# Numerical Calculations

**I want you to complete all numerical calculations using both MS Excel (for questions (1), (2), (3), (4a) and (4b)) and R (for all questions). However, provide me with a “concise” summary document/pdf of your results. Derivations need not be typed as I will accept a pdf containing a scan of your handwritten derivations as long as I can read your writing. (I suggest printing all handwritten material). Make sure your R scripts are clearly documented with comments. Post your excel spreadsheet and R scripts to D2L.**

1. An asset is expected to produce an annual cyclical income stream of $50, $20, $10, $50, $20, $10, $50, $20, $10, …. The annual opportunity cost or discount rate is 6%. Use algebra to find the present value of the infinite cycle.

2. Assume that you are the finance officer for an equipment dealer. The owner has asked you to determine the markup needed for the following financing packages. Assume that you need to net (today) **$500,000 **on a particular machine for the dealership to break even and that the company **requires a rate of return of 6% AR**. All payment plans are to involve monthly payments with discrete monthly compounding. Determine a common list price and rebate plan that will generate a $500,000 net to your company for each financing alternative.

a. ” **No payments for 12 months**, **36 month financing at a 3% AR**.”

b. ” **1% AR financing for 4 years (48 months). **”

c. ” **2% AR financing with 24 monthly payments and no payments or interest**

**for one year.”**

**3. **A business owner initially borrowed $125,000 and made payments of $25,000 at the end of year 1, $30,000 at the end of year 2, $20,000 at the end of year 3, and $50,000 at the end of year 4. The loan balance at the end of year 4 was $100,000 . **What effective interest rate was the business owner charged on their note?**

(over)

4. Suppose that an individual starts with a zero retirement account balance and makes their initial payment one year from today. They wish to save a constant amount (at the end of each year) over the next 35 years to have an amount sufficient to purchase an expected fixed income retirement annuity of $75,000 per year (in today’s purchasing power) for 30 years of possible retirement. During their savings years they can invest into one of two index funds. The first fund is a stock index with expected annual returns of 7.0% per year but with an expected annual volatility of 𝜎𝜎=0.20 over the next 35 years. The second fund is a bond index with expected annual returns of 4.0% and an expected annual volatility of 𝜎𝜎=0.05. Assume both the stock and bond indexes follow the standard log-normal stochastic diffusion process. At retirement, assume that the fixed annuity firm will charge them 4% for a fixed 30-year retirement annuity.

a. Modify the procedures presented in the AGBE/ECNS 345 notes and derive both the present and the future value amortization formulas that can be used to determine expected future accumulations or to repay a note with N discrete payments but with continuous compounding.

b. Ignoring risk and assuming the individual chooses to invest in only one fund or the other, use your results from part (a) to find the annual amount that they need to independently save in each fund to accumulate an **expected amount **sufficient to fund their desired retirement annuity level. Assume that both the investment funds and the final risk-free annuity fund use continuous compounding with discrete payments.

c. Simulate 100,000 accumulated fund balance trajectories for each fund (using the log normal process). Amortize the accumulated amount at the end of each trajectory using the 30 payment continuous compounding PVAF with r = 0.04. Use the simulated values to estimate the probability that they reach their desired expected level of income of $75,000 using the annual savings amounts from part (b).

d. Assume that the individual feels that having their income fall below $45,000 per year would be a serious hardship and that they are only willing to accept a 10% probability of having their retirement income fall below $45,000. Use the simulated values to estimate the 10% VaR for each of the funds and annual savings amounts found in part (b).

e. If the individual saves in only one fund or the other, use the simulated values to estimate the minimal level of annual savings in each fund needed to be to satisfy their 10% VaR constraint of $45,000