1. Problem: Explain the difference between present and future value. 1. The present value (PV) is the equivalent value today of an amount to be received in the future given a discount or interest rate. 1. The future value (FV) is the amount an investment made today will grow to at a specified interest rate and time. 1. The relationship with periodic compounding is given by $$FV=PV(1+r)^n$$. 1. Solving for present value gives $$PV=\frac{FV}{(1+r)^n}$$. 1. Explanation: To move money forward multiply by the growth factor $ (1+r)^n $ and to move money backward divide by the same factor $ (1+r)^{-n} $. 1. Example: If $PV=1000$, $r=0.05$, and $n=3$ then $$FV=1000(1+0.05)^3$$. 1. Intermediate: Compute $(1+0.05)^3=1.157625$ so $$FV=1000\times 1.157625=1157.625$$. 1. Conclusion: Present value discounts future cash and future value compounds present cash, and the two are inverses under the factor $1+r$. 2. Problem: How do interest rates affect business decisions? 2. Interest rates serve both as the cost of borrowing and as the discount rate for future cash flows. 2. Firms use the discount rate when computing net present value (NPV) of a project via $$NPV=\sum_{t=0}^N\frac{CF_t}{(1+r)^t}$$. 2. A higher interest/discount rate reduces the present value of future cash inflows and increases the present cost of borrowing. 2. Consequence: Projects with marginal returns may become unattractive when rates rise because discounted benefits fall relative to costs. 2. Numerical example: A single future cash flow of 100 at $t=1$ has present value $\frac{100}{1+r}$. 2. Compute with $r=0.10$: $$PV=\frac{100}{1.10}=90.9090909$$. 2. Compute with $r=0.05$: $$PV=\frac{100}{1.05}=95.2380952$$. 2. Interpretation: The higher rate (0.10) gives a lower PV (90.909) than the lower rate (0.05) which gives PV 95.238, so higher rates discourage investment and raise the required return for projects. 3. Problem: Discuss steps for calculating compound interest for quarterly compounding. 3. Step 1: Identify the principal $P$, the nominal annual interest rate $r$ (in decimal), the compounding frequency $m=4$ for quarterly compounding, and the time in years $t$. 3. Step 2: Compute the periodic rate $\frac{r}{m}$ and the total number of periods $m t$. 3. Step 3: Apply the compound interest formula $$A=P\left(1+\frac{r}{m}\right)^{m t}$$ to obtain the accumulated amount $A$. 3. Step 4: Work an example with $P=2000$, $r=0.06$, and $t=5$. 3. Compute the periodic rate: $$\frac{r}{m}=\frac{0.06}{4}=0.015$$. 3. Compute the number of periods: $$m t=4\times 5=20$$. 3. Compute the growth factor: $$(1+0.015)^{20}=1.346855007$$. 3. Compute the final amount: $$A=2000\times 1.346855007=2693.710014$$. 3. Interpretation: The principal 2000 grows to approximately 2693.71 after 5 years with quarterly compounding at a 6 percent nominal annual rate. 3. Additional note: The effective annual rate is given by $$EAR=\left(1+\frac{r}{m}\right)^m-1$$ which captures the effect of intra-year compounding. 3. Example EAR for $r=0.06$ and $m=4$ is $$EAR=\left(1+0.015\right)^4-1\approx 0.06136$$.