Math 420 Yield Rates Problem Set 6
Net Present Value(NPV)
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
The interest rate, i, used to compute NPV is called the “cost of capital”, or “opportunity coast of capital.” or “interest preference rate.” This is the rate that is believed to be earned on an alternative investment that is subject to the same degree of risk.
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.
Reinvestment Rates
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.
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Math 420 Yield Rates Problem Set 6
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 ,
Analysis for Loans
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 420 Yield Rates Problem Set 6
Problems
1. 1000 is deposited into Fund X which earns an annual effective rate of 6%. At the end of each year,
the interest earned plus an additional 100 is withdrawn from the fund. At the end of the 10 th year, the
fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y, which
earns an annual effective rate of 9%. Determine the accumulated value of Fund Y at the end of year
10.
2. Karen invests 500 into a bank account at the beginning of each year for 20 years, The account pays
out interest at the end of each year at an annual effective interest rate of i %. The interest is reinvested
at an annual effective rate of ( i /2)%. The yield rate on the entire investment over the 20 year period
is 9% annual effective. Determine i.
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Math 420 Yield Rates Problem Set 6
3. Kevin invests 1800 at the beginning of the year in a fund which credits interest at an annual effective
interest rate of 8%. He reinvests each interest payment in a separate fund that accumulates at an
annual effective interest rat of 7%. The interest payments from this fund are invested in a bank
account that credits interest at an annual effective interest rate of 6%. Determine the sum of principal
and interest at the end of 10 years.
4. Shrek purchases an annuity at a price of 10,000. The annuity makes payments of 500 at the
beginning of every 6 months for 20 years. The payments are reinvested in a fund which earns an
interest at an annual effective rate of i . Interest payments are received every 6 months and reinvested
at a nominal rate of 6%, convertible semiannually. Shrek realizes an overall effective annual yield of
7% on his original investment over the 20 year period. Calculate i .
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Math 420 Yield Rates Problem Set 6
5. Jessica invests 100 at the end of each year for 15 years at an annual effective interest rate of i. The
interest payments are reinvested at an annual effective rate of 6%. The accumulated value at the end
of 15 years is 1850.45. Calculate i .
6. John invests 2,000 at an effective annual interest rate of 17% for 10 years. Interest is payable
annually and is reinvested at an effective annual rate of 11%. At the end of 10 years, John's
accumulated interest is 5685.48.
Judy invests 150 at the end of each year for 20 years at an effective annual interest rate of 14%.
Interest is payable annually and is reinvested at an effective annual rate of 11%. Calculate Judy's
accumulated interest at the end of 20 years.
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Math 420 Yield Rates Problem Set 6
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 PS1, 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, i D W b) Time-weighted rate of return, i TW
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
C t
– 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
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. i TW 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.
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Math 420 Yield Rates Problem Set 6
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. Find the time-weighted and dollar weighted interest
rates if you are given the details for 2009 as follows:
Fund Values
Date
Jan 1
Mar 1
Sep 1
Nov 1
Jan 1, 2010
Amount
1,000,000
1,240,000
1,600,000
1,080,000
900,000
Contributions Received
Date Amount
Feb 28
Aug 31
200,000
200,000
Benefits Paid
Date
Oct 31
Dec 31
Amount
500,000
200,000
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Math 420 Yield Rates Problem Set 6
8 . On Jan 1, 2004, an investment account is worth 100. On May 1, 2004, the value increases to 120 and
D is deposited. On Nov 1, 2004, the value is 100 and 40 is withdrawn. On Jan 1, 2005, the
investment account is worth 65. The time-weighted interest rate is 0%. What is the dollar-weighted
rate of interest?
9. For the following cash flows, the time-weighted yield rate is 13.61%, and the dollar-weighted rate is
12.04%. Calculate T .
1/1/12 3/1/12 4/1/12 T /12 1/1/13
Account Value before W/D
100 108 102 118 130
Deposit(D)
Withdrawal(W) 12
20 X
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Math 420 Yield Rates Problem Set 6
Portfolio Method and Investment Year Method
I) Portfolio Method : an average rate of interest based on the earnings of the entire fund is computed and credited to each account.
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 t
= 1, 2, ….
m , is given by ________ th year of investment, t
Note: For t > m , the portfolio method is applicable.
10.You are given the following table of interest rates:
Calendar Year of
Original Investment, y
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
10.00
10.00
9.50
9.00
8.25
8.50
i
1 y
9.00
9.00
9.25
9.50
Portfolio Rates
(in %) i
5 y i y +5 i
2 y
8.25
8.70
9.00
9.10
9.35
9.50
i
3 y
8.40
8.75
9.10
9.20
9.50
9.60
8.50
8.90
i
4 y
9.10
9.30
9.55
9.70
8.50
9.00
9.20
9.40
9.60
9.70
8.35
8.60
8.85
9.10
9.35
10.00
9.90
9.80
9.70
9.50
9.80
The amount in the fund on January 1, 1997 is $1000. Let the following be the accumulated value of the fund on Jan 1, 2001:
P: under the investment year method
Q: under the portfolio method
R: if the balance is withdrawn at the end of every year and is reinvested at the new money rate.
Determine P, Q, and R.