SAMENVATTING
INTERNATIONAL FINANCE
INTRODUCTION TO INTERNATIONAL FINANCE
TIME VALUE OF MONEY
A fundamental principle within finance is the so-called time value of money. This principle states that an amount of money you
own today is worth more than the same amount in the future. This is because money you have today can be invested,
generating additional income in the form of interest or returns.
Inflation also plays a role: over time, the purchasing power of money decreases, meaning that you can buy less for the same
amount in the future. Also, the risk of not receiving the promised money in the future makes people prefer to receive money
today rather than later. Therefore, lenders are willing to lend money, but expect an interest rate in return.
The value of money over time is calculated using two key concepts:
- The future value indicates how much an amount will be worth today in the future, taking into account a certain interest
rate.
o 𝑉𝑇 = 𝑉0 (1 + 𝑟)𝑇
- Present value then calculates a future amount back to what it is worth today.
o 𝑉0 = 𝑉𝑇 / (1+𝑟)𝑇
When interest is not paid annually, but for example monthly or even daily, we speak of compounding. In this case, you earn
interest not only on the original amount, but also on previously received interest. The more often the interest is calculated, the
larger the final amount will be. In the extreme case, interest is compounded continuously, meaning that interest is added
continuously. The formula for this contains the exponential number e.
• Final value after 1 year (with annual interest r):
o Compound interest formula: (1 + 𝑟/𝑛) 𝑛
o Continuous interest formula: 𝑒 1⋅𝑟
COMPUTING RETURNS
Most investors are risk-averse: when 2 investments have the same expected payoff, they will choose the least risky one. This is
why in general we see that investments have higher reward when their risk is higher. If we want to quantify this reward and risk
and we want to compare it across investments of different value, we have to use relative performance measures, like returns
The return on an investment shows how much your investment has increased or decreased in value over a period of time →
final value −initial value
simple 1-period return =
initial value
1
,But return is not just about profit; it also relates to time and uncertainty. In practice, we never know in advance exactly how
much an investment will yield. That is why we speak of return as a stochastic variable - it depends on future conditions and
contains uncertainty.
Our best prediction for the future return is called the expected return → E[R], where E[] stands for the expectations.
- Example:
- The expected return then becomes: (0.2 × 10%) + (0.5 × 5%) + (0.3 × -20%) = -1.5%
Besides expected return, risk is a second important characteristic. Risk is expressed through variance and standard deviation:
these measures indicate how much the return may deviate from its mean. The higher the standard deviation, the higher the
probability of fluctuations - and thus the riskier the investment.
- Example: need to decide between 3 actions
o Actions A & B:
▪ Both have same expected returns, but B has less risk
▪ Choose B above A
o Actions B & C:
▪ C has a higher expected return, but also a higher risk than B
▪ Choice depends on the risk tolerance
• Risk-averse: choose B, less risk and more stable return
o Action A & C
▪ Both have same risk, but C has a higher expected return
▪ Choose C → C becomes dominant over A and possibly more attractive than B (if you are willing to take
risk)
DIVERSIFICATION
Finally, diversification is a crucial technique to reduce risk without losing returns. Diversification means spreading your
investments across multiple assets rather than putting everything into one stock or product. Spreading across different assets
that do not move perfectly together (are uncorrelated) reduces the overall risk of the portfolio. This is because negative
performance of some assets can be offset by positive performance of others.
The weight of a portfolio can be calculated as follow:
𝑉𝑖,𝑡
- wi,t = , where the portfolio value Vt = V1,t + V2,t + … + VN,t
𝑉𝑡
The expected return of a portfolio E[Rₚ] is the weighted average of the expected returns of individual assets:
- E[Rₚ] = w₁E[R₁] + w₂E[R₂] + ... + wₙE[Rₙ]
- Note: the expected return of a combination of assets is always between the individual returns of those assets.
The risk of a portfolio depends not only on the risk of the individual assets, but also on how those assets move together
(correlation or covariance).
p
- When N=2: Var(R t ) = w12 var(R1,t ) + w22 var(R 2,t ) + 2w1 w2 cov(R1,t , R 2,t )
o Cov(R1,t , R 2,t ) = ρ1,2 var(R1,t ) var(R 2,t )
p
o Var(R t ) = w12 var(R1,t ) + w22 var(R 2,t ) + 2w1 w2 ρ1,2 var(R1,t ) var(R 2,t )
p
- When N=∞: Var(R t ) = ∑N 2 N−1 N
i=1 wi var(R i,t ) + 2∑i=1 ∑j=i+1 wi wj cov(R i,t , R j,t )
2
, → Each combination of assets leads to a point in this chart.
A rational investor will almost never invest in a single risky asset, because the assets need to be correlated. Tif they are not
perfectly correlated, it is possible to achieve the same expected return at a lower risk.
Within modern portfolio theory (Markowitz), the aim is to create portfolios that
offer the highest return for a given level of risk, or the lowest possible risk for a given
return. The curve line connecting all possible combinations of two assets is called the
feasible set or investment opportunity set.
- For example:
o If ρ = +1 → straight line between A and B.
o If ρ < +1 → curved line: convex inwards.
o If ρ = -1 → then it is even possible to construct a portfolio without risk (σₚ = 0).
The collection of these optimal portfolios is called the efficient frontier. Any investor would ideally choose a portfolio that is on
that frontier, because anything below it means more risk for the same or less return, which is irrational.
VALUATION
The value of a financial asset is based on the discounted value of the future cash flows it generates. This principle relies directly
on the concept of the time value of money
CF1 CF2 CFn
→ Value = + +…+ , with CFn = future cashflows and r = required rate
(1+r)1 (1+r)2 (1+r)n
CFet+j
→ Or, Pt = ∑nj=1 j,
(1+r)
→ r = Rf + risk premium
The discount rate (r) plays a crucial role in valuation:
- The higher the risk of the asset, the higher the discount rate.
- A higher discount rate → lower present value of future cash flows.
This again ties in with the risk-return trade-off principle: higher expected returns are needed to compensate for higher risk.
For investors, valuation is essential to:
- Determine whether an asset is over- or undervalued,
- Make investment decisions,
- Weigh returns and risks against other investment options.
3
, 4
INTERNATIONAL FINANCE
INTRODUCTION TO INTERNATIONAL FINANCE
TIME VALUE OF MONEY
A fundamental principle within finance is the so-called time value of money. This principle states that an amount of money you
own today is worth more than the same amount in the future. This is because money you have today can be invested,
generating additional income in the form of interest or returns.
Inflation also plays a role: over time, the purchasing power of money decreases, meaning that you can buy less for the same
amount in the future. Also, the risk of not receiving the promised money in the future makes people prefer to receive money
today rather than later. Therefore, lenders are willing to lend money, but expect an interest rate in return.
The value of money over time is calculated using two key concepts:
- The future value indicates how much an amount will be worth today in the future, taking into account a certain interest
rate.
o 𝑉𝑇 = 𝑉0 (1 + 𝑟)𝑇
- Present value then calculates a future amount back to what it is worth today.
o 𝑉0 = 𝑉𝑇 / (1+𝑟)𝑇
When interest is not paid annually, but for example monthly or even daily, we speak of compounding. In this case, you earn
interest not only on the original amount, but also on previously received interest. The more often the interest is calculated, the
larger the final amount will be. In the extreme case, interest is compounded continuously, meaning that interest is added
continuously. The formula for this contains the exponential number e.
• Final value after 1 year (with annual interest r):
o Compound interest formula: (1 + 𝑟/𝑛) 𝑛
o Continuous interest formula: 𝑒 1⋅𝑟
COMPUTING RETURNS
Most investors are risk-averse: when 2 investments have the same expected payoff, they will choose the least risky one. This is
why in general we see that investments have higher reward when their risk is higher. If we want to quantify this reward and risk
and we want to compare it across investments of different value, we have to use relative performance measures, like returns
The return on an investment shows how much your investment has increased or decreased in value over a period of time →
final value −initial value
simple 1-period return =
initial value
1
,But return is not just about profit; it also relates to time and uncertainty. In practice, we never know in advance exactly how
much an investment will yield. That is why we speak of return as a stochastic variable - it depends on future conditions and
contains uncertainty.
Our best prediction for the future return is called the expected return → E[R], where E[] stands for the expectations.
- Example:
- The expected return then becomes: (0.2 × 10%) + (0.5 × 5%) + (0.3 × -20%) = -1.5%
Besides expected return, risk is a second important characteristic. Risk is expressed through variance and standard deviation:
these measures indicate how much the return may deviate from its mean. The higher the standard deviation, the higher the
probability of fluctuations - and thus the riskier the investment.
- Example: need to decide between 3 actions
o Actions A & B:
▪ Both have same expected returns, but B has less risk
▪ Choose B above A
o Actions B & C:
▪ C has a higher expected return, but also a higher risk than B
▪ Choice depends on the risk tolerance
• Risk-averse: choose B, less risk and more stable return
o Action A & C
▪ Both have same risk, but C has a higher expected return
▪ Choose C → C becomes dominant over A and possibly more attractive than B (if you are willing to take
risk)
DIVERSIFICATION
Finally, diversification is a crucial technique to reduce risk without losing returns. Diversification means spreading your
investments across multiple assets rather than putting everything into one stock or product. Spreading across different assets
that do not move perfectly together (are uncorrelated) reduces the overall risk of the portfolio. This is because negative
performance of some assets can be offset by positive performance of others.
The weight of a portfolio can be calculated as follow:
𝑉𝑖,𝑡
- wi,t = , where the portfolio value Vt = V1,t + V2,t + … + VN,t
𝑉𝑡
The expected return of a portfolio E[Rₚ] is the weighted average of the expected returns of individual assets:
- E[Rₚ] = w₁E[R₁] + w₂E[R₂] + ... + wₙE[Rₙ]
- Note: the expected return of a combination of assets is always between the individual returns of those assets.
The risk of a portfolio depends not only on the risk of the individual assets, but also on how those assets move together
(correlation or covariance).
p
- When N=2: Var(R t ) = w12 var(R1,t ) + w22 var(R 2,t ) + 2w1 w2 cov(R1,t , R 2,t )
o Cov(R1,t , R 2,t ) = ρ1,2 var(R1,t ) var(R 2,t )
p
o Var(R t ) = w12 var(R1,t ) + w22 var(R 2,t ) + 2w1 w2 ρ1,2 var(R1,t ) var(R 2,t )
p
- When N=∞: Var(R t ) = ∑N 2 N−1 N
i=1 wi var(R i,t ) + 2∑i=1 ∑j=i+1 wi wj cov(R i,t , R j,t )
2
, → Each combination of assets leads to a point in this chart.
A rational investor will almost never invest in a single risky asset, because the assets need to be correlated. Tif they are not
perfectly correlated, it is possible to achieve the same expected return at a lower risk.
Within modern portfolio theory (Markowitz), the aim is to create portfolios that
offer the highest return for a given level of risk, or the lowest possible risk for a given
return. The curve line connecting all possible combinations of two assets is called the
feasible set or investment opportunity set.
- For example:
o If ρ = +1 → straight line between A and B.
o If ρ < +1 → curved line: convex inwards.
o If ρ = -1 → then it is even possible to construct a portfolio without risk (σₚ = 0).
The collection of these optimal portfolios is called the efficient frontier. Any investor would ideally choose a portfolio that is on
that frontier, because anything below it means more risk for the same or less return, which is irrational.
VALUATION
The value of a financial asset is based on the discounted value of the future cash flows it generates. This principle relies directly
on the concept of the time value of money
CF1 CF2 CFn
→ Value = + +…+ , with CFn = future cashflows and r = required rate
(1+r)1 (1+r)2 (1+r)n
CFet+j
→ Or, Pt = ∑nj=1 j,
(1+r)
→ r = Rf + risk premium
The discount rate (r) plays a crucial role in valuation:
- The higher the risk of the asset, the higher the discount rate.
- A higher discount rate → lower present value of future cash flows.
This again ties in with the risk-return trade-off principle: higher expected returns are needed to compensate for higher risk.
For investors, valuation is essential to:
- Determine whether an asset is over- or undervalued,
- Make investment decisions,
- Weigh returns and risks against other investment options.
3
, 4
