Cash flows of the same amount is called "annuity". Paying 10,000 yen per month for gift certificate of the department store constitutes an annuity. If cash flows start immediately, it is called an "immediate annuity." If the cash flows start at the end of the current period, rather than immediately, it is call an "ordinary annuity."
Let r be interest rate per month. Suppose you keep paying amount $A each month for a year. Starting today, you pay twelve times. Apply monthly compounding. This is an immediate annuity. FV is given by FV t1
12 A1rt
Next suppose that you will receive amount $A per month 12 times for a year. The first payment is a month from now. What is PV of this ordinary annuity? Apply monthly compounding. PV of the annuity is expressed as a sum of geometric sequence.
PV of the first cash flow = A 1r PV of the second cash flow = A
1r2 ª
PV of the last cash flow = A 1r12 Then PV of this annuity is given by t1
12 A
1rt .
Example
Suppose that you will receive amount $100 per month 12 times for a year. Interest rate per month is 1 percent.
Then PV of this annuity is $1,125.51 as shown below. In terms of PV, having $1,125.51 today is equivalent to receiving $100 each month for a year.
In[92]:= ClearA, r; A 100; r0.01; Print" t1 12 A 1rt ", t1 12 A 1rt t1 12 A 1rt 1125.51 ¤ Mathematica
•Print[ expression you want to show]
For example command if you input Print[ variable name ] then you will see value of variable name. •Print[ "expression as you see " , expression to be calculated]
You have comma here. You use comma to differentiate the end of one expression from the next one.
Chapter 6. How to Analyze Investment Projects
ü
6.3 The Net Present Value Investment Ruleü
6.5 Cost of CapitalCost of Capital is the risk-adjusted discount rate to use in computing a project's NPV. It depends on riskiness of the project. It can vary from project to project.
ü 6.8 Projects with Different Lives
When the lives of projects are different, we can compare their annualized capital costs.
Annualized capital cost: an annual cash payment that has a present value equal to the initial outlay. In other words, annual-ized capital cost is a constant payment per year over the investment period which is calculated so that total of their present values is equal to the cost.
Decision rule based on annualized capital cost: We choose the project with smaller annualized capital cost. ü Example: Two types of machines with different lives
Suppose that we are considering to install a machine to filter water from the hot spring. Two types of machines are available. Which machine type to choose? Type A works 5 years. Type B works 10 years. Type B lasts longer and costs more. Suppose that interest rate to borrow is 10%.
machine type life year price yen Type A 5 years 2106 Type B 10 years 4106
Let ca be annual cash payment for type A machine. Then annualized capital cost is value of ca which solves the following equation.
t1
5 ca
1rt 210 6∫ (1)
Let cb be annualized capital cost of type B. It satisfies the following equation. t1
10 cb
1rt 410
6 ∫ (2)
ü Finding value of annualized capital cost of Type A
We are going to solve equation (1).
In[93]:= Clearca, r, ansa; r0.1; ansaNSolve t1
5 ca
1rt
2106, ca
Out[93]= ca527 595.
In[94]:= caca. ansa1; Print"ca ", ca
ca 527 595.
ü Finding value of annualized capital cost of Type B
Annualized cost of capital of type B is solution to equation (2).
In[95]:= Clearansb, cb; ansbNSolve
t1 10 cb 1rt 4106, cb; cbcb. ansb1; Print"cb ", cb cb 650 982.
Annualized cost of capital of type B is $650,982 .
In[97]:= Print"cacb ", cacb
cacb 123 387.
Conclusion: Type A has lower annualized capital cost. We choose type A.
Mathematically, finding value of annualized capital cost is the same the following: Interest rate is given. Value of annuaity is given. You find value of value of constant payment.
ü 6.9 Ranking Mutually Exclusive Projects
If projects under consideration require exclusive use of unique asset, we call them mutually exclusive projects. "Internal Rate of Return" may not be a good measure for ranking such mutually exclusive projects. You should use NPV method. ü Example: Plans with different scales for the same parcel of land
Suppose you have a peace of land. You have two choices. building office building and making parking lot. 1. office building: initial outlay $20×106. You can sell it for $24×106 in one year.
2. parking lot: initial outlay $10,000. You can expect $10,000 per year forever. We like to compare these two alternatives and choose by using IRR and NPV methods. ü Comparison by Using IRR
Net present value of the project of the office building is NPV 20 24
1x . Internal rate of return is an interest rate which make NPV equal to zero. We solve an equation; 20 24
1x 0.
In[98]:= office building Clearx, ansx, IRRx; ansxNSolve20
24 1x
, x;
IRRxx. ansx1; Print" IRRx", IRRx
IRRx0.2
IRR for the office building is 20% . NPV of the parking lot is equal to t1 10
1yt 10. We solve an equation; t1
10
In[101]:= parking lot Cleary, IRRy, ansy; ansyNSolve t1 10 1yt 100, y;
IRRyy. ansy1; Print"IRRy ", IRRy
IRRy 1.
IRR of Parking lot is 100%. So parking lot has higher IRR. ü Comparison by Using NPV
We want to choose the one which has higher NPV. Suppose that cost of capital is 15%.
In[104]:= office building ClearNPVx, r; r0.15; NPVx
24 1r
20 106;
Print"NPV of the office building is ", NPVx
NPV of the office building is 869 565.
In[107]:= Parking lot ClearNPVy NPVyN
t1
10 1rt
10 103;
Print" NPV of the parking lot is ", NPVy , ", NPVxNPVy ", NPVxNPVy
NPV of the parking lot is 56 666.7, NPVxNPVy 812 899.
As shown above, if cost of capital is 15%, then the office building has higher NPV. You should choose office building.
ü Result can be reversed
However, if cost of capital is higher than 20%, the result is reversed. The result depends on cost of capital. For example, suppose cost of capital is 21%.
ü Cost of Capital = 21%
In[110]:= Clearr, NPV1, NPV2; r0.21; NPV1
24 1r
20 106; Print"NPV of office building when r0.21 : NPV1", NPV1
NPV of office building when r0.21 : NPV1165 289.
In[112]:= NPV2 N
t1
10 1rt
10 103; Print"NPV of parking lot ", NPV2
NPV of parking lot 37 619.
Comparison: NPV1NPV2202 908.0
If the cost of capital is 21%, then the parking lot is the project to choose. ü Switch-over Point
When is the result reversed? Where is "switch-over point"? It is 19.7568% as shown below.
In[115]:= Clearr; NSolve 24 1r 20 106 t1 10 1rt 10 103 0, r Out[115]= r0.197568,r0.00253204
How can such a reversal happen? One of the reasons is different lives of the projects. Consider NPV as a function of cost of capital. It is denoted as r in our example. If cost of capital is below 0.2532%, then parking lot has higher NPV again. ü NPV as a function of cost of capital
Let's draw graphs and see how NPV's changeas cost of capital changes.
In[116]:= NPV of Office Building Clearf1, r, g1 f1r_:
24 1r
20 106;
In[118]:= g1 Plotf1r,r, 0, 0.22, ImageSize200, PlotLabel"NPV of office building"
Out[118]= 0.05 0.10 0.15 0.20 1μ106 2μ106 3μ106 4μ106 NPV of office building
In[119]:= NPV of Parking Lot Clearf2, g2 f2r_:
t1
10 1rt
In[121]:= g2Plotf2r,r, 0, 0.22, ImageSize200, PlotLabel"NPV of parking lot" Out[121]= 0.05 0.10 0.15 0.20 100 000 200 000 300 000 400 000 500 000 NPV of parking lot
In[122]:= Showg1, g2, PlotLabel"comparison of NPV's"
Out[122]= 0.05 0.10 0.15 0.20 1μ106 2μ106 3μ106 4μ106 comparison of NPV's
ü Homework No. 2, Due next class
Q1. p.146, Problem 36. Hint: interest rate per month = APR
12 . Consider initially there were 13 loans. Sammy paid back 12 of
them in a year.
Q2. Suppose cost of capital is 18%. At what scale would the NPV of the parking lot be equal to the office building? Hint: Quick Check 6-8.
Q3. p190, Problem 1 . Also calculate IRR.
Q4. p.193, Problem 18.
