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Time value of money Part 2
| Date post: | 13-Aug-2015 |
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Time value of moneyPart 2
The “Rule-of-72”
Approx. Years to Double = 7272 / i%
QUICK!!How long does it take to double 150,000 Taka at a compound interest rate of 12% per year?
7272 / 12% = 6 years [Actual time is 6.12 years]
The “Rule-of-72”
PV: Compound interest
•Assume that you need 1,000 Taka in 2 years. How much do you need to deposit today at a discount rate of 7% compounded annually?
0 1 227%
Taka 1,000Taka 1,000PV1PVPV00
PV: Compound interest formula
Formula PVPV00 = FVFVn / (1+i)n
PVPV00: Present value (at time 0)
FVFVn:Future value after n time periods
i: Interest rate per period
n: The number of time periods
PV: Compound interest
PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2
= FVFV22 / (1+i)2 = $873.44$873.44
0 1 227%
Taka 1,000Taka 1,000PV1PVPV00
General PV compound interest formula
Formula
PVPV00 = FVFV11 / (1+i)1
PVPV00 = FVFV2 / (1+i)2
etc
General present value formula
PVPV00 = FVFVn / (1+i)n
or PVPV00 = FVFVn (PVIFPVIFi,n) -- See See Table IITable II
Valuation using PV table
•PVIFPVIFi,n is found in this table. – You can find this table in your text book. – I will also provide you with one during tests/midterm etc.
Period 6% 7% 8% 1 .9434 .9346 .9259 2 .8900 .8734 .8573 3 .8396 .8163 .7938 4 .7921 .7629 .7350 5 .7473 .7130 .6806
Valuation using PV table
PVPV2 = Taka 1,0001,000 (PVIF7%,2)= Taka 1,0001,000
(.8734.8734) = Taka 873.40873.40Period 6% 7% 8%
1 .9434 .9346 .9259 2 .8900 .8734 .8573 3 .8396 .8163 .7938 4 .7921 .7629 .7350 5 .7473 .7130 .6806
PV table example #1
Shovon wants to know how large a deposit to make so that the money will grow to 10,000 10,000 TakaTaka in 5 years at a discount rate of 6%.
0 1 2 3 4 55
10,000 Taka10,000 TakaPVPV00
6%
PV table solution #1Shovon wants to know how large a deposit to make so that the money will grow to 10,000 10,000 TakaTaka in 5 years at a discount rate of 6%.
Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00 = Taka 10,000Taka 10,000 / (1+ 0.06)5
= Taka 7,472.58Taka 7,472.58 Calculation based on table:
PVPV00 = Taka 10,000Taka 10,000 (PVIFPVIF6%, 5)
= Taka 10,000Taka 10,000 (.7473)= Taka 7,473.00Taka 7,473.00
PV table example #2Marjan wants to know how large a deposit to make so that the money will grow to 10,000 10,000 TakaTaka in 3 years at a discount rate of 8%.
Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00 = Taka 10,000Taka 10,000 / (1+ 0.08)3
= Taka 7,938.32Taka 7,938.32 Calculation based on table:
PVPV00 = Taka 10,000Taka 10,000 (PVIFPVIF8%, 3)
= Taka 10,000Taka 10,000 (.7938)= Taka 7,938.00Taka 7,938.00
PV table example #3Galib wants to know how large a deposit to make so that the money will grow to 10,000 10,000 TakaTaka in 4 years at a discount rate of 7%.
Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00 = Taka 10,000Taka 10,000 / (1+ 0.07)4
= Taka 7,628.95Taka 7,628.95 Calculation based on table:
PVPV00 = Taka 10,000Taka 10,000 (PVIFPVIF7%, 4)
= Taka 10,000Taka 10,000 (.7629)= Taka 7,629.00Taka 7,629.00
Annuities
An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
Types of annuities
•Ordinary annuity: Payments or receipts occur at the end of each period.
•Annuity due: Payments or receipts occur at the beginning of each period.
Examples of annuities
•Insurance Premiums
•Retirement Savings (Provident Fund)
•Student Loan Payments
•Car Loan Payments
•Mortgage Payments
Parts of an annuity
0 1 2 3
Tk. 100 Tk. 100 Tk. 100
EndEnd ofPeriod 1
EndEnd ofPeriod 2
Today EqualEqual Cash Flows Each 1 Period Apart
EndEnd ofPeriod 3
Ordinary Annuity
Parts of an annuity
0 1 2 3
Tk.100 Tk.100 Tk.100
BeginningBeginning ofPeriod 1
BeginningBeginning ofPeriod 2
Today EqualEqual Cash Flows Each 1 Period Apart
BeginningBeginning ofPeriod 3
Annuity Due
Overview of an ordinary annuity - FVA
FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 +
R(1+i)0
R R R
0 1 2 n n n+1
FVAFVAnn
R = Periodic Cash Flow
Cash flows occur at the end of the period
i% . . .
Example of an ordinary annuity - FVA
FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1
+ $1,000(1.07)0
= $1,145 + $1,070 + $1,000 = $3,215$3,215
$1,000 $1,000 $1,000
0 1 2 3 3 4
$3,215 = $3,215 = FVAFVA33
7%
$1,070
$1,145
Cash flows occur at the end of the period
Hint on annuity valuation
The future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginning of the last cash flow period.
Valuation using Table III
FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000
(FVIFA7%,3) = $1,000 (3.2149) = $3,214.90$3,214.90Period 6% 7% 8%
1 1.0000 1.0000 1.0000 2 2.0600 2.0700 2.0800 3 3.1836 3.2149 3.2464 4 4.3746 4.4399 4.5061 5 5.6371 5.7507 5.8666
Overview of an annuity due - FVAD
FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 +
R(1+i)1
= FVAFVAn n (1+i)
R R R R R
0 1 2 3 n-1n-1 n
FVADFVADnn
i% . . .
Cash flows occur at the beginning of the period
Example of an annuity due – FVAD
FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 +
$1,000(1.07)1
= $1,225.04 + $1,144.90 + $1,070.00 = $3,439.94$3,439.94
$1,000 $1,000 $1,000 $1,070.00
0 1 2 3 3 4
$3,439.94 = $3,439.94 = FVADFVAD33
7%
$1,225.04
$1,144.90
Cash flows occur at the beginning of the period
Valuation using Table III
FVADFVADnn = R (FVIFAi%,n)(1+i)FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)
= $1,000(3.2149)(1.07) = $3,439.94$3,439.94
Period 6% 7% 8% 1 1.0000 1.0000 1.0000 2 2.0600 2.0700 2.0800 3 3.1836 3.2149 3.2464 4 4.3746 4.4399 4.5061 5 5.6371 5.7507 5.8666
Overview of an ordinary annuity – PVA
PVAPVAnn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
R R R
0 1 2 n n n+1
PVAPVAnn
R = Periodic Cash Flow
i% . . .
Cash flows occur at the end of the period
Example of an ordinary annuity – PVA
PVAPVA33 = $1,000/(1.07)1
+ $1,000/(1.07)2
+ $1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32$2,624.32
$1,000 $1,000 $1,000
0 1 2 3 3 4
$2,624.32 = $2,624.32 = PVAPVA33
7%
$ 934.58$ 873.44 $ 816.30
Cash flows occur at the end of the period
Hint on annuity valuation
The present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the end of the first cash flow period.
Valuation using Table IV
PVAPVAnn = R (PVIFAi%,n) PVAPVA33 = $1,000 (PVIFA7%,3)
= $1,000 (2.6243) = $2,624.30$2,624.30Period 6% 7% 8%
1 0.9434 0.9346 0.9259 2 1.8334 1.8080 1.7833 3 2.6730 2.6243 2.5771 4 3.4651 3.3872 3.3121 5 4.2124 4.1002 3.9927
Overview of an annuity due - PVD
PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAPVAn n (1+i)
R R R R
0 1 2 n-1n-1 n
PVADPVADnn
R: Periodic Cash Flow
i% . . .
Cash flows occur at the beginning of the period
Example of an annuity due – PVAD
PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1
+ $1,000/(1.07)2 = $2,808.02$2,808.02
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02 $2,808.02 = PVADPVADnn
7%
$ 934.58$ 873.44
Cash flows occur at the beginning of the period
Valuation using Table IV
PVADPVADnn = R (PVIFAi%,n)(1+i)PVADPVAD33 = $1,000 (PVIFA7%,3)
(1.07) = $1,000 (2.6243)(1.07) = $2,808.00$2,808.00Period 6% 7% 8%
1 0.9434 0.9346 0.9259 2 1.8334 1.8080 1.7833 3 2.6730 2.6243 2.5771 4 3.4651 3.3872 3.3121 5 4.2124 4.1002 3.9927
Steps to solve TVM problems
1. Read problem thoroughly2. Determine if it is a PV or FV problem3. Create a time line4. Put cash flows and arrows on time line5. Determine if solution involves a single CF, annuity stream(s), or mixed flow6. Solve the problem7. Check with financial calculator (optional)
