Sukrit Mittal
Franklin Templeton Investments
A dollar today is worth more than a dollar tomorrow. Why?
How do we compare cash flows occurring at different points in time?
The answer: discount future cash flows to their present value, or compound present values to future equivalents. A typical example is forward contracts.
Consider two offers:
Which is better?
Answer: It depends on the interest rate.
The interest rate is the bridge between present and future values.
We can solve for the break-even interest rate where you'd be indifferent:
\[100(1 + r) = 110 \implies r = 0.10 = 10\%\]
At exactly 10%, both options are equivalent.
Under simple interest, the interest earned is proportional to:
\[A(t) = P(1 + rt)\]
where: \(A(t)\) = amount at time \(t\)
Simple interest ignores the fact that interest earned can itself earn interest.
Example: With $1,000 at 8% simple interest:
But what if you could reinvest the $80 earned in Year 1?
This leads us to compound interest.
"Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it."
Approximate time to double your money:
\[t \approx \frac{72}{r\%}\]
where \(r\%\) is the annual percentage rate.
Examples:
Verification at 8%: \((1.08)^9 = 1.999 \approx 2\) ✓
Solve \((1+r)^t = 2\) for \(t\):
\[t = \frac{\ln 2}{\ln(1+r)} \approx \frac{0.693}{r}\]
For \(r\) as a percentage: \(t \approx \frac{69.3}{r\%}\). The rule of 72 is easier to use and quite accurate for rates between 6-10%.
With compound interest, interest is added to the principal periodically, and subsequent interest is earned on the new total.
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
where:
Each period, the amount is multiplied by \(\left(1 + \frac{r}{n}\right)\). After \(nt\) periods:
\[P \rightarrow P\left(1 + \frac{r}{n}\right) \rightarrow P\left(1 + \frac{r}{n}\right)^2 \rightarrow \cdots \rightarrow P\left(1 + \frac{r}{n}\right)^{nt}\]
| Frequency | Value of \(n\) |
|---|---|
| Annual | \(n = 1\) |
| Semi-annual | \(n = 2\) |
| Quarterly | \(n = 4\) |
| Monthly | \(n = 12\) |
| Daily | \(n = 365\) |
Key insight: As \(n\) increases, the final amount \(A\) increases, but with diminishing returns.
Invest $1,000 at 8% annual interest for 3 years under different compounding frequencies.
Annual compounding (\(n=1\)):
\[A = 1{,}000\left(1 + 0.08\right)^{3} = 1{,}000(1.08)^3 = \$1{,}259.71\]
Quarterly compounding (\(n=4\)):
\[A = 1{,}000\left(1 + \frac{0.08}{4}\right)^{12} = 1{,}000(1.02)^{12} = \$1{,}268.24\]
Monthly compounding (\(n=12\)):
\[A = 1{,}000\left(1 + \frac{0.08}{12}\right)^{36} = \$1{,}270.24\]
Notice: The difference between monthly and quarterly is 2.00, much smaller than the 8.53 difference between quarterly and annual. This diminishing return pattern continues as \(n\) increases.
\[A = 1{,}000\left(1 + \frac{0.08}{12}\right)^{36} = \$1{,}270.24\]
The effective annual rate is the actual annual return accounting for compounding within the year.
\[\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1\]
Example: 8% APR compounded quarterly
\[\text{EAR} = \left(1 + \frac{0.08}{4}\right)^4 - 1 = (1.02)^4 - 1 = 0.0824 = 8.24\%\]
The EAR exceeds the stated rate (APR) whenever \(n > 1\).
Financial institutions often quote APR (nominal rate) but compound more frequently. The EAR reveals the true cost or true return.
Regulation: In many jurisdictions, lenders must disclose the EAR (also called APY - Annual Percentage Yield) to help consumers compare products.
A credit card charges 18% APR with daily compounding. What is the effective annual rate?
As compounding becomes more frequent, we approach continuous compounding.
Starting from periodic compounding:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
Take the limit as \(n \to \infty\):
\[A = P \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt}\]
Rewrite the exponent: let \(m = \frac{n}{r}\), so \(n = mr\) and as \(n \to \infty\), we have \(m \to \infty\):
\[A = P \lim_{m \to \infty} \left[\left(1 + \frac{1}{m}\right)^m\right]^{rt}\]
Using the fundamental limit \(\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m = e\):
\[A = P(e)^{rt} = Pe^{rt}\]
The number \(e\) naturally arises as the base for continuous growth processes.
\[A = Pe^{rt}\]
where:
Invest $1,000 at 8% continuously compounded for 3 years:
\[A = 1{,}000 \cdot e^{0.08 \times 3} = 1{,}000 \cdot e^{0.24} = \$1{,}271.25\]
The stated interest rate, not adjusted for inflation.
The interest rate adjusted for inflation—the actual increase in purchasing power.
\[(1 + r_{\text{nominal}}) = (1 + r_{\text{real}})(1 + r_{\text{inf}})\]
where \(r_{\text{inf}}\) is the inflation rate.
Approximation (for small rates):
\[r_{\text{real}} \approx r_{\text{nominal}} - r_{\text{inf}}\]
You earn 8% on an investment, but inflation is 3%.
Exact: \(r_{\text{real}} = \frac{1.08}{1.03} - 1 = 0.0485 = 4.85\%\)
Approximate: \(r_{\text{real}} \approx 8\% - 3\% = 5\%\)
When comparing investments across time or countries, real rates provide the true comparison of purchasing power gains.
If \(A = Pe^{rt}\), then solving for \(P\):
\[P = Ae^{-rt}\]
This is the present value of a future amount \(A\) under continuous compounding.
Example: What is the present value of $1,500 to be received in 5 years at 6% continuous rate?
\[P = 1{,}500 \cdot e^{-0.06 \times 5} = 1{,}500 \cdot e^{-0.3} = \$1{,}109.96\]
The discount factor \(e^{-rt}\) represents the "weight" we assign to future cash. As \(t\) or \(r\) increases, this weight decreases exponentially.
Invest $1,000 at 8% for 3 years:
| Method | Formula | Amount |
|---|---|---|
| Simple interest | \(P(1+rt)\) | $1,240.00 |
| Annual compounding | \(P(1+r)^t\) | $1,259.71 |
| Quarterly | \(P(1+r/4)^{4t}\) | $1,268.24 |
| Monthly | \(P(1+r/12)^{12t}\) | $1,270.24 |
| Daily | \(P(1+r/365)^{365t}\) | $1,271.22 |
| Continuous | \(Pe^{rt}\) | $1,271.25 |
Observation: More frequent compounding yields higher returns, but the gains diminish.
Given periodic rate \(r_n\) with frequency \(n\), the equivalent continuous rate \(r_c\) is:
\[r_c = n \ln\left(1 + \frac{r_n}{n}\right)\]
Given continuous rate \(r_c\), the equivalent periodic rate \(r_n\) is:
\[r_n = n\left(e^{r_c/n} - 1\right)\]
Given periodic rate \(r_n\) with frequency \(n\), the equivalent continuous rate \(r_c\) is:
\[r_c = n \ln\left(1 + \frac{r_n}{n}\right)\]
Derivation: Set the growth equal over one year:
\[\left(1 + \frac{r_n}{n}\right)^n = e^{r_c}\]
Taking natural log: \(n \ln\left(1 + \frac{r_n}{n}\right) = r_c\)
Given continuous rate \(r_c\), the equivalent periodic rate \(r_n\) is:
\[r_n = n\left(e^{r_c/n} - 1\right)\]
Derivation: From \(e^{r_c} = \left(1 + \frac{r_n}{n}\right)^n\), raise both sides to power \(1/n\):
\[e^{r_c/n} = 1 + \frac{r_n}{n} \implies r_n = n(e^{r_c/n} - 1)\]
So far: single payment at time 0.
In practice, many financial instruments involve multiple cash flows over time:
How do we value these?
Answer: Sum the present values of each individual payment.
The present value operator is linear:
\[PV(C_1 + C_2) = PV(C_1) + PV(C_2)\]
This allows us to value complex cash flow streams by breaking them into components.
Consider cash flows \(C_1, C_2, \ldots, C_n\) at times \(t_1, t_2, \ldots, t_n\).
\[PV = \sum_{k=1}^{n} C_k e^{-rt_k}\]
(using continuous compounding)
Or with periodic compounding at rate \(r\):
\[PV = \sum_{k=1}^{n} \frac{C_k}{(1+r)^{t_k}}\]
Cash flows: 100 at \(t=1\), 200 at \(t=2\), 150 at \(t=3\). Rate: 5% (annual compounding).
\[PV = \frac{100}{1.05} + \frac{200}{(1.05)^2} + \frac{150}{(1.05)^3}\]
\[= 95.24 + 181.41 + 129.58 = \$406.23\]
An annuity is a sequence of equal payments at regular intervals.
Two types:
Formula:
\[PV = C \cdot \frac{1 - (1+r)^{-n}}{r}\]
Derivation:
\[PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^n}\]
Factor out \(\frac{C}{1+r}\):
\[PV = \frac{C}{1+r}\left[1 + \frac{1}{1+r} + \frac{1}{(1+r)^2} + \cdots + \frac{1}{(1+r)^{n-1}}\right]\]
This is a geometric series with first term \(a = 1\), ratio \(q = \frac{1}{1+r}\), and \(n\) terms.
Geometric series sum: \(S_n = a \cdot \frac{1-q^n}{1-q} = \frac{1 - (1+r)^{-n}}{1 - \frac{1}{1+r}} = \frac{1-(1+r)^{-n}}{\frac{r}{1+r}}\)
Therefore:
\[PV = \frac{C}{1+r} \cdot \frac{1-(1+r)^{-n}}{\frac{r}{1+r}} = C \cdot \frac{1 - (1+r)^{-n}}{r}\]
You will receive $500 at the end of each year for 5 years. The interest rate is 6%.
What is the present value?
\[PV = 500 \cdot \frac{1 - (1.06)^{-5}}{0.06}\]
\[= 500 \cdot \frac{1 - 0.7473}{0.06} = 500 \cdot \frac{0.2527}{0.06}\]
\[= 500 \times 4.212 = \$2{,}106.00\]
Formula:
\[FV = C \cdot \frac{(1+r)^n - 1}{r}\]
Derivation: Relate to present value. If \(PV\) grows for \(n\) periods:
\[FV = PV \cdot (1+r)^n = C \cdot \frac{1 - (1+r)^{-n}}{r} \cdot (1+r)^n\]
\[= C \cdot \frac{(1+r)^n - 1}{r}\]
Example: You save $500 at the end of each year for 5 years at 6%.
\[FV = 500 \cdot \frac{(1.06)^5 - 1}{0.06} = 500 \cdot \frac{1.3382 - 1}{0.06}\]
\[= 500 \times 5.637 = \$2{,}818.50\]
Verification: This is \(2,106.00 \times (1.06)^5 = 2,818.50\) ✓
Payments occur at the beginning of each period.
Present Value of Annuity Due:
\[PV_{\text{due}} = PV_{\text{ordinary}} \times (1+r)\]
Future Value of Annuity Due:
\[FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r)\]
Intuition: Each payment earns interest for one additional period.
For an annuity due, the first payment is at \(t=0\), not \(t=1\):
\[PV_{\text{due}} = C + \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^{n-1}}\]
\[= (1+r)\left[\frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^n}\right] = (1+r) \cdot PV_{\text{ordinary}}\]
Using our earlier example ($500/year for 5 years at 6%):
A perpetuity is an annuity that continues forever.
As \(n \to \infty\) in the annuity formula:
\[PV = \lim_{n \to \infty} C \cdot \frac{1 - (1+r)^{-n}}{r}\]
Since \((1+r) > 1\), we have \((1+r)^{-n} \to 0\) as \(n \to \infty\):
\[PV = C \cdot \frac{1 - 0}{r} = \frac{C}{r}\]
Example: A bond pays $50 per year forever at 5% interest.
\[PV = \frac{50}{0.05} = \$1{,}000\]
At 5%, a $1,000 investment yields 50 annually forever—exactly matching the perpetuity's cash flows.
A growing perpetuity has payments that increase at rate \(g\) each period.
First payment: \(C\)
Second payment: \(C(1+g)\)
Third payment: \(C(1+g)^2\), etc.
\[PV = \frac{C}{1+r} + \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \cdots\]
\[= \frac{C}{1+r}\left[1 + \frac{1+g}{1+r} + \left(\frac{1+g}{1+r}\right)^2 + \cdots\right]\]
This is a geometric series with ratio \(q = \frac{1+g}{1+r}\). For convergence, we need \(|q| < 1\), i.e., \(r > g\).
Sum: \(\frac{1}{1-q} = \frac{1}{1 - \frac{1+g}{1+r}} = \frac{1+r}{r-g}\)
Therefore:
\[PV = \frac{C}{1+r} \cdot \frac{1+r}{r-g} = \frac{C}{r - g}\]
Example: First payment $100, growing at 3% annually, discount rate 8%.
\[PV = \frac{100}{0.08 - 0.03} = \frac{100}{0.05} = \$2{,}000\]
A bond pays periodic coupons plus face value at maturity.
\[\text{Bond Price} = PV(\text{coupons}) + PV(\text{face value})\]
Example: A 3-year bond with $1,000 face value pays 6% annual coupons. Market rate is 5%.
Coupons: 60 per year (ordinary annuity)
Face value: 1,000 at \(t=3\)
\[\text{Price} = 60 \cdot \frac{1-(1.05)^{-3}}{0.05} + \frac{1{,}000}{(1.05)^3}\]
\[= 60 \times 2.7232 + 863.84 = 163.39 + 863.84 = \$1{,}027.23\]
The bond trades at a premium because the coupon rate (6%) exceeds the market rate (5%).
Monthly loan payments form an annuity. Given loan amount \(L\), rate \(r\), term \(n\):
\[\text{Payment} = L \cdot \frac{r}{1 - (1+r)^{-n}}\]
This is derived by solving \(L = PV(\text{payments})\).
You borrow $300,000 for 30 years at 4.5% annual rate (compounded monthly).
Monthly rate: \(r = 0.045/12 =
0.00375\)
Number of payments: \(n = 30 \times 12 =
360\)
\[\text{Monthly Payment} = 300{,}000 \cdot \frac{0.00375}{1 - (1.00375)^{-360}}\]
\[= 300{,}000 \cdot \frac{0.00375}{1 - 0.2556} = 300{,}000 \times 0.005067 = \$1{,}520.06\]
Total paid: $1,520.06 \times 360 = $547,220.40
Interest paid: $547,220.40 - $300,000 = $247,220.40
Over the loan's life, you pay 82% of the principal in interest!
First payment breakdown:
Early payments are mostly interest; later payments are mostly principal.
The Net Present Value of a project is the sum of all discounted cash flows, including the initial investment (typically negative):
\[NPV = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+r)^t}\]
where \(C_0\) is the initial investment and \(C_t\) are future cash flows.
A project requires $10,000 investment today and generates 3,000, 4,000, 5,000 over the next 3 years. Discount rate is 8%.
\[NPV = -10{,}000 + \frac{3{,}000}{1.08} + \frac{4{,}000}{(1.08)^2} + \frac{5{,}000}{(1.08)^3}\]
\[= -10{,}000 + 2{,}778 + 3{,}429 + 3{,}969 = \$176\]
Decision: Accept (positive NPV).
The Internal Rate of Return is the discount rate that makes NPV equal to zero:
\[0 = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1+IRR)^t}\]
IRR is the break-even rate of return for the project.
For our earlier project, find IRR where:
\[0 = -10{,}000 + \frac{3{,}000}{(1+IRR)} + \frac{4{,}000}{(1+IRR)^2} + \frac{5{,}000}{(1+IRR)^3}\]
This requires numerical methods (e.g., Newton-Raphson). Solution: IRR ≈ 8.89%
Since IRR (8.89%) > discount rate (8%), the project is acceptable—consistent with positive NPV.
| Concept | Formula |
|---|---|
| Simple interest | \(A = P(1+rt)\) |
| Compound interest | \(A = P(1+r/n)^{nt}\) |
| Continuous compounding | \(A = Pe^{rt}\) |
| PV of ordinary annuity | \(PV = C \cdot \frac{1-(1+r)^{-n}}{r}\) |
| FV of ordinary annuity | \(FV = C \cdot \frac{(1+r)^n-1}{r}\) |
| Perpetuity | \(PV = \frac{C}{r}\) |
| Growing perpetuity | \(PV = \frac{C}{r-g}\) |
These tools form the foundation for:
You win a lottery with two payout options:
At what discount rate are you indifferent between the two options?
Hint: Set the PV of option B equal to 500,000 and solve for \(r\).
You want $2 million when you retire in 30 years. You plan to invest in an account earning 7% annually (compounded monthly).
(a) If you make a single deposit today, how much do you need?
(b) If you make equal monthly deposits, how much per month?
What semi-annual compounding rate is equivalent to 6% continuously compounded?
A growing annuity pays \(C\) in the first period, \(C(1+g)\) in the second, etc., for \(n\) periods total.
Derive the present value formula:
\[PV = \frac{C}{r-g}\left[1 - \left(\frac{1+g}{1+r}\right)^n\right]\]
Hint: Start with the geometric series for the growing perpetuity and subtract the tail.
A 10-year bond with face value $1,000 pays 8% annual coupons. If the yield is 6%, what is the bond price? What if the yield is 10%?
The weighted average time until cash flows are received:
\[D = \frac{\sum_{t=1}^{n} t \cdot PV(C_t)}{\sum_{t=1}^{n} PV(C_t)} = \frac{\sum_{t=1}^{n} t \cdot \frac{C_t}{(1+r)^t}}{PV}\]
Duration measures the sensitivity of a bond's price to interest rate changes:
\[\frac{\Delta P}{P} \approx -D \cdot \frac{\Delta r}{1+r}\]
A bond with duration 7 years will lose approximately 7% in value if interest rates rise by 1 percentage point.
This will be covered in detail in later lectures on fixed income.
Set up: \(500{,}000 = 50{,}000 \cdot
\frac{1-(1+r)^{-15}}{r}\)
Solve numerically: \(r \approx
7.23\%\)
(a) \(PV = 2{,}000{,}000 / (1 + 0.07/12)^{360} \approx 245{,}975\)
(b) Solve: \(2{,}000{,}000 = PMT \cdot \frac{(1+0.07/12)^{360}-1}{0.07/12}\), giving \(PMT \approx 1{,}633\)
\(r_2 = 2(e^{0.06/2} - 1) = 2(e^{0.03} - 1) \approx 6.09\%\)
At 6% yield: Bond price = $1,147.20 (premium)
At 10% yield: Bond price = $877.11 (discount)
When coupon rate > yield, bond trades at premium; when coupon rate < yield, bond trades at discount.