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Financial Math

Financial Math. Financial Math - Agenda. AGENDA. SIMPLE INTEREST Money Market Instruments TIME VALUE OF MONEY Compound Interest Future Value Present Value EFFECTIVE & NOMINAL RATES EXPONENTIAL RATE INTERNAL RATE OF RETURN (IRR) NET PRESENT VALUE ANNUITY FORMULA

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Financial Math

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  1. Financial Math

  2. Financial Math - Agenda AGENDA • SIMPLE INTEREST • Money Market Instruments • TIME VALUE OF MONEY • Compound Interest • Future Value • Present Value • EFFECTIVE & NOMINAL RATES • EXPONENTIAL RATE • INTERNAL RATE OF RETURN (IRR) • NET PRESENT VALUE • ANNUITY FORMULA • BOND PRICING USING YTM

  3. SIMPLE INTEREST – Day Count Convention TWO MAJOR CONVENTIONS FOR DAY COUNT: • Actual/365  GBP, AUD, Canadian Money Markets • Actual/360  US, JPY, EUR, CHF Money Markets

  4. SIMPLE INTEREST • Deposit/Loan products: + Bank time deposits, bank loans, repos • Discount Products: + Commercial paper + Treasury bills + Acceptances • Yield to Discount:

  5. SIMPLE INTEREST – Example of Discount Rate • NOTIONAL: 1,000,000 • INTEREST: 5% • DAYS: 91 • BASIS: 360 • AMOUNT PAID: 1,000,000 * (1-5.00%) * (91/360) = 987,361.11

  6. SIMPLE INTEREST – Example Yield Add theoretical formula (12,638.89/987,361.11) * (360/91*100) = 5.0640%

  7. SIMPLE INTEREST – Money Market Calculations You make a 91 day, USD 1,500,000 loan to your corporate client at 3.30% plus a 50 basis point spread. How much interest do they pay at maturity? Your client deposits GBP 1 million for one year at 5.05%. What is the total amount repaid to them in one year? Your money market desk has $3 million (FV) in excess funds for three months (92 days). They invest the funds in Treasury bills at a discount rate of 2.10%. How much cash do they hand over today and what is the true yield on this investment? You identify some 30 days commercial paper for $2 million face value that looks attractive at a current discount rate of 6.70%. How much do you pay for this investment and how much will you receive back at maturity? Today is May 1st and I have just deposited $10,000 into a bank account at a fixed rate of 5.00% per annum for six months. How much can I withdraw on 1st November? Bank of America says that they are going to give me a $10,000 bonus on 31st December. They say I can have $9,800 today 24th May if I would like it earlier. If I can earn 2.50% on a savings account, what is best for me?

  8. SIMPLE INTEREST – Money Market Calculations • 7a. I buy a 364 day US Treasury Bill at a discount rate of 5.00% with a face value of $1,000,000. What do I pay for the bill and what is my true yield? • 7b. 91 days later the discount rate has fallen to 4.50%. I decide to sell my bill. How much will I receive in cash and what was my yield over the 91 days that I held the bill?

  9. SIMPLE INTEREST – Money Market Calculations ANSWERS 14,408.33 1,050,500 2,983,900 and 2.1113% 1,988,833.32 and 2,000,000 10,255.56 9,800*(1+(0.025*221/360))=9,950.40 A: 949,444.44 Yield=5.2662% B: 965,875 Yield achieved= 6.846%

  10. SIMPLE INTEREST – Money Market Calculations CALCULATE NUMBER OF DAYS IN EXCEL • EXAMPLE = • Start date: 05-24-2009 • End date: 12-31-2009 • Enter  = (12/31/2009) – (05-24-2009) = 221 • The result cell should have number format

  11. TIME VALUE OF MONEY • General Formula: • Where: FV = future value PV = principal (or present value) y = interest rate per annum N = compounding frequency per annum n = number of compounding periods

  12. TIME VALUE OF MONEY – Formula Variation FUTURE VALUE INTEREST RATE PRESENT VALUE

  13. TIME VALUE OF MONEY – Compound Interest Rate Example • Present Value (PV) = GBP 100 • Interest Rate (y) = 8% • Frequency (N) = 4 quarters • Periods (n) = 4 (3.5 years)

  14. TIME VALUE OF MONEY – FV Function Excel • EXCEL  FORMULAS  FINANCIAL  FV

  15. TIME VALUE OF MONEY – Calculating Rate • Invest $1000 with interest compounded quarterly. At the end of 7 years I have accumulated $1,741.02. What is my annual rate of interest?

  16. TIME VALUE OF MONEY – Excel Rate Function • EXCEL  FORMULAS  FINANCIAL  RATE

  17. TIME VALUE OF MONEY – Calculating Rate • Find the present value of an investment that pays GBP 1,000 in five years at a semi annually compounded annual rate of 4.50%.

  18. TIME VALUE OF MONEY – PV Function Excel • EXCEL  FORMULAS  FINANCIAL  PV

  19. TIME VALUE OF MONEY – Exercises What is the future value of an investment of $5000, invested today, for 3 years, withinterest compounded quarterly at a rate of 2.25%? Find the present value of GBP 100,000,000 received in 7 years at an annual rate of 5.85% Find the yield (interest rate) on an investment that requires $200,000 today and pays $1,000,000 out in 10 years. Offered an annual rate of 6.50%, which is more valuable, receiving ￡1,000 today or ￡1,200 in three years? Find the present value of GBP 100,000 received in five years at an interest rate of 5.85% paid semi-annually. Find the yield of an investment that requires $100,000 today and pays out $300,000 in 5 years based on monthly compounding.

  20. TIME VALUE OF MONEY – Answers ANSWERS 5,348.14 67,168,239.20 17.46% 1,207.95 74,953.38 22.17%

  21. TIME VALUE OF MONEY – Effective and Nominal Rates Different compounding periods: Nominal Rate Frequency Per year Effective Rate 9.00% Annual 1 9.0000% 9.00% Semi-Annual 2 9.2025% 9.00% Quarterly 4 9.3083% 9.00% Monthly 12 9.3807% 9.00% Daily 365 9.4162% 9.00% Hourly 8760 9.4174% 9.00% Continuous Infinity 9.4174%

  22. EFFECTIVE AND NOMINAL RATES – Nominal to Effective Rate Where: NR= Nominal rate/100 EY= Effective annual interest rate NP= Number of periods

  23. EFFECTIVE AND NOMINAL RATES – Example Assuming a 10% nominal interest rate payable on a quarterly and semi annual basis, what is the equivalent/effective annual rate? Quarterly = Semi annual =

  24. EFFECTIVE AND NOMINAL RATES – Effective to Nominal Rate To calculate the nominal equivalent of an effective quarterly rate of 9.3083

  25. EFFECTIVE AND NOMINAL RATES – Excel Functions NOMINAL FUNCTION • Periods = 2 • Effective Rate = 0.08160 • Nominal Rate = 0.08000 (annual) EFFECTIVE FUNCTION • Periods = 2 • Nominal Rate = 0.0800 • Effective Rate = 0.08160 (annual)

  26. EFFECTIVE AND NOMINAL RATES – Questions You want to deposit $10,000 for two years. You are offered 5.15%, interest paid quarterly or 5.50% with interest paid annually. What is best? What is the equivalent quarterly effective rate of a semi-annual effective rate of 7.00% I invest £1,500 for 3 years with interest paid quarterly. At the end of 3 years my investment is worth £2,125. What quarterly effective rate have I achieved? Bank of America pays interest to its savings account customers on a semi-annual basis. Wachovia has just introduced a savings account with which pays 5.00% nominal but compounded daily. What rate will Bank of America have to pay on a semi-annual basis to remain competitive?

  27. EFFECTIVE AND NOMINAL RATES – Answers ANSWERS 5.15% is 5.2503% effective quarterly Nominal annual of a s.a. rate of 7.00% is 6.8816%. This nominal expressed as a quarterly effective is 7.0612% 11.78% nominal, 12.31% effective 5.00% compounded daily = 5.12675%. Nominal semi-annual equivalent is 5.062673%

  28. EXPONENTIAL RATE -- Exponential • This is the natural logarithm and has a value of 2.71828 • It is used for continuous compounding especially in derivatives calculations • The equivalent function in Excel is “EXP” • Most calculators have an “e” button • The future value of an investment given a rate of z and t annual periods is: ert • Assuming a nominal rate of 9.00% and a period of five years, and investment of 100 becomes:

  29. EXPONENTIAL RATE -- Exponential • We can use “e” to continuously compound and calculate present value as well • The exponential becomes minus to calculate this: e-rt The present value of an amount of 100 at A continuously discounted rate of 9.00% for 5 years becomes:

  30. EXPONENTIAL RATE – Exponential Function in Excel To use a PV function just put a minus in front of B1 field

  31. EXPONENTIAL RATE – Exercise You are pricing an option for a client using a Black & Scholes option model and calculate a premium for a two year option as $125,500. The formula has the following sign in front of it: e-rt If two year rates are 5.00% what does this mean in terms of premium for the client? Using continuous compounding what is the present value of $126,351.12 received in 3 years using a rate of 3.45%? Continuously compounded what is the future value of 100,000 over 10 years at a rate of 3.76%?

  32. EXPONENTIAL RATE – Answers ANSWERS DF = 0.904837 PV = 113,557.09 2. DF = 0.9017. PV = 113,927.78 3. EXP = 1.4564 FV is 145,644.71

  33. ANNUITY – The Annuity Formula • Future Value Annuity • An investor saves 500 per month, for 3 years at an annual rate of 5.00%. How much will they have accumulated by maturity? • Using the TVM menu on an HP 17 calculator: • Periods per year(other) =12 • 1% YR=5.00 • PV=0 • Pmt=500 • Solve for FV=19,376.67

  34. ANNUITY – The Annuity Formula

  35. ANNUITY – PV of an Annuity • Bond pricing and pension calculation • The amount of money to be invested now to create the required payments • An insurance company needs to know how much to invest today to provide a monthly disability payment of 500 to a client assuming a 4.00% interest rate.

  36. ANNUITY – PV of an Annuity

  37. ANNUITY– PV of an Annuity

  38. ANNUITY – Exercises • I am 60 years old and the insurance assessment of my life span is 85 years old. I have a pension pot of GBP 200,000 and want to take a pension annuity. If rates are at 5.00% and I want a monthly payment how much will I receive at the end of each month? • I can afford to pay $1000 per month on a 25 year mortgage. Interest rates area at 4.00%. How much can I borrow?

  39. ANNUITY – Answers • ANSWERS • 1,169.18 • 189,452

  40. Thank You

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