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Time value of money
1. Cash flow diagrams
It is helpful to use cash flow diagram (CFDs) when analyzing cash flows that occur over several time period (years, months, weeks…).
2. Simple interest calculations
Fn=P (1+ in) P= Present worth F= Future worth i=interest % n=the number of interest periods.
3. Compounding interest calculations
There are two ways to calculate present worth and future worth
- By Formulas
F= P (1+i)n P= F (1+i)-n
- By Tables
F=P(F/P i,n) P=F(P/F i,n)
Ex.
Ali borrows SR 12000 at 10 % compounded annually. What is the future worth of this investment after 10 years?
Solution
Fw=12000(1+0.10)^10 = SR 31124. Or by using tables FW= P (F/P i,n) = 12000(2.59374)= SR 31124.
Ex.
Find the present worth if the future worth 10000 at 5 years and i=10%?
Solution
Pw=10000(1+0.1)-5= SR 6209.
Or by using tables Pw= F (P/F i,n) = 10000(0.62092)= SR 6209.
Ex.3 (57/page 101)
The cash flow profile for an investment is given below, and the interest rate 8 percent compounded annually
EOY 0 1 2 3 4 5 6 Cash Flow 0 $500 $200 $600 $100 -$100 $
- Find the present worth of this series
- Find the future of this cash flow
Solution
Gradient series of cash flows A gradient series of cash flows occurs when the value of a given cash flow is greater than value of previous cash flow by a constant amount G(.
P=G(P/G i,n)
Ex.
Ex.
Ex.7 (98/105)
Consider the following cash flow profile Cash Flow in"$" EOY Cash Flow
EOY Cash Flow
EOY Cash Flow 0 - 75000 3 9000 6 18000 1 3000 4 21000 7 21000 2 6000 5 15000 8 24000
Using a gradient series factor, determine the present worth equivalent for cash flow series using annual compound interest rate of 6% and 7%.
Solution
At i=6%
Pw =-$75,000+3,000 (P|A 6%,8)+3,000 (P|G 6%,8)= $3,154.
At i=7%
Pw= =-$75,000+3,000 (P|A 7%,8)+3,000 (P|G 7%,8)= -$719.
Geometric series of cash flows The Geometric series of cash flows occurs when the size of cash flow increase or decrease by a fixed percent from one period to the next.
o Notes
- IF i ≠ j
P =A 1 (P/A i,j,n) F= A 1 (F/A i,j,n)
- IF i = j
P= nA 1 /(1+i) F=nA1(1+i)n-^1
- When the geometric series decreased from one period to the next must use formals to get present worth or future worth.
Ex.9(Ex2.31/75)
Ali receives an annual bonus and deposits it in a saving account that pays 8 % compounded annually. The size of bonus increase by 10% Each year, his initial deposit is $ 500. How much will be in the fund immediately after 10 year?
Solution
n=10 years A=$500 i=8% j=10%
F= A 1 (F|A i,j,n)=500(F|A 8%,10%,10)=$ 10870.
Ex.10 (24/98)
You have to borrow $ 10,000 which you will pay back in 4 year; your local bank has the following for loan account available
Account Interest Interest types 1 7 Compounded annually 2 7.5 Simple 3 7.5 Compounded annually 4 8.25 Simple
Solution
P= $10,000 i=7, 7.5, 7.5and 8.25 n=4 yrs.
Account 1: F=10000 (1+0.07)^4 = $13,107.
Account 2: F=10000 (1+0.075×4) =$
Account 3: F=10000 (1+0.075)^4 = $13,354.
Account 4: F=10000 (1+0.0825×4) = $13,300.
Ex.11 (90/104)
Determine equivalent annual cash flow if this series at 10 %
EOY 0 1 2 3 4 5 6 7 8 CASH FLOW
Solution
Pw =-$2,500+$3,000 (P|F 10%,1)+$4,500 (P|F 10%,2)-$5,000 (P|F 10%,4) -$1,000 (P|F 10%,6)+$7,000 (P|F 10%,7)+$3,000 (P|F 10%,8) = $ 4958.
Aw= $4,958.43 (A|P 10%,8) = $ 929.
