Time Value of Money

Sukrit Mittal
Franklin Templeton Investments

Outline

  1. Introduction and motivation
  2. Simple interest
  3. Periodic compounding
  4. Continuous compounding
  5. Comparing compounding methods
  6. Streams of payments
  7. Applications

1. Introduction and Motivation

Why Is Money Time-Dependent?

A dollar today is worth more than a dollar tomorrow. Why?

  • Opportunity cost: Money today can be invested to earn returns
  • Inflation: Purchasing power erodes over time
  • Risk: Future payments are uncertain

Core Question

How do we compare cash flows occurring at different points in time?

The answer: discount future cash flows to their present value, or compound present values to future equivalents.

The Fundamental Trade-Off

Consider two offers:

  • Receive $100 today
  • Receive $110 one year from now

Which is better?

Answer: It depends on the interest rate.

  • If you can invest at 12% annually: $100 today grows to $112 next year → take $100 today
  • If you can only invest at 8%: $100 today grows to $108 next year → take $110 next year

The interest rate is the bridge between present and future values.

The Indifference Rate

We can solve for the break-even interest rate where you’d be indifferent:

\[100(1 + r) = 110 \implies r = 0.10 = 10\%\]

At exactly 10%, both options are equivalent.

2. Simple Interest

Definition

Under simple interest, the interest earned is proportional to:

  • Principal amount $P$
  • Interest rate $r$ (per period)
  • Time $t$

Formula

\[A(t) = P(1 + rt)\]

where:

  • $A(t)$ = amount at time $t$
  • $P$ = principal (initial investment)
  • $r$ = interest rate per period
  • $t$ = number of periods

Simple Interest: Example

You invest $1,000 at 8% simple interest per year.

How much will you have after 3 years?

\[A(3) = 1{,}000(1 + 0.08 \times 3) = 1{,}000 \times 1.24 = \$1{,}240\]

Key Feature

Interest is earned only on the principal, not on accumulated interest.

This is a linear growth model:

\[A(t) = 1{,}000 + 80t\]

Practice Questions

  1. If you invest $5,000 at 6% simple interest for 18 months, what is the final amount?
  2. How long does it take for $2,000 to grow to $2,500 at 5% simple interest?

Hint for Q2: Solve $2000(1 + 0.05t) = 2500$ for $t$.

Limitations of Simple Interest

Simple interest ignores the fact that interest earned can itself earn interest.

Example: With $1,000 at 8% simple interest:

  • Year 1: earn $80
  • Year 2: earn $80
  • Year 3: earn $80
  • Total: $1,240

But what if you could reinvest the $80 earned in Year 1?

This leads us to compound interest.

The Power of Compounding

Einstein’s Alleged Quote

“Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn’t, pays it.”

Illustration: The Rule of 72

Approximate time to double your money:

\[t \approx \frac{72}{r\%}\]

where $r\%$ is the annual percentage rate.

Examples:

  • At 6%: $t \approx 72/6 = 12$ years
  • At 8%: $t \approx 72/8 = 9$ years
  • At 12%: $t \approx 72/12 = 6$ years

Verification at 8%: $(1.08)^9 = 1.999 \approx 2$ ✓

Derivation

Solve $(1+r)^t = 2$ for $t$:

\[t = \frac{\ln 2}{\ln(1+r)} \approx \frac{0.693}{r}\]

For $r$ as a percentage: $t \approx \frac{69.3}{r\%}$. The rule of 72 is easier to use and quite accurate for rates between 6-10%.

3. Periodic Compounding

Definition

With compound interest, interest is added to the principal periodically, and subsequent interest is earned on the new total.

Formula

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

where:

  • $n$ = number of compounding periods per year
  • $t$ = number of years
  • $r$ = annual interest rate (APR)

Why Does This Work?

Each period, the amount is multiplied by $\left(1 + \frac{r}{n}\right)$. After $nt$ periods:

\[P \rightarrow P\left(1 + \frac{r}{n}\right) \rightarrow P\left(1 + \frac{r}{n}\right)^2 \rightarrow \cdots \rightarrow P\left(1 + \frac{r}{n}\right)^{nt}\]

Compounding Frequencies

Frequency Value of $n$
Annual $n = 1$
Semi-annual $n = 2$
Quarterly $n = 4$
Monthly $n = 12$
Daily $n = 365$

Key insight: As $n$ increases, the final amount $A$ increases, but with diminishing returns.

Discount Rate and Present Value Relationship

Inverse Relationship

Present value and discount rate move in opposite directions:

\[\frac{\partial PV}{\partial r} = \frac{\partial}{\partial r}\left[\frac{C}{(1+r)^t}\right] = -\frac{tC}{(1+r)^{t+1}} < 0\]

Interpretation:

  • Higher discount rates → lower present values
  • Lower discount rates → higher present values

Financial Implications

Bond markets: When interest rates rise, bond prices fall (and vice versa).

Equity markets: Higher discount rates reduce the present value of future earnings, lowering stock prices.

Example

PV of $1,000 in 10 years at different rates:

  • At 5%: $1,000/(1.05)^{10} = $613.91
  • At 10%: $1,000/(1.10)^{10} = $385.54
  • At 15%: $1,000/(1.15)^{10} = $247.19

Example: Periodic Compounding

Invest $1,000 at 8% annual interest for 3 years under different compounding frequencies.

Annual compounding ($n=1$):

\[A = 1{,}000\left(1 + 0.08\right)^{3} = 1{,}000(1.08)^3 = \$1{,}259.71\]

Quarterly compounding ($n=4$):

\[A = 1{,}000\left(1 + \frac{0.08}{4}\right)^{12} = 1{,}000(1.02)^{12} = \$1{,}268.24\]

Monthly compounding ($n=12$):

\[A = 1{,}000\left(1 + \frac{0.08}{12}\right)^{36} = \$1{,}270.24\]

Notice: The difference between monthly and quarterly is $2.00, much smaller than the $8.53 difference between quarterly and annual. This diminishing return pattern continues as $n$ increases.

\[A = 1{,}000\left(1 + \frac{0.08}{12}\right)^{36} = \$1{,}270.24\]

Effective Annual Rate (EAR)

The effective annual rate is the actual annual return accounting for compounding within the year.

Formula

\[\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1\]

Example: 8% APR compounded quarterly

\[\text{EAR} = \left(1 + \frac{0.08}{4}\right)^4 - 1 = (1.02)^4 - 1 = 0.0824 = 8.24\%\]

The EAR exceeds the stated rate (APR) whenever $n > 1$.

Why It Matters

Financial institutions often quote APR (nominal rate) but compound more frequently. The EAR reveals the true cost or true return.

Regulation: In many jurisdictions, lenders must disclose the EAR (also called APY - Annual Percentage Yield) to help consumers compare products.

Exercise

A credit card charges 18% APR with daily compounding. What is the effective annual rate?

4. Continuous Compounding

What Happens as $n \to \infty$?

As compounding becomes more frequent, we approach continuous compounding.

Derivation

Starting from periodic compounding:

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

Take the limit as $n \to \infty$:

\[A = P \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt}\]

Rewrite the exponent: let $m = \frac{n}{r}$, so $n = mr$ and as $n \to \infty$, we have $m \to \infty$:

\[A = P \lim_{m \to \infty} \left[\left(1 + \frac{1}{m}\right)^m\right]^{rt}\]

Using the fundamental limit $\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m = e$:

\[A = P(e)^{rt} = Pe^{rt}\]

Why $e$?

The number $e$ naturally arises as the base for continuous growth processes.

Continuous Compounding Formula

\[A = Pe^{rt}\]

where:

  • $e \approx 2.71828$ (Euler’s number)
  • $r$ = annual interest rate
  • $t$ = time in years

Example

Invest $1,000 at 8% continuously compounded for 3 years:

\[A = 1{,}000 \cdot e^{0.08 \times 3} = 1{,}000 \cdot e^{0.24} = \$1{,}271.25\]

Real vs. Nominal Interest Rates

Nominal Rate

The stated interest rate, not adjusted for inflation.

Real Rate

The interest rate adjusted for inflation—the actual increase in purchasing power.

Fisher Equation

\[(1 + r_{\text{nominal}}) = (1 + r_{\text{real}})(1 + \pi)\]

where $\pi$ is the inflation rate.

Approximation (for small rates):

\[r_{\text{real}} \approx r_{\text{nominal}} - \pi\]

Example

You earn 8% on an investment, but inflation is 3%.

Exact: $r_{\text{real}} = \frac{1.08}{1.03} - 1 = 0.0485 = 4.85\%$

Approximate: $r_{\text{real}} \approx 8\% - 3\% = 5\%$

Why It Matters

When comparing investments across time or countries, real rates provide the true comparison of purchasing power gains.

Present Value with Continuous Compounding

If $A = Pe^{rt}$, then solving for $P$:

\[P = Ae^{-rt}\]

This is the present value of a future amount $A$ under continuous compounding.

Example: What is the present value of $1,500 to be received in 5 years at 6% continuous rate?

\[P = 1{,}500 \cdot e^{-0.06 \times 5} = 1{,}500 \cdot e^{-0.3} = \$1{,}109.96\]

Interpretation

The discount factor $e^{-rt}$ represents the “weight” we assign to future cash. As $t$ or $r$ increases, this weight decreases exponentially.

Exercise

Verify that $1,109.96$ invested at 6% continuous rate for 5 years yields $1,500.

5. Comparing Compounding Methods

Comparison Table

Invest $1,000 at 8% for 3 years:

Method Formula Amount
Simple interest $P(1+rt)$ $1,240.00
Annual compounding $P(1+r)^t$ $1,259.71
Quarterly $P(1+r/4)^{4t}$ $1,268.24
Monthly $P(1+r/12)^{12t}$ $1,270.24
Daily $P(1+r/365)^{365t}$ $1,271.22
Continuous $Pe^{rt}$ $1,271.25

Observation: More frequent compounding yields higher returns, but the gains diminish.

Converting Between Compounding Frequencies

From Periodic to Continuous

Given periodic rate $r_n$ with frequency $n$, the equivalent continuous rate $r_c$ is:

\[r_c = n \ln\left(1 + \frac{r_n}{n}\right)\]

Derivation: Set the growth equal over one year:

\[\left(1 + \frac{r_n}{n}\right)^n = e^{r_c}\]

Taking natural log: $n \ln\left(1 + \frac{r_n}{n}\right) = r_c$

From Continuous to Periodic

Given continuous rate $r_c$, the equivalent periodic rate $r_n$ is:

\[r_n = n\left(e^{r_c/n} - 1\right)\]

Derivation: From $e^{r_c} = \left(1 + \frac{r_n}{n}\right)^n$, raise both sides to power $1/n$:

\[e^{r_c/n} = 1 + \frac{r_n}{n} \implies r_n = n(e^{r_c/n} - 1)\]

Example: Rate Conversion

Question: What continuous rate is equivalent to 8% compounded quarterly?

\[r_c = 4 \ln\left(1 + \frac{0.08}{4}\right) = 4 \ln(1.02) = 4 \times 0.0198 = 0.0792 = 7.92\%\]

Verification:

  • Quarterly: $1{,}000(1.02)^{12} = 1{,}268.24$
  • Continuous: $1{,}000e^{0.0792 \times 3} = 1{,}268.24$ ✓

6. Streams of Payments

So far: single payment at time 0.

In practice, many financial instruments involve multiple cash flows over time:

  • Bond coupons
  • Loan payments
  • Dividend streams
  • Retirement income

How do we value these?

Answer: Sum the present values of each individual payment.

The Additivity Principle

The present value operator is linear:

\[PV(C_1 + C_2) = PV(C_1) + PV(C_2)\]

This allows us to value complex cash flow streams by breaking them into components.

Present Value of Cash Flow Stream

Consider cash flows $C_1, C_2, \ldots, C_n$ at times $t_1, t_2, \ldots, t_n$.

Present Value

\[PV = \sum_{k=1}^{n} C_k e^{-rt_k}\]

(using continuous compounding)

Or with periodic compounding at rate $r$:

\[PV = \sum_{k=1}^{n} \frac{C_k}{(1+r)^{t_k}}\]

Example

Cash flows: $100 at $t=1$, $200 at $t=2$, $150 at $t=3$. Rate: 5% (annual compounding).

\[PV = \frac{100}{1.05} + \frac{200}{(1.05)^2} + \frac{150}{(1.05)^3}\] \[= 95.24 + 181.41 + 129.58 = \$406.23\]

Annuities

An annuity is a sequence of equal payments at regular intervals.

Two types:

  • Ordinary annuity: payments at the end of each period
  • Annuity due: payments at the beginning of each period

Notation

  • $C$ = payment per period
  • $r$ = interest rate per period
  • $n$ = number of periods

Present Value of Ordinary Annuity

Formula:

\[PV = C \cdot \frac{1 - (1+r)^{-n}}{r}\]

Derivation:

\[PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^n}\]

Factor out $\frac{C}{1+r}$:

\[PV = \frac{C}{1+r}\left[1 + \frac{1}{1+r} + \frac{1}{(1+r)^2} + \cdots + \frac{1}{(1+r)^{n-1}}\right]\]

This is a geometric series with first term $a = 1$, ratio $q = \frac{1}{1+r}$, and $n$ terms.

Geometric series sum: $S_n = a \cdot \frac{1-q^n}{1-q} = \frac{1 - (1+r)^{-n}}{1 - \frac{1}{1+r}} = \frac{1-(1+r)^{-n}}{\frac{r}{1+r}}$

Therefore:

\[PV = \frac{C}{1+r} \cdot \frac{1-(1+r)^{-n}}{\frac{r}{1+r}} = C \cdot \frac{1 - (1+r)^{-n}}{r}\]

Example: Ordinary Annuity

You will receive $500 at the end of each year for 5 years. The interest rate is 6%.

What is the present value?

\[PV = 500 \cdot \frac{1 - (1.06)^{-5}}{0.06}\] \[= 500 \cdot \frac{1 - 0.7473}{0.06} = 500 \cdot \frac{0.2527}{0.06}\] \[= 500 \times 4.212 = \$2{,}106.00\]

Future Value of Ordinary Annuity

Formula:

\[FV = C \cdot \frac{(1+r)^n - 1}{r}\]

Derivation: Relate to present value. If $PV$ grows for $n$ periods:

\[FV = PV \cdot (1+r)^n = C \cdot \frac{1 - (1+r)^{-n}}{r} \cdot (1+r)^n\] \[= C \cdot \frac{(1+r)^n - 1}{r}\]

Example: You save $500 at the end of each year for 5 years at 6%.

\[FV = 500 \cdot \frac{(1.06)^5 - 1}{0.06} = 500 \cdot \frac{1.3382 - 1}{0.06}\] \[= 500 \times 5.637 = \$2{,}818.50\]

Verification: This is $2,106.00 \times (1.06)^5 = 2,818.50$ ✓

Annuity Due

Payments occur at the beginning of each period.

Present Value of Annuity Due:

\[PV_{\text{due}} = PV_{\text{ordinary}} \times (1+r)\]

Future Value of Annuity Due:

\[FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r)\]

Intuition: Each payment earns interest for one additional period.

Derivation for PV

For an annuity due, the first payment is at $t=0$, not $t=1$:

\[PV_{\text{due}} = C + \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^{n-1}}\] \[= (1+r)\left[\frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^n}\right] = (1+r) \cdot PV_{\text{ordinary}}\]

Example

Using our earlier example ($500/year for 5 years at 6%):

  • Ordinary annuity: $2,106.00
  • Annuity due: $2,106.00 \times 1.06 = $2,232.36

Perpetuities

A perpetuity is an annuity that continues forever.

Present Value of Perpetuity

As $n \to \infty$ in the annuity formula:

\[PV = \lim_{n \to \infty} C \cdot \frac{1 - (1+r)^{-n}}{r}\]

Since $(1+r) > 1$, we have $(1+r)^{-n} \to 0$ as $n \to \infty$:

\[PV = C \cdot \frac{1 - 0}{r} = \frac{C}{r}\]

Example: A bond pays $50 per year forever at 5% interest.

\[PV = \frac{50}{0.05} = \$1{,}000\]

Intuition

At 5%, a $1,000 investment yields $50 annually forever—exactly matching the perpetuity’s cash flows.

Growing Perpetuity

A growing perpetuity has payments that increase at rate $g$ each period.

First payment: $C$
Second payment: $C(1+g)$
Third payment: $C(1+g)^2$, etc.

Present Value

\[PV = \frac{C}{1+r} + \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \cdots\] \[= \frac{C}{1+r}\left[1 + \frac{1+g}{1+r} + \left(\frac{1+g}{1+r}\right)^2 + \cdots\right]\]
This is a geometric series with ratio $q = \frac{1+g}{1+r}$. For convergence, we need $ q < 1$, i.e., $r > g$.

Sum: $\frac{1}{1-q} = \frac{1}{1 - \frac{1+g}{1+r}} = \frac{1+r}{r-g}$

Therefore:

\[PV = \frac{C}{1+r} \cdot \frac{1+r}{r-g} = \frac{C}{r - g}\]

Example: First payment $100, growing at 3% annually, discount rate 8%.

\[PV = \frac{100}{0.08 - 0.03} = \frac{100}{0.05} = \$2{,}000\]

7. Applications

Bond Valuation

A bond pays periodic coupons plus face value at maturity.

\[\text{Bond Price} = PV(\text{coupons}) + PV(\text{face value})\]

Example: A 3-year bond with $1,000 face value pays 6% annual coupons. Market rate is 5%.

Coupons: $60 per year (ordinary annuity)
Face value: $1,000 at $t=3$

\[\text{Price} = 60 \cdot \frac{1-(1.05)^{-3}}{0.05} + \frac{1{,}000}{(1.05)^3}\] \[= 60 \times 2.7232 + 863.84 = 163.39 + 863.84 = \$1{,}027.23\]

The bond trades at a premium because the coupon rate (6%) exceeds the market rate (5%).

Loan Amortization

Monthly loan payments form an annuity. Given loan amount $L$, rate $r$, term $n$:

\[\text{Payment} = L \cdot \frac{r}{1 - (1+r)^{-n}}\]

This is derived by solving $L = PV(\text{payments})$.

Application: Mortgage Example

You borrow $300,000 for 30 years at 4.5% annual rate (compounded monthly).

Monthly rate: $r = 0.045/12 = 0.00375$
Number of payments: $n = 30 \times 12 = 360$

\[\text{Monthly Payment} = 300{,}000 \cdot \frac{0.00375}{1 - (1.00375)^{-360}}\] \[= 300{,}000 \cdot \frac{0.00375}{1 - 0.2556} = 300{,}000 \times 0.005067 = \$1{,}520.06\]

Amortization Insight

Total paid: $1,520.06 \times 360 = $547,220.40
Interest paid: $547,220.40 - $300,000 = $247,220.40

Over the loan’s life, you pay 82% of the principal in interest!

First payment breakdown:

  • Interest: $300,000 \times 0.00375 = $1,125.00
  • Principal: $1,520.06 - $1,125.00 = $395.06

Early payments are mostly interest; later payments are mostly principal.

Net Present Value (NPV)

Definition

The Net Present Value of a project is the sum of all discounted cash flows, including the initial investment (typically negative):

\[NPV = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}\]

where $C_0$ is the initial investment and $C_t$ are future cash flows.

Decision Rule

  • NPV > 0: Accept the project (it adds value)
  • NPV < 0: Reject the project (it destroys value)
  • NPV = 0: Indifferent (breaks even)

Example

A project requires $10,000 investment today and generates $3,000, $4,000, $5,000 over the next 3 years. Discount rate is 8%.

\[NPV = -10{,}000 + \frac{3{,}000}{1.08} + \frac{4{,}000}{(1.08)^2} + \frac{5{,}000}{(1.08)^3}\] \[= -10{,}000 + 2{,}778 + 3{,}429 + 3{,}969 = \$176\]

Decision: Accept (positive NPV).

Internal Rate of Return (IRR)

Definition

The Internal Rate of Return is the discount rate that makes NPV equal to zero:

\[0 = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+IRR)^t}\]

Interpretation

IRR is the break-even rate of return for the project.

Decision Rule

  • If IRR > hurdle rate: Accept the project
  • If IRR < hurdle rate: Reject the project

Example (continued)

For our earlier project, find IRR where:

\[0 = -10{,}000 + \frac{3{,}000}{(1+IRR)} + \frac{4{,}000}{(1+IRR)^2} + \frac{5{,}000}{(1+IRR)^3}\]

This requires numerical methods (e.g., Newton-Raphson). Solution: IRR ≈ 8.89%

Since IRR (8.89%) > discount rate (8%), the project is acceptable—consistent with positive NPV.

Limitations

  • Can have multiple solutions if cash flows change sign more than once
  • Assumes reinvestment at IRR (often unrealistic)
  • NPV is generally preferred for ranking projects

Summary Formulas

Concept Formula
Simple interest $A = P(1+rt)$
Compound interest $A = P(1+r/n)^{nt}$
Continuous compounding $A = Pe^{rt}$
PV of ordinary annuity $PV = C \cdot \frac{1-(1+r)^{-n}}{r}$
FV of ordinary annuity $FV = C \cdot \frac{(1+r)^n-1}{r}$
Perpetuity $PV = \frac{C}{r}$
Growing perpetuity $PV = \frac{C}{r-g}$

Key Takeaways

  1. Time value of money is fundamental to all financial decisions
  2. Compound interest dominates simple interest in practice
  3. Continuous compounding is the mathematical limit of periodic compounding
  4. Annuities and perpetuities provide closed-form valuation formulas
  5. All valuations reduce to: sum of discounted cash flows

These tools form the foundation for:

  • Bond pricing
  • Stock valuation (dividend discount models)
  • Derivatives pricing
  • Capital budgeting

Practice Problems

Problem 1: Comparing Options

You win a lottery with two payout options:

  • Option A: $500,000 today
  • Option B: $50,000 per year for 15 years (first payment in 1 year)

At what discount rate are you indifferent between the two options?

Hint: Set the PV of option B equal to $500,000 and solve for $r$.

Practice Problems (continued)

Problem 2: Retirement Planning

You want $2 million when you retire in 30 years. You plan to invest in an account earning 7% annually (compounded monthly).

(a) If you make a single deposit today, how much do you need?

(b) If you make equal monthly deposits, how much per month?

Problem 3: Rate Equivalence

What semi-annual compounding rate is equivalent to 6% continuously compounded?

Practice Problems (continued)

Problem 4: Growing Annuity

A growing annuity pays $C$ in the first period, $C(1+g)$ in the second, etc., for $n$ periods total.

Derive the present value formula:

\[PV = \frac{C}{r-g}\left[1 - \left(\frac{1+g}{1+r}\right)^n\right]\]

Hint: Start with the geometric series for the growing perpetuity and subtract the tail.

Problem 5: Bond Pricing

A 10-year bond with face value $1,000 pays 8% annual coupons. If the yield is 6%, what is the bond price? What if the yield is 10%?

Advanced Topic: Duration (Preview)

Macaulay Duration

The weighted average time until cash flows are received:

\[D = \frac{\sum_{t=1}^{n} t \cdot PV(C_t)}{\sum_{t=1}^{n} PV(C_t)} = \frac{\sum_{t=1}^{n} t \cdot \frac{C_t}{(1+r)^t}}{PV}\]

Why Duration Matters

Duration measures the sensitivity of a bond’s price to interest rate changes:

\[\frac{\Delta P}{P} \approx -D \cdot \frac{\Delta r}{1+r}\]

Example Interpretation

A bond with duration 7 years will lose approximately 7% in value if interest rates rise by 1 percentage point.

Applications

  • Immunization: Match asset and liability durations to hedge interest rate risk
  • Portfolio management: Adjust duration to express interest rate views
  • Risk management: Measure and limit interest rate exposure

This will be covered in detail in later lectures on fixed income.

Hints for Practice Problems

Problem 1 (Lottery Options)

Set up: $500{,}000 = 50{,}000 \cdot \frac{1-(1+r)^{-15}}{r}$
Solve numerically: $r \approx 7.23\%$

Problem 2 (Retirement Planning)

(a) $PV = 2{,}000{,}000 / (1 + 0.07/12)^{360} \approx $245{,}975$
(b) Solve: $2{,}000{,}000 = PMT \cdot \frac{(1+0.07/12)^{360}-1}{0.07/12}$, giving $PMT \approx $1{,}633$

Problem 3 (Rate Equivalence)

$r_2 = 2(e^{0.06/2} - 1) = 2(e^{0.03} - 1) \approx 6.09\%$

Problem 5 (Bond Pricing)

At 6% yield: Bond price = $1,147.20 (premium)
At 10% yield: Bond price = $877.11 (discount)
When coupon rate > yield, bond trades at premium; when coupon rate < yield, bond trades at discount.