Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 1 Lecture 8

1.8 Investments -Net Present Value and Internal Rate of Return When a corporation makes an investment in goods or assets, it wants to predict whether the expenditure will give a good rate of return. One method of comparing costs with expected returns is the use of an equation of value to find the Net Present Value(NPV). As the name suggests, it is the “net”, or difference, between income and calculation is called the cost of capital and generally corresponds with the company’s goals and requirements for investments. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 2

Goal: predict if an investment will give a good rate of return. The Net Present Value (NPV) is the income minus expenses, all moved to the present. The Internal rate of return (IRR) is the rate at which the returns and investments make the NPV zero. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 3

To find NPV : - Determine cash flows - Use plus sign for income (cash inflows) - Use minus sign for expense (Cash outflows) - Move everything to the present - Add them all Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 4

(IRR) IN CASE OF : More than one outflow or More than one inflow Chose a smaller rate if the first rate used had given a negative NPV Chose a larger rate if the first rate used had given a positive NPV One outflow and One inflow IRR=i = I ÷ pt Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 5

Example 8.1 and 8.2 page 36 Find the net present value and internal rate of return at 14% of an investment of $25,000 which is expected to bring returns of $14,000 in 9 months and $15,000 in 14 months. Solution  Draw a time line diagram and write a equation of value for the net present value. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 6

Solution Time Line Diagram Present (-) $25,000 9 mo. (+) 14, mo. (+) 15, % Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 7

There are more than one cash inflow So we will use the interpolation method And since the first rate used had given a positive NPV so we need to pick a larger rate and re compute the NPV

Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 9

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When we write the equation of value to solve directly for the IRR we use a variable x for the unknown rate. The resulting quadratic equation can be solved quickly with the TI 83 SOLVER routine. X= %

Exercise 11 page 37 Marlon agrees to pay $500 now to get a return of $700 in a year. Find his NPV at 30% and his IRR. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 22

Solution NOW (-) $500 1 year $ % Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 14

Exercise 13 page 37 Jon is starting a business that requires $8000 startup money but will return $4000 in 1 year and $6000 in 2 years. Find the NPV at 10% and 20%, and find the IRR to one decimal place. 23 Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance

Solution NOW (-) $8000 NOW (-) $ year $ year $ years. $ years. $ 6000 i% Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 17

Exercise 15 page 37 Jan's Boutique needs some new equipment. The cost of the equipment is $500 now. The returns on that investment are $300 in a year and $800 in 2 years. Assuming all returns are figured at the end of the calendar year, find the NPV at 70% and at 80%, and find the IRR to two decimal places. Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 24

Solution NOW (-) $500 NOW (-) $500 1 year $300 1 year $300 2 years. $ years. $ Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 29

Exercise 17 page 38 An investment requires a cash outlay of $4500 and is expected to bring returns of $2500 in 9 months and $3200 in 18 months. What is the net present value at 15% and what is the internal rate of return? Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 25

Solution NOW (-) $4500 NOW (-) $ year $ year $ years. $ years. $ 3200 Instructors: Lobna M Farid Copy rights for Gary C. Guthnie Larry D. Leman, Mathematics of Interest Rates and Finance 17

20 Thank You