Actuarial Present Value and IRR Analysis

library(abslife)
#> Welcome to the abslife package!
#> WARNING: Under construction.

This vignette provides a clear example of how to calculate the Actuarial Present Value (APV) and risk-adjusted Internal Rate of Return (IRR) for a loan portfolio subject to competing risks (Default and Pre-payment).

We use the built-in aloans dataset from the abslife package, which contains left-truncated and right-censored loan performance records.

Step 1: Hazard Estimation with Competing Risks

First, we estimate the hazard rates for the competing risks (Default vs. Pre-payment) conditioning on left-truncation. We focus on deep subprime loans and specify support_lifetime_rv up to the loan term of 72 months to cover the entire potential duration.

data(aloans)

orig_term <- 72
cur_age <- 7

# Estimate hazards for the competing risk events
hazards <-
  with(subset(aloans, risk.cat == "deep_subprime"),
       estimate_hazard(lifetime = Z,
                       trunc_time = Y,
                       event_type = ifelse(D == 1, "Default", "Pre-payment"),
                       censoring = C,
                       ci_level = 0.95,
                       carry_hazard = TRUE))
#> Warning in check_censored(lifetime, censoring_indicator, support_lifetime_rv):
#> Warning: Detected censored observations at the maximum observed limit of the
#> support (lifetime == max(support_lifetime_rv)). This may lead to
#> identifiability issues or unstable hazard estimates at the tail.

Step 2: Setup Recovery Curve

We define a recovery curve mapping the recovery percentage to each remaining month of the loan’s term. In this example, we construct a dummy recovery curve showing declining recovery over time.

rem_months <- cur_age:orig_term
recov_curve <- data.frame(
  month = rem_months,
  recovery = seq(0.40, 0.15, length.out = length(rem_months))
)

Step 3: Calculate Actuarial Present Value (APV)

We set the loan parameters (original amount, monthly payment, and reference market rate) and compute the APV:

orig_loan_amt <- 50000
mon_pmt <- 1031.84
ref_rate <- 0.05 # Annual market discount rate

apv_res <- calculate_apv(
    x = hazards,
    cur_age = cur_age,
    orig_term = orig_term,
    orig_loan_amt = orig_loan_amt,
    mon_pmt = mon_pmt,
    ref_rate = ref_rate,
    recov_curve = recov_curve,
    orig_apy = 0.15
)
#> Warning in .apv_setup(x, cur_age, orig_term, orig_loan_amt, mon_pmt,
#> recov_curve, : Lifetime observed support does not include the original loan
#> term. Extrapolating hazards.

• Estimated APV: $43659.65 (13677.69)

Step 4: Solve for Risk-Adjusted IRR

Assuming a current loan price/balance of $37,719.47, we calculate the risk-adjusted monthly and annualized Internal Rate of Return (IRR) that equates the loan price to its expected present value:

cur_bal <- 37719.47

irr_res <- solve_irr(
    x = hazards,
    abs0_bal = cur_bal,
    cur_age = cur_age,
    orig_term = orig_term,
    orig_loan_amt = orig_loan_amt,
    mon_pmt = mon_pmt,
    recov_curve = recov_curve,
    orig_apy = 0.15
)
#> Warning in .apv_setup(x, cur_age, orig_term, orig_loan_amt, mon_pmt,
#> recov_curve, : Lifetime observed support does not include the original loan
#> term. Extrapolating hazards.

• Calculated monthly rate: 0.012147 (1.2147%)
• Calculated annualized rate: 0.155909 (15.5909%)