: Johanna wants to expand her café to a second location. Her financial advisor estimates there’s a 70% chance the new location will be profitable each year (independent from year to year). If the new location would generate additional profit of $25,000 annually when successful and requires annual investment of $15,000 regardless of success, what is the expected present value of this expansion over 10 years using an 8% discount rate?

Solution: Expected annual net cash flow: \((70\% \times \$25,000) - \$15,000 = \$17,500 - \$15,000 = \$2,500\)

This is a fixed annuity for 10 years, so: \(PV = 2,500 \times PV(A,10,0.08) = 2,500 \times \frac{1-(1.08)^{-10}}{0.08}\)

\[PV = 2,500 \times \frac{1-0.4632}{0.08} = 2,500 \times \frac{0.5368}{0.08} = 2,500 \times 6.71 = \$16,775\]

The expected present value of the expansion is $16,775.

Loans and Borrowing

Level Payment

Formula: \(PMT = \frac{PV}{PV(A,n,r)}\)

Where:

• \(PMT\) = Periodic payment
• \(PV\) = Principal (loan amount)
• \(PV(A,n,r)\) = Present value of annuity for \(n\) periods at rate \(r\)

To find the periodic payment needed to repay a loan with equal installments, divide the initial loan amount by the present value of an annuity for the number of payment periods.

Example: Johanna’s monthly bank loan payment:

Question 29: Johanna borrows $50,000 to renovate her café. The loan has a 5-year term with monthly payments and an annual interest rate of 7.2%. Calculate her monthly payment.

Solution: Monthly interest rate: \(r = \frac{0.072}{12} = 0.006\)

Number of payments: \(n = 5 \times 12 = 60\)

Present value of annuity: \(PV(A,60,0.006) = \frac{1-(1.006)^{-60}}{0.006} = \frac{1-0.6988}{0.006} = \frac{0.3012}{0.006} = 50.2\)

Monthly payment: \(PMT = \frac{50,000}{50.2} = \$996.02\)

Outstanding Balance

Formula: \(BAL_t = PV(1+r)^t - PMT \times FV(A,t,r)\)

Where:

• \(BAL_t\) = Outstanding balance at time \(t\)
• \(PV\) = Principal
• \(r\) = Interest rate per period
• \(t\) = Number of periods elapsed
• \(PMT\) = Periodic payment
• \(FV(A,t,r)\) = Future value of annuity for \(t\) periods at rate \(r\)

To find the loan balance at any point in time using the retrospective method, subtract the accumulated value of all payments made from the accumulated value of the initial loan amount.

Example: Johanna’s loan balance after 3 years (36 payments):

Key Insight: Johanna’s loan balance is like a mountain - starting with a gentle slope (mostly interest), getting steeper over time (more principal).

Question 30: Johanna took a $75,000 equipment loan at 6.6% annual interest with monthly payments over 7 years. Calculate the outstanding balance after 3 years using the retrospective method.

Solution: Monthly interest rate: \(r = \frac{0.066}{12} = 0.0055\)

Number of months in loan term: \(n = 7 \times 12 = 84\)

Monthly payment: \(PMT = \frac{75,000}{PV(A,84,0.0055)} = \frac{75,000}{\frac{1-(1.0055)^{-84}}{0.0055}} = \frac{75,000}{\frac{1-0.6311}{0.0055}} = \frac{75,000}{\frac{0.3689}{0.0055}} = \frac{75,000}{67.0727} = \$1,118.19\)

Number of payments made: \(t = 3 \times 12 = 36\)

Outstanding balance: \(BAL_{36} = 75,000(1+0.0055)^{36} - 1,118.19 \times \frac{(1.0055)^{36}-1}{0.0055}\)

\[BAL_{36} = 75,000 \times 1.2177 - 1,118.19 \times \frac{1.2177-1}{0.0055}\] \[BAL_{36} = 91,327.50 - 1,118.19 \times \frac{0.2177}{0.0055} = 91,327.50 - 1,118.19 \times 39.5818\] \[BAL_{36} = 91,327.50 - 44,259.10 = \$47,068.40\]

After 3 years, Johanna still owes $47,068.40 on her loan.

Retrospective vs. Prospective Balance

Retrospective (looking backward): \(BAL_t = PV(1+r)^t - PMT \times FV(A,t,r)\)

To find the loan balance looking backward, subtract the accumulated value of all payments made from the accumulated value of the initial loan amount.

Prospective (looking forward): \(BAL_t = PMT \times PV(A,n-t,r)\)

Where:

• \(BAL_t\) = Outstanding balance at time \(t\)
• \(PMT\) = Periodic payment
• \(PV(A,n-t,r)\) = Present value of annuity for remaining periods at rate \(r\)

To find the loan balance looking forward, calculate the present value of all remaining payments.

Example: Johanna’s loan after 3 years (prospective method):

Remaining payments: 60 - 36 = 24

Question 31: Using the prospective method, calculate the outstanding balance of Johanna’s $40,000 business loan after making 30 payments. The loan has a 10-year term with monthly payments and an annual interest rate of 5.4%.

Solution: Monthly interest rate: \(r = \frac{0.054}{12} = 0.0045\)

Number of payments in loan term: \(n = 10 \times 12 = 120\)

Monthly payment: \(PMT = \frac{40,000}{PV(A,120,0.0045)} = \frac{40,000}{\frac{1-(1.0045)^{-120}}{0.0045}} = \frac{40,000}{\frac{1-0.5827}{0.0045}} = \frac{40,000}{\frac{0.4173}{0.0045}} = \frac{40,000}{92.7333} = \$431.34\)

Remaining payments: \(120 - 30 = 90\)

Outstanding balance (prospective method):

\[BAL_{30} = 431.34 \times PV(A,90,0.0045) = 431.34 \times \frac{1-(1.0045)^{-90}}{0.0045}\] \[BAL_{30} = 431.34 \times \frac{1-0.6672}{0.0045} = 431.34 \times \frac{0.3328}{0.0045} = 431.34 \times 73.9556 = \$31,902.12\]

Sinking Fund Method

Formula for interest payment: \(\text{Interest payment} = L \times r_b\)

Where:

• \(L\) = Loan amount
• \(r_b\) = Bond interest rate

To calculate interest-only payments on a loan, multiply the loan amount by the interest rate.

Formula for sinking fund deposit: \(\text{Sinking fund deposit} = \frac{L}{FV(A,n,r_{sf})}\)

Where:

• \(L\) = Loan amount
• \(FV(A,n,r_{sf})\) = Future value of annuity for \(n\) periods at sinking fund rate \(r_{sf}\)
• \(r_{sf}\) = Sinking fund interest rate

To calculate the periodic deposit needed in a separate fund to accumulate the loan principal, divide the loan amount by the future value of an annuity factor at the fund’s interest rate.

Example: Johanna uses sinking fund method for her $100,000 bond repayment over 5 years, with 6% bond interest and 4% sinking fund return:

Annual interest payment: $100,000 × 0.06 = $6,000

Annual sinking fund deposit: \(\frac{100,000}{FV(A,5,0.04)} = \frac{100,000}{\frac{(1.04)^5-1}{0.04}} = \frac{100,000}{5.4163} = \$18,463\)

Total annual outlay: $6,000 + $18,463 = $24,463

Question 32: Johanna issues $150,000 in 5-year bonds at 7% annual interest. Rather than amortizing the debt, she plans to create a sinking fund that earns 4.5% annually to repay the principal at maturity. Calculate: a) The annual interest payment b) The annual sinking fund deposit c) The total annual outlay

Solution: a) Annual interest payment: \(150,000 \times 0.07 = \$10,500\)

b) Annual sinking fund deposit: \(\frac{150,000}{s_5^{0.045}} = \frac{150,000}{\frac{(1.045)^5-1}{0.045}} = \frac{150,000}{\frac{1.2462-1}{0.045}} = \frac{150,000}{\frac{0.2462}{0.045}} = \frac{150,000}{5.4711} = \$27,417\)

c) Total annual outlay: \(10,500 + 27,417 = \$37,917\)

Interest vs. Principal Portions

Interest portion of payment: \(I_t = r \times BAL_{t-1}\)

Where:

• \(I_t\) = Interest portion of payment at time \(t\)
• \(r\) = Interest rate per period
• \(BAL_{t-1}\) = Outstanding balance before payment

To calculate the interest portion of any payment, multiply the interest rate by the outstanding balance just before the payment.

Principal portion of payment: \(PR_t = PMT - I_t\)

Where:

• \(PR_t\) = Principal portion of payment at time \(t\)
• \(PMT\) = Periodic payment
• \(I_t\) = Interest portion of payment at time \(t\)

To calculate the principal portion of any payment, subtract the interest portion from the total payment.

Example: Johanna’s first loan payment:

Interest: $0.005 × $100,000 = $500

Principal: $1,951.51 - $500 = $1,451.51

Question 33: Johanna has a $60,000 loan at 6% annual interest with monthly payments over 5 years. For the 25th payment: a) Calculate the outstanding balance just before the payment b) Determine the interest portion of the payment c) Find the principal portion of the payment

Solution: Monthly interest rate: \(r = \frac{0.06}{12} = 0.005\)

Number of payments: \(n = 5 \times 12 = 60\)

Monthly payment: \(PMT = \frac{60,000}{PV(A,60,0.005)} = \frac{60,000}{\frac{1-(1.005)^{-60}}{0.005}} = \frac{60,000}{51.2473} = \$1,170.79\)

a) Outstanding balance just before the 25th payment: We can use the prospective method with 60 - 24 = 36 payments remaining: \(BAL_{24} = 1,170.79 \times PV(A,36,0.005) = 1,170.79 \times \frac{1-(1.005)^{-36}}{0.005} = 1,170.79 \times 33.4155 = \$39,122.59\)

b) Interest portion of the 25th payment: \(I_{25} = 0.005 \times 39,122.59 = \$195.61\)

c) Principal portion of the 25th payment: \(PR_{25} = 1,170.79 - 195.61 = \$975.18\)

Decision Point: Johanna’s expansion requires $150,000. She compares a 7-year amortizing loan versus a 5-year interest-only loan with balloon payment. Despite lower initial payments on the second option, she chooses the amortizing loan to avoid refinancing risk.

Question 34: Johanna is creating an amortization schedule for a $25,000 loan at 6.9% annual interest with quarterly payments over 4 years. Create the first three rows of the amortization schedule showing the payment breakdown.

Solution: Quarterly interest rate: \(i = \frac{0.069}{4} = 0.01725\)

Number of quarters: \(n = 4 \times 4 = 16\)

Quarterly payment: \(PMT = \frac{25,000}{a_{16}^{0.01725}} = \frac{25,000}{\frac{1-(1.01725)^{-16}}{0.01725}} = \frac{25,000}{12.9536} = \$1,929.25\)

Amortization schedule:

Payment	Beginning Balance	Payment	Interest	Principal	Ending Balance
1	$25,000.00	$1,929.25	$431.25	$1,498.00	$23,502.00
2	$23,502.00	$1,929.25	$405.41	$1,523.84	$21,978.16
3	$21,978.16	$1,929.25	$379.12	$1,550.13	$20,428.03

Calculations: Payment 1:

• Interest: $25,000 × 0.01725 = $431.25
• Principal: $1,929.25 - $431.25 = $1,498.00
• Ending balance: $25,000 - $1,498.00 = $23,502.00

Payment 2:

• Interest: $23,502.00 × 0.01725 = $405.41
• Principal: $1,929.25 - $405.41 = $1,523.84
• Ending balance: $23,502.00 - $1,523.84 = $21,978.16

Payment 3:

• Interest: $21,978.16 × 0.01725 = $379.12
• Principal: $1,929.25 - $379.12 = $1,550.13
• Ending balance: $21,978.16 - $1,550.13 = $20,428.03

Yield Rates and Internal Rate of Return (IRR)

Internal Rate of Return (IRR)

Formula: \(\sum_{k=0}^{n}CF_k(1+IRR)^{-t_k} = 0\)

Where:

• \(CF_k\) = Cash flow at time \(t_k\)
• \(IRR\) = Internal rate of return
• \(t_k\) = Time of cash flow \(k\)

To find the IRR, solve for the interest rate that makes the net present value of all cash flows equal to zero.

Example: For the investor putting $100,000 into Johanna’s business:

• Initial investment: -$100,000
• Annual profits share: $15,000 for 10 years
• Business sale: $70,000 (investor’s share) after 10 years

Solving gives IRR ≈ 12.31%

Question 35: Johanna is evaluating an espresso machine purchase with the following cash flows:

• Initial cost: -$18,000
• Annual savings: $4,500 for 6 years
• Salvage value: $2,000 after 6 years

Calculate the IRR of this investment.

Solution: Using the IRR formula: \(-18,000 + \sum_{t=1}^{6} \frac{4,500}{(1+IRR)^t} + \frac{2,000}{(1+IRR)^6} = 0\)

To solve for IRR, we can use a financial calculator or try different rates:

Let’s try IRR = 12%: \(-18,000 + 4,500 \times PV(A,6,0.12) + 2,000 \times (1.12)^{-6}\)

\[-18,000 + 4,500 \times 4.1114 + 2,000 \times 0.5066\] \[-18,000 + 18,501.30 + 1,013.20 = 1,514.50 > 0\]

Let’s try IRR = 13%: \(-18,000 + 4,500 \times PV(A,6,0.13) + 2,000 \times (1.13)^{-6}\)

\[-18,000 + 4,500 \times 3.9975 + 2,000 \times 0.4665\] \[-18,000 + 17,988.75 + 933.00 = 921.75 > 0\]

Let’s try IRR = 15%: \(-18,000 + 4,500 \times PV(A,6,0.15) + 2,000 \times (1.15)^{-6}\)

\[-18,000 + 4,500 \times 3.7845 + 2,000 \times 0.4323\] \[-18,000 + 17,030.25 + 864.60 = -105.15 < 0\]

Using linear interpolation between 13% and 15%: \(IRR \approx 13\% + \frac{921.75}{921.75+105.15} \times 2\% = 13\% + \frac{921.75}{1,026.90} \times 2\% = 13\% + 1.79\% = 14.79\%\)

The IRR is approximately 14.8%.

Multiple IRRs

Some cash flow patterns can have multiple IRR solutions when cash flows change sign more than once.

Example: Johanna considers an equipment arrangement with:

• Initial payment: -$50,000
• Rebate after 1 year: +$120,000
• Final payment after 2 years: -$80,000

This has two solutions: \(i = 20\%\) and \(i = 100\%\)

David, Johanna’s equipment vendor, explains this mathematical curiosity means IRR alone isn’t enough to evaluate this deal.

Question 36: Johanna is considering a specialty coffee machine with unusual cash flows:

• Initial investment: -$12,000
• Year 1 rebate: +$28,000
• Year 2 final payment: -$18,000

a) Show that this investment has multiple IRRs b) Explain why multiple IRRs occur c) Recommend an alternative evaluation method

Solution: a) The IRR equation is: \(-12,000 + \frac{28,000}{(1+i)^1} - \frac{18,000}{(1+i)^2} = 0\)

This can be rewritten as: \(-12,000(1+i)^2 + 28,000(1+i) - 18,000 = 0\)

Dividing by \((1+i)^2\): \(-12,000 + \frac{28,000}{(1+i)} - \frac{18,000}{(1+i)^2} = 0\)

Let \(y = 1/(1+i)\): \(-12,000 + 28,000y - 18,000y^2 = 0\)

\[-18,000y^2 + 28,000y - 12,000 = 0\]

Using the quadratic formula: \(y = \frac{-28,000 \pm \sqrt{28,000^2 - 4 \times (-18,000) \times (-12,000)}}{2 \times (-18,000)}\)

\[y = \frac{-28,000 \pm \sqrt{784,000,000 - 864,000,000}}{-36,000}\] \[y = \frac{-28,000 \pm \sqrt{-80,000,000}}{-36,000}\]

This has no real solutions due to the negative discriminant.

Let’s verify by testing some values:

At \(i = 25\%\): \(-12,000 + \frac{28,000}{1.25} - \frac{18,000}{(1.25)^2} = -12,000 + 22,400 - 11,520 = -1,120 < 0\)

At \(i = 100\%\): \(-12,000 + \frac{28,000}{2} - \frac{18,000}{4} = -12,000 + 14,000 - 4,500 = -2,500 < 0\)

At \(i = 10\%\): \(-12,000 + \frac{28,000}{1.1} - \frac{18,000}{(1.1)^2} = -12,000 + 25,455 - 14,876 = -1,421 < 0\)

At \(i = 1000\%\): \(-12,000 + \frac{28,000}{11} - \frac{18,000}{121} = -12,000 + 2,545 - 149 = -9,604 < 0\)

This problem actually has no real IRR solutions, which can occur with certain cash flow patterns.

b) Multiple IRRs (or no IRR solutions) occur when cash flows change sign more than once. This is because the NPV function can cross the x-axis multiple times or not at all.

c) Alternative evaluation methods:

• Net Present Value (NPV) at the company’s cost of capital
• Modified Internal Rate of Return (MIRR)
• Profitability Index
• Payback Period

For this case, Johanna should use NPV with her café’s cost of capital to evaluate this investment.

Net Present Value (NPV)

Formula: \(NPV = \sum_{k=0}^{n}CF_k(1+r)^{-t_k}\)

Where:

• \(NPV\) = Net present value
• \(CF_k\) = Cash flow at time \(t_k\)
• \(r\) = Discount rate
• \(t_k\) = Time of cash flow \(k\)

To calculate NPV, sum the present values of all cash flows, discounted at a predetermined interest rate.

Example: If the investor’s required return is 10%:

Since NPV > 0, this is a good investment at a 10% required return.

Decision Point: Johanna must choose between two coffee roasters: one with lower upfront cost but higher operating costs, and another with higher upfront cost but lower operating costs. She calculates NPV for both and selects the second option despite its higher initial investment.

Question 37: Johanna is comparing two coffee suppliers:

• Supplier A: $4,000 upfront cost, $1,200 monthly operating cost
• Supplier B: $8,500 upfront cost, $900 monthly operating cost

Using a 3-year time horizon and 8% annual discount rate, calculate the NPV of each option and recommend the better choice.

Solution: Monthly discount rate: \(r = \frac{0.08}{12} = 0.0067\)

Number of months: \(n = 3 \times 12 = 36\)

Supplier A: \(NPV_A = -4,000 - 1,200 \times PV(A,36,0.0067)\)

\[NPV_A = -4,000 - 1,200 \times \frac{1-(1.0067)^{-36}}{0.0067}\] \[NPV_A = -4,000 - 1,200 \times \frac{1-0.7858}{0.0067}\] \[NPV_A = -4,000 - 1,200 \times \frac{0.2142}{0.0067}\] \[NPV_A = -4,000 - 1,200 \times 31.97 = -4,000 - 38,364 = -\$42,364\]

Supplier B: \(NPV_B = -8,500 - 900 \times PV(A,36,0.0067)\)

\[NPV_B = -8,500 - 900 \times 31.97 = -8,500 - 28,773 = -\$37,273\]

Since \(NPV_B > NPV_A\) (-$37,273 > -$42,364), Supplier B is the better choice despite its higher upfront cost. Johanna would save $5,091 in present value terms over the 3-year period.

Risk Management and Interest Rate Sensitivity

Duration

Formula: \(D = -\frac{1}{PV} \frac{dPV}{dr}\)

Where:

• \(D\) = Duration
• \(PV\) = Price
• \(dPV/dr\) = Rate of change of price with respect to interest rate

To measure the sensitivity of an asset’s value to interest rate changes, calculate the negative of the percentage change in price for a given change in interest rate.

Example: For Johanna’s 5-year bond with $50 annual coupons and $1,000 face value at 6% yield: Duration ≈ 4.12 years

This means a 1% increase in interest rates would reduce the bond’s value by approximately 4.12%.

Question 38: Johanna issues 4-year bonds with a face value of $1,000 and annual coupons of $60. If the market yield is 7%, calculate: a) The bond price b) The Macaulay duration c) The approximate percentage change in bond price if interest rates increase by 0.5%

Solution: a) Bond price: \(PV = 60 \times PV(A,4,0.07) + 1,000 \times (1.07)^{-4}\)

\[PV = 60 \times \frac{1-(1.07)^{-4}}{0.07} + 1,000 \times (1.07)^{-4}\] \[PV = 60 \times 3.3872 + 1,000 \times 0.7629\] \[PV = 203.23 + 762.90 = \$966.13\]

b) Macaulay duration: \(D = \frac{1 \times 60 \times (1.07)^{-1} + 2 \times 60 \times (1.07)^{-2} + 3 \times 60 \times (1.07)^{-3} + 4 \times (60+1000) \times (1.07)^{-4}}{PV}\)

\[D = \frac{56.07 + 104.84 + 147.08 + 811.68}{966.13}\] \[D = \frac{1,119.67}{966.13} = 1.159 \times 4 = 3.64 \text{ years}\]

c) Percentage change in bond price: Using the duration approximation: \(\frac{\Delta PV}{PV} \approx -D \times \Delta r\)

\[\frac{\Delta PV}{PV} \approx -3.64 \times 0.005 = -0.0182 = -1.82\%\]

The bond price would decrease by approximately 1.82% if interest rates increased by 0.5%.

Convexity

Formula: \(CONV = \frac{1}{PV} \frac{d^2PV}{dr^2}\)

Where:

• \(CONV\) = Convexity
• \(PV\) = Price
• \(d^2PV/dr^2\) = Second derivative of price with respect to interest rate

To measure the curvature of the price-yield relationship, calculate the second derivative of price with respect to interest rate, divided by price.

Example: For Johanna’s bond, convexity ≈ 19.8

This means duration slightly underestimates price changes for large interest rate movements.

Question 39: For the bond in the previous example: a) Calculate the modified duration b) Using both duration and convexity, estimate the percentage change in bond price if interest rates increase by 2%

Solution: a) Modified duration: \(D_{mod} = \frac{D}{1+r} = \frac{3.64}{1.07} = 3.40\)

b) Using both duration and convexity: First, we need to estimate the convexity. For a 4-year, 6% coupon bond with 7% yield, the convexity is approximately 14.5 (this would normally be calculated using a formula involving the second derivative).

Percentage change in price: \(\frac{\Delta PV}{PV} \approx -D_{mod} \times \Delta r + \frac{1}{2} \times CONV \times (\Delta r)^2\)

\[\frac{\Delta PV}{PV} \approx -3.40 \times 0.02 + \frac{1}{2} \times 14.5 \times (0.02)^2\] \[\frac{\Delta PV}{PV} \approx -0.068 + 0.5 \times 14.5 \times 0.0004\] \[\frac{\Delta PV}{PV} \approx -0.068 + 0.5 \times 0.0058 = -0.068 + 0.0029 = -0.0651 = -6.51\%\]

Using duration alone would estimate a 6.8% decrease, while including convexity gives a more accurate 6.51% decrease.

Immunization Strategies

Johanna can protect against interest rate risk by matching the duration of assets and liabilities.

Example: If Johanna has a $50,000 obligation due in 5 years, she can invest in a portfolio of bonds with an average duration of 5 years to minimize interest rate risk.

Question 40: Johanna needs to accumulate $75,000 in exactly 4 years for a café expansion. She currently has $60,000 to invest and wants to immunize against interest rate risk. If current interest rates are 5%: a) What should be the duration of her investment portfolio? b) If she invests in zero-coupon bonds, what maturity should she choose? c) Explain why immunization protects her against interest rate changes

Solution: a) To immunize against interest rate risk, the duration of the investment portfolio should equal the time horizon of the liability. Therefore, Johanna should maintain a portfolio duration of 4 years.

b) For zero-coupon bonds, the duration equals the maturity. Therefore, she should invest in zero-coupon bonds with a 4-year maturity.

c) Immunization works because:

• If interest rates rise, the market value of her bonds will decrease (price risk), but she’ll be able to reinvest coupon payments at higher rates (reinvestment risk)
• If interest rates fall, the market value of her bonds will increase (price risk), but she’ll have to reinvest coupon payments at lower rates (reinvestment risk)
• When the duration equals the time horizon, these two effects offset each other, protecting the total accumulated value at the target date

The required amount in 4 years: \(75,000 = 60,000 \times (1+r)^4\)

Solving for \(r\): \(\frac{75,000}{60,000} = (1+r)^4\)

\[1.25 = (1+r)^4\] \[r = 1.25^{1/4} - 1 = 0.0574 = 5.74\%\]

As long as Johanna maintains a portfolio with a 4-year duration, she will accumulate approximately $75,000 regardless of how interest rates change.

Arbitrage-Free Valuation

Formula: \(PV(t,T) = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^T r_s ds}\right]\)

Where:

• \(PV(t,T)\) = Price at time \(t\) of a zero-coupon bond maturing at time \(T\)
• \(\mathbb{E}^{\mathbb{Q}}\) = Expected value under risk-neutral probability measure
• \(r_s\) = Instantaneous short rate at time \(s\)

To price financial instruments in a way that prevents risk-free profits, calculate the expected value of discounted cash flows under a risk-neutral probability measure.

Example: Priya, Johanna’s risk manager, uses this approach to value coffee price derivatives that Johanna might use to hedge against commodity price fluctuations.

Decision Point: Johanna anticipates interest rates rising over the next year. She restructures her debt to use more fixed-rate financing and less variable-rate financing to protect against higher future costs.

Question 41: Johanna is considering hedging against coffee price fluctuations using derivatives. The current spot price of premium coffee beans is $8.50 per pound, the risk-free rate is 4%, and the cost of storage is 5% of the spot price per year. Using arbitrage-free pricing: a) Calculate the theoretical 1-year forward price b) If a supplier offers a 1-year forward contract at $9.25, should Johanna accept?

Solution: a) Theoretical 1-year forward price: The cost of carry model for a storable commodity: \(F_{0,1} = S_0 \times (1 + r + s)\)

Where:

• \(F_{0,1}\) = 1-year forward price
• \(S_0\) = Current spot price
• \(r\) = Risk-free rate
• \(s\) = Storage cost rate

\[F_{0,1} = 8.50 \times (1 + 0.04 + 0.05) = 8.50 \times 1.09 = \$9.27\]

b) Since the offered forward price of $9.25 is less than the theoretical price of $9.27, Johanna should accept the supplier’s offer. She’s getting a slight discount compared to the arbitrage-free price.

Capital Budgeting and Investment Decisions

Comparing Investment Options

Johanna is deciding between different equipment options:

1. Machine A: $30,000 cost, $8,000 annual savings for 5 years
2. Machine B: $45,000 cost, $12,000 annual savings for 5 years

Using NPV at 8% required return:

Machine A: \(NPV = -30,000 + 8,000 \times PV(A,5,0.08)\) \(NPV = -30,000 + 8,000 \times 3.9927 = \$1,941.60\)

Machine B: \(NPV = -45,000 + 12,000 \times PV(A,5,0.08)\) \(NPV = -45,000 + 12,000 \times 3.9927 = \$2,912.40\)

Machine B has the higher NPV and would be the better choice.

Question 42: Johanna is evaluating three possible locations for her second café:

• Location X: Initial cost $175,000, expected annual profit $45,000
• Location Y: Initial cost $220,000, expected annual profit $52,000
• Location Z: Initial cost $250,000, expected annual profit $58,000

Assuming a 10-year time horizon and 9% discount rate, which location should Johanna choose based on NPV?

Solution: Present value of annuity for 10 years at 9%: \(PV(A,10,0.09) = \frac{1-(1.09)^{-10}}{0.09} = \frac{1-0.4224}{0.09} = \frac{0.5776}{0.09} = 6.4178\)

Location X NPV: \(NPV_X = -175,000 + 45,000 \times 6.4178 = -175,000 + 288,801 = \$113,801\)

Location Y NPV: \(NPV_Y = -220,000 + 52,000 \times 6.4178 = -220,000 + 333,726 = \$113,726\)

Location Z NPV: \(NPV_Z = -250,000 + 58,000 \times 6.4178 = -250,000 + 372,232 = \$122,232\)

Based on NPV, Johanna should choose Location Z as it has the highest NPV of $122,232.

Risk-Adjusted Returns

Different financing options carry different risks:

• Bank loan (lower risk): 6% interest
• Bond (medium risk): 7% coupon
• Equity investment (higher risk): 12% expected return

The risk premium compensates investors for the uncertainty of future cash flows.

Question 43: Johanna is evaluating three investment opportunities for her excess cash:

• Treasury bonds: 3.5% return with negligible risk
• Corporate bonds: 5.8% return with moderate risk
• Small business loan: 8.5% return with high risk

If Johanna requires a 2% risk premium for moderate risk and a 5% risk premium for high risk, calculate the risk-adjusted returns and recommend the best option.

Solution: Risk-adjusted returns:

• Treasury bonds: 3.5% - 0% = 3.5%
• Corporate bonds: 5.8% - 2% = 3.8%
• Small business loan: 8.5% - 5% = 3.5%

On a risk-adjusted basis, the corporate bonds offer the highest return (3.8%), so Johanna should choose that option.

The Power of Compounding

The value of Johanna’s café if sold after:

2 years: \(FV = 100,000 \times (1.15)^2 = \$132,250\)

5 years: \(FV = 100,000 \times (1.15)^5 = \$201,136\)

10 years: \(FV = 100,000 \times (1.15)^{10} = \$404,556\)

This illustrates the power of compound interest over time.

Key Insight: Think of Johanna’s business value like a bamboo plant - seemingly growing slowly at first, then dramatically accelerating as compounding takes effect.

Question 44: Johanna invests $25,000 of her profits into an account earning 7.5% compounded annually. Calculate the account value after: a) 5 years b) 10 years c) 20 years d) Calculate how many years it will take for the initial investment to triple in value

Solution: a) After 5 years: \(FV = 25,000 \times (1.075)^5 = 25,000 \times 1.4356 = \$35,890\)

b) After 10 years: \(FV = 25,000 \times (1.075)^{10} = 25,000 \times 2.0610 = \$51,525\)

c) After 20 years: \(FV = 25,000 \times (1.075)^{20} = 25,000 \times 4.2479 = \$106,198\)

d) To find when the investment triples: \(3 \times 25,000 = 25,000 \times (1.075)^n\) \(3 = (1.075)^n\) \(\ln(3) = n \times \ln(1.075)\) \(n = \frac{\ln(3)}{\ln(1.075)} = \frac{1.0986}{0.0723} = 15.19 \text{ years}\)

It will take approximately 15.2 years for Johanna’s investment to triple in value.

Term Structure and Yield Curves

Bond Yield Curve

The relationship between bond yield and maturity:

• If Johanna issues 1-year bonds at 4% yield
• 2-year bonds at 4.5% yield
• 5-year bonds at 5.5% yield
• 10-year bonds at 6% yield

This upward-sloping yield curve suggests markets expect rising interest rates.

Question 45: Johanna observes the following yields for Treasury bonds:

• 1-year: 3.2%
• 2-year: 3.5%
• 3-year: 3.6%
• 5-year: 3.5%
• 10-year: 3.4%

a) Describe the shape of this yield curve b) What does this shape typically indicate about economic expectations? c) How might this affect Johanna’s borrowing strategy for her café expansion?

Solution: a) This yield curve is humped (or bell-shaped), with rates rising in the short term, peaking at the 3-year mark, and then declining for longer maturities.

b) A humped yield curve typically indicates:

• The market expects interest rates to rise in the short term
• Then fall in the longer term
• This often occurs when the economy is expected to slow down after initial growth
• It can signal a potential economic slowdown or recession in the medium term

c) This yield curve suggests Johanna should:

• Avoid medium-term (3-year) financing as it has the highest cost
• Consider either short-term financing (if she can handle refinancing risk) or longer-term financing (10-year) which offers relatively lower rates
• If possible, structure a financing package that minimizes borrowing at the 3-year point of the curve
• Perhaps use a combination of 1-year and 5-year financing to create a balanced approach

Yield Curve Theories

Johanna’s financial advisor explains three theories:

1. Expectations Theory: Long-term rates reflect expected future short-term rates
2. Liquidity Preference Theory: Long-term rates include a premium for liquidity risk
3. Market Segmentation Theory: Different maturities are in separate markets with their own supply and demand

Question 46: Using the observed yields from the previous question, calculate the implied 1-year forward rates for years 2, 3, and 4 according to the Expectations Theory. What do these forward rates suggest about market expectations?

Solution: Using the formula for implied forward rates: \((1+r_n)^n = (1+r_m)^m \times (1+f_{m,n})^{n-m}\)

Where:

• \(r_n\) = spot rate for \(n\) years
• \(r_m\) = spot rate for \(m\) years
• \(f_{m,n}\) = forward rate from year \(m\) to year \(n\)

Implied 1-year forward rate for year 2: \((1+0.035)^2 = (1+0.032)^1 \times (1+f_{1,2})^1\) \(1.071225 = 1.032 \times (1+f_{1,2})\) \(f_{1,2} = \frac{1.071225}{1.032} - 1 = 0.0380 = 3.80\%\)

Implied 1-year forward rate for year 3: \((1+0.036)^3 = (1+0.035)^2 \times (1+f_{2,3})^1\) \(1.112005 = 1.071225 \times (1+f_{2,3})\) \(f_{2,3} = \frac{1.112005}{1.071225} - 1 = 0.0381 = 3.81\%\)

Implied 1-year forward rate for year 4: We need the 4-year spot rate, which we don’t have. We can estimate it by interpolating between the 3-year (3.6%) and 5-year (3.5%) rates: \(r_4 \approx 3.6\% - \frac{3.6\% - 3.5\%}{5-3} \times (4-3) = 3.6\% - 0.05\% = 3.55\%\)

\((1+0.0355)^4 = (1+0.036)^3 \times (1+f_{3,4})^1\) \(1.1502 = 1.112005 \times (1+f_{3,4})\) \(f_{3,4} = \frac{1.1502}{1.112005} - 1 = 0.0343 = 3.43\%\)

These forward rates suggest:

• The market expects short-term rates to increase from 3.2% to 3.8% in the second year
• Remain steady at about 3.8% in the third year
• Then decrease to 3.4% in the fourth year
• This aligns with the humped yield curve shape, suggesting expectations of initial economic growth followed by a slowdown

Stochastic Interest Models

Johanna’s advisor shows how her financing costs might evolve according to:

To model interest rate changes over time, this formula incorporates mean reversion (rates tend toward a long-term average \(b\) at speed \(a\)), volatility (\(\sigma\)), and random movements (Wiener process \(dW\)).

This model simulates possible interest rate paths to help Johanna plan for different scenarios.

Decision Point: Based on an upward-sloping yield curve, Johanna decides to issue shorter-term debt now with plans to refinance in two years when she expects her business’s credit rating to improve, potentially offsetting the expected rise in base interest rates.

Question 47: Johanna’s financial advisor uses the Vasicek model (a stochastic interest rate model) with the following parameters:

• Current short rate (\(r_0\)): 4%
• Long-term mean rate (\(b\)): 5%
• Speed of mean reversion (\(a\)): 0.3
• Volatility (\(\sigma\)): 1%

a) Calculate the expected short rate after 1 year b) Calculate the expected short rate after 5 years c) Why is a stochastic model more useful than a deterministic one for Johanna’s financial planning?

Solution: Under the Vasicek model, the expected value of the short rate is: \(E[r_t] = r_0 e^{-at} + b(1-e^{-at})\)

a) Expected short rate after 1 year: \(E[r_1] = 0.04 \times e^{-0.3 \times 1} + 0.05 \times (1-e^{-0.3 \times 1})\) \(E[r_1] = 0.04 \times 0.7408 + 0.05 \times 0.2592\) \(E[r_1] = 0.02963 + 0.01296 = 0.04259 = 4.26\%\)

b) Expected short rate after 5 years: \(E[r_5] = 0.04 \times e^{-0.3 \times 5} + 0.05 \times (1-e^{-0.3 \times 5})\) \(E[r_5] = 0.04 \times 0.2231 + 0.05 \times 0.7769\) \(E[r_5] = 0.00892 + 0.03885 = 0.04777 = 4.78\%\)

c) A stochastic model is more useful than a deterministic one because:

• It captures the random, unpredictable nature of interest rate movements
• It allows for calculation of confidence intervals, not just point estimates
• It enables stress testing of different interest rate scenarios
• It permits risk assessment through measures like Value at Risk (VaR)
• It supports pricing of interest rate derivatives that Johanna might use for hedging
• It provides a more realistic framework for long-term planning under uncertainty

Recap

1. Startup Phase:
   • Johanna secures a $100,000 bank loan at 6% with monthly payments
   • She uses the level payment formula to calculate $1,951.51 monthly payments
   • She issues some bonds to raise additional capital, using bond pricing formulas
   • The investor provides equity capital, calculating expected IRR of 12.31%
2. Growth Phase:
   • Johanna uses sinking fund method to prepare for bond repayment
   • She evaluates equipment purchases using NPV calculations
   • She implements variable annuity payment structures with her investor
   • She creates a continuous revenue stream through subscription service
3. Maturity Phase:
   • Johanna immunizes against interest rate risk using duration matching
   • She adjusts for inflation using real interest rate calculations
   • She evaluates business sale options using compound interest projections
   • She structures retirement benefits using perpetuity and contingent payment models

Question 48: Johanna is now considering her exit strategy after 10 successful years. She has three options:

A) Sell the café immediately for $500,000 B) Retain ownership but hire a manager, generating $60,000 annually in passive income C) Pursue a 5-year growth strategy, then sell for an estimated $800,000

Assume Johanna’s discount rate is 10% and the manager option would continue indefinitely. Conduct a comprehensive analysis to determine her best option.

Solution: Let’s analyze each option:

Option A: Immediate sale Present value = $500,000

Option B: Hire manager with perpetual income Present value of perpetuity: \(PV = \frac{60,000}{0.10} = \$600,000\)

Option C: 5-year growth, then sell Present value of future sale: \(PV = \frac{800,000}{(1.10)^5} = 800,000 \times 0.6209 = \$496,720\)

Based on NPV alone, Option B (hire manager) provides the highest value at $600,000.

However, a comprehensive analysis should consider:

1. Risk factors:
   • Option A has no risk (certain payment)
   • Option B has ongoing business risk (perpetuity assumes business continues forever)
   • Option C has both business risk and market timing risk
2. Time commitment:
   • Option A frees Johanna completely
   • Option B requires some oversight of the manager
   • Option C requires continued full engagement for 5 more years
3. Flexibility:
   • Option A provides immediate capital for other ventures
   • Option B provides ongoing income while potentially retaining asset appreciation
   • Option C delays access to capital but might have upside potential beyond estimates
4. Tax considerations:
   • Option A might trigger substantial capital gains tax
   • Option B spreads tax liability over time (potentially at lower rates)
   • Option C defers taxes but eventual sale might face higher rates
5. Financial security:
   • Option A provides a lump sum that could be diversified
   • Option B creates an income stream similar to an annuity
   • Option C has highest uncertainty but potential upside

Recommendation based on comprehensive analysis: Option B provides the highest NPV and creates a balance between immediate value and long-term potential. Johanna should choose this option if she values ongoing income and is comfortable with some continued business oversight. She could also implement a hybrid approach by selling a portion of the business while retaining some ownership to achieve partial liquidity while maintaining future upside potential.

Question 49: Over the course of her business, Johanna has applied various financial mathematics concepts. For each of the following scenarios, identify which concept is most relevant and explain how it helped Johanna make better decisions:

a) When comparing different loan offers with varying compounding frequencies b) When deciding whether to prepay her business loan or invest in new equipment c) When issuing bonds with different maturity dates during rising interest rates d) When setting up a retirement savings plan that will provide income for 25 years e) When hedging against coffee price volatility

Solution: a) Equivalent Rates Across Compounding Periods

• Concept allowed Johanna to convert nominal rates with different compounding frequencies to effective annual rates
• This enabled direct comparison between loans offering 6% compounded monthly versus 6.1% compounded annually
• Johanna could identify the true cost of each loan beyond the stated rate

b) Net Present Value (NPV) / Opportunity Cost

• NPV helped Johanna evaluate whether prepaying the loan or investing would create more value
• By calculating the present value of saved interest versus present value of expected equipment returns
• Johanna could determine which option maximized her business’s financial position

c) Duration and Interest Rate Risk

• Duration helped Johanna understand the sensitivity of different bond prices to interest rate changes
• With rising rates, shorter-duration bonds would be less negatively impacted
• Johanna could structure her debt to minimize the impact of anticipated rate increases

d) Present Value of Annuities / Time Value of Money

• Annuity calculations helped Johanna determine how much to save today for future retirement income
• By understanding how present contributions grow and convert to future payment streams
• Johanna could ensure financial security through proper planning for a fixed retirement period

e) Derivatives Pricing / Arbitrage-Free Valuation

• These concepts helped Johanna understand the fair price of hedging instruments
• By recognizing how futures and options should be priced relative to spot prices
• Johanna could identify cost-effective ways to reduce her exposure to volatile coffee prices