Quick Tip - Common Financial Formulas

Some common and some not so common formulas used in the finance:

Determining the Yield of a Discounted Security -- Discounted securities, such as United States Treasury Bills (T-bills), commercial paper, repurchase agreements, etc., do not have a stated return as other bond instruments do (i.e. a return of X%). Discounted securities are purchased at a price less than the amount the investor will receive upon maturity and thus have a rate of return built into their pricing. For example, you purchase a $1,000, 13 week T-bill for $990 and in 13 weeks you will receive $1,000. The rate of return is calculated as:

1. Formula: ([Selling Price - Purchase Price] / Purchase Price) x (365 days in a year / [days investment held]
2. Example: 
• ([$1,000  - $990] / $990) x (365 / [13 weeks * 7 days in a week])
• ($10 / $990) x (365 / 91)
• 0.01 x 4.01
• 0.0401 or 4.01% return
3. Note: Some calulcations use a 360 day year rather than a 365 day year. Typically, you should use the following:
• 365 Day Year - T-bills
• 360 Day Year - Commercial Paper and Repurchase Agreements
  

Present Value of a Bond -- This is the current value of a bond (or what you should pay for it today based on its expected return). Remember that with a bond you pay now for what you expect to receive in coupon payments over the years and a final payment of principal at the end of the bond. Also, all of the coupon payments and the final principal payment needs to be adjusted to account for inflation.

1. Formula - (assuming constant coupon payments):  
• Present Value of Bond = Present Value of Coupon Payments + Present Value of Principal Payment
• PV of Bond (Present Value of An Annuity x Coupon Payment) + (Present Value of Single Sum x Principal Payment)
  
2. Example 1 - 10% coupon bond with annual coupon payments. $1,000 par value and 3 years to maturity:
• First, determine your discount rate. 
• What is your required return? 
• Consider interest rates and returns of similar investments. 
• Let's assume average inflation of 3% per annum and that a current Certificate of Deposit with a maturity of 3 years would return 4.5%. Assuming that the bond is a high quality bond with good liquidity and no expectation of default, a return of 5% - 7% may be appropriate. Or whatever you determine the appropriate required return to be. 
• For our purposes, let's go with 7%.
  
• Now, determine the future value of your coupon payments
• First, determine present value interest factor (PVIFA). Using the discount rate determined above and the maturity of 3 years, use the standard Present Value of an Annuity Tables (can be found via Google search or in an accounting or finance textbook) to determine your PVIFA. For example find the 3 years on the left column and follow that over to 7% required return. The number is: 2.6243.
• Next, Determine your Coupon Payment
  
• Coupon Payment = quoted bond yield x par value of bond
• 10% x $1,000 = $100 per year
• Then, multiply your coupon payment by the PVIFA: 
• $100 x 2.6243 = $262.43, 
• This is the current value of the coupon payments over the three years (assuming that you require 7% return). 
• Another way to think of it is that this is the most you should pay for the future value of the coupon payments in order to return 7% on them.
• Now determine the future value of your final principal payment (par value)
  
• First determine your Present Value Interest Factor (PVIF). Using the Present Value Interest Factor tables (again, a Google search or textbook may be helpful in finding these common tables). Find the 3 on the left and follow it over to the 7%, the number is 0.8163.
• The multiply the par value (amount you will receive at maturity) by the PVIF
  
• $1,000 x 0.8163 = $816.30
• This is the amount that the final payment is worth today (assuming a 7% required return)
• Another way to think of it is that this is the most you should pay for the future value of the final bond payment in order to receive a return 7% on that amount (remember that the cash flows from the bond interest payments are calculated above).
• Now Add the Present Value of the Coupon Payments and the Present Value of the Final Payment (Par Value)
• $262.43 + $816.30 = $1,078.73
• This is the maximum amount that you should pay for this bond in order to receive a return of 7%
3. Example 2 - 10% coupon bond with semi-annual coupon payments. $1,000 par value and 3 years to maturity:
•  Interest payments may be made more frequently than annually (as in the previous example). You can use the following to adjust the above calculation accordingly. Note that I will use semi-annual payments for this example but if it is quarterly just adjust everything by 4 instead of 2.
• Adjust the above by:
1. When calculating the coupon payment divide the annual payment by the number of payments per year
• Annual coupon payments of $100 per above / 2 semi-annual payments in a year = $50 per payment
• $100 / 2 = $50
2. Multiply the required rate of return by the number of periods in the year
   
• 7% required return x 2 semi-annual payments in a year = 14%
• 7% x 2 = 14%
3. Multiply the number of periods to maturity (i.e. years) by the number of periods in the year
• For our purposes there are 2 semi-annual periods in a year, so
• 3 years x 2 = 6 periods
4. Determine the new PVIFA  using the annualized data
• For our purposes this is 6 periods at 6% per period (as determined above)
• Using the PVIFA tables noted above this yields a PVIFA of 4.917
  
5. Calculate the PVIF using the new annualized required return and number of periods
• For our purposes this is 6 periods at 6% per period (as determined above)
• Using the PVIF tables noted above this yields a PVIF of 0.7050