Math 325
Ch 7 – Yield Rates
Discounted Cash Flow Analysis
The profitability of investment projects can be compared via cash flow analysis, i.e. by finding the Net
Present Value (NPV) of the project.
NPV = P(i) = PV of cash inflows – PV of cash outflows
=
Internal Rate of Return(IRR): The interest rate for which NPV is zero. It is also known as the “yield
rate”
•
•
•
In general, higher the yield rate (IRR), the better for the investor(lender). The lower the IRR,
the better for the borrower.
Yield rates can be negative, indicating that the investor lost money on the investment.
If is valid to use yield rates (IRR) to compare alternative investments only if the period of
investment is same for all the alternatives.
For example, consider investing a sum of money under 2 options – Option A that credits 7%
effective for 5 years, and Option B that credits 6% effective for 10 year.
Since the periods of investments are not the same, we cannot say that Option A is better than B,
without further computation or investigation.
Notation:
Ct = Contributions to the investment, by the investor
• Ct > 0 implies cash outflow from the investor to the investment
• Ct < 0 implies cash inflow from the investment to the investor
Rt = Returns to the investor, by the investment
• Rt > 0 implies cash inflow from the investment to the investor
• Rt < 0 implies cash outflow from the investor to the investment
Of course, Rt and Ct are equivalent except that these cash flows are in opposite directions.
Rt = – Ct
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Math 325
Ch 7 – Yield Rates
Example 1. Consider two investing options. Option A earns 7% effective for 5 years. Option B earns
6% effective for 10 years. After the 1st 5 years, at what rate should we reinvest option A for the next 5
years, such that it is equivalent to the investment under option B for the entire 10 years?
Example 2. A certain 6-yr project requires an initial investment of $10,000 and a maintenance fee of
$1000 at the end of each year for the last 3 years. The project is expected to provide a return of $5,000
at the end of the 1st year, $500 at the end of the 4th year, and $9000 at the end of the 6th year.
a) Find the NPV of this investment, assuming a cost of capital(effective rate of interest) of 7%.
b) Find the internal rate of return (IRR) for this investment.
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Math 325
Ch 7 – Yield Rates
What conditions would guarantee a unique IRR?
1. IRR will be unique in an investment or project if all cash flows in one direction happen before all
cash flows in the other direction.
2. IRR will be unique if the outstanding balance of an investment is positive at all points throughout the
period of investment.
Note: It is possible for IRR to not have any real values, or to have more than one possible value.
However, these cases rarely occur.
Reinvestment Rates
Up until now, we have always assumed that the lender can reinvest payments received from the
borrower at a reinvestment rate equal to the original investment rate.
We now consider a situation in which our reinvestment rate is different from the rate at which payments
are made.
a) Consider an initial investment of 1 at time 0 that earns an interest of i per period. The interest earned
every period is reinvested at a rate of j per period.
b) Consider an investment of 1 at the end of each period where interest is paid at a rate i per period.
The interest earned every period is reinvested at a rate of j per period.
Note: For anuity-due with payments of 1 at the beginning of each period for n periods at a rate i with
reinvestment rate j,
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Math 325
Ch 7 – Yield Rates
Example 3. Payments of $1000 are investment at the beginning of each year for 10 years. The
payments earn an interest at 7% effective and interest can be reinvested at 5% effective.
a) Find the amount in the fund at the end of 10 years.
b) Find the purchase price an investor should pay to produce a yield rate of 8% effective.
Analysis for Loans So far, we have worked on loans with level payments with equation of value ____.
Now consider a position where lender invests an amount L at the rate i, receives n periodic payments of
R in return, and reinvests the payments at a rate j.
What is the new yield rate (IRR) i' for this investment?
Analysis for Bonds
Suppose the coupons from a bond are reinvested at rate j, where i ¹ j. Then to find the yield rate of the
bond considering reinvesting, we have
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Math 325
Ch 7 – Yield Rates
Example 4. A loan of $10,000 is being repaid with 25 level annual payments with interest charged at
8%/yr effective. Find the yield to the lender if he is only able to reinvest the payments received at 4%
per year.
Example 5. A $100 par value 10-yr bond with 8% semi-annual coupons is selling for $90. The bonds
are reinvested at only 6% convertible semiannually. Find the yield rate taking into account
reinvestment rates.
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Math 325
Ch 7 – Yield Rates
Interest Measurement of a Fund :
A common requirement in practical work is the determination of the yield rate earned by an investment
fund.
To use the fundamental principles developed in Ch.1, we must assume that the principal is remains
constant throughout the period and that all interest is paid at the end of the period. In practice, this is
often not what happens.
There are two standard methods for measuring the annual rate of return:
a) Dollar-weighted rate of return, iDW
b) Time-weighted rate of return, iTW
Goal : Find the effective rate of interest over one measurement period.
Given :
A – Amount in the fund at the beginning of the period
B – Amount in the fund at the end of the period
I – Total interest earned during the periodic
Ct – deposits (positive) or withdrawals (negative) at any time t
C – Net sum of deposits and withdrawals (net principal contributed) during the period
Then,
B=A+C+I
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Math 325
Ch 7 – Yield Rates
Example 6. At the beginning of the year, an investment fund was established with an initial deposit of
$1000. A new deposit of $500 was made at the end of 4 months. Withdrawals of $200 and $100 were
made at the end of 6 and 8 months, respectively. The amount in the fund at the end of the year is $1272.
Find the dollar-weighted rate of return earned by the fund during the year.
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Math 325
Ch 7 – Yield Rates
Time-Weighted Rates of Return
An investment fund manager generally does not have control over the timing or amounts of cash
inflows or outflows for the fund.
iTW is often used to compare the relative performance of various investment fund managers since this
method eliminates the impact of cash flows in and out of the fund.
Example 7. A pension fund receives contributions and pays benefits from time to time. The fund value
is reported after every transaction and at year end. The details for 2009 are as follows:
Fund Values
Date
Amount
Jan 1
1,000,000
Mar 1
1,240,000
Sep 1
1,600,000
Nov 1
1,080,000
Jan 1, 2010
900,000
Contributions Received
Date
Amount
Feb 28
200,000
Aug 31
200,000
Benefits Paid
Date
8
Amount
Oct 31
500,000
Dec 31
200,000
Math 325
Ch 7 – Yield Rates
Example 8. For the data in Ex. 7, show that dollar-weighted return for 2009, assuming each month is
1/12 th of the year, is 0.17391304.
Fund Values
Date
Amount
Jan 1
1,000,000
Mar 1
1,240,000
Sep 1
1,600,000
Nov 1
1,080,000
Jan 1, 2010
900,000
Contributions Received
Date
Amount
Feb 28
200,000
Aug 31
200,000
Benefits Paid
Date
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Amount
Oct 31
500,000
Dec 31
200,000
Math 325
Ch 7 – Yield Rates
Portfolio Method and Investment Year Method
Consider an investment fund that is being maintained for a number of individuals or companies. The
investment fund is commingled, i.e. each account has a pro rata share of the entire fund.
How do you credit interest to various accounts? Two methods:
I) Portfolio Method : an average rate of interest based on the earnings of the entire fund is computed
and credited to each account.
This method is unfavorable to new deposits when interest rates have risen significantly in the recent
past.
II) Investment Year Method : developed to address this problem by recognizing the date of
investment, as well as the current date, in crediting interest.
“New money” is segregated from the rest of the fund every year for several years and earns a specified
“new money rate” before being added to longer held money which earns the portfolio rate.
Notation: Let y be the calendar year of deposit, and let m be the number of years for which the
investment year method is applicable. Then
the rate of interest credited for the tth year of investment, t = 1, 2, ….m , is given by ________
Note: For t > m, the portfolio method is applicable.
Example 9. Using the Investment Year Method for an investment club, and the interest rate information
given in the table, answer the questions that follow. Assume a member joins on Jan 1 st of the year in
question.
Year of original
investment, y
Portfolio Rates, Calendar Year of
i y+3
Portfolio Rate, i y+3
2005
7.0%
7.8%
7.3%
7.4%
2008
2006
9.0%
8.1%
8.5%
8.7%
2009
2007
6.0%
6.4%
6.7%
6.5%
2010
a) What rate of interest would be credited to members in 2005 - 2010, who joined on Jan 1, 2005?
b) Ted joins the investment club on Jan 1, 2007. On Jan 1, 2008, he has $1000 in his account. On Dec
31, 2011, he has $1290.30 in his account. He makes no deposits or withdrawals after Jan 1, 2008.
Determine i2011.
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