FV10 = $3,483.71 * FVIF7%,6 FV10 = $5,228.11 (ii) Each cash flow separately at a 7% discount rate
End of Year 1 Year 2 Year 3 Year 4 750 800 700 900 750 * FVIF7%,3 = 918.78
800 * FVIF7%,2 = 915.92
700 * FVIF7%,1 = 749.00 FV4 = 3,483.70 FV10 = $3,483.71 * FVIF7%,6 FV10 = $5,228.11
PRESENT VALUE (i) One Year at a time at a 7% discount rate
End of
0 Year 1 Year 2 Year 3 Year 4 750 800 700 900 /1.07 841.12 1,541.12 /1.07 1,440.30 2,240.30/1.07 2,093.74 2,843.74 /1.07 2,657.70 PV = $2,657.70 (ii) Each cash flow separately at a 7% discount rate
End of
0 Year 1 Year 2 Year 3 Year 4 700 900 750 * PVIF7%,1 = 700.93 800 * PVIF7%,2 = 698.75
700 * PVIF7%,3 = 571.41
900 * PVIF7%,4 = 686.61 PV = $2,657.70
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750 800
EXAMPLE
You have just received your first job offer from a large investment banking firm. They want to test your skills and have offered you either a $200,000 signing bonus with no salary for 4 years or a $60,000 salary at the end of each year for 4 years.
Ignoring taxes, which would you prefer if you could earn 9% on investments?
PVbonus = $200,000
PVsal = (60,000 *PVIF9%,1)+ (60,000*PVIF9%,2) +(60,000 *PVIF9%,3) + (60,000 * PVIF9%,4)
= (60,000 * .91743) + (60,000 * .84168) + (60,000 * .77218) + (60,000 * .70843)
= 55,046 + 50,501 + 46,331 + 42,506 = = $194,384
The signing bonus of $200,000 has a greater present value.
IMPORTANT: TIMING OF PAYMENTS All preset calculations assume that payments occur at the end of each period. In our example, the assumption is that the annual salary occurs at the end of each year. Since many payments, such as rent payments, occur at the beginning of a period it is very important to pay attention to the timing of the payments…
ANNUITIES :
a level stream of cash payments for a fixed period of time.
Examples: ? Car loans ? Mortgages ? Rent payments ? Fixed costs
? Pension payment
Present Value of = C * 1 – PVIF an Annuity r PVA (r,t) = C * 1 - (1/( 1 + r )t) r Notation PVIFA(r,t) = 1 - 1/( 1 + r )t r
PREVIOUS EXAMPLE
? Salary of $60,000 per year for 4 years with a 9% discount rate.
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PVA (9%,4) = 60,000 * ( 1 – (1/(1.09)4) .09 = 60,000 * 3.23972
PVA = $194,383.19
Three Methods:
? compute the equation
? use present value of an annuity table
? use PV function on a business calculator
EXAMPLE:
You win the “million dollar” lottery but find that you will receive $50,000 per year for 20 years. You could earn 7.5% on an investment. What is the present value of your winnings?
EXAMPLE: CALCULATING THE MONTHLY PAYMENT
? You’ve graduated, have a new job, and have the student loan paid off.
? Now, the fine car! Used but fine.
? Price: $37,500
? Interest rate: 1% compounded monthly
? Term: 60 months
WHAT WILL YOUR MONTHLY PAYMENT BE?
EXAMPLE: CALCULATING THE INTEREST RATE ON AN ANNUITY
? More on the car
? Price: $37,500
? Term: 60 months; compounded monthly
? Payment per month; $950
WHAT MONTHLY INTEREST RATE WILL YOU BE PAYING?
FUTURE VALUE OF AN ANNUITY
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FVA(r,t) = PVA * FVIF(r,t) OR FVA(r,t) = C * FVIF – 1 r FVA(r,t) = C * (1 + r)t –1 r Notation FVIFA(r,t) = (1 + r)t –1 r EXAMPLE You won $50,000 per year for 20 years. You can earn 7.5% per year on an investment. We showed earlier that the Present Value is $509,724.57. What is the Future Value of this Annuity?
EXAMPLE You set aside $3,500 per year for 40 years in an RRSP in preparation for your retirement. If you earn 7.32% per annum, what is the future value of this constant stream of contributions?
PERPETUITY
An annuity in which the cash flow continues forever. PVIFA = 1 – 1/(1 + r)t r
- if t becomes very large, then 1/(1+r)t
approaches zero
So: PVIFAperpetuity = 1/r PVperpetuity = C * 1/r = C/r
EXAMPLE A preferred stock has a $7.50 dividend in perpetuity and the interest rate is 6.9%. What is the value of the preferred stock? PVperpetuity = C/r = 7.50/.069 Price of the preferred stock = $108.70
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