L6. The time value of money Flashcards
1
Learning Outcomes
- interpret interest rates as required rates of return, discount rates or opportunity costs
- explain an interest as sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk
- calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding
- solve time value of money problems for different frequencies of compounding
- calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only) and a series of unequal cash flow
- demonstrate use of a time in modeling and solving time value of money problems
Time value of money
Deals with equivalence relationships between cash flows with different dates
Interest rate
EG. if pay $9500 today, and receive $10,000 next year; interest rate or the required compensation stated as the rate of return is $500/$9500 = 0.0526 or 5.26%
Interest rate can be thought in 3 days
- Required rate of return; the minimum rate of return an investor must receive in order to accept the investment
- Discount rates; eg 5.26% is the rate at which we discounted that $10,000 future amount to find its value today
- Opportunity costs; value that investors forgo by choosing a particular course of action. Eg if decided to spend $9,500 today, he would have forgone earning 5.26% on the money.
To determine r, interest rate composed of a real risk-free interest rate plus a set of 4 premiums that are required returns or compensation for bearing distinct types of risks
r = Real risk-free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium
(Market determined interest rate = real risk free rate + inflation premium + risk premiums for default risk, liquidity and maturity)
- Real risk free interest rate; single period interest rate for a completely risk-free security if no inflation were expected. Reflects time preferences of individuals for current vs future real consumption
- Inflation premium; compensates investors for expected inflation and reflects average inflation rate expected over maturity of debt. Inflation reduces purchasing power of a unit of currency. Sum of real risk free interest rate and inflation premium = nominal risk free interest rate
- Default risk premium; compensates investors for the possibility that borrower will fail to make a promised payment at contracted time and amount
- Liquidity premium; compensates investors for risk of loss relative to an investment’s fair value if investment needs to be converted to cash quickly (eg Tbills do not have liquidity premium because large amount can be bought and sold without affecting market price but small issuers’ bonds trade infrequently after they are issued = interest rate would include liquidity premium)
- Maturity premium; compensates investors for increased sensitivity of market value of debt to a change in market interest rates as maturity is extended
Relationship between initial investment, which earns rate of return and its future value
Initial investment (PV), earns rate of return (r) and its future value (FV) which will be received in N years or periods from today
EG. invest $100 (PV = $100), interest bearing bank account paying 5% annually (r=5%). At end of first year, you will have $105 (5% of 100 plus PV)
For N = 1, formula as below
FV = PV(1+r)
in eg. FV = 100(1+0.05)
FV = 100(1.05)
FV = $105
If N=2, FV = PV(1+r)
FV = 105(1+0.05)
FV = 110.25
In other words, the simple interest is $5 earned year on year. In 2 years, earn $10 simple interest from principal (100). The extra 0.25 is earned from the interest of 5% (0.05*5)
Compounding interest
Is the interest earned on interest. Importance of compounding interest increases with the magnitude of the interest rate
Eg. 100 invested today would worth about 13,150 after 100 years compounded annually at 5% but worth more than 20 million if compounded annually over the same period at 13%
FV = PV(1+r)^n
*if N is in months, rate need to be monthly interest rate
Use of BA calculator to find answer (FV of single cash flow)
EG.
PV = 5 million
R = 7% annually
N = 5 years
What is FV?
Always 2nd FV to clear previous records, check RCL to see if all functions cleared
Key 5,000,000 and press PV
Key 7 and press I/Y
Key 5 and press N
Key CPT FV to find FV
Ans should be 7,012,758.65.
Examining payment interest more than a year
Many banks offer monthly interest rate that compounds 12 times a year IE monthly referred to stated annual interest rate or quoted interest rate
Eg. bank pays CD 8% compounded monthly; stated annual interest rate equals the monthly interest rate multiplied by 12. Monthly interest rate = 0.08/12 = 0.67% therefore it means (1+0.0067)^12 = 1.083
FV(n) = PV (1 + rs/m)^mN
where rs = stated annual interest rate
m = compounding periods per year
n = number of years
Use of BA calculator to find answer (Non annual Compounding)
EG.
CD offers 2 year maturity, stated annual interest rate of 8 percent compounded quarterly
PV = 10,000
What will the CD be worth at maturity?
Key 10,000 and press PV
Key 8/4 and press I/Y (8% divide by 4 periods; quarterly)
Key 4*2 and press N (4 periods x 2 years)
Key CPT FV to find FV
Ans should be 11,716.59
Use of BA calculator to find answer (Non annual Compounding)
EG
Australian bank offers to pay 6% compounded monthly
PV = 1,000,000
What is the FV?
Key 1,000,000 and press PV
Key 6/12 and press I/Y (6% divided by 12 periods; monthly)
Key 12*1 and press N (12 periods x 1 year)
Key CPT FV to find FV
Ans should be 1,061,677.81
Continuous compounding
Number of compounding periods per year becomes infinite, then interest is said to compound continuously
FV(n) = PVe^rsN
where e^rs^n is the transcendental number e=2.7182818 raised to the power rsN. On calculator it is e^x
Use of BA calculator to find answer (Continuous compounding)
PV = 10,000
8% compounded continuously for 2 years
FV(n) = PVe^rsN
= 10,000e^(0.08x2)
Type 0.08 x 2 = 0.16
Type 2nd LN 0.16 = 1.17
Type 10,000 x 1.17
Ans should be 11,735.11
Effective annual rate (EAR)
The amount of which a unit of currency will grow in a year with interest on interest included
EAR = (1 + periodic interest rate) ^m - 1
The periodic interest rate is the stated annual interest rate divided by m
or
EAR = e^rs - 1
Practice qns for EAR
1) Stated annual interest rate is 12.75%. If frequency of compounding is monthly, the EAR is?
EAR = (1 + periodic interest rate) ^m - 1
EAR = ( 1 + 0.1275/12)^12 - 1
= 13.52%
2) If the stated annual interest rate is 9% and the frequency is compounding daily, the EAR is?
EAR = (1 + periodic interest rate) ^m - 1
EAR = (1+ 0.09/365)^ 365 - 1
= 9.42%
Practice Qns for continuous compounding
Given a 1,000,000 investment for 4 years with a stated annual rate of 3% compounded continuously, the difference in its interest earnings compared with the same investment compounded daily is?
When compounded continuously; FV(n) = PVe^rsN PV = 1,000,000 rsN = 0.03 x 4 = 0.12 Type 0.12 2nd LN Type x 1,000,000 = 1,127,496.85 Interest = 1,127,496.85 - 1,000,000 = 127,496
When compounded daily; FV(n) = PV (1 + rs/m)^mN Type 1,000,000 PV Type 3/365 I/Y Type 365*4 N FV = 1,127,491.29 Interest = 1,127,491.29 - 1,000,000 = 127,491
Difference in interest = 127,496 - 127,491 = 5.56
Series of cash flow
- Annuity; finite set of level sequential cash flows
- Ordinary annuity; first cash flow that occurs one period from now (indexed at t = 1)
- Annuity due; first cash flow that occurs immediately (indexed at t = 0)
- Perpetuity; set of level near-ending sequential cash flows, with the first cash flow occurring one period from now
Equal cash flow - General future value annuity
FV(n) = A [ {(1+r)^N - 1} / r]
- Eg if 5% ordinary annuity annually, deposits of 1,000 for 5 years
(1+r)^N - 1 / r
(1 + 0.05)^5 - 1 / 0.05 = 5.525641
A x 5.525641 = 1000 x 5.525641 = 5,5525.63
- Eg. if 9% ordinary annuity annually, deposits 20,000 for 30 years
(1+r)^N - 1 / r
(1+0.09)^30 - 1/0.09 = 136.3075385
A x 136.3075385 = 20000 x 136.3075385 = 2,726,150.771
Unequal cash flow
Eg. Interest rate = 5%
Year 1 - 1,000 Year 2 - 2,000 Year 3 - 4,000 Year 4 - 5,000 Year 5 - 6,000
FV using calculator Type CF (cash flow) CF0 = 0, Enter CF1 = 1,000, F01 = 1, Enter CF2 = 2,000 and so on..
Once done, double check entries then press NPV where I = 5% press down and CPT = 15,036.45968
Press CE/C ONCE, press PV, press 5 = I/Y and press 5 = N then CPT FV = 19,190.76
FV with working Y1 FV = 1,000(1+0.5)^4 = 1,215.51 Y2 FV = 2,000(1+0.5)^3 = 2,315.25 Y3 FV = 4,000(1+0.5)^2 = 4,410.00 Y4 FV = 5,000(1+0.5)^1 = 5,250.00 Y5 FV = 6,000(1+0.5)^0 = 6,000
Sum = 19,190.76
Practice qns for FV of delayed annuity
Eg. 2 years from now, client will receive 3 annual payments of 20,000 from a small project. Earn 9% annually and plans to retire in 6 years, how much will the 3 project payments be worth at time of retirement?
Timeline 0 (0) 1 (0) 2 (20,000) 3 (20,000) 4 (20,000) 5 (0) 6 (0)
FV(n) = A [ {(1+r)^N - 1} / r]
[ {(1+r)^N - 1} / r]
(1+0.09)^3 - 1 / 0.09 = 3.27810
A x3.27810 = 20,000 x 3.27810 = 65,562
Next you find the FV of 65,562
FV = PV(1+r)^n
= 65,562 (1+0.09)^2
= 77,894.21
Practice qns for FV of uneven cash flow
Saver deposits following amounts in account paying stated annual rate of 4%, compounded semiannually
Year 1 - 4,000
Year 2 - 8,000
Year 3 - 7,000
Year 4 - 10,000
At the end of year 4, value of account?
Note: stated annual rate at 4%, therefore compound semiannually = 2%
First year n = 6 because 3 years x 2 semi annual = 6 periods (last year does not earn interest payment)
Year 1 - 4000 x (1.02)^6 = 4504.65
Year 2 - 8000 x (1.02)^4 = 8659.46
Year 3 - 7000 x (1.02)^2 = 7282.80
Year 4 - 10,000 x (1.02)^0 = 10,000
Total = 30,446.91
Practice qns for FV delayed annuity
Consultant starts project today that last for 3 years. Compensation package includes
Year 1 - 100,000
Year 2 - 150,000
Year 3 - 200,000
Expects to invest these amounts with annual interest rate of 3%, compounded annually until her retirement 10 years from now, the values at end of 10 years is?
Find PV of 3 cashflow first PV = FV(n) / (1+r)^n PV = 100,000 / (1+0.03)^1 = 97,087 PV = 150,000 / (1+0.03)^2 = 141,389 PV = 200,000 / (1+0.03)^3 = 183,028
Total = 421,504
FV = PV(1+r)^n
= 421,504 (1+0.03)^10
= 566,468
Finding PV of Single Cash flow
PV = FV(n) (1+r)^-N
- For a given discount rate, the farther in the future the amount to be received, the smaller the amount’s present value
- Holding time constant, the larger the discount rate, the smaller the present value of a future amount
Finding PV for non-annual compounding; increased frequency of compounding
PV = FV(n) (1+ (rs/m)) ^ -mN
Finding PV for series of equal cash flow; an annuity due as PV of an immediate cash flow plus an ordinary annuity
PV = A [1-1/(1+r)^N / r]
Eg. Given 2 options, 2million lump sum given today, or 200,000 annuity 20 payments starting today. Interest rate = 7% compounded annually. Which option better?
Option 1; Annuity
A = 200,000
r = 0.07
N = 19 (instead of 20 because payment start today, so 1 period of 200,000 is already in PV)
Sub into equation, PV = 2,067,119.05 + initial payment of 200,000 = 2,267,119.05 vs 2 million, PV of annuity greater than lump sum.
Finding PV for series of equal; projected present value of an ordinary annuity
Eg. German pension fund manager anticipates 1million per year must be paid to retirees. Retirements will not occur until 10 years from now at time t = 10. Once benefits given, will extent till t = 39 for total of 30 payments. What is PV of pension liability if annual discount rate = 5% compounded annually
Solution 1: Using PV = A [1-1/(1+r)^N / r] where A = 1,000,000 N = 30 R = 5% PV = 15,372,451.03
Next need to find the PV today since 15,372,451.03 = t9 Using PV = FV(n) (1+r)^-N where FV = 15,372,451.03 N = 9 R = 5% PV = 9,909,219.00
Solution 2:
Using calculator
PMT = 1,000,000
N = 30
I/Y = 5
CPT PV = 15,372,451.03
Next FV = 15,372,451.03 N = 9 I/Y = 5 CPT PV = 9,909,218.996
Finding PV for series of unequal cash flow
When having unequal cash flows, need to find PV of each individual cash flow and sum of the respective PVs
Eg. Period 1 = $1,000 Period 2 = $2,000 Period 3 = $4,000 Period 4 = $5,000 Period 5 = $6,000
PV = FV(n) (1+r)^-N
PV = 1000(1+0.5)^-1 = 952.38 PV = 2000(1+0.5)^-2 = 1,814.06 PV = 4000(1+0.5)^-3 = 3,455.35 PV = 5000(1+0.5)^-4 = 4,113.51 PV = 6000(1+0.5)^-5 = 4,701.16 Total = 15,036.46
If PV = 15,036.46
FV = PV(1+r)^n
= 19,190.76
Practice qns for PV of annuity
Eg. Client have to choose between 10 annual 100,000 retirement payments, starting 1 year from today or receive a lump sum today. R = 5% annually, he has decided to take the lump sum. What lump sum = to future annual payment?
N = 10
PMT = 100,000
I/Y = 5
CPT PV = 772,173.49
Practice qns for PV of delayed annuity
Eg. 2 instruments to choose; 1st pays nothing for 3 years, but will pay 20,000 per year for 4 years. 2nd will pay 20,000 for three years and 30,000 in 4th year. R = 8%, what should you be willing to pay for (1) 1st instrument, (2) 2nd instrument?
1st instrument Solution PMT = 20,000 N = 4 R = 8 CPT PV = 66,242.54
FV = 66,242.54
N = 3
R = 8
CPT PV = 52,585.46
2nd instrument PMT = 20,000 N = 4 R = 8 PV = 66,242.54 However, additional 10,000 in year 4
So using PV = FV(1+r)^-n or FV = 10,000 N = 4 R = 8 PV = 7350.30
Add with 66,242.54 = 73,592.84
Practice qns for PV of ordinary annuity
Eg. Send daughter to college 3 years later for 10,000/ year for 4 years. R = 8%, how much should you set aside now to cover these payments?
PMT = 10,000
N = 4
R = 8
CPT PV = 33,121.27
FV = 33,121.27
N = 2 ( because send to college 3 years later and first annuity payment is one period away, therefore n=2)
R = 8
CPT PV = 28,396.15
Practice qns for PV of ordinary annuity
R = 5 %
PV of 10 year annuity with annual payment 2,000 is 15443.47 The PV of a 10 year annuity due with same rate and payment is?
Deriving 15443.47 as below PMT = 2,000 N = 10 R = 5 CPT PV = 15443.47
However for annuity due, T0 = 2,000 = PV already So, PMT = 2,000 N = 9 R = 5 CPT PV = 14,215.64 Plus initial 2,000 = 16,215.64
Practice qns for PV of annuity due
Eg.
Investment pays 300 annually for 5 years, first payment occurs today. what is PV when r = 4%?
PMT = 300
N = 4 (4 because 1 period is immediate so 300 is already PV)
R = 4%
CPT PV = 1088.97
Need to add back T0 = 300
Therefore, 1088.97+ 300 = 1388.97
Practice qns for PV of ordinary annuity
Grandparents funding newborn’s future tuition cost estimated at 50,000/year for 4 years, with first payment due as lump sum in 18 years. If R = 6%, deposit required today is?
PMT = 50,000
N = 4
I/Y = 6
CPT PV = 173,255.28
FV = 173,255.28
N = 17
I/Y = 6
CPT PV = 64,340.85
Practice qns for PV of uneven cash flow
Year 1 - 100,000
Year 2 - 150,000
Year 5 - 10,000
R = 12%
What is PV?
Y1 = 100,000 (1.12)^-1 = 89,285.71 Y2 = 150,000 (1.12)^-2 = 119,579.08 Y5 = 10,000 (1.12)^-5 = -5,674.27 Total = 203,190.52
Finding PV of a perpetuity and indexed at times other than T = 0
PV = A/r
Only valid for perpetuity with level payments
Eg. consol bond paid 100 per year in perpetuity, what is the worth today if required rate were 5%?
100/0.05 = 2,000
However, if 100 payment start in Y2, we find that PV1 = 100/0.5 = 2,000. Bring forward to T0
FV = 2000
N = 1
I/Y = 5
CPT PV = 1,904.76
Finding PV of a projected perpetuity
Eg. Level perpetuity of 100 per year with first payment starting at T=5, what is PV when 5% discount rate?
PV = A/r PV = 100/0.05 = 2,000
Find PV at T0
FV = 2000
N = 4 (rmb that for perpetuity and ordinary annuity has first payment one period away thats why 4 instead of 5)
PV = 1,645.40
Finding PV of ordinary annuity as the PV of a current minus projected perpetuity
Eg. Given 5% discount rate, find PV of 4 year ordinary annuity of 100 per year starting in Y1 as the difference between the following 2 level perpetuities
Perpetuity 1 - 100 per year starting in Y1 (first payment at t1)
Perpetuity 2 - 100 per year starting in Y5 (first payment at t5)
Solution For Perpetuity 1, PV0 = 100/0.05 = 2,000 For Perpetuity 2, PV4 = 100/0.05 = 2,000 For Perpetuity 2, PV0 = FV = 2,000 N = 4 I/Y = 5 CPT PV = 1,645.40
Difference = 2,000 - 1,645.40 = 354.60
Practice qns for PV of projected perpetuity
Eg. perpetual preferred stock makes first quarterly dividend payment of 2 in 5 quarters. if annual rate of return is 6% compounded quarterly, PV is?
PV = A/r 2/(0.06/4) = 133.33
PV = FV (1+r)^-n PV = 133.33 (1+0.015)^-4 PV = 125.62 or 126
Solving for interest rates, growth rates and number of periods
Eg. 100 bank deposit known to generate 111 in 1 year. Sub FV = PV (1+r)^n FV = 111 PV = 100 N = 1 What is R?
R = 11%
Means that interest rate that equates 100 at t = 0 to 111 at t = 1 is 11 percent. Which also refers to growth rate
g = (FV/PV)^1/n - 1
Calculating growth rate
Eg. Sales increased from 14,146.4 billion in 2012 to 19,166 billion in 2017. But net profit decreased from 796.4 billion in 2012, 727.5 billion in 2017. What is the sales growth rate and profit growth rate?
Solution Sales growth rate In calculator Step 1: press 2nd P/Y, 1 Enter) Step 2: Then press 2nd Quit Step 3: -14,146.4 PV Step 4: 19,166 FV Step 5: 5 N Step 6: CPT I/Y
I/Y = 6.3%
Net profit growth rate Step 1: press 2nd P/Y, 1 Enter Step 2: Then press 2nd Quit Step 3: -796.4 PV Step 4: 727.5 FV Step 5: 5 N Step 6: CPT I/Y
I/Y = -1.8%
Eg 2
Sales of 8.96 million in 2018 and 7.35 in 2012. What is the growth rate?
Step 1: press 2nd P/Y, 1 Enter Step 2: Then press 2nd Quit Step 3: -7.35 PV Step 4: 8.96 FV Step 5: 6 N Step 6: CPT I/Y
I/Y = 3.35%
Solving number of periods
Eg. How long will it take to double 10,000,000 in value with current interest rate 7% compounded annually?
Solution Using Calculator 2nd 5 = growth function Old = 10,000,000 Enter down New = 20,000,000 Enter down % Ch = 7 Enter down #PD = CPT = 10.24 years
Practice qns for solving number of periods
Lump sum investment of 250,000 invested at annual rate of 3% compounded daily, number of months needed to grow to 1,000,000 is?
First compute the EAR
EAR = (1+period interest rate)^m - 1
EAR = (1 + 0.03/365)^365 - 1
EAR = 0.0304533 or 3.0453
Then use calculator and type 2nd 5 = growth function Old = 250,000 Enter down New = 1,000,000 Enter down % Ch = 3.0453 Enter down #PD = CPT = 46.212 but x 12 months cause monthly = 554.5 months
Practice qns for solving growing rate
Investment 500,000 today that grows to 800,000 after 6 years has a stated annual interest rate closest to?
Step 1: press 2nd P/Y, 1 Enter Step 2: Then press 2nd Quit Step 3: -500,000 PV Step 4: 800,000 FV Step 5: 6 N Step 6: CPT I/Y = 8.14
Solving for size of annuity payments on fixed rate mortgage
Eg. Planning to purchase 120,000 house by making a down payment of 20,000 and borrowing the remainder with 30 year fixed rate mortgage with monthly payments. First payment due at t = 1, current mortgage interest rates quoted at 8% with monthly compounding. What will be monthly mortgage payment?
In calculator, solve for PMT PV = 100,000 (120,000-20,000) I/Y = 8/12 N = 360 (30*12) CPT PMT = 733.76
Solving for projected annuity amount needed to find a future annuity inflow
Eg.
Jill is 22 years old (t=0) planning retirement at 63 (t=41). Plans to save 2000 per year for next 15 years (t1 to t15). Wants to have retirement income of 100,000 per year for 20 years with first payment starting at t=41. How much to save each year from t=16 to t=40. 8% return
Solution
1. Calculate FV of 2000CF at T=15
PMT 2000, I/Y 8, N 15, CPT FV = 54,304.23
- Find FV of FV=54,304.23 to t=40
PV = 54,304.23, I/Y 8, N 25 (40-15) CPT FV = 371,901.16 - Calculate PV of 100,000 at t=40
PMT 100,000, I/Y 8, N 20 CPT PV = 981,814,47 - Difference in CFs
981,814,47 - 371,901.16 = 609,913.31 - We have now the FV of the unknown CFS and can find annuity amount
FV 609,913.31, I/Y 8, N 25, PMT = 8,342.87
Cash flow additivity principle
Amounts of money indexed at the same point in time are additive ie. 2 sets of cash flows denoted by series A and B. Interest earn in series A is then reinvested into series B, adding to the combined asset
Finding value of combined assets (Additivity principle)
Client invests 20,000 CD that pays annual interest of 3.5%. Annual CD interest payments automatically invested in separate savings account at stated annual interest rate of 2% compounded monthly. At maturity, what is the combined asset closest to?
Solution
Interest payment of CD = 20,000 * 3.5% = 700 (which is reinvested)
EAR of reinvestments = (1+period IR)^m - 1
EAR = (1+0.02/12)^12 -1 = 2.0184356%
1st payment PV = 700 I/Y = 2.0184356% N = 3 CPT FV = 743.25
2nd payment PV = 700 I/Y = 2.0184356% N = 2 CPT FV = 728.54
3rd Payment PV = 700 I/Y = 2.0184356% N = 1 CPT FV = 714.13
4th Payment
Paid at end of Y4 = 700
Total = 700 + 714.13 + 728.54 + 743.25 + 20,000 = 22,885.82
