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Yields

Yields. Future Value Present Value Annuities Fixed-Rate Mortgages. Future Value. Single Period Multiple compounding periods within a year Other Compounding Intervals Examples. Future Value. Example: Deposit $10,000 in bank Interest rate = 6%

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Yields

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  1. Yields Future Value Present Value Annuities Fixed-Rate Mortgages

  2. Future Value • Single Period • Multiple compounding periods within a year • Other Compounding Intervals • Examples

  3. Future Value Example: Deposit $10,000 in bank Interest rate = 6% How much money will you have at the end of the year?

  4. Future Value • PV=$10,000 • i=.06 • FV=10,000(1.06) = $10,600

  5. Future Value -- Compounding Periods • What if the bank offers to “compound” your interest monthly? • How much money will you have?

  6. Future Value • What happened here? • Mathematically: • Intuitively -- we are earning interest on interest with monthly compounding

  7. Periodic Compounding • Consider :

  8. Periodic Compounding • t=0: Invest $10,000 at 6% nominal rate for 1 year--semiannual compounding • t=.5; receive $300 interest • .06/2*(10,000) • t=5; invest $300 @ 6% nominal rate for half year

  9. Periodic Compounding • At t=1; we receive another $300 interest from the original investment plus interest on the $300 investment ($9.0) plus all the principal: • $300+$9+$10,000+$300 • =10,600+9 • Note that

  10. Periodic Compounding • To find the value of the bank account at the end of the year for any number of periods: • divide the interest rate by the number of periods to get the periodic rate • Raise the quantity 1+ periodic rate to the power equal to the number of periods

  11. Observations • Value always increases with increasing number of compounding periods • Same “nominal” annual rate • Getting money sooner is always better than getting it later • There is not much difference between monthly and continuous compounding

  12. Effective Annual Yield • Different Compounding Periods Different Terminal Wealth Values • Effective Annual Yield is an alternative but equivalent measure. For One Year:

  13. Example • For semiannual compounding: • For daily compounding:

  14. Effective Annual Yield & Nominal Yield • When discussing interest bearing investments, people usually speak in terms of the “nominal” annual interest rate. • The nominal rate dos not completely describe the investments return -- you must know the compounding period too. • The compounding period is standard (and so presumed to be known) for certain kinds of investments: • Bonds: Semiannual • Mortgages; monthly • Eurobonds: Annual

  15. More examples • Calculate the EAY for a15% Treasury bond and a 15% mortgage: • Bond: .1556 • Mortgage:.16075 • The mortgage has an EAY more than 50 basis points higher than the bond.

  16. One More Time... • Calculate the EAY for a 5% Treasury bond and a 5% mortgage: • bond: .0506 • mortgage: .0512 • For 5% rates, the difference between the two is 6 basis points • The difference in EAY depends a lot on the level of the interest rate

  17. Other Compounding Periods • What if we plan to leave the investment in the bank for more than one year? • For annual compounding:

  18. Other Compounding Periods • Example: The future value of $10,000 invested at 6% for • 5 years: $13,382.26 • 10years: $17,908.48 • Using a calculator: • n=5 • i=6 • pv=10000 • fv=???

  19. Other Compounding Periods • With more frequent compounding: • Let n= number of years • i = nominal annual interest rate • p = number of periods per year for compounding • I/p=“periodic interest rate”

  20. Other Compounding Periods • Redo the previous problem for monthly compounding • n=5*12 • I=6/12 • pv=10000 • fv=13,488.50

  21. Calculator Tips • Some calculators let you set a “mode” feature for a certain number of compounding periods per year (e.g., 2 or 12 or 365) • The calculator then takes care of dividing the interest rate by p and/or multiplying the years by p. • Some calculators do both, some only one. As a result, using this feature is one of the most common sources of errors. • My HP 19B will divide the interest rate by p but does not multiply the years by p • I strongly recommend keeping your calculator in the 1 payment per period mode until you are comfortable with these calculations

  22. Present Value • Present Values and Future Values are related by the same formula we have been using -- we just solve for PV not FV. • For annual compounding and a single year:

  23. Present Value • Present values can be viewed as: • The amount you would have to put in the bank today (I.e., invest today) in order to generate a future wealth equal to FV, for a specific interest rate, or, • The price you would be willing to pay for the right to receive a cash flow of FV at some point in the future, when your opportunity cost of capital is equal to the specific interest rate

  24. Present Value v. Future Value • To calculate a future value from a present value we compounded the value forward at a specified interest rate. • To calculate a present value from a future value, we discount the future value back to the current time using a specified interest rate. In this case, we call the rate the discount rate.

  25. Present Value v. Future Value • Discounting is the inverse or opposite of compounding. • For a given rate of interest and time horizon, there is a 1:1 relationship between present values and future values. • Given one, you can calculate the other • There is one and only one present value that can generate a fixed future value and vice versa.

  26. Present Value • The calculation of present value from a future value can be extended to multiple compounding periods in a single year and longer time horizons in the same way we extended future value calculations.

  27. Present Value • Let n= number of years • I = nominal annual interest rate • p = number of periods per year for compounding • I/p=“periodic interest rate”

  28. Calculation of Present Value • To use a financial calculator to calculate a present value, we need to know, the nominal rate, the number of compounding periods, the future value, FV and the number of years,n. • N=n*p • i/yr=nominal annual rate/p • FV=FV (often zero) • Solve for PV

  29. Calculate the PV of $419.1 received 12 years from now with a nominal annual interest rate of 12% and monthly compounding. N=12*12=144 i=.12/12=.01 fv=419.1 pv=??? (=100) Calculate the PV of $419.1 received 12 years from now with nominal annual interest rates of 12% and annual compounding. N=12*1=12 i=.12 fv=$419.1 pv=??? (107.57) Example

  30. Calculator Tips • You may have noticed by now that the “sign” of the PV or FV you calculate is negative. • The calculator uses the sign to distinguish between money you invest (pay out)and money you receive (comes in to you). • If you put in either the fv or the pv as a positive number, the calculator assumes that the other cash flow must be in a different direction, and thus applies the opposite sign. While this does not matter a whole lot to you when calculating pv or fv, it will matter a lot when we begin calculating yields. • Most calculators require that you enter interest rates as % ( 6%) not decimals (.06)

  31. What if? • What if you know the PV, the FV and the time horizon and want to know the interest rate that relates the two. • For a single period and annual compounding, the answer is easy:

  32. What if? • Now what if we were using semiannual compounding?

  33. What if? • Some may remember the quadratic formula and be able to solve this equation for i, but if we go to monthly compounding the equation includes terms to the power of twelve and there is no simple equation for i. • Trial and error search is the best way to find i.

  34. What if? • The interest rate we are looking for is the one that makes the right hand side of the equation equal to the left. • One can simply guess an value for the interest rate, calculate the following equation and see if it is equal. • If it is, you are done! • If not guess again and try over.

  35. What if? • Fortunately, most financial calculators do this for you. • Example (annual compounding): • PV=100 • FV= -389.6 • n=12 • Now hit the interest rate key and it should give you 12% • Note carefully that I had to indicate one of either the PV or the FV as a negative number. See what happens on your calculator if you do not do this. Can you explain what happens?

  36. Calculating Yields • More generally, the present value formula shows the relationship that must hold among PV, FV, n and i. • As long as you know three of the four, you can calculate the missing value. • We first calculated FV, given PV, i and n • We then calculated PV, given FV, i and n • Now, we see how to calculate i given PV, FV and n • Can you solve for n, given PV, FV and i?

  37. One Last Extension: Annuities • Up to now, we have been dealing with cash flows where there is a single lump sum payment at t=0 followed by a single lump sum payment at some future date. • Zero coupon bonds • We will now consider present values and future values when there is a sequence of payments made over time.

  38. Annuities • In the first section, we asked what the balance would be in our bank account (after five years) if we deposited $10,000 today and could earn 6% nominal interest rate. • Now we ask what our bank account would be if we deposited $2,000 /year into the bank account--starting at the end of the first year?

  39. Annuities • The formula for the future value is: • Notice that the formula simply reflects the repeated application of the initial FV formula: • The FV of each of the periodic deposits is calculated and added together to get the overall future value.

  40. Annuities • While calculating the future value of a long series of payments by repeatedly applying the original formula, is straightforward, it could be rather tedious. • The financial calculator makes this process as simple as calculating a single future value:

  41. Annuities • Calculate the future value of $2,000 deposited at the end of each year into a bank account earning 6% compounded annually. • N=5 • i=6 • PMT=2000 • FV=??? ($11,274.19) • Caution! You must either “Clear” your calculator before doing this or enter PV=0 to make sure there is nothing left in that register of your calculator.

  42. Annuities • As with single cash flows, it is straightforward to switch to calculating the present value of an annuity. First mathematically: • As with future values, the present value of an annuity is simply the sum of the present values of each individual payment.

  43. Annuities • We can obtain a condensed mathematical expression for the present value of an annuity: • Note that if n is very large: • The second factor adjusts (P/i) for the fact that the payment is only received for a specific period of time.

  44. Annuities • Using a financial calculator, calculate the present value of an annuity of $2,000 received at the end of each year if the discount rate is 6% annually. • N=5 • I=6 • PMT=2,000 • FV=0 • PV= ???($8,424,73) • Note here I set FV=0 to make sure there was nothing left over in the calculator’s memory.

  45. Extensions • When working with annuities, you can change compounding periods from annual (as we have used here) to any other number of periods per year in the same way we did it for a single PV or FV calculation. • What is the present value of a sequence of monthly payments of 166.67 received for five years, when the nominal interest rate is 6%? • N=5*12 • PMT=166.67 • I=6/12=.5 • PV=???

  46. Extensions • Annuities and lump sum payments can be readily combined by combining the use of the PV keys and FV keys • What is the future value of your bank account if you deposit $5,000 now and deposit $2,000 per year for five years - interest rate is 6% annual? • N=5 • i=6 • PMT=$2,000 • PV=5,000 • FV=???($17,965.31)

  47. Yields • Most of the discussion above has focused on calculating present values and future values, given a specified rate for compounding forward or discounting back. • Frequently, we know the current price of a security and what cash flows we can expect to receive in the future. A summary measure that is often calculated is the “yield” or “rate of return” of the investment

  48. Yields • Consider a general form of the present value/future value equation:

  49. Yields • When calculating a yield, we are asking to find a value of i that makes this equation hold as an equality -- given known values for the payments, the price (or present value) and the lump sum payment at the end (FV). • Mathematically, we are solving for the roots of an nth order polynomial. • With a financial calculator, we are solving for i.

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