The Time Value of Money
Suppose a $10,000 investment will earn 8 percent compounded continuously for two years.
PV=$10,000 r s =8%=0.08 N=2 FV N =PVe r s N =$10,000e 0.08(2) =$10,000(1.173511) =$11,735.11
Future Value of a Lump Sum with Quarterly Compounding Suppose your bank offers you a CD with a two-year maturity, a stated annual interest rate of 8 percent compounded quarterly, and a feature allowing reinvestment of the interest at the same interest rate. You decide to invest $10,000. What will the CD be worth at maturity?
PV=$10,000 r s =8%=0.08 m=4 r s /m=0.08/4=0.02 N=2mN=4(2)=8 interest periods FV N =PV(1+r s m ) mN =$10,000(1.02) 8 =$10,000(1.171659) =$11,716.59
Future Value of a Lump Sum with Monthly Compounding An Australian bank offers to pay you 6 percent compounded monthly. You decide to invest $1 million for one year. What is the future value of your investment if interest payments are reinvested at 6 percent?
PV=A$1,000,000 r s =6%=0.06 m=12 r s /m=0.06/12=0.0050 N=1 mN=12(1)=12 interest periods FV N =PV(1+r s m ) mN =A$1,000,000(1.005) 12 =A$1,000,000(1.061678) =A$1,061,677.81
effective annual rate
The periodic interest rate is the stated annual interest rate divided by m, where m is the number of compounding periods in one year. EAR = (1 + Periodic interest rate)to the m - 1
The Present Value of a Series of Unequal Cash Flows
We could calculate the future value of these cash flows by computing them one at a time using the single-payment future value formula. FV N =PV(1+r) N
Stated versus Effective Annual Rates A stated annual rate (rs), which is often referred to as the:
annual percentage rate (APR)
real risk-free rate
delaying consumption
Compounding More Frequently than Annually
future value increased when we compounded more frequently than once per year BUT.. the more frequently we compound, the lower the present value
How many years will it take for $197,000 to grow to be $554,000 if it is invested in an account with a quoted annual interest rate of 8% with monthly compounding of interest?
i = 0.666667 (8% annually divided by 12 comp. periods per year) PV = -197000 FV = 554000 solve for n (answer on calculator = 155.61) Since the interest rate was entered as a monthly rate, the answer for n is in months. The number of years equals the number of months divided by twelve. Number of years = (155.61)/12 = 12.97 years
How many years will it take for $136,000 to grow to be $468,000 if it is invested in an account with an annual interest rate of 8%?
i = 8 PV = -136000 FV = 468000 solve for n (answer = 16.06 years)
The Frequency of Compounding To handle interest payments made more than once a year, we can modify the present value formula (Equation 8) as follows. Recall that rs is the quoted interest rate and equals the periodic interest rate multiplied by the number of compounding periods in each year. In general, with more than one compounding period in a year, we can express the formula for present value as....
m = number of compounding periods per year rs = quoted annual interest rate N = number of years present value and future value factors are reciprocals. Changing the frequency of compounding does not alter this result. The only difference is the use of the periodic interest rate and the corresponding number of compounding periods.
If you wish to accumulate $197,000 in 5 years, how much must you deposit today in an account that pays a quoted annual interest rate of 13% with semi-annual compounding of interest?
n = 10 (5 years times 2 comp. periods per year) i = 6.5 (13% annually divided by 2 comp. period per year) FV = 197000 solve for PV (answer = $104,947.03)
You are offered an annuity that will pay $24,000 per year for 11 years (the first payment will occur one year from today). If you feel that the appropriate discount rate is 13%, what is the annuity worth to you today?
n = 11 i = 13 PMT = -24000 Make sure you are in end mode. solve for PV (answer = $136,486.59)
If you deposit $16,000 per year for 12 years (each deposit is made at the end of each year) in an account that pays an annual interest rate of 14%, what will your account be worth at the end of 12 years?
n = 12 i = 14 PMT = 16000 Make sure you are in end mode. solve for FV (answer = $436,331.98)
You plan to accumulate $450,000 over a period of 12 years by making equal annual deposits in an account that pays an annual interest rate of 9% (assume all payments will occur at the beginning of each year). What amount must you deposit each year to reach your goal?
n = 12 i = 9 FV = 450000 Make sure you are in begin mode. solve for PMT (answer = $20,497.98)
What will $153,000 grow to be in 13 years if it is invested today in an account with a quoted annual interest rate of 10% with monthly compounding of interest?
n = 156 (13 years times 12 comp. periods per year) i = 0.833333 (10% annually divided by 12 comp. periods per year) PV = -153,000 solve for FV (answer = $558,386.38)
You are told that if you invest $11,000 per year for 23 years (all payments made at the end of each year) you will have accumulated $366,000 at the end of the period. What annual rate of return is the investment offering?
n = 23 FV = 366000 PMT = -11000 Make sure you are in end mode. solve for i (answer = 3.21%)
You plan to borrow $389,000 now and repay it in 25 equal annual installments (payments will be made at the end of each year). If the annual interest rate is 14%, how much will your annual payments be?
n = 25 i = 14 PV = -389000 Make sure you are in end mode. solve for PMT (answer = $56,598.88)
You are offered an investment with a quoted annual interest rate of 13% with quarterly compounding of interest. What is your effective annual interest rate?
n = 4 (number of comp. periods in one year) i = 3.25 (13% annually divided by 4 comp. periods in one year) PV = -100 solve for FV (answer = 113.65) Subtract the 100 (percent) you initial had to get the EAR. EAR = 113.65 - 100 = 13.65%
You are offered an annuity that will pay $17,000 per year for 7 years (the first payment will be made today). If you feel that the appropriate discount rate is 11%, what is the annuity worth to you today?
n = 7 i = 11 PMT = 17000 Make sure you are in begin mode. solve for PV (answer = $88,919.14)
At what quoted annual interest rate must $134,000 be invested so that it will grow to be $459,000 in 15 years if interest is compounded weekly?
n = 780 (15 years times 52 comp. periods per year) PV = -134,000 FV = 459,000 solve for i (answer on calculator = 0.157972) Since the number of periods was entered as weeks, the answer for i is the weekly rate. The annual rate equals the weekly rate times 52. Annual rate = (0.157972%)(52) = 8.21%
What will $247,000 grow to be in 9 years if it is invested today in an account with an annual interest rate of 11%?
n = 9 i = 11 PV = -247000 solve for FV (answer = $631,835.12)
If you deposit $15,000 per year for 9 years (each deposit is made at the beginning of each year) in an account that pays an annual interest rate of 8%, what will your account be worth at the end of 9 years?
n = 9 i = 8 PMT = 15000 Make sure you are in begin mode. solve for FV (answer = $202,298.44)
At what annual interest rate must $137,000 be invested so that it will grow to be $475,000 in 14 years?
n=14 PV = -137000 FV = 475000 solve for i (answer = 9.29%)
A bank is considering making a loan to a local business and uses the following data to compute the appropriate interest rate: Real risk-free rate = 3.0% Expected inflation = 2.5% Default premium = 1.5% Liquidity premium = 0.5% Maturity premium = 0.1% Using the simple build-up approach, the nominal interest rate (r) is closest to: r = 3.0 + 2.5 + 1.5 + 0.5 + 0.1 = 7.6% Compute the same rate on a compounded basis:
r = (1.03)(1.025)(1.015)(1.005)(1.001) − 1 = 0.078 = 7.8% Note that the compounded rate is slightly higher than the simple addition method. Compounded interest rates will always be higher than simple interest and the difference gets larger as the time period gets longer.
liquidity and maturity risk
risk of tying up one's money
APR is calculated by:
simply multiplying the periodic rate by the number of periods in a year.
inflation premium
the loss of purchasing power
default risk
the risk of getting stiffed
For a given discount rate, the farther in the future the amount to be received...
the smaller that amount's present value.
Holding time constant, the larger the discount rate....
the smaller the present value of a future amount.
FVn = PV(1 + r)n As the number of compounding periods increases...
...the future value increases
FVn = PV(1+r)n As the interest rate rises...
...the future value increases
FVn = PV(1+r)n As the present value increases...
...the future value increases.
To calculate the continuously compounded rate (the stated annual interest rate with continuous compounding) corresponding to an effective annual rate of 8.33 percent, we find the interest rate that satisfies Equation 6:
0.0833=e r s −1 1.0833=e to the r s To solve this equation, we take the natural logarithm of both sides. (Recall that the natural log of e to the r s is ln e to the r s =r s .) Therefore, ln 1.0833 = rs, resulting in rs = 8 percent. We see that a stated annual rate of 8 percent with continuous compounding is equivalent to an EAR of 8.33 percent.
What is an Interest Rate?
1) The price of money or required rate of return 2) An opportunity cost 3) A discount rate
PV of a Lump Sum with Discrete Compounding PV = FVn/(1+r)n As the discount rate (r) gets larger,
1) the denominator gets larger 2) the present value (PV) gets smaller
PV of a Lump Sum with Discrete Compounding PV = FVn/(1+r)n As the number of periods (N) gets larger,
1) the denominator gets larger 2) the present value gets smaller.
If you wish to accumulate $140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%?
1. n = 13 i = 14 FV = 140000 solve for PV (answer = $25,489.71)
An investor is offered a two-year lump sum investment of $500 with annual compounding at four percent. The future value of this investment is closest to: A.$460 B.$525 C.$541
500 = PV; 4=i; 2=n FV = 540.80
Future Value Time Line Using an illustration, an investor buys a two-year, annual-pay bond for $1,000 today with an interest rate of six percent. After two years, he adds another $1,000 to the proceeds of the bond and earns eight percent on the balance for two more years. Assume all cash flows are reinvested at the corresponding bond's interest rate. What is the future value of the entire four-year investment?
At time zero (today) the investor spends $1,000 to buy the six percent bond, which is worth $1,123.60 two years later [$1,000(1.06)2]. At time two, which is time zero for the second bond, he adds another $1,000 for a total investment of $2,123.60 and ends up with $2,476.97 at the end of four years [$2,123.60(1.08)2].
As the number of discrete compounding periods increases (from annual to semiannual to quarterly to monthly to daily), the future value increases. If we take this progression to the infinite extreme, we get...
Continuous Compounding
to find a periodic rate that corresponds to a particular effective annual rate, We can reverse the formulas for EAR with discrete and continuous compounding to find a periodic rate
EAR=e to the r s −1
PV of a Lump Sum with Discrete Compounding An investor has estimated the asset base required to meet his retirement goals to be $1.75 million. He plans to stop working ten years from today and currently has $800,000 in various retirement accounts. If he expects an average return of seven percent per year, his current retirement asset value shortfall is
FV = 1,750,000 i = 7.0% n = 10 PV = 889.61126
A manager needs to make a $50,000 lump sum payment two years from today. She wants to fund the liability today and has two investment vehicles to choose from. The first pays five percent compounded semiannually and the second pays five percent compounded monthly. Both vehicles mature in two years. Which investment should she choose?
FV = 50,000 i = 5/2, =2.5 n = 2x2 =4 PV = 45,297.53224 or FV = 50,000 i = 5/12, = 0.41667 n = 2x12 = 24 PV = 45251.2712
The Projected Present Value of a More Distant Future Lump Sum You own a liquid financial asset that will pay you $100,000 in 10 years from today. Your daughter plans to attend college four years from today, and you want to know what the asset's present value will be at that time. Given an 8 percent discount rate, what will the asset be worth four years from today?
FV N =$100,000 r=8%=0.08 N=6 PV=FV N (1+r) −N =$100,0001/(1.08) to the 6 =$100,000(0.6301696) =$63,016.96
Finding the Present Value of a Single Cash Flow a Guaranteed Investment Contract (GIC) that promises to pay $100,000 in six years with an 8 percent return rate. What amount of money must the insurer invest today at 8 percent for six years to make the promised payment?
FV N =$100,000 r=8%=0.08 N=6 PV=FV N (1+r) −N =$100,000[1/(1.08) to the 6 ] =$100,000(0.6301696) =$63,016.96
The Present Value of a Lump Sum with Monthly Compounding The manager of a Canadian pension fund knows that the fund must make a lump-sum payment of C$5 million 10 years from now. She wants to invest an amount today in a GIC so that it will grow to the required amount. The current interest rate on GICs is 6 percent a year, compounded monthly. How much should she invest today in the GIC?
FV N =C$5,000,000 r s =6%=0.06m=12r s /m=0.06/12=0.005 N=10 mN=12(10)=120 PV=FV N (1+r s m ) −mN =C$5,000,000(1.005) −120 =C$5,000,000(0.549633) =C$2,748,163.67
Frequency of Compounding equation
FV N =PV(1+r s /m ) to the mN rs = stated intrest rate m=number of compounding periods N=number of years
Continuous compounding
FV N =PVe to the (r s N) e r s N = the transcendental number e ≈ 2.7182818 raised to the power rsN
For example, the future value of a $100 investment earning five percent over three years using continuous compounding would be:
FVn = PVe rn or 0.05 x 3 = 0.15 g ex = displays 1.16183 100 x = displays 116.18342
Future Value (FV) of Lump Sum with Discrete Compounding
If you invest a certain amount of money today, the present value (PV), over a specified period of time (N), using a specified interest rate (r), the amount you end up with is the future value.
After a car accident, Carl Jones won a After a car accident, Carl Jones won a structured settlement that pays him $1,500 every six months for five years. The first payment is due at the end of this year. He plans to retain all the proceeds in a savings account that will earn an average interest rate of four percent. The future value of this settlement is...
PMT = $1,500 i = 4.0/2 = 2.0 n = 5 × 2 = 10 FV = $16,424.58
Ordinary Annuity Consider an ordinary annuity paying 5 percent annually. Suppose we have five separate deposits of $1,000 occurring at equally spaced intervals of one year, with the first payment occurring at t = 1. Our goal is to find the future value of this ordinary annuity after the last deposit at t = 5.
PMT = $1000 i = 5% n = 5 yr Fv = 5525.6313
Klaus Deutsch intends to retire in five years. He has a tax-deferred retirement account with a $400,000 balance as of today. He plans to make contributions of $25,000 at the end of each year until he stops working. If his expected investment return is seven percent per year, his ending account balance will be...
PV = $400,000 PMT = $25,000 i = 7.0 n = 5 FV = $704,789.17
