Single Cash Flow Yield Calculations
In the world of investment math, a vehicle offering a 3.97% return can offer a higher return than an investment offering a 4.05% return. This is because yields for different vehicles are quoted assuming different compounding conversions and calendar conventions. The next few chapters discuss investment math. The people who often have the hardest time working with investment math are mathematicians. This is because many of the investment math calculations done on Wall Street are mathematically incorrect.
The place to start any discussion of investment math is with a discussion of present value (PV) / future value (FV) calculations used for investments that only have two cash flows: one out and one in. There are six formulas that are used to perform PV/FV calculations on problems with two cash flows.
SIMPLE INTEREST:
FV = PV × (1 + (R × T))
PV = FV / (1 + (R × T))
FV = PV × (1 + R)n
PV = FV / (1 + R)n
CONTINUOUS COMPOUNDING:
FV = PVert
PV = FV / ert
While the formulas are fairly simple, the difficulty is in using the right formula at the right time.
The term simple interest means that all of the interest (I) is paid when the investment matures. Thus, there is no opportunity to reinvest any intermittent interest payments in order to earn interest on interest (IOI). When an investment pays simple interest, the amount of interest that is earned can be calculated via the formula shown in Figure 2.1:
FIGURE 2.1
Calculating Simple Interest
I = PV × R × T
Where:
I = the amount of interest earned
PV = the amount of principal invested (the present value)
R = the rate at which the principal is invested
T = the amount of time the investment lasts
The total value at the conclusion of the investment (the future value) is equal to the sum of the PV + I. By rearranging this equation, an equation can be derived to solve for the PV that has to be invested today in order to accumulate a certain FV over a given time frame. First, replace the I with the formula for I.
FV = PV + I
FV = PV + (PV × R × T)
Second, factor out PV on the right side of the equation:
FV = PV × [1 + (1 × R × T)]
Third, within the parentheses, the 1 can be eliminated since anything multiplied by 1 equals itself, as shown in Figure 2.2.
FIGURE 2.2
Simple Interest Calculation of Future Value
FV = PV × [1 + (R × T)]
Fourth, divide both sides of the equation by the quantity: [1 + (R × T)] resulting in the equation shown in Figure 2.3.
FIGURE 2.3
Simple Interest Calculation of Present Value
PV = FV / [1 + (R × T)]
While the formulas appear to be very simple and straight-forward, they hold a trap for the unwary. The trap in the above equations is the T component. There are at least three alternative conventions for measuring time, and different investment vehicles use different conventions. Before any of the above equations can be solved, it is essential to determine which calendar convention is appropriate for the investment vehicle. Consider the next example, which illustrates the significance of the investment vehicle’s calendar convention.
Suppose an investor invests $1MM for 9 months at 10% in a vehicle which pays simple interest. How much interest will the investor earn? The formula for interest is:
I = PV × R × T
The PV ($1MM) and R (10%) components are self-explanatory. The T component, however, can be expressed using a variety of alternative calendar conventions. If the investment vehicle uses an actual/actual (A/A) calendar convention then the “T” component would equal the:
Actual Number of Days in the Investment Period
Actual Number of Days in a Year
If there were 273 days in the 9-month period, the time component would be 273/365 (or 273/366 if the year is a leap year). The amount of interest earned would therefore be:
I = $1MM × .10 × 273 / 365
I = $74,794.52
However, if the investment vehicle uses a 30/360 calendar convention, every month is assumed to have 30 days—regardless of the actual number of days in the month. Thus, in February, the investment pays, and the investor receives, 30 days of interest—even though the month normally has only 28 days. Likewise, in December the investment pays and the investor receives the same 30 days of interest—even though the month has 31 days. Since, in a 30/360 calendar convention, every month has 30 days, a year is assumed to have 360 days. Thus, the time component would be:
Number of Days in the Investment Period Assuming a 30-Day Date Count
360
The amount of interest earned would therefore be:
I = $1MM × .10 × 270 / 360
I = $75,000
A third commonly used calendar convention in the United States is Actual/360 (A/360). Under this calendar convention, the time component is equal to:
Actual Number of Days in the Investment Period
360
It is obviously inconsistent to use an actual-day count in the numerator while using 360 in the denominator. Yet, this is a very commonly used convention particularly with money market instruments. If the investment uses an A/360 calendar convention, the interest calculation would be:
I = $1MM × .10 × 273 / 360
I = $75,833.33
Thus, depending upon the calendar convention, the amount of interest earned in this example could be $74,794.52, $75,000, or $75,833.33—a difference that any investor would regard as significant. The calendar conventions for some of the common US investment vehicles are listed in Figure 2.4.
FIGURE 2.4
US Calendar Conventions and Associated Vehicles
|
Calendar Convention |
Applicable Vehicles |
|
A/A |
US Treasury bonds |
|
30/360 |
US corporate bonds Municipal bonds US agencies Eurobonds Mortgages CMOs Fixed side of swaps |
|
A/360 |
Repurchase agreements Bankers’ acceptances Commercial paper Treasury bills Floating side of swaps |
While the three calendar conventions in Figure 2.4 are the principal conventions used in the United States, there are a wide variety of other calendar conventions used in other countries, including 365/365, 365/360, and A/365. The only difference between an actual-day count and a 365-day count is that the actual-day count accounts for the extra day in a leap year. Thus, in 3 years out of 4, a 365-day count and an actual-day count are the same. Only if the investment extends into leap year do the two conventions vary. Figure 2.5 lists the calendar conventions of many common non-dollar instruments.
Calendar Conventions for Common Non-US Instruments
|
Calendar Convention |
Representative Markets |
|
A/365 |
Japanese government bonds (JGBs) British government bonds (Gilts) |
|
30/360 |
German government bonds (Bunds) Dutch government bonds (Guilders) |
|
A/A |
French government bonds (OATS) Most emerging markets |
As you can see, in order to calculate interest, it is often necessary to calculate the number of days between two dates using either an A- or 360-day count. Fortunately, both Excel and the HP-12C calculator include functions that calculate the number of days between two dates using either convention. The procedures to calculate the number of days between two dates are:
IN EXCEL
FOR THE HP-12C
For example, suppose on January 6, 1995, you invest $1MM in an investment that yields 10% and matures on October 10, 1995. How much interest will you earn assuming the investment uses an A/A, 30/360, and A/360 calendar convention?
Calculate the number of days between the two dates on both an actual- and 30-day count basis:
KEYSTROKES:
1.061995 [ENTER]
10.101995 [g] [ΔDYS] = 277 (actual)
[X><Y]= 274 (360)
Once the number of days has been determined, that number can be used to calculate the amount of interest earned assuming different calendar conventions.
Using an A/A Calendar: I = $1MM × .1 × 277 / 365 = $75,890.41
Using an A/360 Calendar: I = $1MM × .1 × 277 / 360 = $76,944.44
Using a 30/360 Calendar: I = $1MM × .1 × 274 / 360 = $76,111.11
Before proceeding on to the next section, confirm your understanding of calendar conventions and simple interest by reviewing these next sample problems.
PROBLEM 2A
What is the PV of a $1 return in 6 months assuming an 8% 30/360 simple interest rate?
ANSWER:
PV = FV / [1 + (R × T)]
PV = $1 / [1 + (.08 × 180 / 360)]
PV = $1 / 1.04
PV = .9615384615
PROBLEM 2B
What is the PV of a $1 return that will be received in 9 months (273 days) assuming a 12% A/360 simple interest?
ANSWER:
PV = FV / [1 + (R × T)]
PV = $1 / [1 + (.12 × 273 / 360)]
PV = $1 / 1.091
PV = .9165902841
What is the PV of a $1 return received in 1 year assuming that the dollar is discounted first by 8% simple interest 30/360 for 3 months and then by 9% simple interest 30/360 for 9 months?
ANSWER:
Discount the $1 for the 3-month period:
PV = FV / [1 + (R × T)]
PV = 1 / [1 + (.08 × 90 / 360)]
PV = 1 / 1.02
PV = .9803921569
Discount the PV in Step 1 for the additional 9 months:
PV = FV / [1 + (R × T)]
PV = .9803921569 / [ 1 + (.09 × 270 / 360)]
PV = .9803921569 / 1.0675
PV = .9184001469
PROBLEM 2D
How long will it take to earn at least $100,000 in interest if you invest $1MM in an investment which yields 12% simple interest on an A/A basis?
ANSWER:
I = PV × R × T
$100,000 = $1MM × .12 × A / 365
A = ($100,000 / $120,000) × 365
A = 304.17 = 305 days
How long will it take to earn $100,000 in interest if you invest $1MM in an investment which yields 12% simple interest on a 30/360 basis?
ANSWER:
I = PV × R × T
$100,000 = $1,000,000 × .12 × A / 360
A = $100,000 / $120,000 × 360
A = 300 days
PROBLEM 2F
How much interest will you earn if you invest $1MM on January 12, 1997, until August 27, 1997, at 8% simple interest assuming the yield is expressed on an A/360 basis?
ANSWER:
I = PV × R × T
I = $1,000,000 × .08 × 227 / 360
I = $50,444.44
PROBLEM 2G
How much interest will you earn if you invest $1MM on January 12, 1997, until August 27, 1997, at 8% simple interest assuming the yield is expressed on a 30/360 basis?
ANSWER:
I = PV × R × T
I = $1,000,000 × .08 × 225 / 360
I = $50,000.00
What rate will allow $1MM to grow to $1.1MM in 1 year assuming simple interest quoted on an A/360 basis?
ANSWER:
I = PV × R × T
$100,000 = $1,000,000 × R × 365 / 360
R = 9.86%
PROBLEM 2I
You want to earn $50,000 of interest on a $1MM investment over a 9-month period that has 273 days. What rate would you have to earn on a 30/360 calendar?
ANSWER:
I = PV × R × T
$50,000 = $1MM × R × 270 / 360
R = $50,000 / $1MM × 360 / 270
R = 6.67%
PROBLEM 2J
You want to earn $50,000 of interest on a $1MM investment over a 9-month period that has 273 days in it. What rate would you have to earn on an A/360 calendar?
ANSWER:
I = PV × R × T
$50,000 = $1MM × R × 273 / 360
R = $50,000 / $1MM × 360 / 273
R = 6.59%
You want to earn $50,000 of interest on a $1MM investment over a 9-month period that has 273 days in it. What rate would you have to earn on an A/A calendar?
ANSWER:
I = PV × R × T
$50,000 = $1MM × R × 273 / 365
R = $50,000 / 1MM × 365 / 273
R = 6.68%
PROBLEM 2L
How much would you have to invest today in order to earn $500K in interest over 7 months, assuming the 7-month period had 213 days and the investment yields 12%, quoted on an A/360 day basis?
ANSWER:
I = PV × R × T
$500,000 = PV × .12 × 213 / 360
PV = $500,000 / (.12 × 213 / 360)
PV = $7,042,253.52
PROBLEM 2M
What simple interest rate would you have to earn on an investment quoted on an A/360 calendar in order to earn the same amount of interest you would earn if you invested for 1 year at 10% simple interest quoted on a 30/360 calendar?
ANSWER:
Choose any notional amount—I used $1 in this example.
PV × R × T = PV × R × T
$1 × .1 × 360 / 360 = $1 × R × 365 / 360
.1 = R × 365 / 360
R = (.1 × 360 / 365)
R = 9.86%
PROBLEM 2N
What simple interest rate of return would you have to earn on a 9-month (273 days) investment, quoted on an A/360 calendar, in order to earn the same amount of interest as you would earn on a 7.53% 30/360 investment over the same time period?
ANSWER:
Choose any notional amount—I used $1 in this example.
PV × R × T = PV × R × T
$1 × .0753 × 270 / 360 = $1 × R × 273 / 360
.0753 × 270/360 = R × 273 / 360
.056475 = R × 273 / 360
R = 7.44%
PROBLEM 2O
How much interest accrues daily on a $100MM position of 10% Treasuries that mature on April 15, 1998, during the month of February 1987?
ANSWER:
I = PV × R × T
I = $100,000,000 × .05 × 1 / 182
I = $27,472.53
How much interest accrues daily on a $100MM position of 10% Treasuries that mature on April 15, 1998, during the month of July 1986?
ANSWER:
I = PV × R × T
I = $100MM × .05 × 1 / 183 = $27,322.40
PROBLEM 2Q
Suppose on January 15, 1997, you settle the purchase of $1 billion worth of 10% US Treasuries that mature in 1 year, priced at par. If you reinvest the coupon at 4% 30/360, what is your FV?
ANSWER:
I = PV × R × T
I = $1,000,000,000 × .1 × 181 / 365
I = $49,589,041.10
I = $1,000,000,000 × .1 × 184 / 365
I = $50,410,958.90
IOI = $49,589,041.10 × .04 × 180 / 360
IOI = $991,780.82
TOTAL = $100,991,780.82
PROBLEM 2R
Suppose on July 15, 1997, you settle the purchase of $1 billion worth of 10% US Treasuries that mature in 1 year priced at par. If you reinvest the coupon at 4% SA 30/360, what is your FV?
I = P × R × T
I = $1,000,000,000 × .1 × 184 / 365
I = $50,410,958.90
I = $1,000,000,000 × .1 × 181 / 365
I = $49,589,041.10
IOI = $50,410,958.90 × .04 × 180 / 360
IOI = $1,008,219.18
TOTAL = $101,008,219.18
PROBLEM 2S
What simple interest rate expressed on an A/360 basis would you have to earn in order to quadruple your money assuming you invested on January 1, 1990, and the investment matured on January 1, 2000?
ANSWER:
FV = PV + (PV × R × T)
$4 = $1 + ($1 × R × 3,652 / 360)
$3 = ($1 × R × 3,652 / 360)
$3 = $10.144444 × R
R = .295728
R = 29.57%
PROBLEM 2T
What rate would you have to earn in order to double your money in a year, assuming an A/A, A/360, and 30/360 calendar convention simple interest?
ANSWER:
For A/A:
I = PV × R × T
R=100% A/A
For 30/360:
I = PV × R × T
1 = 1 × R × 360 / 360
R=100% 30/360
For A/360:
1 = 1 × R × 365 / 360
R = 98.63% A/360